Compact Stars in the Nonsymmetric Gravitational Theory by Lyle … LM 1988.pdf · NGT takes this...

Compact Stars in the

Nonsymmetric Gravitational Theory

by

Lyle McLean Campbell

A Thesis subm itted in conformity with the requirements

for the Degree of D octor of Philosophy in the

University of Toronto

© Lyle McLean Campbell 1988

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Abstract

Stable w hite dwarfs and neutron stars axe shown to exist in the Nonsym-

m etric G ravitational Theory (NGT). They are modelled as static spherically

sym m etric bodies of charge-neutral perfect fluid m atter, w ith s tandard equa

tions of state. The particle num ber model for the NGT conserved current,

S 11, is used.

The effects of N G T reduce the stability of these compact stars com

pared to similar stars modelled using General Relativity or Newtonian grav

ity. There is a decrease in the maximum mass of bo th white dwarfs and

neutron stars. The central densities are greater and the radii are smalls- In

all these ways, it can be seen th a t NGT produces a greater gravitational force

in compact stars than General Relativity. N GT also decreases the surface

gravitational redshift.

From exam ination of the solutions for compact stars, constraints are

placed on the £2 charges, ( /" - f /" ) and f 2, of protons, neutrons and electrons.

If m atter composed only of these particles is considered, the constraints keep

the £2 charge of the Sim so small th a t the N GT effects it produces in the

solar system are unobservable a t present. Similarly, the NGT term s in the

periastron precession of eclipsing binary s ta r systems, such as DI Herculis,

would be so small th a t NGT could not explain the anomalies found there.

Extended models for S 11 are considered. One of these, based on cosmions

(wimps), m ight allow £2 charges for the Sun and DI Herculis to be large

enough to be interesting while keeping the £2 charges of compact stars small.


T ab le o f C o n te n ts

A b s t r a c t ............................................................................................................ i

A cknow ledgem ents............................................................................................... »

C h a p te r 1 Introduction to the Nonsymmetric G ravitational Theory 1

C h a p te r 2 Modelling Stars in NGT ......................................................10

Section 1: Setting Up the P r o b l e m ...............................................................10

Section 2: The Model for S ........................................................................... 15

Section 3: Derivation of the Numerical Equations ................................ 21

Section 4: Density and Pressure V a r ia b le s ................................................. 36

Section 5: Initial D ata for the I n t e g r a t i o n ................................................. 43

Section 6: Derivation of the Stability C o n d i t io n s .................................... 47

C h a p te r 3 W hite Dwarf Stars in N G T ................................................... 59

Section 1: The W hite Dwarf Equation of S t a t e .......................................59

Section 2: GR W hite Dwarf Stars ........................................................64

Section 3: NGT W hite Dwarf S t a r s ........................................................70

C h a p te r 4 Neutron Stars in N G T ....................................................... 82

Section 1: The Mean Field Equation of S tate ............................................82

Section 2: GR N eutron Stars ............................................................94

Section 3: NGT N eutron S t a r s ..........................................................100

C h a p te r 5 C o n c lu s io n s ................................................................ 112

Section 1: Approximations ............................................................112


Section 2: Consequences of the B o u n d s .....................................................116

Section 3: The Perihelion Precession of M e r c u r y ....................................113

Section 4: The Anomalous Periastron Shift of DI Herculis . . . . 120

Section 5: Extended Models for ..............................................................124

Section 6: Summary ....................................................................................... 129

A p p e n d ix 1 The Perfect Fluid in NGT ................................................ 131

A p p e n d ix 2 Conservation Laws and the Total E n e r g y .......................145

R e f e r e n c e s ............................................................................................................. 161


C hapter 4


Figure 4 . 5 ....................................................................................................................95

Figure 4 . 6 ....................................................................................................................96

Figure 4 . 7 ....................................................................................................................97

Figure 4 . 8 ....................................................................................................................98

Figure 4 . 9 ....................................................................................................................99

Figure 4.10 101

Figure 4.11 103

Figure 4.12 104

Figure 4.13 106

Figure 4.14 107

Figure 4.15 108

Figure 4.16 109

Figure 4.17 110


I

To my wife Jane, who has infinite patience, and to my parents.


C H A P T E R 1

In tro d u ctio n to th e N o n sy m m etr ic G ra v ita tion a l T h eo ry

The Nonsymmetric G ravitational Theory (NGT) t1-3! is a theory of

gravity which generalizes the structure of General Relativity (GR). In GR,

gravity arises from the geometry of spacetime which is described by the

m etric tensor, a sym m etric tensor. In NGT, this g is extended to

a nonsym m etric fundam ental tensor by dropping the sym m etry restriction,

thus including additional fields into the theory. The connection, W£v , is also

generalized to include an antisym m etric p art called the torsion tensor.

This nonsymmetric structu re has its origins in E instein’s unified field

theory, M which tried to include electrom agnetism into the framework of GR.

In the unified field theory the antisym m etric p art of g was in terpreted as the

electromagnetic field strength tensor. The hope was th a t this would produce

a theory explaining bo th forces as p art of a single coherent whole. This

program failed and was abandoned because of its inability to reproduce the

Lorentz force ^ in the equation of m otion of a charged test body. A lthough

the structu re produced a consistent classical field theory, it failed to live up

to the in terpretation given the extra fields.

N GT takes this same structure, bu t changes the in terpreta tion of the

antisym m etric fields. This avoids the problem encountered by Einstein, since

there are now no preconceived ideas of how the ex tra fields should behave.

There are o ther im portan t differences between NGT and E instein’s uni

fied field theory besides interpretation. NGT has an additional m atter field,


5 ^ , as well as the usual This was introduced into the theory for the

following reason. One of the field equations in the unified field theory shows

th a t * yj—g g ^ v'- is a conserved current of th a t theory, b u t one not generated

by any m atter source. It seems natura l to provide such a source, S ^ . The

a n t is y m m e t r y of \J—g <7^ then ensures th a t S'1* itself is conserved.

The charge associated with S ^ is

I 2 = f y / = f S ' d3x. (1.1)J b od y

It has units of area, and is defined to be i 2 so th a t £ has units of length.

Note, however, th a t £2 cam be either positive or negative. In NGT, any

bodvrs interaction with the gravitational field is governed by its I2 charge

as well as its mass. Choosing a model for S*. and therefore £2, is extremely

im portant. The predictions which the theory makes depend crucially on the

£2 charges of the bodies involved. The standard model of S'* takes it to be

a linear combination of conserved particle numbers. The NGT charges of

the elem entary particles are then the only degrees of freedom in the model.

More will be said about this in Section 2.2.

In order tha t the reader become familiar w ith the fields and equations

of N GT a short m athem atical review is in order. The theory can be derived

from the variation of the following Lagrangian: ^

C = y f T g ^ R ^ W ) - 8 - g ^ T ^ + y (1. 2)

* Throughout this work the following convention is used for symmetric

and antisym m etric parts of a tensor: X^u = -f , where X ^ ^ =

oiX^u + X VI1) and = ^ [ X ^ — _Y„M]. Also, units w ith c = G = 1 are

used throughout.


Here. g>iU is the inverse of g ^ defined by

g*x g»A = gx>L g\u = 8$. (1.3)

Note th a t this is only one of two possible definitions for g ^ . The lack of

sym m etry of g ^ and in NGT forces constant care in tbe ordering of

indices. Tbe g in tbe scalar density yj—g is tbe determ inant of tbe full g^v,

mixing tbe symmetric and antisym m etric parts of tbe field. Tbis density is

used to relate tensors and tensor densities in tbe theory, tbus = y/—g

and S* = yj—g S*.

Tbe contracted curvature, R fll/(^W). is defined in term s of the connection,

by

R ^ W ) = W ^ x - \ 4- IVX ) - Wp\W ?x 4 W;\W£U. (1.4)

Tbis is a com bination of tbe Ricci contraction of tbe curvature, R°au (or

—R°ua)- and the second contraction, -R®^. Here, the curvature is defined

by

R i^ w ) 3 w * - wi,,„ + w ;fw*. - wz,w>„. (1.5)

Note th a t, because W£v is not symmetric in (i and u, the only sym m etry of

tbe curvature is tbe antisym m etry of the final two indices. is tbe vector

torsion, defined by aj.

Tbe field equations of tbe theory can be derived from a Palatini-like

variation of tbis Lagrangian with respect to g^v and W£„. Tbe variation

w ith respect to gMi/ gives

G ^ i W ) = 8ttT ^ , ( 1.6)


where G ^ = R — \ g v.uga3R a3 - This has the same form as in GR but

there are six m ore equations here, since neither G ^ nor are constrained

to be symmetric.

The variation w ith respect to W£v gives

( ^ 9 ~ 9 n , x + ^ 9

+ f [ S ^ ~ = 0. (1.7)

One of the consequences of this equation is

( V = F 9 M ) u = ( 1 .8 )

which shows the current y '—g g ^ u]- w ith its NGT m atter source. As previ

ously m entioned, taking the divergence of equation (l.S) gives

= 0, (1.9)

thus S* is a conserved m atter current of the theory.

To simplify equation (1.7), it is useful to define a new connection

r ; „ = (1 .1 0 )

which has the property tha t = 0. The F connection is still not sym m et

ric, bu t contains no vector torsion. The 64 degrees of freedom in are

thus split up into which has 60 degrees of freedom, and the four degrees

of freedom of W^. The contracted curvature satisfies

RAW) = RA?) + ( i . i i )


Using this transform ation, equation (1.7) can be rew ritten as

=

- Y [ + «J K - «5 S ' ] S ' , (1.12a)

or

9ltv,\ 9Pa^Xu 9av^p \ =4_2~ [ 9pX9ai/ 9Xv9iia T <7/xj/<7[Aa] ] S • (1.126)

These equations are the NG T equivalent of the com patability relations for

the T connection, although neither the T nor the W connection is metrically

compatible. In providing a source term for equation (1.8), the W ^ S 11 term in

the Lagrangian also causes T and to be directly influenced by m atter,

ra ther than being purely geometrical. The consequences of this perm eate the

whole theory.

There does exist a connection in NGT which depends only on geometry,

the Christoffel symbol. It is defined in term s of g(^u) exactly as in GR,

S 9(ncr),i/ + 9(<rv)lP. — 9(pir),ir } > (1-13)

where 7 ^ ") is the inverse of g(p„). It is w orth emphasizing tha t 7 ^ ") is

not the same as . 7 ^ " ) is a function only of bu t is a non

trivial function of bo th g(ttu) and g[pV]- The operations of index raising and

sym m etrization do not com mute unless the raising of indices is done with

9{n»)-

Varying the Lagrangian w ith respect to and throwing away the to tal

divergences th a t arise gives the Bianchi identities for NGT: ^

[ y f T g ^ G ^ V ) + \ f —g~9 vaGt,p(T)\ a = 0. (1.14)


T he m a tte r response equations can be derived from this, using field equations

(1.6) and (1.8). This gives

gap T a0,? + + T a 0 [gailj + g ^ , a - gaa,^\

2 3

See A ppendix 2 for a more complete discussion of this.

+ \ w M S a = 0. (1.15)

From the m atter response equations for perfect fluid m atter, the equa

tion of m otion for a test body 21 is derived:

, f lJLy(A/*)p dx_d r 2 T I p j d r i r 2 m / ^ dr 1 ^

where m p and £2 are the mass and €' charge of the test b o d y . Even test

bodies deviate from geodesic m otion in NGT, because of the force on the

right-hand side of this equation. NGT therefore violates the equivalence

principle.

Much other work has been done to develop NGT to its present state.

M any exact and approxim ate solutions to the theory have been found: exact

interior and exterior sta tic spherically symmetric ^ solutions, several cos

mological i9-12l solutions, linear i13'14! and post-New tonian approxim ate

solutions. T he m otion of m atter in NG T gravitational fields I16-24) has been

studied and predictions made for the bending of light P51 about the Sim, the

perihelion precession of M ercury I26-28! and other solar system tests [25>29~341

of gravitational theory. Comparison to observations of eclipsing binary star

system s I35-3'! have also been done. The NGT predictions for gravitational

rad ia tion f38'39l have been com pared to observations of the b inary pulsar


7

P S R l9 1 3 -rl6 and the cataclysmic variable 4U1S20—30 t41L T he energy-

m om entum tensor of perfect fluid m a tte r in NGT ^ has been derived from

a variational principle. A solution of the Cauchy initial value problem I43’44!

has been completed. The linearized N G T Lagrangian has been determ ined

[14,45] to b e £j-ee 0f ghost fields. N G T has been reform ulated in term s of a

hyper-com plex geometrical structure, i46-49]

N GT is able to satisfy all the current observational tests of gravitational

theory. Doing so determines the i 2 charges of various astrophysical objects:

from the red shift of spectral lines in the Sim t25! I q < 2 x 104 km, from

the deflection of light rays near the limb of the Sun ■£© < 6 x 103 km,

from the Viking da ta from M ars [25’°°! £q < 104 km, and from the perihelion

precession of Mercury, -2‘]

' - ^ ( 4 r £4

|3 1..< 3.5 x 10 km, (1.1T)

where M q and £q refer to the Sun, and m p and refer to Mercury.

O utside of the solar system, b inary s ta r systems provide the m ajor tests

of gravity. If the anomalous periastron precession of the eclipsing binary

system DI Herculis is to be explained entirely as an NG T effect, this would

[37]require

(m i + m 2){i\ - i \ ) ( — ------\ m ! 7712/_

i~ 2 x 104 km, (1.18)

where m i, m 2, t \ and are the masses and NGT charges of the two stars

in the system. Results have also been found for several o ther binary star

svstems.


8

The b inary pulsar, PSR.1913+16, has some of the m ost precisely m ea

sured orbital param eters 151_531 in any astronom ical system. N G T ran ac

count for the ra te of periastron precession and the energy loss, presum ed to

be due to gravitational radiation, as long as the I? charges for the pulsar and

its com panion are bo th several orders of m agnitude smaller th a n the solar

constraints, t25l

A n im portan t question for the theory is w hether stars and planets can

exist w ith the I 2 charges predicted above. The present work a ttem pts to

answer this question by numerically modelling stars as static, spherically

symmetric bodies of perfect fluid m atter. A study of compact stars, white

dwarfs and neu tron stars, then pu ts constraints on the size of the ir NGT

charges.

The theoretical work is presented in C hapter 2. The standard model for

S 11 is assum ed in Section 2.2, based on a com bination of conserved particle

num bers. T he N GT field equations are reduced to num erically integrable

form in Section 2.3. In doing this, it is found tha t the m a tte r variables in

Tm„ are not simply the pressure and density of the fluid. Each contains an

additional self-energy term for the <SM current. The appearance of this term

can be traced to th e direct response of th e connection to in equation

(1.12). This is explored in Section 2.4. T he initial d a ta required a t the

centre of a s ta r are considered in Section 2.5. Stability criteria are necessary

to decide which solutions of the equations represent stable stars. These are

discussed in Section 2.6.

C hapters 3 and 4 present the results of the com puter calculations. Chap

ter 3 deals w ith white dwarf stars. Section 3.1 describes the equation of


9

state used for m a tte r in the density regime below p ~ 1011 g /cm 3. Section

3.2 presents the results of using this to build GR. white dw arf stars. These

m atch previous numerical solutions of this type, giving confidence in the

com puter program . Section 3.3 then exhibits the solutions for N GT white

dw arf stars. The differences between the GR and NGT stars are discussed in

detail. Several constraints are found which lim it the size of the NGT charge

for white dw arf s ta r m atter. C hapter 4 follows the same pa tte rn as C hapter

3. except th a t it deals with neutron stars and the equation of sta te in a much

higher density regime.

Finally, C hapter 5 deals w ith the consequences of the bounds on the

NGT charges derived in the previous two chapters. If only the NGT charges

of norm al m a tte r are im portant, then the £2 charge of the Sun and of the

stars of the binary system DI Herculis are restricted to be quite small. NGT

effects would not then play an im portant role in the solar system; nor could

these effects explain the anomalies found in the orbital motion of the binaries.

An extended model, I3 ‘ 1 which includes hypothetical particles called cosmions

(or wimps), is discussed. This model allows the I 2 charges of m ain sequence

stars to be large enough to be interesting, while keeping the £2 charges of

compact stars small.

rft-'Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

10

C H A P T E R 2

M od ellin g S tars in N G T

S ectio n 1: S e ttin g up th e P rob lem

In this work, stars are modelled as static, spherically symmetric bodies

of charge-neutral perfect fluid. These idealized stars should be sufficiently

realistic to allow an understanding of the changes th a t NGT introduces into

stellar structure. It also makes it possible to pu t constraints on the maximum

possible t 2 charges of different kinds of stars.

The basic idea is to solve the field equations of NGT w ithin the star,

for perfect fluid m atter w ith a specified equation of state, and then m atch

this interior geometry to the geometry of the static, spherically symmetric

vacuum solution at the surface of the star. From the param eters of the

external m etric the s ta r s mass, radius and £2 charge can then be extracted.

Each different solution is characterized by two param eters, which can

be taken to be the central density of the s ta r and an N GT param eter, /" ff

, which is discussed more fully in Section 2.2. By solving for the geometry

of m any solutions, covering a wide range of the param eter space, all the

interesting solutions w ith the specified equation of s ta te can be found.

Not all of these solutions will correspond to stable stars. Applying rea

sonable stability criteria, the stable solutions can be separated out. The

masses, radii, and i 2 charges of these stable stars, along with each solution’s

param eters make up a numerical ‘database’ of inform ation about stars w ith


one particu lar equation of state. From this inform ation N G T stars can be

com pared to those of GR and the differences understood.

In NGT, the form of the fundam ental tensor for any static, spherically

symmetric problem is M

/-o r(r) 0

09—r-

0 0

-w (r) 0

6 amd x 4 _ J. —

0

0

w(r)

0

0

7 (r)

\

(2.1.1)

/

O utside m atter an exact solution of the field equations of this form exists,

[54]

( . \0

— r *

V

0

0 0

0

0

0

Lir 2

0

-r2 sin2 0 0

q ft 2 M \ r t , L4 '( 1 - “ ■)(! + p r)

. (2.1.2)

/Note th a t L 2 m ay be either positive or negative. It is defined conventionally

as the square of a length, L, because it has units of area.

These two sets of g^u s describe the geometry of spacetime inside and

outside of the star. They m ust m atch a t the surface,

= ( i -

7 (R) = ( l -

2 M R

2 M

- l

R(2.1.3)

u;

!&:


12

so th a t the geom etry remains non-singular a t the edge of the star. This

allows the mass, M , and NGT charge, L 2, to be determ ined if the values of

the functions a , 7 and u a t the s ta r’s edge are known:

R?uj(R)L- = - =-.= (2.1.4)

y / a (R )7 ( R ) - u ( R ) *

Note th a t only a ( R ) and the combination need to be known for

this, ra ther them all three quantities separately.

These functions are determ ined by solving the field equations of NGT

w ithin the star, in the presence of m atte r which is described by v and S^ .

For a perfect fluid in NGT, t42- the ener:y mom entum tensor is

Tpv = P ) u ° u p - P g (2.1.5)

This has been derived using two different Lagrangian-based variational prin

ciples and appears to be the natu ra l extension in NGT of the perfect fluid

energy-m om entum tensor of GR. It describes m atter in term s of two fields,

the energy density, p. and the pressure, P, of the fluid. T he exact type of

m a tte r is determ ined by specifying the equation of state, P = F(p), for the

fluid. More will be said in Sections 3.1 and 4.1 about the equations of sta te

used here.

The NGT field equations can be reduced to a system of first order,

ordinary differential equations in the radius variable r. The m ethod followed

here is based on calculations originally performed by P. Savaria I55]. Even

w ith spherical sym m etry and no time dependence the equations are still so


13

complicated th a t no analytic solution has been found for a realistic equation

of state. Instead, a com puter program called NSTAR has been developed

to solve the problem numerically. I t is based on an ordinary differential

equation solving package called LSODE This program takes the system

of equations

| ^ ( * ) = (2 .1 .6 )

for dependent variables, y.'(r), w ith the corresponding Jacobian m atrix, J^-,

and a set of initial d a ta {y;(xi)} at some value of the independent variable,

x i , and from it produces values for the variables at some other vaiue xo.

LSODE is a predictor-corrector algorithm which is designed to allow

usage in a wide variety of modes of operations, depending on the nature of

the equations and user choice. It is used here in a mode of operation which is

based on backward differentiation formulas, w ith chord iteration, using the

supplied Jacobian m atrix, in the corrector phase of the procedure.

To build stars w ith this, the NGT field equations are reduced to the

form of equations (2 .1 .6 ) w ith independent variable taken to be the radius,

r , and the Jacobian of the system worked out. For this, a model for the

NGT conserved current m ust be specified in terms of the other variables of

the problem. Initial d a ta are then specified a t the centre, r = 0. It is shown

in Section (2.5) th a t only two param eters are required to determ ine all the

dependent variables a t the centre. NSTAR then uses LSODE to integrate

the equations out by some chosen step to a new radius, r, finding the s ta r ’s

geometric and m atter variables there. These new data are then used as the

initial d a ta for a further step away from the centre. In this way the s ta r is

iReproduced with permission of the copyright owner. Further reproduction prohibited without permission.

14

built up from tbe centre to the edge, which is defined to be the radius where

the m atter variables, P, p and <?**, vanish.

A variable step size was chosen, equal a t any radius to one ten th of tha t

radius, w ith provision made for an upper limit on the step if such was found

to be necessary. This upper lim it was found to be useful only for cases where

volume integrations of the mass were being done. In these cases the accuracy

of the integrals depended crucially on keeping the step size small compared

to the overall radius. In all other cases this lim it was not used.

This increasing step reduces the com puting c.p.u. tim e bu t means tha t

when the edge is first reached it is overshot quite a b it and the radius is quite

imprecise. To correct this, once the edge is reached NSTAR steps back to the

previous radius, cuts the last step size by a factor of ten and proceeds again.

W hen it reaches the edge again, the over-shooting is less and the radius more

precise. Repeating this procedure a number of times allows the radius to be

found to any required precision. Note th a t this precision differs from the

accuracy to which the radius is known.

T he accuracy of the radius is limited by the accuracy of the numerical

m ethod used by LSODE and by the accuracy to which the equation of state

portrays the low density regime of m atter at the s ta r’s edge. The la tte r effect

does not influence the neutron s ta r results significantly, as the density only

becomes this low in the outer m etre or two of the star, bu t in white dwarf

stars the outer few hundred metres are affected. The other effect is harder

to quantify but the inaccuracy may lie near the 1 % level.

The mass and I 2 charge of the solution are unaffected by this inaccuracy

of the radius. Both quantities can be found either by m atching of the interior


15

and exterior solutions or by direct integration. In bo th integrations the

contribution from the region of the solution near the surface is negligible,

because of the low density there. The m atching m ethod for finding M and

Or depends on R, bu t it produces exactly the same num bers as th e integration

m ethod in all cases checked. It is therefore ju s t as trustw orthy.

Section (2.2) discusses the model for the conserved NGT current which

is necessary to relate to the other variables of the problem . Then, in

Section (2.3), the field equations of NGT are reduced to the form of the

system of equations (2.1.6). In the process, new m atte r variables p and P

are introduced which simplify the equations and are then shown, in Section

(2.4), to be the true energy density and pressure of the fluid.

In Section (2.5), the reduced equations are expanded in a Taylor series

about r = 0 to determ ine the values of the dependent variables near the

origin in term s of the two initial data, pc and Finally, in Section (2 .6 ),

the stability conditions used to classify stable and unstable solutions of the

equations are found.

S ectio n 2: T h e M o d el for S M

W ith the perfect fluid energy-m omentum tensor and a given equation of

state, is fully determ ined in term s of the m atter variables, p and P, and

the fundam ental tensor g ^ . Before calculations can be done, the same m ust

also be true for the other m a tte r variable, S'*.

There is nothing in the theory which forces an in terpreta tion on it. The

only constraint is tha t it is a conserved vector current. Its integral over any


volume, V, gives the £2 charge of the m a tter in th a t volume:

£2 = f S 4 d3x. (2.2.1)J v

It is possible th a t is not a presently known conserved current, bu t, if this

were assumed, it would make any further discussion very difficult. It seems

best to first try the m ost obvious possibilities, the known conserved currents

seen in nature.

W hat known conserved currents are there which S p might represent?

The conservation of energy and m om entum are already accounted for in TMI/.

One possibility is the conservation of baryon num ber, or one of the several

types of lepton num ber which are independently conserved in strong, weak

and electromagnetic interactions. The most general possibility of this type

is a linear combination of all of these conserved num ber densities. If is

the fluid’s four velocity, n Q the rest num ber density of type-a particles, and

f 2 the £2 charge of a single type-a particle then, w ith such a general particle

num ber interpretation,

S 11 = (2 -2 .2 )a

This applies only to different types of fermions. The num ber of bosons is not

conserved in most interactions. All bosons are therefore assigned zero f 2. In

particular, photons do not contribute to S ’*.

This definition of is consistent w ith the derivation of the perfect fluid

energy m om entum tensor f42l in NGT. In the variational Lagrangian used

there, an explicit form for S ** is required. A restricted form of definition


17

(2.2.2) is used with only a single type of NGT charge carrier. This work ran

be extended (see Appendix 1) to a fluid composed of m any types of particles,

using a more general form for the N G T current, S 1* = y/—g u^Sij ia) . The

perfect fluid energy-momentum tensor remains the same as in equation (1 .6 )

if the following constraint is satisfied:

£ " * ! : - 5<"»)• (2-2-3>or

This condition is satisfied by definition (2.2.2).

Note th a t the variational principle used in Appendix 1 is not unique. It

is an extension to N GT of a variational principle used in GR, and as such

contains some ambiguity. There are fields present in N G T which exactly

vanish in GR and which might be included in different ways in the variational

Lagrangian. It is not too surprising, then, th a t the p and P of the fluid are

different from the p and P which appear in by term s which vanish in

GR. More will be said about this in Section 2.4.

The charges f 2 are not all independent of one another. The conserva

tion of I2 m ust hold during a microscopic scattering process as well as on

macroscopic scales. From this constraint there cam be only one independent

f 2 for each conserved num ber density, f B , / £ e, /£ , f \ r and possibly oth

ers depending on how the standard model of the other three interactions is

extended. In norm al m atter, formed just of protons, neutrons and electrons,

there will only be two, f 2B and f \ (or equivalently f 2 and ( f 2 + f 2) if these

constants are w ritten in term s of the elementary Z2 charges of different types

of fermions.)


18

Since stars axe to a very good approxim ation charge neu tra l (so th a t

rip = n e) and formed of norm al m atter, definition (2 .2 .2 ) reduces to

S M [ f 2 n n + ( f 2 + f ; ) n p] . (2.2.4)

In the dense cores of neutron stars there exists the possibility th a t m atter

which is normally unstable could exist in a stable state. This occurs when

the fermi energy of the normal m atter to which the unstable particles usually

decay is so high tha t it is energetically favorable for the unstable particle to

remain. In this case, there could be a sufficient adm ixture of muons, strange

hadrons and so on th a t /£ and other constants could be im portan t. It is

expected th a t the num ber of such particles would be far lower than for norm al

m atter (after all, it is the normal m atter th a t is filling up the fermi levels).

These particles would, therefore, only contribute significantly to the final I 2

of the neutron s ta r if these exotic f 2 ’s were many orders of m agnitude larger

th an f 2 and ( f 2 -f f 2). This seems unlikely bu t cannot be ruled out a priori.

A nother possible exception to the assum ption th a t only ordinary m atter

contributes to the £? of stars is the group of cold dark m atter candidates

known as weakly interacting massive particles (W IM Ps) or cosmions I57-59].

These particles are conjectured to be captured by stars as they sweep along

their paths through a background cloud of cold dark m atter. They may then

provide a possible solution to the solar neutrino puzzle [6 0 >611 by transporting

heat from the core of the Sim as they orbit through the star. This would

slightly decrease the central tem perature of the Sun, reduce the burning rate

of 8B and thus reduce the emission ra te of detected solar neutrinos.


19

Although the num ber of cosmions would be very small compared to

norm al m atter, a recent fit to observational d a ta has shown th a t if their

N GT charge, / | , is large enough they can contribute in an im portant way.

In this case

^ = V'= F ’ [ / 'n n + (fp + f e ) np + f c n c] • (2.2.5)

More will be said about this in C hapter 5.

In most cases, the details of how the composition varies throughout

a s ta r axe not known well enough to make it possible to incorporate this

variation into the calculations. The stars are then assumed to have uniform

composition throughout. This makes each of np/ n , n e/n , n n/ n and n c/n

(where n = n a ) constant within the star.

A further assum ption was made, since da ta for the neutron star equation

of s ta te was only available for p and P , but not n. The only m atter variables

in term s of which S 11 could be usefully w ritten were p and P . It was assumed

therefore that

p = Po = '^2i m an a = m effn, (2.2.6)QI

where m eg = This works best for m atter a t low densities, where

the internal energy, (p — po)/po, is small, and P <C p- There should be no

problem for white dwarf stars. For neutron stars, the approxim ation is less

well justified.

For norm al m atter, equation (2.2.4) then reduces to

= y / Z f u ^ & L p (2.2.7)TTln


•where the effective N G T charge, f^s , is given by

TTlrflix = + / « - 1

Tt TL J TTZgJf

[ f n ± ( f p ± f ! ) n p / n n]( n i j T7i, - n p/ n nj

(2.2.8)

p Inclusion of cosmions leaves equation (2.2.7) intact bu t changes to

I 2 [ / n + (fp + f e ) np /n Ti + f * n c/ n n)I eff 1 - /_ I m / 1 ’ )? 1 + mri nP/Tin + m^ n cj n nItI, . The effective N GT charge will vary from star to s ta r w ith changes inI|; composition. Main sequence stars like the Sun are composed almost entirelyiI

of hydrogen and helium, w ith about 2.5 times as much hydrogen as helium,r| by mass, for young stars. This gives np/ n n = 5.5. As a s ta r ages the helium£)

and heavy element abundances increase at the expense of hydrogen, so n p/ n n

decreases.

A white dwarf s ta r is formed from helium, carbon, oxygen and other

elements, bu t little hydrogen. All the elements heavier th an hydrogen have

close to the same num ber of protons and neutrons so np/ n n = 1 for a white

dw arf star, or a planet. For a neutron star the process goes even further

until there are far more neutrons than protons or electrons. For neutron

stars w ith a simple equation of state based on three non-interacting zero

tem perature fermi gases, in be ta decay equilibrium, np/ n n varies from 0 . 0 0 2

to 1 /8 t62l. An estim ate of 0.004 < np/ n n < 0.05 is derived in Section 5.1,

for the equation of s ta te used to build neutron stars in C hapter 4.

This variation in n p/ n n causes / 2ff to differ from star to s ta r and to be

very different for stars as different as neutron stars and the Sun. Young stars


21

would be dom inated by ( /" + 7 ? ) , neutron stars would be dom inated by / 2,

and white dwarf stars would be influenced by each of these in equal amounts.

S ectio n 3: D erivation o f th e N u m erica l E quations

As seen in Chapter 1, the NGT field equations are:

= 8~ (2.3.1)

~ ( g ^a gxu ~ 9liX 9av ~ [Aa] 9nv ) = 0? (2.3.2)

and.

(>/=? 9 ^ ) ^ = ^ J = ~ 9 S K (2.3.3)

The fields also satisfy the m atter response equations,

9 ct(iT ,j3 + 9 l ia T : ,/? T T P \9an,P + 9li0,a ~ 9a/3,n\

+ 1 ^ 5 “ = 0. (2.3.4)

These equations simplify enormously because of the symmetries of the sit

uation. All tim e derivatives vanish, and all the unknown functions depend

only on r . Since the problem is static, the usual Schwarzchild coordinates

are co-moving with the m atter, so u l = 0 , 5 ' = 0 and, from g ^ u ^ u " = 1 ,

u 4 = 7 -1 / 2. The r j „ ’s and R ^ ’s depend only on a, 7 , ui and S 4. T

depends on p , P , a, 7 and oj. The only other unknown functions in addition


22

$ to tlie six m entioned here are the components of the vector torsion, , only-

one of which, W 4 , tu rns out to be relevant.

W ith the g^u given in equation (2.1.1),

and

9** =

\ / —g = r 2 sin Q\/a~f — u - ,

0 0

0

0

(2.3.5)

cry—w-

0

a-*—u/2

0

0

\or'y—w*

r2 sin- 9

0 0

0

or*Y—w

(2.3.6)

The four equations (2.3.3) reduce to

r~ui= 47TT-2 \ / 0 7 — u; 2 5 4. (2.3.7)

^ -^<*7 ~ w2

Prim es denote derivatives with respect to r. This allows the calculation of

the NGT charge ^2, integrated from the centre of the star out to radius R:

l 2(R) = f S 4(r ) <?r J s R

rR________________= / 4 /rr2\ / a 7 — a;25 4(r) d r

Jo

= [ s ( ^ V a -\ v a 7 —Op-J

r~uR

(2.3.8)■\/a7 — w2

where 5 k is the spherical volume of radius R. This function vanishes at

r = 0 assum ing th a t a, 7 and u> are all well behaved there. In a physical


23

star, the centre is ju s t the same as any other place in the fluid so ui should

not diverge, nor should y / c r f~ ^ u ^ vanish there. The NGT charge, integrated

out to radius r is therefore,

i 2 {r) = r u (r ) _ (2.3.9)y/a(r)- j(r) - w ( r ) 2

Recall equation (2.1.4). This is the same form for I 2 th a t is found by m atching

the interior and exterior geometries a t the s ta r’s edge. It holds throughout

the star.

It is convenient to make the following definition:

so tha t.

s = 2 ~ r u S 4, (2.3.10)

I 2' - or f = 4 r 3 s. (2.3.11)

Through this derivative of £2, s enters into all of the other equations.

It will become obvious tha t the only combinations of a, 7 and ca which2

appear in the field equations are a itself and which is quite closely related

to i 2. The static natu re of the problem makes the tim e coordinate unim por

tan t and eliminates one of the functions. An analogous thing happens in

the vacuum case leaving g w ith only two independent param eters. It is

therefore also convenient to define,

,2LJ0 :7

(2.3.12)

for which,

.£4 e 14= ^ e = (2.3.13)r 4 1 — e kt + r 4


24 Iand,

, 4(1 - e )e (s — e — es). (2.3.14)

This is the first equation of (2.1.6), the system of NGT equations of stellar

structure.

Inverting equation (2.3.2) gives T*u in term s of a, e, and s as well as

some rem aining 7 *s and u/s (which all disappear in the final equations). The

only non-zero components of are:

T(2 3 ) = cot 9

rg 3 = — sin 9 cos 9

r i _ EL —11 2a 3 r

r (12) = r (13) = “ ( X + 3 )

r p 4] = r [3 4] = ^ ( l + a ) e - | a ) (2.3.15)

r 1 = 1 ( e' 4- - - t14l u V 2(1 - e ) + 3 r

T4 = e' + lL + ±(14) 2(1 - e ) 2 7 3r

t,i 7 f e' l ' sr « = a r ( W ) + 2 7 + r

Using equation (2.3.14), the la tte r half of these can be rew ritten as

2 7 / .. . 2

r uj^[14] — 7 77 ( e( l + s ) ~ os

2e(l + s) — | s , 7 'r (i4 ) = r ' + (2.3.i 6 )

r i _ 7 ( 4e(l + s) - 3s 4 4 o: \ r 2 7


Note th a t r^24j = T®34j = — f r j ^ j so th a t = 0. This shows explicitly

th a t the vector torsion of the T connection vanishes in this case, as it m ust

in all cases because of its definition in term s of the W connection.

These connection components allow the calculation of From

definition (1.3), the only non-zero components axe:

Ru = - ( 2r ’12)+ r ‘u y - 2 ( r fn )y - (r j14, ) 2 + 1^ ( 21^ , + r ; I4))

V , 2 - s + 2 e ( l- f s ) V27

2r J r 2

Y 2 — 3 4- 2e(l -j- s ) 2 7 T r

7 ' , 2e(l 4- s) — f s

(2.3.17a)

R 22 — r 22-x — rj23)2 — (r f23)) 4 - r 22 (Vjx + T(X4

r ( l T i ) V r ( l4 - s )a a

a2a

2e(l -f s) — 2s27

(2.3.176)

— r^ 3.x 4 - r 53 2 4- I * 3 ( r ^ x 4- r^ X4 — p 2 *n3 1 33 (2 3 )

= sin" 9 (2.3.17c)

Ra = r L . t + I (rfuiy + r ‘4 ( r j , + 2TfI2) - rfw )

4e(l 4- s) — 3s[27

4- - a

4e(l -I- 5 ) — 3i

+ 3 7 / 2e(l + 3) - | s 2 ae \ r

_y _ y_ 2 ( 1 4 - s ) ( l - e)2 a 27 r

(2.3.17 d)

- r [l4],l + 3 r [14]r (12)

' 7 ^ 2 e (l 4-s) - f s ^ l ( (3 4 - s ) 7 /"2e(l 4 -s) - f sa;

(2.3.17e)


The prim ary field equation (2.3.1) also requires T^v. For a perfect fluid

in NG T M

Ty.u = g ^ g QV(p + P ) u av.P - g ^ P . (2.3.18)

In the interior of a static, spherically symmetric s tar the only non-zero com

ponents of T,,.,, are:

oTn = —— { p - r P ) -i- a P

7T22 = r2P

T33 = r2P sin2 9 (2.3.19)

T4 4 = 7 p

T14 = — T n = up,

with

T = g^Tft , , = p - 3 P . (2.3.20)

P u ttin g this into equation (2.3.1) then gives

^ [ 1 ,2] = W[l,3] = ^ 2 ,3] = ^ [ 2 ,4] = [ 3 ,4] = 0 (2.3.21)

and

2

R u = —8 ~ ^ - ( p +■ P ) -f 4rro:(p — P)

J?22 = 4tt r2(p — P)

R 3 3 = 4?rr2(p — P) sin2 9 (2.3.22)

i?44 = 4 ttj(p + 3 P )

•R[i4] = 47ru>(p + 3P ) - — W[lt4] (2.3.22e)


A comparison of the R 22 and .R33 equations here and in equation (2.3.17)

above reveals th a t the R 33 equation is ju st a trivial m ultiple of the R 02

equation. W ith this symmetry, the prim ary field equation contains only four

independent equations ra th e r than the full sixteen.

These fields also satisfy the m atter response equation (2.3.4) -which gives

(2.3.23)

This can be combined w ith the field equation, (2.3.22e), to eliminate

W[li4j and give an equation for P ' ,

(2.3.24)

The equation can be used instead of one of the prim ary field equations. This

is easier th an using the complicated equations th a t arise from combining

equations (2.3.17) w ith equations (2.3.22).

This means th a t it is only necessary to take two combinations of the re

m aining i? n , R 22 , and JZ4 4 equations to solve for a and 7 . The combinations

th a t are used here come from isolating P and p from equations (2.3.22),

and

( R 1 1 / & + R 4 4 / 7 ) , 2- + —r i t 22Ti( 1 - e )

( i 2 n / a + .R44/ 7 ) 2 n

( T ^ j ------------

= 8 ~ p

= 87rP.

(2.3.25)

(2.3.26)

These combinations of the R ^ ’s give equations which are considerably sim

pler than any of the individual equations are.


Fitting the „’s of equation (2.3.17) into equations (2.3.25) and (2.3.26)

and simplifying gives

and

(2.3.27)

8 ' F =H z - '

f l + sl /7* 4--------- ( ------—(eC1 T s) - s)ar \ ~f t

a r - \ 3e (2.3.2S)

Consider equation (2.3.27). Expanding it out and regrouping term s gives

= ( ! _ C L ± i £ ) + ' ( x _ f l ± i ) i y + i i l . (2.3 .29)V a J ( 1 — 3 ) \ a J 3 ae

At this point it is natu ra l to define

. . X - (2.3.30)

which inverts to give

a = ( 1 + A)2 (2.3.31)( 1 - v ) '

In GR, s = 0 and the variable v becomes 2m where m(r) is the volume

integral of the density out to radius r, 4~r ' 2 p(r') dr1. In NGT, because of

the presence of s in equation (2.3.30), the situation is more complex.

It will be useful to have the following definitions:

4ttG 2 2 s2b— — r p - --cz

4txG3 ef2

l n 2 s2b P — ------- .

Z e t 2

(2.3.32)


Using these, equation (2.3.29) simplifies to,

v ' = - £ [ » - 2 -] . (2.3.33)

If it were not for the factor of t in this equation it would be possible to

solve the equation in exactly the same way as in GR, yielding v{r) = — ~

where m (r) = JQr 4~r'2p(r') dr' w ith modified density p = p — 3 7 ^ 7 7 7 . The

additional t factor, however, makes v more complicated:

. , 2 m (r) 2 e - J^ r) f r x fr,x . ,v(r) ---------- i--------------- / e 's ( rr r J 0

4tt r'~p(r') —m(r ')

dr'. (2.3.34)

where I 3(r) = dr ' . Note tha t m (r) is not the mass integrated out to

radius r, as it is in GR.

It is not an efficient use of com puter time to introduce v into program

NSTAR in this form. It would be necessary to update the m and I 3 integrals

at each outw ard radius step. This would force the size of the r step to rem ain

small in order to protect the accuracy of the integrals and this would increase

the running c.p.u. time of the program dramatically. Instead, equation

(2.3.33) becomes the second equation in the system of equations, (2.1.6).

Now consider equation (2.3.2S). W hen it is expanded out and regrouped

it giveso,' 1 r +

(2.3.35)± = 1

7 r6 s — 4ef + \ { v + 2 q)

b

This equation m ay be included in the system of equations, (2.1.6), bu t it is

not used in the version of NSTAR which generates most of the results seen

in C hapters 3 and 4. Its use can be avoided because 7 only appears in the

equations of (2 .1 .6 ) as and thus it can be eliminated from the system using


30

equation (2.3.35). T he values of M and L 2 in the exterior solution axe set by

v ( R ) and e(R) so it is not necessary to have 7 (R) to solve for the external

geometry of the star. This is due to the static natu re of the problem.

Furtherm ore, equation (2.3.35) does not relate the value of 7 a t r = 0 to

any of the other initial data. If equation (2.3.35) is included in system (2.1.6),

a separate piece of initial data, 7 0 , m ust be provided and, when the surface

is reached, this m ust be rescaled to satisfy 7 (R) = ( l — ^1 4 -

The only place where 7 (r) is necessary is in performing the volume

integral of the conserved energy-momentum pseudo-tensor, which can be

done as a check on the value of M . From Appendix 2, the to ta l energy of

a static, spherically symmetric star in NGT seen by an observer at spatial

infinity is exactly M and it can be shown tha t

R t_________'p -r 3 PM ^ 1 + f a r 2 ^ a - f ( l - e)

0 + +dr, (2.3.36)

where p and P are defined by,

1 s2b * 1 s2 6 ,P ~ P f a r 2 et2 P - P Qirr2 et2, (2-3-3 0

or p = p — y~S 2 and P = P — ^f-S2. Note th a t z = f a r 2p and q = f a r 2 P. It

is these same modified m atter variables which appear in equations (2.3.33)

and (2.3.35).

In a wide variety of cases, the integral in equation (2.3.36) was num eri

cally evaluated, along with the integral in equation (2.3.S), as a check on the

boundary-m atched values of M and L 2. In these cases, it was necessary to


31

include equation (2.3.35) in system (2.1.6) and to trea t 7 as an additional

variable, weakly coupled to tbe others. After it had been verified th a t in

every case the boundary-m atched and integrated values of M and L~ were

identical, this feature was dropped from the program . This saves c.p.u. time

in the program ’s running by reducing the size of the system of equations,

elim inating calls to the integration subroutine and allowing a wider spacing

of interm ediate radii (which was not otherwise possible as it decreased the

accuracy of the integrals).

R eturning to the P' equation. (2.3.24), and using equations (2.3.10) and

(2.3.17e),

R[U]S4 = 72 ~ ru u:

, ,1

U!( 2 + s ) \ (2 ef — |s )

! ^ \ 2 e't \ U e 3 ] s' (2.3.38)

From definitions (2.3.12) and (2.3.32),

w e' v' 7 ' s'IT = 2 l ' r 2 6 T 2^ + 7 ’

(2.3.39)

so th a t, using equations (2.3.14), (2.3.33) and (2 Q.35),

sbR [14]S 4 =

4 s' e! 2+ 7 r + 3 s

- f v' , 2 ( 2 + »)

+ .7 ie ( - | S (2.3.40)

W hen this is substitu ted into equation (2.3.24), it yields

P' = — ^~{p + P) + - ( p + 3P) 2 7 r

sb

+7 ' i t '

~ ~ T7 0

2 (2 + s)4n~r2 et2

e t - \ s

4 s' e' ( 2- 5 7 + 7 6 4 + 3 s

(2.3.41)


32

To get this into the form of (2.1.6), the derivatives of the other depen

dent variables m ust be eliminated from it. Equations (2.3.14), (2.3.33) and

(2.3.35) can be used to replace e', v' and by combinations of r , v, e, s, p

and P. This leaves only s' unreplaced.

The s' in this equation can be eliminated by using the model for 5^ ,

equation (2.2.6), along with equation (2.3.10) and u4 = (7 )-1 / 2,

s == 0 -VT m n

= (1m

- e r f

^ r v '

Untangling s from this yields

(2.3.42)

So1 - ■>0

(2.3.43)

where

V o(2.3.44)

This gives s as a function of the other variables. From this

1 . 1 1 e ' v ' p 'S — S t ( - - r — 4--TT + —

r 2 e 2 o p^2.3.45)

Substitu ting this into equation (2.3.41) gives

P ' = 7 \ p + P) + - ( p + 3 P ) + 52 7

so4~r 2 ef2

+

3~r2 et2 p

If + T(25-ei) + T ( et + t s

(2.3.46)


The equation of s ta te gives P as a known function of p , so this equation ran

be rew ritten as

' d P s2bdpP =

sb4 ~r 2 et2

i

3~r2 t t 2p

4 s3 r +

- l~ ^ 0 > + P ) + ;0> + 3J>)

v . e! ( 4

r + 2 ( 2 + »)(2.3.47)7 r / v 3 / J j

This is a complicated and ugly equation. There is feedback because p1 (or

P ') depends on s ' which itself depends on p1. This contorts the equation so

much tha t p’ is divergent if

dP 1 s2b 4 - 2

dp 3 ~r2 et2 3 (2.3.48)

or. equivalently, if ^ = 0. For large enough /" ff this condition will always

be satisfied for some p. This puts an a priori lim it on the space of initial

param eters, / and pc.

Furtherm ore, it is necessary tha t p and P be monotonically decreasing

throughout a star. The closer a fluid element is to the s ta r ’s centre the more

mass it has pressing down on it. There m ust then be a larger pressure within

the fluid element to support it against the greater weight and this increases

the local energy density. Equation (2.3.47), however, does not respect this

necessity, p1 and P ' are not negative definite. W hat is wrong?

A clue can be found by going back to equation (2.3.41) and absorbing

the term with s' into a new total derivative term which can be included with

P ' on the left-hand side of the equation,

P -67rr2 et2 J

s2b

sb4<r r 2et

’■b V 7 ,= — o~ ^— W I22 j \ 3~r^ei2

4s f e' v' 7 + e +r \ e 0 7

' + 2 ( 2 + s ) \ ](2.3.49)


or, using equations (2.3.14), (2.3.33) and (2.3.35),

* - - £ » + *> + ; ( » * + * - 5 5 ? ) (2.3.50)

This is a dram atic simplification over equation (2.3.47), and yet has exactly

the same content and problems. The divergences and changes of sign have

been absorbed inside the new m atter variables, p and P.

It is perhaps surprising tha t although the new to tal derivative term was

formed for a totally different purpose, and is the unique way to accomplish

th a t purpose, it is also exactly the term necessary to convert P' into P ' ,

as defined previously in (2.3.37). Moreover, every other place in equation

(2.3.41) where either p or P appears, a corresponding term appears which

converts it into p or P, while at the same time simplifying the equation. Thet

v' and equations are also simplified by their use, in the form of r and q

(since 2 = 4~r~p and q = 47r r 2 P ). Again, it is using these variables that

the volume integral of the conserved energy, equation (2.3.36), takes on its

simplest form. All this is suggestive tha t there is more to p and P than

simply convenience. In the next section this is explored in more detail.

In the GR limit (p —► p, P —>• P, s —► 0, t —» 1 , e —» 0 , i 2 —> 0 ,

v —* —T™ and b —* ( l — — )) equation (2.3.50) becomes the Oppenheimer-

Volkoff equation:

" - - J -e ± ^ 7 $ f n - P-3.51)

The NGT modifications of this equation come from m any sources, changes

in ^ (which in the Newtonian limit is the gravitational potential) and the


35

addition of several completely new term s. The meaning of the changes will

be explored further in the next section.

To finish this section, the system of equations of (2.1.6) is presented in

the two final forms in which it entered computer program NSTAR. In both

cases, it was found to be easier to work with p as a variable ra ther than P so

equation (2.3.50) is rew ritten here as an equation for p'. In the first version,

7 is included as a variable along w ith u, e, and p:

v' = - - ( v - 2 z)

e' = ^ - et)

V

P =

6,5 — 4 et + 7 ( u - r 2 ff) o

¥ f

(2.3.52)

2^ )In the shorter system 7 has been eliminated by substitu ting equation (2.3.35)

into the p' equation. This gives

v' = (u — 2 z)T_

(2.3.53)

+ + - 2* ( P + P ) + 2 s ( P + ^ )

1.Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

S ectio n 4: D en sity and P ressu re V ariables

To understand the meaning of the new m atter variables consider first

the behaviour of test particles. Take as a ‘test particle’ a th in spherical shell

of perfect fluid at radius ro and thickness A r. A shell is considered rather

than a particle to preserve spherical symmetry. Initially it will be considered

to be in vacuum bu t la ter will surround a star. The thickness of the shell

can be taken to be small enough th a t p and S are constant across it and tha t

— ■ -C 1. Also, the mass, m t, of the shell and its NGT charge, £%, can be

made small enough tha t and are bo th extremely small com pared to

1 . Thus a = 7 = 1 bu t, while s. e, v and 7 ' are all small, they are not zero.

If the density of the shell is in the range of normal m atter then P can be

neglected compared to p.

From equation (2.3.8), £2 {r) is zero for r < r 0 and 4 - 7’;; A rS (which is

then £%) for r > r 0 -f A r. For r 0 < r < r 0 — A r, it grows linearly,

£2{rQ-rSr) = 4-nrlSSr = £2t ^ ~ ,A r

where 8 r = r — ro. Terms smaller than leading order by factors of £ \ j r \ ,

S r / r 0 and A r/ro are dropped throughout these expressions.

From equation (2.3.11), s is zero everywhere outside the shell and

s '2A2>w ithin the shell. Since this is always very small, t = 1 still.

From definition (2.3.13), e is zero for r < ro and £\!{£\ + r 4) (= £ ? /r4)

for r > ro + A r. W ithin the fluid of the shell

e(r° + Sr) = f ( 2 A 3 )


37

From equation (2.3.34), v = 0 for r < r0 and for r > r 0 + A r.

W ithin the shell itself

m (r0 + 6 r ) = 4 7 rr;;p<5r and I 5 ( r 0 -f- 5r) = (2.4.4)4r04(Ar)2

and, therefore,

v (r0 + 6 r) =2 m (r0 4 - 6 r)

ro(2.4.5)

Again, this is small, so 6 = 1 . M atching equation (2.4.5) to the exterior

solution then shows that

m t = m (r 0 -f A r) = 47rrjjA rp = pV (2.4.6)

where Vt — 4 r r jA r is the volume of the shell. The mass of a test shell of

m atter is the integral over the shell of p.

The to tal energy of this shell is also m t and, from equation (2.3.36),

mt = ■A , f(A r ) 2

8r3 p _ 4pr o.

| d(Sr). (2.4.7)

The second term in the square brackets (due to v) is much smaller than 3p

so, elim inating it and integrating,

m t ( 1 + = 4" ro P { A r + F (A r ) 2 (2.4.S)o / t r o (A r ) 2 3 '

or m t = 4tttq A rp , in agreement with the result from v above.

It is p, not p, which enters into the mass of a ‘test particle’ such as this.

This suggests th a t it is p and not p which is really the energy density of the

fluid. Such an identification would carry over to the pressure as well. In

A ppendix 1 , it is shown th a t if p is the density of the fluid then P is the


corresponding pressure. The variables in T ^ , bearing the symbols p and P ,

would then be the density and pressure plus additional self-interaction term s

due to the other m a tte r field, S'1.

This can be reconciled w ith the derivation of in Appendix 1 since the

variational principle used there is the generalization to NGT of a variational

principle used for the same purpose in GR. Inevitably, since there are fields

in the NGT variational Lagrangian which vanish in GR, the generalization

is not unique. It is possible to rewrite equation (A1.14) so tha t the pressure

and density variables are clearly p and P.

The variables, p and P , behave like a real pressure and density would

be expected to. They vanish at the edge of a star. The P' equation is

negative definite, as is required. This has not been shown analytically, bur

in numerous runs of NSTAR, even vh en v ' , e' and. in some cases, 7 ' become

negative towards the edge of a star, P ' and p' never change sign.

Consider again the test shell of m atter, but now surrounding a s ta r of

mass M , N GT charge L 2 and radius R. If P is the pressure in the shell then

P' gives the density of force acting on the shell. The acceleration of the shell

thus predicted should be in agreement w ith the NGT test particle equation

of motion.

Note th a t P' is a gradient of purely local variables. The local variables,

p, P and S, depend only on conditions in the local fluid element while e, u,

s and 7 depend m ainly on the s tar a distance ro away. Both P' and (y~S2) '

are gradients of local fluid quantities and so both act as pressures. This

suggests th a t it is P , not just P , which acts as the pressure of the fluid.


39

For the test shell of fluid surrounding a star, £2 = L 2 for R < r < ro,

£ 2 = L 2 + £% for r > ro + A r and

£2 (r0 + S r ) = T 2 + £2-^ -a r (2.4.9)

for ro < r < r 0 + A r. Then,

2rg A r

and

e(r0 + Sr) = ( i 2 + q & f

(2.4.10)

(2.4.11)( £ 2 + * ? & ) ' + r ‘

w ithin the shell of m atter.

Considering the s ta r alone for a moment, v ( R ) = so equation

(2.3.34) gives

»R= m ( R ) + e - 7' (i?) [ eJ' (r)s(r)

Jo4 - r 'p ( r ) +

m O ' ) ’ dr. (2.4.12)

Combining this w ith the integrals

Strh(ro + Sr) = rh(R) + m*——

A r

and

Is{ro + Sr) = I S(R) +L 2£2t 6 r 2r„ A r

(2.4.13)

(2.4.14)

gives, to leading order,

v(r0 + A r) =2 ( M + m t f e )

7-0(2.4.15)

w ithin the shell of m atter. Out beyond the shell, v(r) =


40

Now consider the P ' equation, (2.3.50). If P ' is the force density on the

fluid shell a t r = ro + A r, then the force, F , on the shell is, in leading order,

F =Atttq A r f iv - o

{ 2 bP ~ '2 etp -r

sv 'I 2 ttr2 J

M 2 X4

r5 ( l - ) <■> ( l + 1 0

Thus the acceleration of the test m atter is

M ( 2A f \ _ 1 2M L 2 f L 2a ---- - 1 -------

r o V r o

2 L 2 i 2M(2.4.16)

/ (2.4.17)Tq \ M — ' v J

to first order in the N GT effects. This agrees w ith the previously derived

equation of motion for a test particle in an SSS background metric. This

agreement adds support to the idea tha t P' and not P ' is the force density

in a test body and th a t its mass density is p ra ther than p.

Some additional understanding of the two sets of m atter variables may

be derived from an exam ination of where the difference term =f-S2 comes

from. W i enters into the m atter response equations and therefore the P '

equation. There, it provides the term s which tu rn (p -i- P ) into (p + P ) and.

as well, the other term s proportional to p. It also enters into the energy-

m om entum pseudo-tensor and therefore into E to t. Again, it is W± which

alters (p + 3P ) into (p + 3P ) and also provides the ex tra terms in equation

(2.3.36). It does not enter into the other prim ary field equations (2.3.22)

however, and the v' and 7 ' equations also contain p and P , so the answer

cannot rest entirely w ith W 4 .

Still, consider the solution for W4 . Equation (2.3.22e) can be combined

with equation (2.3.17e) to give

W't = - 1 2 i r u ( p + 3P) + 3


41

(3 + -s)“ VT^e - Y ^ S }

= - 8 * ( V l S y - 12—a; [3p + P -2~r* J (2.4.1S)

and so

1^4 — — 8 —-v/yS 4- .4 4 , (2.4.19)

where

-44Jo

07 3p + P - ar. (2.4.20)2 —r 2

The W 4 field is made up of a purely local p art, which can be w ritten

as — 8 — , and a non-local field. .4 4 . The non-local part interacts w ith

m atter in locations distant from the field point and couples to the conserved

NGT current. It is analogous to the electromagnetic field. The local piece

depends only on the m atter at the position of the field point. It is like the

the electric charge density. The W ^ S 11 term in the Lagrangian breaks up

into .4M<S/i, which is analogous to the electromagnetic field-current coupling,

and yj—g S 2, which is the self-energy of the NGT charge density and is of

exactly the form of the the extra term s in the density and the pressure. It is

the local term in W4 which shifts the p and P to p and P in E tot and in the

P' equation, while provides the extra term s in each case.

This does not explain the p and P term s within .4 4 itself or in the

v' and 7 ' equations. To see this, consider where the local term in W4 came

from. Equation (2.3.22e) relates W4 to i?[i4] and equation (2.3.17e) puts th a t

almost exclusively in terms of which can be rew ritten from equation

(2.3.16) as

rM - ^ T s) • <2-4-21)


Thus the local term in W 4 is traceable back to this local term in r t V

An exam ination of equations (2.3.15) and (2.3.16) shows th a t all the

connection coefficients contain similar sorts of term s, those proportional to

p. They arise from two sources, from e' and directly from the S 4 term in

field equation (2.3.2). The p term s arising from e' can be traced through

equation (2.3.11) to the secondary field equation (2.3.3), which is itself a

m anifestation of equation (2.3.2).

Field equation (2.3.2) relates not only to the derivatives of the fun

dam ental tensor but also direcily to S^. This extra dependence is due to the

W p S 11 coupling in the Lagrangian.. T hat coupling was introduced to provide

the source in equation (2.3.3) and therefore to give an explanation for the L 2

constant of integration in terms of a conserved current. It does more than

this however, producing a source in equation (2.3.2) as well. Frum tha t, it

produces the local term s which arise throughout the field equations.

Derivatives and bilinears of the connection coefficients enter R ^ u, con

tributing S ' and S 2 term s to the prim ary field equations. In the S'

term gives the local term in W 4 . In R n and R 4 4 , the S' term s cancel off

against parts of the ~ terms. It is this which creates the factor of t 2 in

definition (2.3.31) for v. The S 2 terms, both explicit and those arising from

bilinears of 7 ', v' and e! and 5 , combine to cancel the S 2 term s in T ^ .

W ith p and P as the density and piessure it is these variables which are

connected by the equation of state. It is p and P which are positive definite

ra ther than p and P. The feedback in the P' equation does not occur.


Hereafter, the symbols p and P are used for p and P. This simplifies the

subsequent notation and should cause no confusion since the p and P from

never appear again.

S e c t io n 5: I n it ia l D a ta for th e In te g r a t io n

The boundary conditions for the system of differential equations (2.3.52)

are given by specifying a minimum amount of initial da ta at the centre of

the s ta r and by the condition tha t the m atter variables vanish a t the star 's

surface.

The factor of p in each of the equations means th a t to avoid v. e and p

diverging at the s ta r s centre v, z, s, e, and q m ust all vanish there. This is

expected of z and q because of the r- in their definitions and the finiteness

of the m a tte r variables p, P and 5 there. The initial data set m ust therefore

consist of p (from which P can be found through the equation of state), jfy

(from which S can be found) and 7 if it is required. All else vanishes, except

th a t b = t = 1 .

It is not possible, however, to s ta rt the com puter program at exactly

r = 0 because the com puter cannot handle the p in the equations w ithout

help. To get around this problem all the unknown functions can be expanded

out as Taylor series in r about r = 0 and the program started slightly away

from the s ta r’s centre. Equations (2.3.52), along w ith definitions (2.3.10),

(2.3.13) and equation (2.3.11), give relations amongst the coefficients of these

Taylor series so th a t the values of all the variables can be specified by the

central d a ta and the radius r.


The coefficient of r n in the Taylor series of a function, y(r), will be

labelled here as y„, except th a t the central values of the m atter densities are

labelled pc, P c and S c. The equation of state, represented here as P = P (p ),

relates the coefficients of p and P , thus Pc = F (p c), Pi = p '( p c)p1, P 2 =

P'(Pc)P2 + j F " ( p c)pi and so on. The definitions of z and q give z2 = 4ttpc,

q2 = 4ttPc and, in general, zn+2 = 4~pn and qn +2 = 4ttPn.

In order tha t the physical variables be analytic functions of r a t the

s ta r’s centre yi m ust vanish for each variable. By spherical symmetry, y'

m ust be the same for any 9 and p, bu t for the derivative to make sense as

r —*• 0, y'(9, 4>) m ust be the negative of y'{—9, —o). Thus, for any physical

variable, y '(0) = 0 and therefore y1 = 0. The vanishing of the odd coefficients

propagates from here so th a t all of the functions v, e. s, £4, z, q, p, P , 7 and

S have only even coefficients. A few integrated variables like m and i 2 have

only odd coefficients.

The results of solving the Taylor expanded equations can be expressed

most conveniently in term s of the five coefficients z2, 3 2 , 6 2 , -4 and 3 4 . These

in tu rn are expressible in term s of pc and f 2s as: z2 = 4?rpc, q2 = 4tzF(pc),

z4 = (P'CPc) ) - 1 3 4 ,

and

34 = 4trP2 =

Note tha t P2 is always negative regardless of the starting param eters,

so th a t the pressure decreases towards the edge of the star, a t least as far as

the series solution is valid. Since P is a monotonically increasing function of

( -2 + 32) ( + 32 ) + ^232 (2.5.2)


p, the density does the same. This is w hat is intuitively expected and what

is not found for the m atter variables in

In term s of these five coefficients then.

To build any particular s tar it is only necessary to specify the equation of

is also needed. The extra param eter, 7 0 , can be set to 1 initially and its true/

value determ ined by m atching <744 at the boundary. Since only occurs in

the equations the absolute scale of 7 never enters the equations.

D uring an actual run of com puter program NSTAR, the initial data

set is obtained by choosing some small initial r for which the Taylor series

q = Q2 r2 + q^r•4 + . . .

(2.5.3)

sta te (the function F) and the two param eters pc and / e2ff. The solutions

form a two param eter family.

If 7 is included and the longer system of equations is used,

(2.5.4)


46

approxim ation is valid, picking pc and f^s and using equations (2 .5 .3 ) to the

order shown. The results of NSTAR have been shown to be totally insensitive

to the initial radius as long as the chosen r is not larger th an about 1 % of

the final radius of the s tar being built.

In GR, only pc and the equation of s ta te are needed to specify a star.

The additional input of / 2ff completely governs the NGT contributions to

the star. As / 2ff — 0, &2 vanishes, the purely NGT variables e, s and t 2 all

disappear and the usual GR results are obtained. This has been confirmed

by com paring the solutions of program NSTAR, run w ith very small / 2ff , to

the solutions of a similar program used to solve the GR equations.

On the other hand, as / 2ff is increased, all the NGT effects increase until,

when t 2 — - 2 they begin to dom inate the equations. If / e2ff is increased much

further the solutions become unstable, as will be seen in the results shown

in Chapters 3 and 4. Thus, a constraint on / 2ff can be found from < ro:

/eff < 3 m n V 4 ~ f ^ - (2-5*5)

For neutron stars, central densities are in the range 1014 —1016 g /cm 3, so the

m axim um possible coupling is expected to be about 10- 4 6 — 10- 4 5 cm2. For

w hite dwarf stars, the densities are more like 1 0 6 —1 0 10 g /cm 3, which gives a

m axim um coupling of 10- 4 4 — 10- 4 2 cm2. For norm al stars such as the Sun

and DI Herculis the central density is more like 100 g /cm 3 so couplings up

to 10- 3 9 cm2 could give stable solutions. These limits on the couplings will

be examined more carefully in C hapters 3 and 4.

Constraint (2.5.5) can be rew ritten as


47

This can be interpreted as saying th a t NGT stars rem ain stable until the self-

energy term which occurs in the m atter variables in becomes as large as

p itself.

S ection 6: D eriva tion o f th e S tab ility C ond ition s

Not all of the solutions found by program NSTAR represent stable stars.

Although static, some of the solutions would collapse if perturbed slightly.

Stability criteria are developed here which are used la ter to distinguish be

tween stable and unstable solutions found in C hapters 3 and 4. The m ethod

followed here is to allow small periodic deviations from the static equilibrium

configuration. Expanding the deviation out in term s of its normal modes, if

the frequency of oscillation of each of the modes is stable, then the solution

is real. If one or more frequency is imaginary, then the solution grows away

from the equilibrium sta te and is unstable. This procedure will be shown for

Newtonian theory t64-60!, where it is relatively simple.

Consider a fluid for which conservation of rest mass holds,

in the Newtonian case. Here, $ is the gravitational potential which satisfies

Poisson’s equation,

(2 .6 .1)

where p* = y/—g u°po. Newton’s equation in the fluid (the Euler equation)

is

(2 .6 .2 )

§ ’\ i = 47rpo. (2.6.3)


48

In NGT, the equation equivalent to (2.6.2) comes from putting the perfect

fluid energy m om entum tensor into the m atter response equations and is

far more complicated. For more detail, see Appendix 1 , especially equation

(A1.52).

The idea is to take the equilibrium configuration of these therm ody

nam ic fields (for which = 0) and perturb it slightly. Consider the linear

response of the fluid to this small perturbation. If the perturbation grows

away from equilibrium, then the equilibrium state is unstable. If the per

turbation oscillates about equilibrium then the equilibrium state is stable.

From this analysis simple rules for identifying stable stellar solutions arise.

Assume tha t the perturbation displaces a fluid element from its equilib

rium position, r , to a new position x - j - £(x, f). The Lagrangian variation of

any fluid field, Q (x , f), is then

A Q(x, <f, t ) = Q(x -r f , t) - Qo(x, i ). (2.6.4)

where Qo is the unperturbed function. This variation follows the change

in Q for a particular fluid element (as opposed to the Eulerian variation,

6Q {x , t ) = Q ( x , t ) — Q o (x , t ) which looks at the change in Q at position x).

In order to perform the Lagrangian pertu rba tion of equation (2 .6 .2 )

consider the variations of the quantities it contains. F irst,

d r, dx df A „ = - ( x + « - T t = T t (2 .6 .= ).

For an integral quantity over a volume V in the unperturbed fluid,

l y = [ Q0 (x ,i) d3 x, (2 .6 .6 )Jv


49

the pertu rbed integral is,

A I v = I Q ( x , t ) d 3x, (2.6.7)J V + & V

where x —► x -f f shifts volume V to V + A V. It may be shifted back to V by

changing the integration variable to x ' = x — £(£, t). The Jacobian of this

transform ation is \4xr\ = 1 4 - <f‘,i to first order in £ so,

A I v - (A Q 4- Q C , i ) d3 x . (2.6.3)

Since the rest mass of any fluid element is not changed by the pertu r

bation,

A f p* d3x = 0, (2.6.9)J v

which imphes, from equation (2.6.S) and the arbitrariness of the volume V

tha t,

A p ' = - p T . i - (2.6.10)

The to ta l energy density of the fluid is p = po [1 -f e(po5 -s)], where e is

the in ternal energy of the fluid per unit rest mass and s is the rest specific

entropy of the fluid. It is assumed th a t the perturbation occurs isentropically

so th a t A s = 0 and therefore,

A p = —A p0, (2.6.11)Po

where P = Po-g^- For a Newtonian fluid A p ~ Apo — ~PoC,i-

The equation of state of the fluid gives P(po) so tha t

dPA P = -7 —A p0. (2.6.12)

dpo


For a polytropic equation of state, P = K p ^ , this gives = T P / pq. Such

an equation of state holds for white dw arf stars in bo th the non-relativistic

and ultra-relativistic perfect gas regimes. It is also a good approxim ation to

portions of the neutron star equation of state. (See Sections 3.1 and 4 . 1 for

more details.)

To perturb the N G T SSS solution, it is necessary as well to have the

variations of which could be found by perturbing the field equations.

This is far from simple. The perturbation is more than ju st a coordinate

transform ation taking x to x + f . Not only does the location of a fluid ele

m ent change as the fluid oscillates bu t the density, pressure and gravitational

potential of the fluid element all change as the fluid stretches and contracts.

In Newtonian theory, where p* = po, the gravitational potential for a

spherical s ta r and radial perturbations, has the solution

$ (r, t) = — m (r ). + f 4 - T< ^ (2.6.13)r J o

where m (r) = / Qr 4~r,2po(r') dr' is the rest mass integrated out to radius r.

The variation of this is

A<$(r) = m ^ £(r) - f 4<r£(r/)po(r /) d r ', (2.6.14)r do

where <f(r) is the r component of if. It will also be useful to note tha t,

= aad A ( 4 ') = - ^ ) f ( r ) . (2.6.15)r 2

It is also necessary to know how the variations affect derivatives:

(A Q Y = t^ ; [ Q ( r + f ( r ,f ) ,f ) - Qo(r,t)]


51

: Q ( rd(r + f) v ' ' dr dr '

= e ' ( r + e , i ) + < ? '(r+ e ,< )e ' - Q i M

= A (Q‘) + g 'e ', (2.6.16 a)

to first order in £. Similarly,

(AQ),4 = A (Qi4) 4- g'<f,4 (2.6.166)

and, combining these,

d dO- ( A Q) = A - (2.6.16c)

Now, consider the effect of a radial perturbation on equation (2.6.2),

0 = A t F + L p ' + * 'at po(2.6.17)

Using all the results from (2.6.4) to (2.6.16),

0 = ^ p - - ^ P ' + — [(A P)' - P '£ 'j + A ($ ') a t Pg po

- e ( • ' - ;<

(2.6.1S)

Everything in this equation except for £ is evaluated at its unperturbed

equilibrium value, so P ' 4- po§' = 0 and ^ = Jy.

Now consider perturbations of the form,

£(r,f) = f ( r )e IU' t . (2.6.19)

Solutions for f (r , t) w ith real w will oscillate about equilibrium and are thus

stable. Solutions w ith imaginary u will collapse and axe unstable. Equation

(2.6.IS) becomes,

n 2 c ^ P ' ^0 = - u p 0 £ + — P -r r- r p o ^ . (2-6'20)


This equation is in one of the standard Stunn-Liouville forms.

[p(x)y']' + q(x)y + Xw(x)y = 0, (2.6.21)

w ith x = r, y — f ( r ) , X = u 2, w = r2p0, p = r2p0j ^ and q =

2r 2 — ArP' . The weight function, w, is positive definite as required.

The boundary conditions are th a t £(0) = 0 (no discontinuity a t the s ta r’s

centre) and A P (R ) = 0 (the s ta r’s edge, denned as the radius where P = 0, is

correctly displaced from r = R to R~-E,{R)). This last condition is equivalent

to £'(R) + (i?) = 0. These are proper boundary conditions for a Sturm-

Liouville equation over the interval 0 < r < R.

From the theory of the Sturm-Liouville equation there exist eigenfunc

tions, of this equation with eigenvalues, w2, where n = 0 ,1 .2 ,3 ... The

satisfy,r R/ r2p0( m£n dr = 0 m # n, (2.6.22)

Jo

and form a complete set of functions on the interval 0 < r < R. The mode

number, n, counts the number of nodes in f n in the range 0 < r < R. The

LJ2 are all real and satisfy u>2 < oj2+1 for all n.

Note th a t the w2 and their corresponding ^ ( r ) depend on the equilib

rium functions po(r) and P (r ) th a t are being perturbed about. Consider a

series of equilibrium solutions of equations (2.6.2) and (2.6.12), the Newto

nian equivalent of the NGT field equations. For a given equation of state,

they can all be param eterized by p c , the central density of the star. The £„

and w2 generated by these equilibrium solutions can then also be param e

terized by pc. Division of the equilibrium solutions into stable and unstable

stars 'can then be done by examining u;2 as a function of pc.


Any perturbation, £(r), can be w ritten as a linear combination of the

f„ (r) . A particular <f„ is stable if u>„ is real, thus if is positive. The

fundam ental mode of oscillation, fo, is therefore the least stable mode, as

utfi is the smallest of all the eigenvalues. As the mode num ber increases the

modes become increasingly stable. Thus, if wj > 0 for a star, th a t s ta r is

stable. On the other hand, if ujq < 0, the <fo p art of the pertu rbation grows

exponentially, swamping all other modes in if and the star is unstable.

Solutions for which 10% = 0 are critical solutions between stability and

instability. Here fo O', i) = fcr(^) with,

0 = - Z c r P ' ~ 2 ( — J ^ ) £cr - 4 rr V r “Po J r~

2 d P ? ' "Podpo^cr

(2.6.23)

Note th a t the normalization of f cr is not set by this equation and so there

exists a range of such solutions param eterized by this norm alization constant.

Each such solution is static and spherically symmetric and is therefore an

equilibrium solution w ith the same mass as the unperturbed solution but

w ith a different value of pc (since A pc = —pc [f£r + r^ cr] )• Thus, a t a

critical point between stability and instability,

d M= 0. (2.6.24)

dpc

This means tha t, as pc increases through a series of equilibrium solutions, a

turn ing point from stable solutions to unstable or vice versa always occurs

a t a solution which is a maximum or minimum of M .

Note th a t the above argum ent also holds true for modes other than

fo- Every time tha t any mode changes stability els pc increases, condition


54

(2.6.24) is true. If the mode changing stability is not £0 this will no t result in

a change in the stability of s ta r being perturbed. Thus, not every maximum

or minim um on an M vs pc plot is a place where the stability of the solutions

changes.

This condition is in accordance w ith a simple intuitive argument. If a

stable s ta r has a small am ount of mass added to it one expects th a t the

density and pressure within the s ta r should increase. There is more m atter

pressing down on any fluid element, so there must be more pressure within

the fluid to support it. Also, under the added weight of the new m atter the

fluid should compress to a higher density. Thus the central pressure and

density should increase. For any stable star, > 0 and > 0 and

therefore > 0. If fgy < 0 for a solution of the stellar equations, that

solution does not represent a stable star. If pc is increased through the series

of equilibrium solutions, starting from a region of stable solutions, the first

point where = 0 is the point where the solutions become unstable.

This does not mean, however, th a t all solutions with 4^- > 0 axe stable.’ Opc

In general, for a series of solutions with a given equation of s ta te M will rise

and dip several times as pc increases. Not every rise is a group of stable stars,

although every dip is a group of unstable ones.

A further stability criterion can be discovered by looking a t the radii

ot the stars. Consider a perturbation mode, <fn(r), near the centre of a

star. W hen (jn is positive near the s ta r :s centre the m atter near the centre

has m om entarily expanded outward and thus pc has mom entarily decreased.

Now, has n nodes between the s ta r’s centre and its edge. If n is even,


i£n(R) is also positive and therefore the s ta r’s radius has increased. < 0

in this case. If n is odd, is negative, the radius has decreased and

f £ > ° -

This holds in general, but it is not in general true th a t the perturbed

solutions for the s ta r’s structure are also equilibrium (static) solutions. The

exceptions to this are the critical solutions for which one or another of the

up vanish. In a s tar with up = 0 a small perturbation, fn M , of the star

takes it from one equilibrium configuration into another. Thus, the above

radius conditions apply to the equilibrium solutions at critical points, where

At each critical point consideration of allows the determ ination of

which mode is changing its state of stability and whether it is becoming

stable or unstable. If > 0 then an odd num bered mode, £n, has changed

stability. If < 0 then an even numbered mode has changed stability.

S tarting from a solution at some pc which is known or assumed to be stable,

and using the fact th a t the modes m ust change stability in order (because

up < ^n+i so ^ modes with negative ui~ are unstable and all modes with

positive up are stable), it is possible to pick out at each critical point which

mode is becoming unstable or stable.

Consider an example. Assume th a t it is known th a t for pc ju st below

a critical point w? < 0 but > 0. This means th a t modes £o> £i und

£2 are unstable bu t th a t all other modes, £ 3 and higher are stable. At the

nearby critical point only one of two things can happen; either up becomes

positive, in which case £2 becomes stable at the critical point, or up becomes


56

negative, in which case £3 becomes unstable. If < 0 at the critical point

an even num bered mode is changing stability so £ 2 is becoming stable there.

If > 0 then an odd num bered mode is changing stability so £3 is becoming

unstable. Note tha t, in this example, £0 remains unstable across the critical

point so tha t there is no change in the stability of the s tar there even though

there is a change in the stability of one of its norm al modes of perturbation.

Thus, given a series of equilibrium solutions to the Newtonian equations

of stellar structure param eterized by pc, it is possible to determ ine which

of these solutions represent stable stars by examining at each point in

the series where j y = 0. The only other piece of inform ation required

is the stability of a single solution in the series. In the case of the white

dwarfs and neutron stars dealt with in this thesis, it is assumed th a t solutions

begin stable at low densities and only become unstable as pc increases. This

assum ption leads to the prediction of two groups of stable stars, white dwarfs

at lower densities and neutron stars at much higher values.

All of the preceding work, leading up to these stability criteria has been

done for Newtonian theory only. It has been shown to apply as well to the

more complicated case of GR It has not yet been shown whether or not

the perturbation of the NGT field equations leads to a Sturm-Liouville equa

tion, a feature on which most of the previous analysis is based. Nevertheless,

certain conclusions can be drawn from the above analysis.

The intuitive argum ent mentioned above, which leads to the condition

th a t solutions with < 0 are unstable, certainly still applies. Also, the

entire analysis should rem ain valid in the lim it th a t the elementary NGT


charges, /g , are taken to be small. As the NGT charges increase, the pc’s a t

which the solutions become unstable m ay (if the N GT field equations do not

pertu rb into a Sturm-Liouville equation) deviate further and further from

the pc's indicated by the above analysis (which will themselves depend on

/ 5)-

For a given set of / " , the region of stable white dwarfs is given as all

solutions with pc less than the first critical point. Beyond this point < 0

for a v hiie sc these solutions are definitely unstable. This does not rule out

the possibility tha t some of the white dwarfs considered to be stable by the

above criteria are in fact not stable. I t does rule out the possibility th a t

any thought to be unstable are actually stable. The bounds on the NGT /g

charges set in Chapter 3 by the white dwarf analysis might therefore be more

stringent, but not less so.

The situation with neutron stars is more complex. Here, bo th the bound

aries of the region of stability might shift, although only towards reducing

the stable region. As the / 2 increase it becomes more and more likely th a t

a solution is incorrectly classified as stable or unstable. As the effective neu

tron s ta r NGT charge, / 2ff , approaches about 2 x 1 G- 4 5 cm2, the region

labelled as stable by the criteria above closes off completely. There are no

stable neutron stars for this or any larger effective charge. Since the correct

stability boundary can only reduce the size of the stable region it can only

tighten this constraint, if anything. Thus, bounds based on the above s ta

bility analysis will s ill hold. They will simply not be as tight as they might

other, wise be.


58

For the purposes of drawing conclusions about the stability of the stars

being generated by NSTAR, the stability conditions outlined above are used

to eliminate a large class of unstable solutions and to give the approxim ate

position and shape of the exact stability boundaries.


C H A P T E R 3

W h ite D w a rf Stars in N G T

S ectio n 1: T h e W h ite D w arf E quation o f S ta te

W hite dwarf stars are end products of stellar evolution. They have ex

hausted their nuclear fuel and no longer burn. W ithout the pressure of out

ward flowing radiation to support them they have shrunk down to a fraction

of their form er size and are many times their former density. The support

ing pressure which keeps them from collapsing further is electron degeneracy

pressure.

The Pauli exclusion principle says th a t there can be no more than one

fermion in any given state. Thus fermions of the same type jam m ed into

nearly the same volume cannot all have the same energy and momentum.

The density of states in phase space is given by = h f i E ) for a fluid

of sim ilar fermions at tem perature T and chemical potential p, where

= e(E-»)/kT + i (3.1.1)

is the average occupation of a state in phase space at position x and momen

tum p (and therefore energy E = y/p2 -r rnr for fermions of rest mass m).

Note th a t f ( E ) is independent of position and direction of motion.

For a degenerate fermi gas k T ■C (E , p) so f ( E ) reduces to 0 for E > p

and 1 for E < p. All states below energy p are full and all states above axe

empty. T he transition energy, p, between occupied and unoccupied states


60

(the ‘surface’ of the Fermi ‘sea’ in phase space) is called the Fermi energy of

the fluid, E p . A white dwarf sta r is formed of such degenerate m atter.

W ith this simple distribution of energies the num ber density of fermions

isf dAf 2 fPF o , 8~ ,

n = J #Td~P d p = v I i!rp' dp = W PF' (3'L2)where the Fermi momentum, pp , is simply y / E p — m 2 . Similarly, the energy

density is

/ ^ V 9 fPF ________E ¥ ^ i i p = 4 "p V p 2 + m2 dp

Pf E f (p p -t- Ejr j — m 4 In ( PF ' E fh?

ana the pressure is

I f d t f ,3 2 f PFP = z j p VM t d p = w j , 4 i r r

m (3.1.3)

a / p 2 -i- m 2

■jfz Pf E f ( ^ p2f - E 2f ) + m 4 In ‘ E f

dx>

m (3.1.4)

It is this pressure th a t supports the star.

The equation of state used in this thesis for densities below 1011 g /cm 3

is a variation of the above, due to Chandrasekhar. It is a fluid composed

of electrons, protons and neutrons trea ted as three noninteracting degenerate

fermi gases. The to ta l density and pressure of the fluid are ju s t the sum of

the densities and pressures of the three component fluids.

The num ber densities are related by physical constraints. Charge neu

tra lity makes np = n t and ppF = p F. For all atomic m atte r more complex

th an hydrogen, the numbers of neutrons and protons are almost the same, so

LReproduced with permission of the copyright owner. Further reproduction prohibited without permission.

61

101 0 * 1 0 ° 1 0 '

F ig u re 3 .1 The Chandrasekhar equation of state used for white dwarf stars. It is joined to the Mean Field equation of state at 2 X 1010 g /cm 3. The dashed line is the continuation of the Chandrasekhar equation of state to higher densities.

n n = nP and p F = pPp. All three degenerate fermi gases will be described by

a single num ber density, n = n e = np = n n , and a single Fermi mom entum.

P F = P p = f F = P f -

The equation of s ta te is exhibited in figure 3.1. For all densities in the

range shown the rest masses of the protons and neutrons provide over 99%

of the energy density while the degenerate electron gas provides over 98% of

the pressure. This is because, from equation (3.1.3), pa ~ m apZp for p C m a

while Pa ~ p5p / m a. The electron density is suppressed by compared

to the neutron (or proton) density, but the electron pressure is enhanced by

tha t am ount over the pressure due to the protons and neutrons.

A lthough the proton and neutron gases are non-relativis;ic {i.e., p F <

0.1mn)throughout the density range below 10n g /cm 3 the electron gas is only


62

non-relativistic for densities below about 106 g /cm 3. This is unim portan t for

the density since p = (m p + m n)n bu t the behaviour of Pe changes a t this

point. For lower densities P £= P e ~ p~p ~ n3 so, to a good approxim ation,

the equation of s ta te is polytropic, P = K p r , w ith T = | . This T is called

the adiabatic index. It is related to the polytropic index, n, by T = s ^ l .

For densities higher than about 10s g /cm 3, the electron gas has become

relativistic so P = Pe ~ p4F ~ n£ and the equation of state once again

becomes polytropic, this time with T = j .

2

,5 7 i s 10

p ( g / c m 3)

F igu re 3 .2 The local adiabatic index for the Chandrasekhar equation±

3 — 3 •of state compared to T = §■ and F = =■.

Locally, any equation of state is approxim ately polytropic, although K

and r will in general depend on p. The local adiabatic index can be found

from r = d in P / d l n p . taken a t constant entropy. Here, zero tem perature


is assumed so this condition is autom atically m et. F igure 3.2 displays the

evolution of T for the Chandrasekhar equation of state, showing clearly its

change from the non-relativistic | to the relativistic y.

A t p ~ 2 x 1010 g /cm 3, the equation of state shown in figure 3.1 begins

to fall below T = | (the continuation of the T = y behaviour is shown as a

dashed line in that figure). There, the Chandrasekhar equation of state joins

onto the M ean Field equation of state, a description of the denser m atter

found in neutron siam. More will be said about this equation of state in

Section 4.1.

One approxim ation of the Chandrasekhar equation of s ta te is the crude

treatm ent of the strong and weak interactions. The nuclear forces are treated

only peripherally, by making the num ber of neutrons equal to the num ber of

protons. This simulates the confinement of protons and neutrons to nuclei

w ith roughly equal numbers of each type of particle.

Protons and neutrons are nor, confined to nuclei in this model, bu t are

represented by free degenerate fermi gases. This is a good approxim ation

below 1010 g /cm 3, since the free nucleon gases do not contribute significantly

to the pressure and alm ost all of the density comes from the nucleon rest

masses. The largest correction to the density would be the binding energy

of the nuclei bu t this is at most one part in 102.

A nother approxim ation is tha t electrostatic corrections have been ig

nored. These are im portant only at densities lower than about 104 g /cm 3.

This can be seen be comparing the Coulomb energy, E c of the fluid to its

fermi degeneracy energy, E p (excluding the rest mass which is much greater


than both). The Coulomb energy behaves as — where r c is a typical

electron-nucleus separation, say 1 /n = yf-r3 and therefore rc = “ n - 3.

The fermi degeneracy energy, excluding the rest mass, behaves as - in the

non-relativistic regime. The largest contribution comes from the electron gas.

Since p f ~ n? it follows tha t ~ n~ 3 . As the energy density and num ber

density increase the Coulomb energy becomes less and less im portant.

At p = 104 g /cm 3, ^ 7 = —0.02Z. For m atter composed of carbon,

oxygen and lighter elements, which is expected in white dwarf stars, the

degeneracy energy is more im portant than the Coulomb energy down to

about this density. Below this density, the Coulomb corrections become

increasingly im portant and can result in a large correction to the equation of

state. This correction is not taken into account here, and this causes some

inaccuracy in the white dwarf radii. The affected region is restricted to the

outer few hundred m etres of the stars, so the inaccuracy is not too large.

The problem is greatest for stars of low mass and central density, since

a larger fraction of the mass is at low densities. The interesting results from

the num erical modelling occur at the high end of the mass range, however,

from the maximum mass as a function of / ' ff and the redshifts of relatively

massive stars. Thus, the inaccuracy in the radii does not compromise the

conclusions.

S ectio n 2: G R W h ite D w arf Stars

To test tha t the com puter program , NSTAR, with the Chandrasekhar

equation of s ta te gives reasonable results for white dwarf stars, a GR version

with permission of the copyright owner. Further reproduction prohibited without permission.

65

of the program , called NGRSTAR, was w ritten. It differs from th e NGT

version only by having the GR system of equations, along w ith its Jacobian

and series expansion, replacing the more complex NGT versions. It is other

wise identical. This tests the NGT program by showing tha t it can generate

s tandard G R model white dwarf stars. It also defines a standard behaviour

against which the la ter NGT white dwarf results can be compared.

2

0

i i

0104 105 10s 107 108 109 1010 1011

Pc ( g /c m 3)

F ig u re 3 .3 The variation of mass with pc in GR white dwarf stars. The dashed line indicates the region of unstable solutions.

The system of equations (2.3.52) reduces to

v' = — - (u — 8 ~r2p) r

i - = \ { v + Stt r 2P ) (3.2.1)7 ro


66

54 6 7 9 10 11

p c (g /c m 3)

Figure 3.4 The variation of radius with pc in GR white dwarf stars. As befor:- the dashed line indicates the region of unstable solutions.

= - i f ) ’ f i + ^ 2n p + p ) .

Here, the v' equation can be explicitly evaluated as v(r ) = rUtill -with m (r) =

f Qr 4~r/2p(r') dr' . As in the NGT case, however, v was left as a variable in

the system ra the r than being evaluated as a separate integral. This allows

larger radius steps to be taken w ithout losing accuracy in the integrals. It is

also a more valuable test of the program as it more closely resembles the full

NGT version.

The GR solutions cure param eterized by a single initial value, pc, from

which the variables at the starting radius near the centre are determ ined to


1 0 z 1 0 3 1 0 4 1 0 s

R (k m )

F ig u re 3 .5 The mass-radius relation for GR white dwarf stars. For masses less than about 0.5 M q it obeys M ~ R~ 3. Again, the dashed line represents the region of unstable solutions.

be

v =

P

P

y / V 2 - y (-F'(pc)) 1 (Pc -T P c ) ( ~ p c -T P c ) r4

pc - 2x ( r ( p c) ) - 1 (pc d -P c) Q p c ^ P c ) r 2 +

P c - 2 r ( p c ^ P c) Q p c + P c ) r 2 + ...

(3.2.2)

NGRSTAR was used to generate solutions with pc varying from 104 to

1011 g /cm 3. The results are shown in figures (3.3) to (3.7). Figures 3.3

and 3.4 show how the mass and radius vary as pc increases. Figure 3.5

shows M vs R as pc changes. For pc < 106 g /cm 3, where the electron gas

is non-relativistic, the so-called mass-radius relation for w hite dwarf stars,

M ~ P - 3 , is obtained. This applies only to white dwa’fs w ith masses lower


1-3

i~4

i-5

i-8

F ig u re 3 .6 The variation of the gravitational redshift with pc for GR white dwarf stars. The dashed line indicates unstable solutions. Data from two systems. Sirius B and 40 Eridani B, are shown for comparison.

than about 0.5 M q . Also in this regime M ~ plJ~ and R ~ ■ These

dependences axe as expected from Newtonian polytropic analysis w ith T = | .

Above 106 g /cm 3, mass grows less rapidly, peaking a t 2.7 x 1010 g /cm 3.

At this point the stability condition jjj- > 0 is no longer satisfied and all

further solutions are unstable. As pc approaches the m axim um stable density,

the radius decreases more rapidly than before, els p ^ 1 4. The m axim um mass

and minim um radius predicted for this equation of s ta te are

M < 1.42-V/q and R > 975 km. (3.2.3)

This m axim um mass is the Chandrasekhar limit for white dwarf stars with

this equation of state.


69

1.0

0.8

0.2

0.2 0 . 4 0.6 0.8 1.0r / R

a$via

F ig u re 3 .7 Density profiles of three GR- white dwarf stars. The solid line represents a 1.4 M© star, the dashed line 1 JW© and the dotted line 0.5 A/©.

The gravitational redshift of light as it leaves the surface of a white

dwarf s ta r is given in GR by

x = f e « w r ,/2 - 1 = § • (3.2.4)

This increases w ith pc, as pV^ in the non-relativistic regime and as p~J^ at

higher densities. This redshift can be m easured and in cases where bo th

fairly precise masses and radii axe available (usually only for white dwarfs in

binary systems) equation (3.2.4) can be tested. The crosses in figure 3.6 Eire

da ta for two such stars, Sirius B t68-• °1 and 40 Eridani B I69’' 1*'2]. These

redshifts are well described by GR.

The density profiles within three different GR white dwarf stars are

shown in figure 3.7. The solid line shows a white dwarf s ta r near the Chan

drasekhar lim it, 1.4 M q . The dashed line shows a 1 M q star, w ith radius


5000 km and pc = 4.4 x 10' g /cm 3, like Sirius B. The dotted line shows a 0.5

| M q star, w ith radius 10,000 km and p c = 1.7 x 106 g /cm 3, like 40 Eridani B.

? The densities drop off w ith increasing rapidity w ith radius as the mass and

j/ pc increase, giving smaller core regions and larger atmospheres.

i All of this agrees w ith previous calculations I62-63! of white dw arf struc

ture in GR. This gives confidence in both the com puter program and the

equation of state. The next section examines the modifications tha t NGT

\ makes to this structure.

S ection 3: N G T W h ite D w arfs

This section examines in detail the solutions for white dwarfs in NGT

generated by program NSTAR. In figure 3.8, the param eter space in which

stable white dwarfs are found is shown. The three basically horizontal lines

show different series of results for three different fixed masses, 0.5 M q , 1.0

M q and 1.4 M q , as / 2ff is varied. In this figure the G R results are found on

the pc axis.

At each value of f~s there exists a series of solutions akin to the GR re

sults discussed previously. Each series has a m aximal stable solution exactly

as the GR results do. This occurs a t p c = 2.73 x 1010 g /cm 3, as it does in

GR, for values of / / ff up to 5 x 10-44 cm2. As / 2ff increases further, p£*ax

decreases as / 2ff to a power which varies from —1.27, soon after the decrease

begins, to —3/2 in the lim it of large / 2ff . The line of all maximal stable

solutions can be seen in figure 3.S as a heavy solid line. All stable white

dw arf solutions are found below this line.


71

P"3

6

£

1 . 4 M , UNSTABLE

1.0 M, SXABLE.

0 . 5 M ,

B L U E S H IF T E D -^

1 0 ~ 7 1 0 ~ s 1 0 ~ 5 10~4 1 0 ' 3 1 0~ 2 1 0 ~ L 1 0 ° 1 0 1f2eiI (lO"40 c m 2)

F ig u re 3 .8 The parameter space for NGT white dwarf stars. The heavy line shows the maximum stable pc for each f ^ . The dashed line shows solutions with AX/X = 0. The three lighter lines are series of solutions with constant masses 0.5 M q , 1.0 M q and 1.4 M q .

The decrease of the maximum pc and M with shows th a t NGT

decreases the stability of stars. Less and less mass can be supported as /" ff

increases. Recall that all solutions above the stability line are unstable. Not

all solutions below the line, however, are stable. A true dividing line between

stable and unstable solutions m ust therefore lie below the shown, line. The

destabilizing effects of NGT are therefore stronger than seen here.

This destabilization can be understood by considering the forces on test

particles. For a test particle of the same composition as the star, as would

be expected for a fluid element at the s ta r’s surface, most of the NGT effects

cancel out of equation (2.4.16), leaving

Ma = —■ 1 -

2 M -1 2L & f l + J A ' _1(3.3.1)


72

The NGT term s increase the gravitational force, although only slightly. A

s ta r which, in GR, is a t the edge of stability, ju s t barely supporting itself,

would be subject to more inward force and would therefore collapse. Thus,

NGT stars are less stable than GR stars and the maximum stable mass is

lower.

The larger force also means tha t a larger pressure is required to support

a s ta r against gravity, which implies a larger pc and Pc than in GR. This

is borne out by the upturn of the constant mass lines in figure 3.8. It also

means smaller radii for a given mass as the larger force crushes the m atter

further inward.

r<

1

1

1

-2REDSHIFTS OF SOLUTIONS WITH MAXIMAL NGT CONTRIBUTION

1 .4 M.-3

1.0 M,

-4 0 .5 M,

0 ~ 7 1 0 ~ 6 1 0 ~ 5 I Q ' 4 1 0 ~ 3 1 0 ~ 2 1 0 - 1 1 0 °

elf (10~40 cm 2)

F igure 3.9 The variation of the gravitational redshift with / 2ff in NGT white dwarf stars. Shown are the same three series of constant mass solutions as seen in Figure 3.8. The heavier line shows the redshifts of the solutions corresponding to the largest stable pc for each / e2ff .


GR10 " 1 . 4 M ,1.0 M <

/<< 1

0 . 5 M ,

i - 5 SOLUTIONS WITH MAXIMAL NGT CONTRIBUTION

i-S

1 0 ’ 1 0 ' 1 0 °

P c ( g / c m 3)

F ig u re 3 .1 0 The variation of the gravitational redshift with pc. Shown are the dark, thick line of maximal stable solutions, the GR redshifts (dashed line), the three previously seen lines of constant mass solutions and two data points (crosses) from data on Sirius B and 40 Eridani B.

The dashed line in figure 3.S shows the series of solutions for which the

gravitational redshift, AA/A, vanishes. Between this line and the stability

boundary, the solutions all have gravitational blueshifts ra th e r than redshifts.

In GR the gravitational redshift is always positive, given by equation (3.2.4).

In NGT, AA/A is given by the same function of g±± as in GR bu t g±± itself

has changed, so

As / | ff increases, the second term increases until it overwhelms the first and

gives a white dwarf s ta r w ith a gravitational blueshift.

Physically, this means th a t light em itted from the white dw arf’s surface

would increase in energy as it climbed out of the s ta r 's gravitational well.


74

Indeed, the s ta r would have a gravitational ‘hill’ ra ther than a ‘well’. This

would be disastrous for m atter a t the surface of the s ta r were it no t subject

to additional inward attraction because of its NGT charge. I t would be

energetically favourable for it to fall away from the star, leaving behind a

smaller s ta r w ith a gravitational redshift. This does not happen because the

NGT force from W4 (which is not present for photons because photon = 0)

cancels off the other term leaving only the small additional attractive force

seen in equation (3.3.1).

Figures 3.9 and 3.10 show the gravitational redshift as functions of

and pc. The same three lines of constant mass are shown as in figure 3.S,

however, in figure 3.10 the 1.4 M q curve is almost impossible to see. (It is

very short and near pc = 1010 g /cm 3 and AA/A = 10~3.) The dashed line

in figure 3.10 is the line of GR redshifts previously seen in figure 3.6. The

redshifts of NGT stars lie between this line and the line produced by the

m aximum mass solutions. These la tte r solutions represent the m ost extreme

deviation of NGT from the GR results for white dwarf stars.

The redshift curves for stars of each mass begin at the GR curve and

term inate either a t the line of m axim um masses or at AA/A = 0. It is the

low mass stars, represented here by the 0.5 M q stars which allow negative

redshifts. This is because ^ for these stars is smaller and therefore easier

to counterbalance. The 1.0 M q stars have only positive redshifts, decreasing

as /gff increases. The curve for 1.4 M q stars, however, increases from its

GR value, as can seen most clearly in figure 3.9. This is because i 2 does

not become large enough for the NGT term to cause a noticeable decrease


75

in AA/A and thus the decrease in R which occurs near the m axim um pc for

large white dwarfs dominates.

Also shown in figure 3.10 are two crosses, d a ta for white dw arf stars

Sirius B and 40 Eridani B, shown previously in figure 3.6. In the case of

Sirius B, the largest /" ff which gives a redshift which fits the data , AA/A =

(2.97 ± 0.53) x 10- 4 , to within the quoted error is _/ ff = 2.3 x 10-42 cm2.

Doubling the allowed error loosens this to / 2ff = 3 x 10-42 cm2.

In the case of 40 Eridani B, observation has produced AA/A = (7.97 ±

0.43) x 10-0 . The GR prediction fits this at the low end of the range a t twice

the quoted error. N G T redshift predictions are even smaller. To fit within

twice the quoted error would limit / 2ff to / 2ff < 2 x 10-42 cm2. Combining

these two da ta and using the limits for twice the quoted errors gives a bound:

/ 2ff < 2 x 10"42 cm2. (3.3.4).

Note tha t this bound rules out all of the gravitationally blueshifted solutions,

which occur in the stable region only for

/ 2ff > 5.33 x 1CT42 cm2 (3.3.3)

for this equation of state.

Figure 3.11 shows the masses of NGT white dwarf stars as a function of

/ 2ff . Again, the line of maximum masses, the line of zero redshifts and the

three constant mass lines are present in the figure. Note th a t for a s ta r of

1 M q , such as Sirius B, which has M = (1.053 ± 0.02S)Mq, to exist, would

require

/ 2ff < 6 x 10-42 cm2. (3.3.4)


76

2

Q'

01 0~7 1 0~G 1 0~5 1 0~4 1 0~3 1 0~2 1 O '1 1 0 ° 1 0 1

ft* n o - 40 c m 2)

F ig u re 3 .11 The variation of M with / 2ff in NGT white dwarf stars.Seen again are the (heavy solid) line of maximal stable solutions, the(dashed) line of zero redshift solutions and three constant mass curves.

This bound allows for one standard deviation in the mass determ ination. If

the mass were exact the bound would be tighter, / 2ff < x 10-42 cm2. If

the mass is lower by several times the quoted error the bound is loosened to

/ 2ff < 10-41 cm2. This bound is roughly the same as the previous bound.

The precise observational determ ination of a larger w hite dw arf mass would

fu rther tighten this bound.

The radii of NGT white dwarf stars are slightly less than in GR, as can '

be seen in figure 3.12. This is because the negative term s in P ‘ are larger

th a n in GR. driving initially identical p and P down faster. Alternatively, as

m entioned before, the increase in the inward force over GR causes the sta r to

contract. This decrease in radius is not noticeable, however, until very close

UNSTABLE

- 1 . 4 Mt

- 1.0 M,

STABLE: 0 . 5 M<

BLUESHIFTED


77

1 0BLUESHJFTED

SXABLE1 0B Z 1 .0 M,

- 0 .5 M,

1 0i o ~ 7 i o ~ 6 i o t 5 i o ~ 4 i o ~ 3 i o

fix (1 O'40 c m 2)

-2 -1

F ig u re 3 .1 2 The variation of R with r in NGT white dwarf stars.Shown are the same five lines seen in Figure 3.11.

to the maximum stable pc. The radius never decreases by more than about

10% of its G R value, so this cannot be used to pu t a bound on the possible

values of f 2s through the observed mass-radius relation.

Figure 3.13 shows the behaviour of L 2 w ith changing / e2ff . For a given

mass L 2 ~ / 2ff right up until the stability limit is reached. This was to

be expected from equation (2.2.7) where S 11 ~ / 2ffpuM with / 2ff constant

throughout the star. The only feature of this graph tha t could not have been

predicted before this work is the pc stability cutoff and consequent maximum

L 2 value for any given / ' ff . This m aximum L 2 varies w ith and is shown

as the upperm ost line in figure 3.13.

Figure 3.14 shows how L 2 varies with pc. Again the three constant mass

lines, the line of m axim a and the line of zero redshift are shown. W ithout any

IReproduced with permission of the copyright owner. Further reproduction prohibited without permission.

1 0 e

§ 1 0 4

^ 1 0 2

10°1 0~71 0~6 10~5 10~4 1 0~3 10~2 10~x 10° 10 1

f i r (1 0 -40 cm*)

F igure 3.13 The variation of L2 with f 2s in NGT white dwarf stars. Shown again are the (heavy, solid) line of maximal stable solutions, the (dashed) line of zero redshifts and three (lighter, solid) lines of constant mass solutions. Here, the 1.4 curve is so close to the line of maximal stable solutions that it cannot be distinguished.

constraints on , the largest Zr values occur in small white dw arf stars

w ith m axim al f l s and gravitational blueshifts. km. T he constraint that

/gff < 2 x 10~42 cm2 puts a bound on the maximum L 2, making _Lmax = 530

km.

Figures 3.15 and 3.16 shows w hat happens within an NGT w hite dwarf

star. In figure 3.15, the density profiles of white dwarf stars w ith masses

0.5, 1.0 and 1.4 M q , w ith the m aximum possible NGT contribution for those

masses, axe com pared to the same masses of GR stars, seen previousl3r in

figure 3.7. There is very little change. Once again as the mass increases the

profile deflates, dropping more quickly a t smaller radii and then tailing off

more slowly. Larger mass white dwarfs have a smaller core and thicker crust.

BLUESHIFTED

UNSTABLE

STABLE0.5 Mj

■1.0 u .


STABLE

10 i o 4 105 10s 107 108 109 1010 1011 Pc (g /c m 3)

F ig u re 3 .1 4 The variation of L2 with pc in NGT white dwarf staid.The same lines as before are shown from a different direction in parameter space.

The change from GR to NGT has had seemingly paradoxical effects on

the density profiles, inflating the 1.4 M q profile, deflating the 1.0 M q profile

and changing the 0.5 M q profile very* little. W hich way the shift goes depends

on which of several competing effects wins out. The radius decreases in each

case, but more so for the higher masses. An overall decrease in radius pushes

the whole profile outward, inflating it. As well, though, the profile is deflated

by an increase in p c . At low masses both changes are very slight and so the

0.5 M q profile is very much like GR. At the highest masses the increase in

density wins out, bu t a t interm ediate masses it goes the other way.

Figure 3.16 shows four of the variables used in program NSTAR, p(r),

e(r), v ( r ) and s (r) as they vary throughout the 1 M q star seen in figure 3.15.

Up un til about a ten th of the radius, e, v , and s each grow as r ~ , as predicted


80

1.0

0.8

^ 0 . 4a .5

1.01 .40.2

0.2 0 . 4 0 . 3 0.8 1.0r / R

F ig u re 3 .1 5 The density profiles within three NGT white dwarf stars (solid lines) compared to three GR white dwarf stars of the same masses (dashed lines). The masses are 0.5, 1.0 and 1.4 A/@, as in Figure 3.7.

by the Taylor expansions seen in Section 2.5. After this point the density,

which had rem ained close to constant until this point, begins to decrease.

Since this decrease is faster than the quadratic decay used in the expansions

they lose their validity.

After a period of transition in which each of e, v and s level out, they

begin to decrease. Near the edge of the s ta r v is expected to act as M j r

for constant M , and in fact does decrease as r - 1 . Similarly, e is expected

to act as L 4/ ( L 4 + r 4) w ith constant L 4, and does decrease as r - 4 . Also as

expected, s plum m ets dram atically near the edge in exactly the same way as

p does.

In summary, N G T white dwarf stars behave very similarly to GR white

dwarf stars except th a t NGT produces a slightly larger gravitational force on

'£■ '3I i


81

CD

1 0

F igu re 3 .16 Within a 1 M q NGT white dwarf star with maximal NGT contribution. The solid line shows e(r), the dashed line shows v(r), the dotted line shows s(r) and the dot-dashed line shows p(r).

the s ta r’s m atter. This results in higher central densities, smaller radii and a

decrease in stability. These effects increase with / 2ff , although the first two

axe always small. The destabilization becomes increasingly significant w ith

/■ff . Eventually, the maximum stable white dwarf s ta r mass drops below

observed white dwarf masses. The gravitational'redshift is always lower than

in GR, bu t negative values are ruled out by bounds on / 2ff which arise from

comparison with observed white dwarf masses and redshifts. The tightest of

these bounds is / 2ff < 2 x 10-42 cm2. The ramifications of this bound will

be considered in C hapter 5.


82

C H A P T E R 4

N e u tr o n S ta rs in N G T

S e c tio n 1: T h e M e a n F ie ld E q u a tio n o f S ta te

For stars larger than the Chandrasekhar mass limit, degenerate elec

tron pressure is never large enough to counterbalance the force of gravity.

Such stars collapse through the white dwarf stage, becoming ever smaller

and denser until, if they are not too large, the collapse is halted by the in

creasingly im portant pressure of degenerate neutrons. Such stars, typically

stellar masses crushed to little more than ten kilometres of radius, are called

neutron stars.

If even this pressure is not enough to stop the collapse, as is expected

for very massive stars, the collapse continues until the s ta r’s m atter has

crossed its event horizon. The star has become a black hole. A lthough

nothing further can be seen, because all escape from inside an event horizon

is impossible, the collapse can be followed theoretically to its conclusion.

W ithin the event horizon all time-like trajectories converge inevitably on

r = 0 in finite proper time. The s ta r’s m atter gets crushed beyond any

f recognition into a singularity.

1 Such extrem e relativistic objects present a simple exterior geometry un-

i cluttered by the form of the m atter w ithin the black hole. T he structu re of

v neutron stars, however, depends strongly on the equation of sta te for m at-

j. ter a t densities above 1013 g /cm 3. There is no single accepted equation off!tt

I'I;Ii , iReproduced with permission of the copyright owner. Further reproduction prohibited without permission.

state for densities above this. The complications in modelling the interac

tion between nucleons have resulted in m any equations of sta te and varying

predictions of neutron sta r structure.

W hat follows is a brief description of m a tte r in the different density

regimes found in a neutron star t64’. In the ‘surface’ region, whei'j p < 106

g /cm 3, m a tte r is m ade up of a Coulomb lattice of nuclei immersed in a non-

relativistic electron gas. Therm al and m agnetic effects can play an im portant

role in this region. It is not necessary to take these carefully into account,

however, because the equation of state there is essentially irrelevant to the

gross structure of neutron stars. The surface region is typically restricted to

the outerm ost few metres of the star.

The outer crust region contains densities between 10s and 4.3 x 1011

g /cm 3. It is composed of a Coulomb lattice of increasingly neutron-rich nuclei

in a relativistic electron gas. The nuclei become neutron-rich because, els the

‘fermi sea’ of electron phase space states fills up, increasingly energetic beta

decays from neutron to protons and electrons are blocked while the inverse

reactions can proceed normally.

At p ~ 4.3 x lO 11 g /cm 3, called the ‘neutron drip ’ density, it first becomes

energetically favourable for neutrons to escape from nuclei and exist as a free

neutron gas. Ju st above the neutron drip point, the neutron gas contributes

about 20% of the pressure but this soon increases until, by 2 x 1013 g /cm 3,

it accounts for more than 80%. The neutron gas remains non-relativistic up

to p ~ 1015 g /cm 3.

The inner crust lies between thee neutron drip point and p ~ 2 x 1014

g /cm 3. It is composed of neutron-rich nuclei in a relativistic electron gas with


84

the increasingly im portan t neutron gas. In this region the equation of s ta te is

‘softer’ (i.e., has lower pressure a t a given density) than the Chandrasekhar

equation of state. This is because the average inter-nucleon separation is such

th a t nucleons experience mainly the attractive p a rt of the nucleon-nucleon

potential. This lowers the energy density compared to the non-interacting

Chandrasekhar model.

At p ~ 2 x 1014 g /cm 3, roughly nuclear density, the nuclei are crushed

so tightly together th a t the nuclei effectively cease to exist, leaving only a

neutron gas w ith a small m ixture of protons and electrons. Above this den

sity is the ‘neutron gas’ region. The inter-nucleon separation is small enough

here tha t the repulsive p art of the nucleon-nucleon potential dominates, in

creasing the pressure and stiffening the equation of state compared to the

non-interacting Chandrasekhar model.

There are many complications at densities above 2 x 1014 g /cm 3. At

about 8 X 1014 g /cm 3 the electron states in phase space are filled up to 100

MeV and it becomes possible for muons to exist stably in the star. Similarly

the A, A and S hadrons are expected to become stable at such high densities.

The presence of these particles increases the num ber of possible states in

phase space and thus allows denser m atter a t a given pressure th an would

be possible w ith ju st nucleons. This softens the equation of state.

Calculations I73-' 5] have shown, however, tha t the hvperons have only

a small effect on P (p ). O ther calculations t ' 6 have shown tha t neutrons re

m ain the dom inant constituent of m atter a t least up until p = 1016 g /cm 3.

Finally, the effective masses of the hyperons in a dense medium are larger


85

than their masses in vacuum so th a t their appearance is delayed until den

sities higher than previously thought, possibly beyond 10 16 g /cm 3. Thus,

it is not surprising th a t Bethe and Johnson found th a t their equation

of state which incorporated hyperons was very similar to their equation for

pure neutron m atter.

Except in the youngest neutron stars, the tem perature is expected to

be low enough tha t the free neutron gas becomes superfluid. The protons

remaining after the nuclei have broken up in the ‘neutron gas’ regime are also

expected to pair up in a superfluid state. This reduces the energy density

of the fluid by only about 1%, however, so it does not significantly influence

M , £ ' and R. the gross properties of neutron stars which are examined in

this thesis.

Pions appear at densities about twice nuclear density. This softens the

equation of state, adding density w ithout increasing the degeneracy pressure.

Pions are bosons, so if the tem perature is low enough they will undergo

Bose condensation into the zero m om entum ground state, further softening

the equation of state. A reliable calculation of the density of condensation,

and the energy decrease it causes, depends on the detailed nucleon-nucleon

interaction and has not yet been accomplished.

The possibility exists of a phase transition to a state of m atter consisting

of a free quark plasm a co-existing with the nucleon fluid. Although quarks

are confined to mesons and hadrons at low densities they are ‘asymptotically

free’ a t higher densities (smaller separations). It has been suggested tha t at

high enough densities m atter could become a degenerate fermi gas of quarks.


86

Calculations have shown th a t this would lead to a softening of the equation

of state, but the phase transition has been shown [ '' ' 93 to occur well above

the maximum density for stable neutron stars.

Although there is much controversy about the equation of state in the

neutron gas region,, standard equations of s ta te exist describing the surface,

outer and inner crust regions. The recommended equation of state for 7.9 <

p < 104 g /cm 3 is usually taken from calculations by Feynm an, Metropolis

and Teller (FM T)t80l There, m atter is described as a Coulomb lattice of

2 g Fe, the most tightly bound of all nuclei, w ith a non-relativistic electron

gas, using the Thomas-Fermi-Dirac model for the atomic structure.

This model clumps the electron gas into a spherical cloud around each

nucleus, with the nucleus and cloud forming a neutral cell. The distribution

of the electron gas in each cell is given by a local electrostatic potential, V (r),

which is given by the Poisson equation with source —ene. This equation is

solved with boundary conditions tha t V(r) Z e / r as r —► 0 and th a t

the electric field vanishes on the surface of the sphere. Electron exchange

corrections are also taken into account. The electron gas fermi mom entum ,

Pf , can then be solved for as a function of r through E p = — eV (r) -f

P F 2 ( r ) / 2 m e . Note tha t E f remains constant throughout the electron gas.

From this p can be found as the sum of energy density of the electron gas

(calculated from equation 3.1.3), the Coulomb energy within each cell and

the rest mass and binding energy of the iron nuclei. This m ust be solved

numerically.

At densities from 104 g /cm 3 to neutron drip the equation of s ta te of

Baym, Pethick and Sutherland (BPS) is usually used. This is a continua-


87

tion of the FM T equation of s ta te up to 8 x 106 g /cm 3 where the equilibrium

nuclide begins to change from The idea is th a t a t any given density

there is a single type of nucleus which is energetically favoured. An em pir

ical formula is postulated for the energy density as a function of Z, A, n e,

n,v and n n, including Coulomb energy, binding energy of the nuclei and the

degeneracy energies of the free electron and neutron gases. The inclusion of

a free neutron gas allows the neutron drip density to be calculated in this

model.

Below neutron drip all neutrons are in nuclei, so n ,v = n B/A , where n q

is the num ber density of baryons. n e = Z n g /A and n„ — 0. By specifying

rig, a particular density is picked out and the energy density is minimized by

trying nuclei with all possible A and Z (whose binding energies are known).

The density for tha t n g can then be calculated. There is a phase transition

and a discontinuity in p whenever the equilibrium nuclide changes. The

pressure can then be calculated as in Appendix 1, P = po Y^a with

n a the num ber density of type a particles, pu the rest mass density, and

« = (.P ~ Po)/po-

In the inner crust, there are several fairly standard calculations, by

Baym, Bethe and Pethick (BBP) t82l and by Negele and Vautherin (NV)

I83l. The basic idea in both of these is similar to the BPS equation of state

described above. A semi-empirical mass formula is used for the binding en

ergies of the neutron-rich nuclei, based on complex m any-body calculations.

Care is taken tha t in the limit tha t the density of the free neutron gas equals

tha t of the nuclei, the surface energy of the nuclei vanishes and that the

energies of the two components match.

>s


88

The compressed liquid drop model of the nuclei is used, which includes

the volume of a nucleus as a variable to be varied to minimize the energy

density. This and other variations guarantee tha t the equilibrium state at

each density has a minimum energy per nucleon in the nuclei, obeys beta

equilibrium, has the neutron gas in equilibrium with the neutrons in the

nuclei and has pressure balance between the neutron gas and the nuclei. It

is in calculating the exact energy per nucleon for bulk nuclear m atter that

the detailed manv-body calculations m ust be used.

In the neutron gas region, the various model equations of state begin to

diverge from one another. The repulsive core of the nucleon-nucleon potential

is im portant in this region and very little information is available about

it. This results in a -variation in the strength and range of the repulsion

chosen in different models, which can change predicted neutron s ta r structure

considerably.

At about 1015 g /cm 3, non-relativistic many-body Schrodinger equation

calculations break down. It is no longer possible to think of separated nu

cleons interacting via two-body forces. In this region forces are modelled by

the exchange of scalar pions and heavier vector mesons, p and w. The Mean

Field equation of state, which is used in this thesis, is based on this approach.

It assumes that the attraction between nucleons is due to an effective

scalar meson, which is treated in the mean field approxim ation. Although

tensor parts certainly exist in the pion exchange force, a calculation has

shown t84l th a t the combination of all tensor potentials is only weakly spin

and isospin dependent and could therefore be approxim ated by a scalar field.


89

Tlie nucleons move in this mean scalar field and interact via central potentials

created by exchange of tr, p and w mesons. Param eters in these central

potentials are fitted to nuclear scattering data.

Equations of s ta te vary from the ‘soft’ Reid equation of state, based

on phenomenological nucleon-nucleon potentials which are fitted to nuclear

scattering data, to the ‘stiff’ Mean Field equation of state described above.

In general, a stiffer equation of state means a larger maximum neutron star

mass. In individual neutron stars it also means a larger radius, lower central

density and thicker crust.

16120 a 1 0 1 0 1 C

P ( g / c m 3)10 ' 1 0

F ig u re 4 .1 The equation of state for neutron star matter created by joining the Chandrasekhar equation of state, for p < 1011 g/cm , to the higher density Mean Field equation of state. The dashed line is the continuation of the Chandrasekhar equation of state to high densities. The three squares are the boundaries between the surface, outer crust, inner crust and neutron gas density regimes.

In part, the stiff Mean Field equation of state was chosen in order to


90

find the largest possible neutron star masses in NGT. A bound on for

neutron stars is la ter found, based on the m aximum mass being a t least as

large as observed neutron sta r masses. A softer equation of sta te would make

this bound more severe, so the bound found for this equation of sta te should

remain true regardless of which equation of sta te describes the m atter in

neutron scars most accurately.

,3 2

n ,30

12 15 10

F igu re 4 .2 The Mean Field equation of state (solid line) was interpolated from the 21 points shown as small dots. The dashed line shows the continuation of the Chandrasekhar equation of state to high densities.The squares show the boundaries between the outer crust, inner crust and the neutron gas.

The equation of state, as used in com puter program NSTAR, is shown

in figure 4.1. It was generated by talcing the Chandrasekhar equation of

sta te below p = 2 x 1010 g /cm 3. This includes the surface region and most

of the outer crust. More will be said about this choice later. D ata for the


91

C\2

5; i f f 8

12

F ig u re 4 .3 This figure shows the joining of the Chandrasekhar equation of state (dashed line) to the Mean Field equation of state (solid line) by means of a cubic polynomial (dotted line). The smail dots are the first few of the points used to interpolate the Mean Field curve.

M ean Field equation of state (861 were used in the density range 1011 < p <

1016 g /cm “ and interpolated to cover the whole region. These 21 points are

shown as small dots on the curves in figures 4.2 to 4.4. The density range

between the Chandrasekhar and Mean Field curves was bridged by a cubic

polynomial, shown in figure 4.3, which smoothly matches the two curves.

The 21 Mean Field ‘d a ta ’ points are spread not only over the neutron

gas and inner crust density regimes, where the mean-field treatm ent of the

neutron-neutron interactions is used, but also over the upper end of the BPS

equation of state in the outer crust and the NV equation of s tate in the low

density p art of the inner crust. The NV equation of state is based on the

Reid nucleon-nucleon potentials, considered too soft by the creators of the


92

4

INNERCRUST

NEUTRONGAS3

2

1

012 1411 16

p (g / c m 3)

F igu re 4 .4 The adiabatic index as a function of p for the Mean Field equation of state. The dotted line is T = 4 /3 . The small dots show the locations of the points from which the solid curve was interpolated.

Mean Field model. In the region 1013 < p < 1014 g/cm 3. therefore, the NV

nuclear energies are gradually modified by slowly mixing in the Mean. Field

nucleon m atter energies. Only above p = 1014 g/cm 3 does the pure neutron

gas Mean Field equation of state occur.

In figure 4.4, the adiabatic index corresponding to this equation of state

is shown, compared to T = 4/3 . Calculations have shown 64 th a t stability

occurs when the average value of T throughout the s ta r is greater than this

value. Thus, it is not at all surprising tha t there are no stable stars found with

central densities in the range from 10n to 1013 g /cm 3. Once the neutron

gas takes over, the pressure shoots up and the average T quickly rises to

stability. The region of stable neutron stars corresponds closely to the large


93

peak in T in figure 4.4, although for large f~s the stable region contracts and

eventually does not exist at all.

The surface and outer crust equations of state both make little difference

to neutron s ta r structure. The surface layer, in all cases, forms only the outer

m etre or two of the s tar so its equation of state is essentially irrelevant to

the gross structu re of the star. It is less easy to justify the approxim ations

made in using the Chandrasekhar equation of state through most of the outer

crust. The best justification is a posteriori. The results found in the next

section for GR neutron stars, using this approxim ate equation of state, are

in all aspects the same as those found in the references for the Mean Field

equation of state, where a more rigorous equation of state was used in this

density region.

This is not too surprising, since the outer crust density is small enough

tha t it contributes little to the mass. Also it occupies only the outer 2% of

the star's radius for neutron stars with masses above 2 M g , although this

can range up to 17% for a 0.5 M g star. Except in the smallest of neutron

stars the outer crust has little influence on the radius as well.

One of the m ain advantages of using the Chandrasekhar equation of

state is th a t it enables the generation of bo th white dwarf and neutron star

solutions w ith the same com puter program at the same time. This allows a

stability analysis of the whole density regime which confirms th a t there are

indeed onlv two regions of stable solutions. In addition, it is more readih/

calculable than the FM T and BPS equations of state.


94

S ection 2: G R N eu tro n Stars

As w ith the white dwarf equation of state, the first use of the neutron

star equation of sta te is to generate GR neutron stars. This serves to show

tha t the equation of state combined with the program produces standard GR

neutron stars, a test of the program and equation of state. It also produces

a series of GR solutions with which the NGT results can be compared.

Program NGRSTAR, previously discussed in section 3.2, was used to

generate solutions with pc ranging from 104 to 1016 g /cm 3. The results

for the range between 104 and 1011 g /cm 3 are the GR white dwarf stars

examined in section 3.2. The solutions for the entire range are exhibited in

figures 4.5 to 4.S. The entire range is shown so tha t figures 4.5 and 4.7 would

show all the extrem a of M , each of which is necessary to use the full stability

conditions derived in Section 2.6.

Applying the naive stability condition th a t > 0 would suggest tha t

there are three stable regions of stars, since in figure 4.5 there are three rising

sections of the curve. In order to determine stability, it is necessary to look

at figure 4.7 and consider 4^- at each extremum of M . There are five of= d p c

these.

Recall, from section (2.6), tha t a t each extrem um of M a single norm al

mode of oscillation is changing stability. The modes are arranged in a rigid

sequence beginning with £o- Each succeeding £n is more stable than the

previous one (and each has a higher frequency of oscillation under small

perturbations than the previous one). As instability is approached, it is

mode <fo which becomes unstable first and as long as it remains unstable the


103 105 107 109 1011 1 0 13 1015 Pc ( g / c m 3)

F igure 4.5 The variation of mass with p5 in GR stars. The solid parts of the curve are stable solutions, white dwarfs to the left and neutron stars to the right. The dashed parts of the curve are unstable solutions.

solution as a whole is unstable. Finally, it was shown th a t if R is increasing

at an extrem um of M , then an odd numbered mode is changing stability

there. If R is decreasing, then an even numbered mode is changing stability.

The first extrem um , at pc = 2.73 x 1010 g/cm 3, has R decreasing so an

even num bered mode is changing stability. If the solutions below this density

are assumed to be stable, then it must be mode £o which changes stability

here. This means th a t the solutions beyond are unstable.

The second extrem um occurs at pc = 1.59 x 1012 g /cm 3 with R increas

ing. There are only two possibilities, either mode £o is becoming stable again

or mode is becoming unstable. Since R is increasing at the extremum, it

is th a t is becoming unstable. This is the beginning of the second set of


96

1 0 5

?ct:

i o 2

i U .10

1 0 4 1 0 8 1 0 a IO10 1 0 1Z 1 0 14 1 0 16P c (g /c m 3)

F ig u re 4 .6 The variation of radius with p c in GR. stars. The dashedparts of the curve are unstable solutions and the solid parts of the curveare white dwarf stars and neutron stars.

increasing mass solutions seen in figure 4.5. Far from being stable they are.

in fact, even less stable than were the solutions before the extremum.

The th ird extremum, at pc = 5.37 x 1013 g /cm 3, also has increasing R

so mode £i is changing stability again, this time becoming stable. This does

not change the overall stabilitv of the solutions, however.O *> *

In the region between pc = 1.2 x 1014 and 1.37 x 1014 g /cm 3 a very

slight change in pc leads to a large change in both M and R. The masses of

the solutions drop by an order of m agnitude, while the radii first double and

then drop by an order of magnitude. This is a further sign of instability in

this region.

The fourth extremum is at pc = 1.37 X 1 0 14 g /cm 3 w ith R decreasing.

An even num bered mode is changing stability and since the only possibilities


©

^ 1 Cl

i o 1 0 1 0

F igu re 4.7 The mass-radius relation in GR stars. pc increases generally from right to left. The dashed parts of the curve are unstable solutions and the solid parrs are white dwarf stars to the right and neutron stars to the left.

axe £o becoming stable and becoming unstable, it is £o finally becoming

stable again. At this point sill the modes are stable again and the solutions

beyond this extrem um are stable solutions. This is the beginning of the set

of neutron star- solutions.

Finally, a t pc = 1.47 x 1015 g /cm 3 there is an extrem um with R decreas

ing so once again £o and the solutions become unstable. It is not yet possible

to probe w hether there are stable solutions beyond this density. The physics

of m a tte r at such high densities is not well enough understood.

From this set of neutron star solutions, the masses can be seen to range

between 0.1 Mg and 2.7 Mg while the radii range from 390 km to 14 km.

Only the lowest mass neutron star solutions have large radii. By the time M


98

r<<3

12 1 0 14 1 01 0

F igure 4.8 The variation of the gravitational rcdshift with pe in GR stars. The dashed parts of the curve are unstable solutions. The solid parts of the curve are white dwarf stars and neutron stars.

rises to 0.2 JV/q the radius is already down to 20 km and for M > 0.5JV/©,

R = 15 km. Only very near the stability limit does the radius change from

this value and then it only drops to slightly under 14 km.

Evolutionary calculations tend to favour higher mass neutron stars. The

reasoning is that, to become a neutron star, the mass of the original s tar must

be large enough tha t it does not get stopped at the white dwarf stage during

collapse. For this, its core m ust be larger than the C handrasekhar mass limit

of 1.4 M q . Although some of this mass will be ejected in the collapse, it is

not expected tha t so much would be ejected that a neutron s ta r with such a

tiny mass would be left.

The gravitational redshifts are two to three orders of m agnitude higher

than white dwarf stars of the same mass, since the radii are correspondingly


99

1.0

0.8

0.2

0 . 40.2 0.6 0.8r / R

F ig u re 4 .9 Density profiles of three GR. neutron stars. The solid line is a 2.7 A/© star, the dashed line is 1.4 A'/© and the dotted line is a 0.5 A/© star. The square on each curve shows the boundary between the neutron gas core and the inner crust.

smaller. The largest redshift is ^ = 0.55, found at the maximum mass. All

neutron stars larger than 1 M q have redshifts of at least 0.12. These stars

are all highly relatives Lie, an ideal testing ground for gravitational theories.

Figure 4.9 shows the density profiles of three different GR neutron stars.

The solid line shows a maximum mass neutron star. The dashed line shows

a 1.4 M q star with p c = 4.3 x 1014 g /cm 3. The dotted line shows a smaller

neutron star w ith M = 0.5M q and p c = 2.6 x 1014 g /cm 3. Each curve shows

the point where the boundary between the neutron gas and the outer crust

occurs, a t density 2 x 1014 g/cm 3. Except for very small neutron stars the

neutron gas occupies well over half the radius of the star, far more for larger

stars. Beyond th a t radius is the inner crust. The point where each curve


100

appears to merge with the r / R axis corresponds quite well to the boundary

between the inner and outer crusts, at density 4.3 x 10xl g /cm 3. Both the

inner and outer crust layers get compressed into a smaller fraction of the

radius in larger stars. The surface layer, w ith density less than 106 g/cm 3,

is restricted in each star to less than 0.01% of R or a t m ost the outer 1.5 m.

Comparison of these results to previously derived results [64>861 using the

Mean Field equation of state shows th a t the maximum mass and the central

density at which it occurs are the same to w ithin a few percent. The shapes

of figures (4.5) and (4.7) also compare favourably with the equivalent figures

in the references. In the next section, they are com pared to NGT neutron

stars produced with this equation of state.

S ec tio n 3: N G T N e u tro n S ta rs

Program NSTAR was used to generate solutions with pc ranging from

104 to 1016 g /cm 3. The solutions for the range 104 < pc < 1011 g /cm 3 are

the white dwarf stars examined in section 3.3. The solutions for the range

1014 < pc < 1016 g /cm 3 are neutron stars.

Figure 4.10 shows the param eter space for NGT neutron stars. At each

value of / 3ff there is a series of solutions as in the GR case. For <

1.S4 x 10~45 cm3 the stability analysis for each such series is basically the

same as tha t presented in the last section. There turns out to be a region

of stable white dwarf stars, discussed in chapter 3, and a region of stable

neutron stars bounded by a minimum and a maximum mass. The heavy line


101

101S

UNSTABLE

1.0 Ma_____________

0 .5 Mn STABLE

14 ____ i__ i_i i i i i i I____ i__ i_i i i i i i I t 1*111

1 0 ~ 4

f2a (IO-40 c m 2)

F ig u re 4 .1 0 The parameter space for NGT neutron stars. The heavy line surrounds the region of stable neutron stars. The dashed line surrounds a region of solutions with surface gravitational blueshifts. The three lighter lines are series of solutions with constant masses 0.5 M g,1.4 M g and 2.5 M g.

in figure 4.10 shows both of these stability boundaries. As / 2ff approaches

1.S4 x 10-45 cm2, these bounds converge until at tha t specific value of the

effective NGT charge, they meet. For all / 2ff above l.S4x 10-45 cm2, stability

analysis shows th a t no stable neutron stars exist.

For example, at / 2ff = 1.S5 x 10-45 cm2 the stability analysis goes as

follows. There axe seven extrem a where = 0. The first, at pQ = 2.73x 1010

g /cm 3, has R decreasing and mode fo becoming unstable. This is the upper

edge of the stable white dwarf region. The second extremum is at pc =

1.59 x 1012 g /cm 3 with R increasing. An odd mode, £i, is changing stability

so these solutions axe even less stable than those before the extremum. The

th ird extrem um occurs a t pc = 7 x 1013 g /cm 3 with R increasing again, so


102

fi becomes stable again, although these solutions are still unstable. All of

this is exactly as found before in the stability analysis of the GR staxs.

A fourth extremum now occurs at pc = 9 x l0 13 g /cm 3 w ith R increasing,

thus <fi once again becomes unstable. The fifth extremum, a t pc = 1.2 x

1014 g /cm 3, turns stable again since R is once again increasing a t this

exxremum. The mass of the solutions now decreases dram atically from 1 M q

to 0.03 M q reaching a minimum at pc = 1.35 x 1015 g /cm 3 with R increasing.

In series of solutions with lower / 2ff , R is decreasing here, so <fo becomes

stable and the stable neutron stars begin. Here, though, it is f i which once

again becomes unstable. The final extremum, near pc = 4 X 1015 g /cm 3, has

R decreasing and therefore fo becoming unstable. Clearly there axe no stable

neutron sta r solutions here.

For / “ff = 1.84 x 10-45 cm2, only slightly smaller, the story is the same

up to the fifth extremum where became stable leaving £o still unstable.

Then, although the mass again drops from 1 M q to 0.06 M q , it now bottom s

out at pc — 2.16 x 1014 g/cm 3, w ith R decreasing. Thus <fo becomes stable,

the solutions become stable and represent stable neutron staxs. The seventh

extrem um then occurs when the mass peaks at pc = 2.32 x 1014 g /cm 3 with

R again decreasing. This sends £o unstable again and ends the series of stable

neutron stars. There axe two further extrem a, near pc = 2 x 1015 and 4 X IQ15

g /cm 3 which cause first and then £2 to also become unstable.

For this extreme case, the largest / 2ff which produces potentially stable

neutron stars, the masses of these few stable stars all lie a t 0.064 M q . Clearly,

this / 2ff cannot produce the whole range of observed neutron staxs. If a single


103

is to explain all neutron stars it m ust be smaller. Still, this case provides

the first bound on f~s for neutron stars:

/ e2ff < 1.S5 x IO-45 cm2. (4.3.1)

Recall th a t not all neutron stars w ithin the boundary in figure 4.10 are

necessarily stable, although all those outside the boundary are unstable. The

true stability boundary, which would include only stable solutions w ithin it,

lies somewhere inside the displayed curve. If the true boundary were known,

however, it could only make the above bound tighter, so tha t this constraint

on / 2ff would rem ain valid.

■BOUNDARY OF THE REGION OF STABLE NGT NEUTRON STARS

2 .5 M,1.4 M,

- 4-5

F ig u re 4 .1 1 The gravitational redshift of NGT neutron stars. The heavy line shows the redshifts of the solutions corresponding to the boundary of the region of stable stars. The same three series of constant mass solutions as in Figure 4.9 are also shown.


104

2.5 M(GR. 1.4 M.

'REDSj

BOUNDARIES OF THE REGION 'OF STABLE NGT 'NEUTRON STARS- 3

0.5

1 4 16

F igu re 4-12 The variation of the gravitational redshift with pc. Shown are heavy lines corresponding to the boundary of the region of stable stars, the GR redshifts (dashed line), and three series of solutions with constant masses.

The dashed line in figure 4.10 encloses the region of neutron star pa

ram eter space in which solutions have gravitational blueshifts rather than

redshifts. It only intercepts the region of possibly-stable solutions in the low

mass region, M < 0.5M q , with

f l s > 6.S7 x 10-46 cm2. (4.3.2)

These solutions lie at the very edge of the possibly-stable region and axe

therefore more likely to be unstable than solutions further from the stabil

ity /instability line. It is shown below th a t bounds on / e2ff for neutron stars

rule out all such solutions.

Figures 4.11 and 4.12 show the variation of the gravitational redshift

with / 2ff and with pc. Again, the heavy line is the boundary of the possibly-


104

1 0

2.5 Af,

GR. 1.4

'REDSHIFTS

BOUNDARIES OF THE REGION 'OF STABLE NGT 'NEUTRON STARS

• — 0.5 Af<- 4

14

F ig u re 4 .1 2 The variation of the gravitational redshift with pc. Shown are heavy lines corresponding to the boundary of the regiop of stable stars, the GR redshifts (dashed line), and three series of solutions with constant masses.

The dashed line in figure 4.10 encloses the region of neutron star pa

ram eter space in which solutions have gravitational blueshifts ra ther than

redshifts. It only intercepts the region of possibly-stable solutions in the low

mass region, M < 0.5M q , with

/" ff > 6.87 x IO-46 cm2. (4.3.2)

These solutions lie a t the very edge of the possibly-stable -:gion and are

therefore more likely to be unstable than solutions further from the stabil

ity /instab ility line. It is shown below th a t bounds on / 2fr for neutron stars

rule out all such solutions.

Figures 4.11 and 4.12 show the variation of the gravitational redshift

w ith / 2ff and with pc■ Again, the heavy line is the boundary of the possibly-


I 105

stable region, the line of maximal NGT contribution for each . The

dotted line in figure 4.12 shows the GR redshifts. All NGT redshifts lie

between these two lines. The lines plunge off the bottom s of bo th of these

graphs in the region where blueshifted solutions occur.

The three lighter solid lines in figures 4.10-4.12 are series of solutions in

which and pc are both allowed to vary but the mass of the solutions is held

fixed. The lowest line has M = 0.5M q , the middle line has M = 1.4.V /q and

the upper line has M = 2.5M q . As in the case of white dw arf stars, pc rises

from its GR value as /" ff increases, bu t here the rise is more dram atic. The

central density increases by 48%, 65% and 78% in these series of solutions

(from highest to lowest mass). This is to be expected since neutron stars are

more relativistic than white dwarf stars.

Consider a star of a given mass and let the effective NGT charge vary.

As f 2s is increased from GR to the highest possible value which allows NGT

solutions, there are three distinct behaviours for the gravitational redshift. In

low mass stars it decreases as f 2 increases since the increase in L~ outweighs

the decrease in R, so tha t the L4/2 R 4 term in A A/A grows faster than M /R .

For the 1.4 M q case, there is a decrease of 28% from the GR value. For

the smaller 0.5 M q star, as / e2ff approaches its m axim um possible value for

solutions of tha t mass, the NGT term in AA/A passes the GR term in size and

the solutions become blueshifted. In the 2.5 M q case, the redshift decreases

slightly, by S% at maximum. Here the decrease in R dom inates over the

increase in L 2.

Figures 4.13-4.15 show the variation of M , R and L 2 w ith /" ff . As

mentioned before, as /" ff increases for constant mass, R decreases and L2


106

3

^ ~

010~ 7 10~s 10~5 10~4

f 2eIf (1Q~40 c m 2)

F igure 4 .1 3 The variation of M with / 3ff in NGT neutron stars. Seen again are the (heavy solid) line bounding the region of stable solutions, giving a minimum and maximum mass for every , and three lines of constant mass solutions.

increases. As w ith white dwarf staxs, L 2 increases linearly with /" lT right up

until the stability boundary. The maximum L 2 possible for any neutron star

occurs near the middle mass line, at M = 1.551V/©, pc = S x 1014 g /cm 3

and / 2ff = 3 x 10~46 cm2. It is L max = 8.17 km. Although L 2 increases as

the mass of the series of solutions, higher mass series of solutions reach their

stability lim it too soon to have larger L 2. Lower mass series of solutions

rem ain stable longer but are increasing from a lower L2 value and thus still

do not reach this value.

The radii of stars remain almost constant initially, as / 2ff increases. The

smaller mass stars rem ain constant longer before beginning to decrease. This

leads to a greater decrease in radius for the smaller stars as the minimum

: 2 .5 M { UNSTABLE

- 1 .4 M.

STABLE- 0 .5 M,


I 107

1 7

STABLE15

2 .5 M.s 1.4

0.5

UNSTABLE/ \ J ' .. BLUESHIFTED

tla (10-*° cm*)

Figure 4 .1 4 The variation of R with in NGT neutron stars. Shown again are the boundary of the stable region, three constant mass series of solutions and the dashed line below which solutions have AA/A < 0.

radius has dropped off considerably a t high . The 0.5 M q stars decrease

up to 31% in radius, while 1.4 M q staxs decrease up to 19% and for the 2.5

M q stars R decreases at most S%. All of these decreases are larger than the

decreases seen in white dwarf stars, but occur for the same reason.

The NGT contribution to the force for m atter of uniform composition is

attractive. The increase in inward force compresses the s ta r further, increas

ing pc and decreasing R compared to a GR star of the same mass. These

effects, along with the destabilizing effect and lower maximum mass, have

the same signature as tha t produced by softening of the equation of state.

NGT effectively softens the equation of state.

From figure 4.13 another limit on /~ff arises. The pulsar in the binary

pulsar, PSR19134-16, has a mass of 1.4 M q so tha t the effective NGT charge


108

STUNSTABLE

i -7 - 5 - 4

F ig u re 4 .1 5 The variation of I? with in NGT neutron stars. The region of stable solutions is surrounded by the heavy solid line, within which are seen three series of solutions with constant masses 0.5. 1.4 and 2.5 A/0 .

for neutron, staxs m ust allow for stars of at least th a t mass. This makes

/ e2ff < 5 x 10"46 cm2. (4.3.3)

This uses the mass for the binary pulsar derived using the GR equation

of motion. The mass would be somewhat different if calculated using the

N GT equations for orbital motion, but not very different because the NGT

modifications enter as L4/ r 4 where L < 8 km and r is the separation between

the two components of the binary system. A large 30% error has been allowed

to account for this change, however.

W ithin NGT neutron stars the deviations from GR are greater than for

white dw arf stars. Figure 4.16 shows density profiles within three neutron


109

Ei

1 . 0

N G T 1 .4 M,0.8

GR 1 .4 U tN C T

2 .7 M.o 0.6X^ 0.4

G R 2 .7 U ,

0.2N G T

0 .5 U (GR

0 .5 U<

12 16r (km )

F igu re 4 .1 6 The density profiles within three NGT neutron stars (the heavier solid, dashed and dotted lines) compared to GR neutron stars of the same masses (lighter lines). The solid lines are 2.7 M q, the dashed lines are 1.4 M q and the dotted lines are 0.5 M q . The squares show the boundary between the neutron gas and inner crust density regions. The squares on the axis show the edges of the stars.

stars with masses 0.5, 1.4 and 2.7 M q , each with the maximal NGT contri

butions, compared to staxs of the same masses in GR. Each of the solid lines

are 2.7 M q stars, each of the dashed lines axe 1.4 M q staxs and both the

dotted lines axe 0.5 M q staxs. The squaxes on the r-axis show the edges of

the six stars. The squares within the body of the graph show the boundary

between the neutron gas and inner crust density regimes.

There are several things to notice. F irst, the radii of the N GT staxs

are considerably lower than the the radii of their GR counterparts. In the

highest mass case, the difference between the profiles begins deep inside the

star, after which the profiles run roughly parallel to the edge. In the other

1i*[S

&Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

110

a

-1

r / R

F igu re 4 .1 7 W ithin an NGT neutron star. Shown are e(r), the solid line, v(r), the dashed line, s(r), the dotted line, and p(r), the dot-dashed line.

two cases, the difference is spread throughout the star more evenly.

The locations where each of the curves disappear from sight correspond

quite well to the boundary between the inner and outer crusts. The outer

crust and surface density layers lie between these points and the squares

tha t m ark the surfaces of the stars. These density regions occupy the outer

kilometer or less of the stars, less for the higher mass stars. The neutron gas

makes up more and more of the star as the mass increases, due mainly to

the higher initial radius.

Figure 4.17 shows the variables e(r), v ( r ) and s(r) as well as the density

w ithin a 1.4 M q neutron star. As in the white dwarf case, e, v and s all grow

as r 2 in the interior of the star, but here this behaviour lasts much further

out through the star, until about half the radius. This is because the density


is almost constant within this region and only begins to decrease much after

this point. Both M and I r grow as r 3 in this region.

Once p begins to decrease it drops far more rapidly than in the white

dwarf case. As expected, s turns over and follows it down. As in the white

dwarf case, v is decreasing as r~ 1 and e is decreasing as r ~4 a t the s ta r’s

edge. This is consistent with v = 2M / r and e = L4/ ( r 4 + X4).

In summary, neutron stars deviate from GR more than w hite dwarf stars

but in the same ways. The radii are smaller, the densities are higher, the

maximum mass is smaller, the minimum mass is larger and they are less

stable. All these effects increase as / 2ff increases until there are no stable

neutron stars at all. Again there are blueshifted solutions at the edge of the

stable region.

The cores of neutron stars can be described well by the Taylor expan

sions given in section 2.5, since the density is essentially constant in this

region. The / ' lT for neutron stars is bounded by much the same criteria as

those for white dwarf staxs, t ' t here the bounds are about four orders of

m agnitude tighter. The strongest bound is < 5 x 10 46 cm2. In the next

chapter, this bound, and tha t for white dwarf stars, will be examined for

their consequences.


112

C H A P T E R 5

C o n clu sio n s

S e c tio n 1: A p p r o x im a tio n s

For white dwarf stars, / 2ff < 2 x 10-42 cm2. For neutron stars, / 2ff <

5 x 10“ 46 cm2. These bounds can be converted into bounds on the elementary

NGT charges, / 2 and ( / 2 4- /p ), using equation (2.2.9), which defines / 2ff.

These bounds then constrain the £2 charges of others types of stars, most

im portantly the Sun and the young B type stars in the binary star system

DI Herculis.

These constraints follow from analyses which incorporate various ap

proximations. The m ajor ones are: charge neutrality, static structure, spher

ical symmetry, perfect fluid m atter, constant composition, p ~ po, and the

approxim ations built into the equations of state.

The first four are well justified and are commonly made in such mod

elling. The equations of s ta te have been justified in detail in Chapters 3 and

4. The other approxim ations are necessary in order to write <5 in terms

of the m a tte r variables, p and P. The model for S 11 as a combination of

fermion num ber densities, equation (2.2.2), is treated only approximately in

this work.

Even though np/ n n does change throughout a star, for both white dwarf

stars and neutron stars, it is possible to define a useful average / 2f f . For white

dwarf stars, the equation of state used here itself assumes np = n n, which


115

models the binding of nucleons into atoms w ith equal num bers of protons

and neutrons. Thus, np/ n n = 1 exactly for these model white dwarf stars.

For neutron stars the situation is more complex. Inform ation on np/ n n

was not available for the Mean Field equation of state, bu t it clearly varies

from density regime to density regime. The surface region is composed

of ordinary atomic m atter with npf n n ~ 1. The outer crust is composed

of neutron-rich nuclei. The equilibrium nuclide varies from 56Fe with

np/ n n = 0.S7 a t lower densities to liaK r w ith n p/ n n = 0.44 at the neutron

drip density.

In the denser inner crust, a free neutron gas coexists w ith neutron-rich

nuclei. The nuclei themselves continue to become more neutron-rich than

in the outer crust, and there is also a free eutron gas so np/n„ decreases

further. In the neutron gas region, there will be very few remaining protons

and electrons so np/ n n tends to zero.

To quantify this, consider a simple model of the neutron gas density

regime. This is the same model used in the Chandrasekhar equation of state,

except tha t np/ n n is set by demanding beta decay equilibrium,. /

^ T n l + (pnF)‘ = y j m j + {pvF)2 + ^ m 2 + ( p eF )2 , (5.1.1)

where p F = ^(3tt2n a )1//3, instead of modelling ordinary atomic m atter by

enforcing n n — np. This model can be solved exactly, giving

n? 1 [ (P f)4 + 2 m \{pnFf + m j ynn S [ (P f)2((.Pf)2 m n) J ’ ‘

where = m* —m 2p — m \ and m i = [(mn — m p)2 — + m p)2 — m"].


Exam ination of this shows th a t a free neutron gas is only possible above

p = 1.2 x 107 g/cm 3. As p increases, the num ber of neutrons then increases

until np/ n n reaches a m inimum of 1.35 x 10-4 at p = 7.85 x 10u g /cm 3.

This corresponds to the neutron drip point in this model. At p = 2 x 1014

g /cm 3, where the neutron gas region begins, np/rin = 0.004. It rises from

there to 0.05 by p = 1016 g /cm 3. As the density further increases, np/ n n

continues to rise and asym ptotically approaches 1/8 as p —► co.

For neutron stars, then, the surface and outer crust regions have np/ n n

declining from 1 to 0.44, set by the equilibrium nuclide. At the low density

boundary of the inner crust, m a tte r is still entirely in the form of nuclei.

As the density increases further, the fraction of m atter in the form of nuclei

declines and the fraction in the form of a free neutron gas increases. Accom

panying this, there is a sm ooth decline of np/ n n until, at the high density

end of the inner crust, no nuclei are left, and np/ n n is given by its neutron

gas value of 0.004. In the neutron gas region, the pure neutron gas result

can be used, so np/ n n increases from 0.004 to 0.05 a t p = 1016 g /cm 3.

All stable neutron stars are composed mainly of neutron gas and inner

crust m aterial. Their central densities range from just below the lower bound

ary of the neutron gas region, which has np/ n n ~ 0.004, to pc = 1.5 x 101S

g /cm 3, which has np/ n n ~ 0.02. The average density of the s ta r is lower than

the central value, bu t the density remains almost constant throughout much

of a neutron star and then, when it drops, drops sharply. The lower density

regions occupy comparatively little of the s ta r’s volume. Also, there are less

particles in a given volume of the less dense regions. The average of n p/ n n

is weighted very strongly in the neutron gas region and it can be expected,


115

therefore, th a t the average n?/n „ is not much higher than its central value.

The range assumed here is

0.004 < < 0.05. (5.1.3)V n n J jV S

The conclusions do not depend critically on this assumption.

The final approxim ation is p = po. This holds quite well through all of

the white dwarf density regime, which corresponds also to the surface and

outer crust regions in neutron stars. At higher densities, the approxim ation

becomes less exact. The pressure, which is caused by internal energies not

accounted for in po, can be used to m onitor the change. At p = 1011 g /cm 3,

P j p = 0.001 so p is still very close to po- Even at p = 1014 g /cm 3 this has only

grown to P / p = 0.004. By the tim e the density reaches 1015 g /cm 3, however,

the pressure has grown to P j p = 0.33 and, a t p = 1016 g /cm 3, P j p = 0.73.

The increase in P j p between p = 1014 and 1015 g/cm 3 corresponds to the

peak in the adiabatic index, T, seen in figure 4.4.

Since p < po, in general, because of negative binding energies, is

underestim ated at densities higher than 1014 g /cm 3 and NGT effects in neu

tron stars are therefore suppressed by this approxim ation. The N GT effects

should be stronger for a given / 2ff , w ith a smaller / 2ff required to decrease

the maximum neutron star mass to 1.4 M q or to produce a surface gravita

tional blueshift. A calculation which took better account of this would find

a tightening of the neutron star bounds.

The highest central density for which stable neutron stars occur is about

1.5 x 1015 g /cm 3, the GR density upper bound. From this, P j p < 0.42 and


116

one might estim ate th a t th a t p/po is no smaller than 1/2. <SP is therefore

too small by, at worst, a factor of 2. The values of f~s a t which the bounds

occur would not be expected to decrease by more than this factor. Thus, the

neutron sta r bound which would be found from a more accurate treatm ent

in expected to be tighter, bu t not by more than a factor of 2.

I t would be possible to improve on this work by taking the num ber

densities, n„ and np, as m atter variables, instead of p and P. It would be

necessary to calculate p and P in terms of these variables. This would give

an /" ff which varied appropriately with the density and model (2.2.2) could

be trea ted exactly. This is a more difficult problem, and has not been treated

in this thesis.

S e c tio n 2: C o n se q u e n c e s o f th e B o u n d s

Using equation (2.2.9), which defines / 2ff for m atter composed only of

protons, neutrons and electrons, the bounds can be rewritten:

f l + ( / e2 + / p2 )

1 + 771

I n + J / ( / e + f p )

< 2 x 10"42 cm2

1 + ym< 5 x 10 -46 cm '

(5.2.1a)

(5.2.16)

where y is n p/ n n for neutron stars and m = . Here, absolute value

signs have been used since, despite the notation, /" ff need not be positive.

There is no guarantee, for example, tha t neutron stars have the same sign of

NGT charge as the Sun. In fact, if ( / ' + / ' ) is positive and / ' is negative


then it would be quite natu ra l to expect the Sun to have positive I 2 and

neutron stars to have negative I 2.

Equations (5.2.1) can be inverted to give

| / ? + / e2| < 1 f f t s Q + m y ) + J w d O- + m ) ] / ( l “ y) (5.2.2a)

and

If n | < + m y ) + I w d O- + m y ] / ( i — y ) - ( 5 . 2 . 2 b)

Here, / 2v s = 5 x 10-46 cm2 and fw D — 2 x 10~42 cm2.

At the upper bound on y, y = 0.05, | f 2 + f 2 1 < 4.2 x 10-42 cm2 and

| / 21 < 2.1 x 10-43 cm2. For the lower bound, y = 0.004, it gives | / 2 -j- f 21 <

4.0 x 10-42 cm2 and | / 2 | < 1.7 x 10-44 cm2. In the limit as np/n „ —* 0, the

constraint on | / 2 + f 2 1 remains at 4.0 X 10-42 cm2 while the other drops to

| / 2 | < 5 x 10-46 cm2. It is evident tha t |f 2 + f 21 is insensitive to y, bu t | / 2 |

is very sensitive to it.

Regardless of the value of y, the two constraints will always give ele

m entary NGT charges which axe smaller than about 5 x 10-42 cm2. To get

anything much larger than the larger of the two bounds requires a precise

cancellation between two charges of opposite sign. This can work only for

one specific value of y. If a neutron star with a different mass, and therefore

a different average np/ n n , were considered, this cancellation would not take

place and the neutron s ta r bound would be violated.

These limits on the elementary NGT charges in tu rn place limits on the

C2 charge of other types of stars. Of particular interest are the San and the

stars making up the binary s tar system DI Herculis. These are m ain sequence


118

stars, made mostly of hydrogen, w ith at least 20% by mass of helium and

all heavier elements, which makes n?/n „ < 9 . If / 2 and (/* + f l ) are taken

to be a t their maximum values and both positive, then /*ff for these stars is

maximized at 3.8 x 10“ 42 cm2. This, in turn, means th a t

£q < 680 km and I m Her < 1530 km. (5.2.3)

Even if y was allowed to grow as large as 0.25, this would only increase these

bounds to £© < 770 km and £di Her < 1740 km.

Terrestrial type planets are composed of low density atomic m atter so

np/ n n = 1, as for white dwarf stars. Thus, | / 2ff| < 2 X 10~42 cm2 for these

planets as well. This gives

^Mercury < ^ ^ < ° ‘9 k m : (5.2.4)

using £2 = f ; s M / m n which is true to a good approxim ation for equation

(2.2.8) in low density m atter.

S ectio n 3: T h e P erih elion P recession o f M ercury

T he perihelion precession of Mercury about the Sun is observed to be

in agreement w ith the predictions of GR, if these predictions are calculated

assuming a spherical Sun. The rotation of the Sun flattens it, however, and

the resulting mass quadrupole moment enters the precession equations. If

this is large enough, it disturbs the agreement between observation and the

GR predictions.


119

In NGT, the rate of precession of a p lanet’s perihelion is calculated to

be, I88) in radians per orbital period,

Au; = (5.3.1)clp

where,

with,

\ c«(l + eV4)+ 2 G.U2,p {GM q p Y Gj” ^

Here, M q , R g and £q are the mass, radius and £2 charge of the Sun, M p

and 12 axe the mass and £2 charge of the planet, and Jo is the dimensionless

mass quadrupole moment constant of the Sun. Also, p = a(l — e2), where e

is the eccentricity of the orbit and a is its semi-major axis.

From observation, Shapiro. Councelman and King have found tS9l \ =

(1.003 ± 0.005). Anderson and his coworkers find t90J A = (1.007 ± 0.005). In

GR, where I\ QP = 0, inverting equation (5.3.2) then gives J 2 = (1-0 ± 1.7) x

10 " 6 .

There is much controversy surrounding the measurem ent of J 2 ■ Different

measurements have, over the years produced values ranging from 10“ ' to

2.5 x 10- °. From a recent review t91- which tabulates the m easurem ents of

Jo, an average of Jo = (5 ± 6) x 10-6 is found. This can be translated ,

through equation (5.3.2), into

A ’eM ercu ry = ( 1 - 2 ± 1 . 3 ) X 1 0 34 cm4. ( 5 . 3 . 4 )


//©Mercury depends on the difference in the I2/ M ratio between the Sun and

Mercury. Note th a t AT©Mercury m ust be positive to ensure th a t the NGT

effect cancels off the effect of Jo.

Since (/■ + f 2) dominates f 2 in the Sun. planets and white dwarf stars,

all of these bodies m ust have the same sign of I 2 as ( /" -f- / j ). The sign of

i 2 for neutron stars is then determined by the sign of / " , which dominates

there. Since np/ n n is greater for the Sun than for Mercury, it follows that

f 2s for the Sun will then be larger than /" ff for Mercury. This is reflected

in the larger bound on (/gff)© than on {f^s )\vD- This, in turn, means tha t

i 2 j M for the Sim is greater than for M ercury and /{©Mercury is positive, as

required.

Even if the elementary NGT charges occur in such a way as to maximize

//©Mercury (note K q p < £©), it is restricted to

//©Mercury < 2 X 1031 Cm4. (5.3.5)

W ith this //©Mercuryr the NGT term in equation (5.3.2) cannot account for

any significantly larger Jo than GR can.

S ectio n 4: T h e A n om alou s P er iastron Sh ift o f D I H ercu lis

DI Herculis is a binary system of type B main sequence stars, both

around 5 M©. It, too, has a large anomalous periastron precession. Al

though the two stars cannot be visually separated, the spectrum of the sys

tem clearly shows the presence of two stars. Sharp, periodic decreases in


intensity of the spectrum show th a t the orbit of the stars is inclined a t al

most 90° to the E arth , so th a t the stars regularly eclipse one another. Much

more information can be extracted from the spectrum , the radii of the stars

relative to the size of the orbit, the inclination of the orbit, the velocities

of each star perpendicular to the E arth at different points in the orbit, the

relative luminosity of the two stars, their spectral types, and the precession

rate of the periastron of the orbit.

The observed periastron precession rate, based on timing of the prim ary

and secondary eclipse minima with da ta spread out over an interval of 84

years, is = (0.65 =: 0.18)°/century t92 . A different average of the

data , which weights the recent photom etric d a ta more heavily produces

a larger result, Au 0b3 = (0.9 ± 0.2)°/century.

The lowest of these is 50 times the precession ra te of Mercury. More

than half of this due to relativistic effects whereas, in M ercury’s case, the

relativistic precession is only a small fraction of the classical precession. This

system should therefore provide a good test of gravitational theory. Careful

observation I92-96! of the system has determined the im portant details of the

system 's structure.

The theoretical precession rate is a generalization of equations (5.3.1)-

(5.3.3). The classical effects of rotational flattening and tidal distortion be

come more complex for two similar stars in orbit about one another. Thus,

[97-99]

a . . c , = t* 1’ ( A ' 5

+ ^ ( A y

, V i 1 ^ r 1’ f , . M ,la / (e )M 2 + ( T T 5 ) 2 - ^ - ( 1 + i ^

M 2 1 iO{r2) / M l15 /(e )— - + — rr^- I 1 + — -

iV /j (1 — e 2 ) - cl?o y Mo( 5 . 4 . 1 )


12

where the stars are labelled 1 and 2, w ith Mi the mass of s ta r i (i = 1 or 2),

R i its radius, k2 its coefficient of composition t100J (related to the J2 used

above) and its rotational angular velocity. Here also, uj0 is the orbital

angular velocity and

/(e ) = (1 - e5) - 5 ( l + | e 2 + g e1) . (5.4.2)

The relativistic part of the precession remains the same as in equation

(5.3.1) and (5.3.2), except tha t K qp is generalized to ^

K* - + ® ( j f c - j l ; ) - (5-4 '3)

Note th a t for two stars with the same mass and composition K \ 2 = 0 and

the periastron prediction reduces to its GR form.

This la tte r prediction is independent of the i 2 charges of the stars, and

can be tested by looking at eclipsing binary star system where the masses

of the two stars are equal and their spectral types are the same. Since the

stars are the same type and mass, and have evolved together in the same

system it can be presumed tha t they have equal £2 charges too. Three such

systems have been observed, V1143 Cygnus, VSS9 Aquila, and V541 Cygnus

[1 0 1 - 106] _ •j'kg p eriastron precession in each of these systems agrees w ith GR,

and therefore also with NGT.

The classical precession is Auici = (1.93±0.26) ° / century. This assumes

the following da ta for the system: T = 10.55 days, e = 0.489 ± 0.002, M i =

(5.15±0.10)M q, M 2 = (4.52±0.06)iV/o , a = 43.2 AU, Ri = (0.0621±0.001)a,

R 2 = (0.0574 ± 0.001)a, = (3.5 ± l.l)w 0> ^ = (3.8 ± 1.3)w0, k (2x) =


0.0083 ±0.0010 and = 0.0078 ±0.0010. This d a ta comes from references

92-96.

The theoretical GR contribution is A l j g r = (2.34 ± 0.15) °/century.

The to tal prediction in GR is therefore A l j g r t o t = (4.27 ± 0.30) °/century.

This is much larger than the observed periastron shift. Even if the classical

effects were entirely absent, the relativistic shift is still much too large. NGT

produces a retrograde piece in the periastron shift which can make theory

and observation m atch for

Since the masses of the two stars in DI Herculis are not equal, their com

positions axe not expected to be quite the same either. The core tem perature

of the larger star should be slightly higher and burning should proceed at a

slightly quicker rate there. Thus, the larger of the two stars should be more

evolved and possess a higher helium mass fraction, which means a slightly

lower n p/ n n. As zero age main sequence stars, bo th should possess less he

lium per hydrogen atom than the Sun, which has (np/ n n) ~ 7. T he values

assumed here are (np/ n „)i = 7.5 and (npj n n)2 = 8 I10'), values chosen to

agree with stellar model calculations. Combined w ith w ith the m aximum

values of / " and ( / 2 + /" ) , the bound in equation (5.2.3) is more accurately

i i = 1510 km and l 2 = 1420 km. This makes

Note th a t K \ 2 is negative, because the smaller, less evolved s ta r has

larger f~a and i 2 / M . Not only is K \ 2 the wrong sign here bu t it is seven

K 12 = 1.5 x 1037 cm4. (5.4.4)

I \ 12 = -S .7 x 1029 cm4. (5.4.5)


124

orders of m agnitude too small to be the explanation for the anomalous peri

astron precession of DI Herculis.

S ection 5: E x ten d ed M odels for <SM

In the dense m atter a t the hearts of neutron stars the fermi sea is filled

up high enough th a t muons cannot decay, and they therefore exist as stable

particles. They form another degenerate fermi gas in the core of the star,

which adds an ex tra degree of freedom, / “, to the neutron star problem.

Since np = n e + now,

f p nP + + f l n P = ( / p + / e ) nr + ( f t ~ f l )** - (5-6.1)

This makes muons a potential explanation of the low /" ff of neutron stars.

This fails for several reasons.

Calculations have been performed which include muons as an extra

free fermi gas in addition to proton, neutron and electron gases. Charge

neutrality and beta-decay equilibrium relate the num ber densities of the p a r

ticles. Ju st above the threshold density a t which they first appear, the num

ber of muons increases sharply and peaks, bu t n M/ n e never exceeds 0.1, and

remains less than 10-3 . Thereafter n ^ /n e and n ^ /n „ decrease. Muons

never become significant in num ber in the cores of neutron stars.

In order for them to contribute significantly to the neutron s ta r , / 3

would have to be a t least 10 times ( / 3 + f%) or 103 times / 3. To cause an

increase in H? for the Sun and DI Herculis it is necessary to increase ( / 3 -f / 3),


■since those stars axe mostly hydrogen. There m ust be a cancellation between

a large -r f I ) and an even larger f 2 of opposite sign. This causes a

fine tunin 'r problem. The cancellation might occur for one y value, bu t y is

expected to vary between large and small neutron stars. For other values of

y the cancellation would not occur and the bound on for neutron stars

would be violated. Muons cannot, therefore, be called upon to loosen the

bounds on and H e r -

Finally, muons exist only in the cores of neutron stars, not in the outer

regions. If were large enough th a t it made an im portant contribution in

the cores of neutron stars, then there would be a dram atic change in f 2s at

the radius beyond which muons could no longer be stable. The approxim ation

of constant composition used in this work would break down. In th a t case,

the present work can say little about such stars. If the m uon contribution

is small enough tha t the approxim ation is still valid, then their presence

becomes uninteresting, and can be ignored.

Similar arguments hold for other particles, such as the A resonance, the

A baryon and the E baryons. which are unstable under norm al conditions

but may exist in a stable sta te in the core of neutron stars. None of these

particles can contribute to increasing the £2 charges of m ain sequence stars.

Another extension of the model for S 11 was discussed in Section 2.2, the

inclusion of cosmions, or wimps t5 ‘ o9,ios,io9] ^ cosmj0n is the neutrino of a

hypothesized fourth generation of m atter. As the lowest mass lepton of th a t

generation of particles it is stable unless it annihilates with an anti-cosmion,

or weakly interacts with a fourth generation electron (almost all of which are


126

expected to have decayed weakly into cosmions). Thus, cosmions interact

very weakly w ith norm al m atter.

Cosmions could act as the cold dark m atter t108-109) thought by some

to invisibly make up a large traction of the m atte r in the universe. This

is thought necessary because dynamical determ inations of the am ount of

m atte r in the universe show about 10 times more m atter than is implied by

observations of luminous stars. The remaining 90% of the universe m ust be

made up of some kind of dark m atter. If the cosmion mass is in the range

4 Gev < m c < 10 Gev, cosmions could be the missing dark m atter particles.

As a s ta r travels through a cloud of this dark m atter, some of the cos

mions are trapped gravitational!}' in orbits about its core. Although, some

of these annihilate with anti-cosmions and some scatter out of the s ta r’s

gravitational pull, new ones are constantly swept up so tha t the star always

m aintains an equilibrium num ber density of cosmions.

This equilibrium num ber density depends on the composition of the

star. The more heavy nuclei present, the larger the cosmion scattering and

the fewer cosmions can be trapped by the star. Thus, young stars, such as

the stars of DI Herculis, should contain more cosmions per baryon than older

stars, such as the Sun. Compact stars should contain far fewer still, since the

cross-section for collisions with norm al m atte r goes up dram atically as the

density increases. W hite dwarfs and neutron stars are expected, therefore,

to have a much lower cosmion content than main sequence stars.

If cosmions exist in the Sun with n c/ n = 10-11, then enough heat is

transferred out of the core by them to cool the core by several percent. This


127

is not much, but it is enough to decrease the burning ra te of 8B and slow the

production of solar neutrinos. Thus, cosmions may provide a solution to the

solar neutrino problem [6 0 >611.

One problem with this explanation is tha t the annihilation cross-section

of cosmions with anti-cosmions seems to be too large. The num ber of cos

mions is then too low and the heat transfer is not enough to reduce the

output of solar neutrinos. NGT could provide help here.

Looking at the NGT force on a test particle, equation (2.4.16), it is evi

dent that the force between a particle and anti-particle (which has — partjcie)

is - 7 5 - at close approach, while for like particles it almost cancels, leav

ing only . Thus, two cosmions are slightly a ttracted by their NGT

force, while a cosmion and an anti-cosmion are more strongly repelled. This

repulsion would reduce the cosmion/anti-cosmion annihilation rate.

In this extended model for S 11, cosmions are given an NGT charge 1011

times larger than norm al m atter. This enables them to have a significant

effect, even though their numbers are small. A fit can then be found for

the t 2 charges of the Sun and eight eclipsing binary systems, including DI

Herculis, which allows NGT to explain their anomalous periastron preces

sions while respecting the bounds found for white dwarf and neutron s ta r £2

charges.

This fit uses ( / 2 + / 2) = 6.47 x 10- 4 6 cm2, / 2 = 4.6S x 10- 4 6 cm2

and / 2 = S.75 x IO- 3 0 cm2. These values are chosen to give the best fits

to the binary star data. The NGT charges of the norm al m atter could be

set lower and not change this fit very much as they have an effect only on


128

compact stars. The t 2 charge of m ain sequence stars is set almost entirely by

the ir cosmion content, while the i 2 charge of white dwarfs and neutron stars

depends on ( / 2 + / ; ) and f 2. The neutron NGT charge could be lower than

the sta ted value, which comes close to the upper bound, w ithout significantly

altering the fit.

W ith n c/ n = 10- n for the Sun, K q = 1.1 x 1034 cm4. If J2 for the Sun

is as large as 5 x 10-6 , this would fit the observed perihelion precession of

Mercury, whereas GR would conflict with the data. If J2 is smaller, then a

slight decrease in f~ still allows a fit.

The stars of DI Herculis, being younger, axe expected to have a higher

cosmion content. The more massive of the pair, being slightly more evolved,

should have a slightly lower n c/n than the less massive star. The fitt37!

assumes (nc/n ) i = 5.5 x 10-10 and (nc/ n ) 2 = 7 x 10-1 °. This gives —

1.5 x 105 km, l 2 = 1.6 x 10s km and K = 3 x 1039 cm4. The smaller star

has the larger I 2 charge, because of its larger cosmion content. This makes

K positive, which is necessary for the NGT term to explain the anomalous

precession.

The NGT predictions for the three systems, V I143 Cygnus, V889 Aquila

and V541 Cygnus, remain in agreement with the observations because of the

equal masses and compositions of each pair of stars. Four other systems,

w ith unequal masses, have been studied: AS Camelopardus, EK Cepheus,

AG Perseus and a Virginis (also known as Spica). Each is an eclipsing

binary s ta r system with an anomalous periastron shift. In each case, with

appropriate choices for the cosmion abundances, the periastron shift can be


129

fitted by the NGT prediction with f 2 = 8.75 x 10 30 cm2. F urther details

can be found in reference 37.

It should be emphasized tha t cosmions rem ain at this tim e purely spec

ulative. Although their existence would solve several outstanding problems

of cosmology and astrophysics, there is a t present no evidence for their exis

tence.

S e c tio n 6: S u m m a ry

Stable white dwarfs and neutron stars have been shown to exist in NGT,

with the particle num ber model for S ^ . They have lower £2 charges than main

sequence stars. The Sun, for example, may have £q = 3000 km, bu t a one

solar mass white dwarf star can have no more than £w d — 700 km and a one

solar mass neutron star can have at most Avs = S km.

The NGT effects reduce the stability of stars. They decrease the maxi

mum mass of both white dwarfs and neutron stars. They increase the central

densities and decrease the radii. In all these ways NGT acts as if it were in

creasing the gravitational force within the stars. NGT also decreases the

surface gravitational redshift, although the possibility of a blueshift is ruled

out by comparison to data.

If m a tter composed only of protons, neutrons and electrons is considered,

the tight constraints placed on ( / 2 + f 2) and f 2 make the £2 charge of the

Sun so small th a t the NGT effects it produces in the solar system cannot be

observed a t present. Similarly, the NGT term s in the periastron precession


130

of DI Herculis would be so small tha t this could not explain the anomaly.

The inclusion of muons and other unstable particles in the model will not

change this conclusion.

A speculative extension of the model, the inclusion of cosmions, allows

for a fit of both m ain sequence and compact stars, given certain assumptions

about cosmion content of different stars. This could provide an explana

tion for the anomalous periastron precessions of severed eclipsing binary star

systems and, should Jo for the Sun prove to be large, provide a possible

explanation for the perihelion precession of Mercury.


131

A P P E N D IX 1

T h e P erfect F luid in N G T

To derive the energy-momentum tensor for a perfect fluid in NGT, Vin

cent f42l generalized a procedure used successfully in GR. The work presented

here is a further generalization of tha t work. The idea is to write down the

NGT Lagrangian, replacing gliUTfiU by the fluid energy density, y/—g p. and

to include a group of Lagrange multiplier term s which enforce reasonable con

straints on the fluid. The variation of all this will give a prim ary field equation

which looks like, Gflu(W ) = X ^ , from which the energy-momentum tensor

can be identified as, = j ^ X ^ .

T h erm od yn am ic P relim inaries

The fluid under consideration is one composed of many different types

of fermions, labeled by a , with different rest masses, m a, different NGT

charges, / 2, and different rest number densities, n a . A consideration of the

test particle equation of motion, (1.17), would lead to the expectation th a t

particles with different /■ ’s move along different paths and have different four

velocities. It is assumed here, however, tha t strong, weak and electromagnetic

interactions within the fluid act to bind together fluid elements composed of

different kinds of fermions and keep the fluid flowing in a homogenous fashion.

All particles in the fluid are therefore assumed to be moving together w ith a

single four velocity, u'L

A perfect fluid is one characterized by two internal therm odynam ic de

grees of freedom, taken in Vincent’s treatm ent to be the rest energy density


132

of the fluid, po, and the rest specific entropy of the fluid, s. The n a ’s, how

ever, enter into po and in different linear combinations so it is better to

take the entire set of n Q’s as thermodynamic variables rather than simply p 0 .

Treating the fluid in this fashion is more general than V incent’s treatm ent

of it and will lead to the same energy-momentum tensor.

W ith this generalization, the first law of thermodynamics for fluids ele

ments becomes,

Here, p„ is a density which is exactly conserved in the fluid,

w = r = °> o41-2)

making,

m* = / \J—g p .u 4 d3x, (.41.3)J v

constant over each fluid element (the volume V ). s. is the entropy per unit

m«, and therefore per fluid element. Similarly, p /p « is the energy per fluid

element, 1/p* is the volume per fluid element and n Q/p . is the num ber of

type a particles per fluid element. In a Newtonian fluid either po or the

to ta l num ber density of particles, n (= 2 a n “ ) would do for p .. It is only

the conservation of the density which is im portant. Here, p, will be left

undefined for now. It could be a conserved number density or perhaps the

conserved NGT density S , defined by = S u 11, with m , = i 2. One further

constraint will be imposed on p« below.

The first law of thermodynamics relates the change in the energy per

fluid element to changes in the volume, entropy and the num ber of particles of


133

each type in the fluid element, w ith the pressure, tem perature and the various

chemical potentials defined as the constants of proportionality. Rearranging

equation (A l.l) gives,

dp = ( p + P — ^ 2 V-*n a ) + p .T d s , + J 2 ^ d n aV or J P* a

= E E 0 dpmf ce P*t OTIq

where E q is defined as,

dna + n t E ° ^ p *p* o s ,

ds«, (-41.4)

Eo = ^ / i Qn a — p — P. (.41.5)or

The definition of p in term s of the therm odynamic variables is

2?rP = P o ( n a ) [1 + e(s„ ,na )] + ^ - i 5 2(n a ), (.41.6)

where po = ]T)a m an a is the rest mass density of the fluid, e is t l i internal

energy of the fluid per unit rest mass, and S ( n a ) is the conserved NGT

density. The first term , po(l + e), is the energy density due to all forces not

explicitly taken account of in the NGT Lagrangian. It contains all strong,

weak and electromagnetic effects, nuclear binding, Coulomb forces and the

effects of be ta decay. It will be called the non-gravitational energy density.

It is this energy density which is related to pressure by the equation of state.

The final term is not conventional, nor is it included in V incent’s analy

sis. It is included here with adjustable constant k to include the possibility,

explored in Chapter 2, tha t a self-energy term for the S density should be

included in the full energy density of an NGT fluid.


This definition produces an alternative form for dp,

ri \ 4tt d Sm “(1 + e) + m f t ^ + T k s d Z d n a + (p 0 ^ ) d s . . (.-11.7)

Comparison of equations (A l.4) and (A l.7) shows that,

T = — + E ° dpmp, ds , p, d s , ’ (Al.S)

and,

„ , n , de , 47rr n d S , Eo dp,p a — mQ(l + e) + p 0— + - k S - z — + — -r— .o n a 3 d n a p, d n a (.41.9)

M ultiplying equation (A l.9) by n a , summing over a and using equation

(A1.5) gives,

a

Eo, , r ' 9p,+ — > ~— n a - p.P* V „ d n Q

(.41.10)

These equations simplify if

( A l . l l )

w ith the coefficients m* independent of n a and s, . This happens if p, is any

of po, n or S. In this case

Po deT =

p , ds,

F = M E + T ks ( 2 S ^ n° ~ 5 ) (’“ ' 12)

. de 4tt d S m*p.a = m a( \ + e) + Po t.— + ~7rkS-^ d* ^ 0 .^ v 9 n a 3 d n a p,


135

Note th a t the equations also simplify if E q = 0, taking the same form

as in equation (A1.12) bu t with even simpler [±a . This would leave p . un

constrained bu t would eliminate the coefficient of dp, in equation (.4.1.4). A

change in p , would then result in no direct change in the fluid’s energy den

sity (however, p , cannot change without either s , or one of the n a changing

so there would be an indirect change).

The first law of thermodynamics (A l.l) could then have been -written

with the fluid variables taken per unit volume rather than per unit m ,. This

would mean tha t p , is not only conserved but constant. Since [y/—g

would vanish, every conserved quantity would then be a constant in space

and time. To avoid this situation and with 5 in mind as a candidate for p„,

equation (A l . l l ) is assumed from here on.

W ith the model for <5 introduced in Section (2.2) and used throughout

this work

E = s ■a

From equation (Al.12) it can then be seen th a t the k S 2 term in p produces an

identical term in P. This implies that if the non-gravitational energy density

is p (the k = 1 case) then the corresponding non-gravitational pressure is P.

T h e V a r ia tio n a l D e r iv a tio n o f T ^

One begins with the Lagrangian,

£ = y f T g ^ R ^ W ) +

+ 16i?\/—g~p(s*,nQ) -|- 16~yj—g Ai [ g ^ u ^ u " - l]

+ 16-A2 \ \ / —g u^p . (n a )l + (.41.14)U j tfX


136

•where the A’s are Lagrange multipliers. The X term found in the Lagrangian

in V incent’s work has been left out here for simplicity. It allowed the general

ization to ro tating fluids in GR bu t was found by Vincent to be unnecessary

in NGT due to the presence of the WpS* term in the Lagrangian.

The variation of £ w ith respect to each of the Lagrange multipliers y i e l d s

one of the imposed constraints: the norm alization of the four velocity,

g ^ y u ^ u 1' = 1, (.4.1.15)

co-moving constancy of the entropy per unit m ,,

= Sn^ u11 = 0, (.41.16)d'T

and the conservation of m ,,

W = sT u>iP*),n = °- (.41.17)

The variation of £ w ith respect to gives,

g _ j.

0 = — y/—g W ^ S + 32TTyf—g~X\g{iiv)uU + yj—g X^s*^

- 16 x y / - g p . A2(M. (.41.IS)

The variation of £ w ith respect to n a gives,

+ ™ «(i + e) + p o ^ + Y k S ^ a (AL19}

The variation of £ w ith respect to s , gives,

{ V - 9 X3 u M) ^ = lQ~y/~g~~q^T- (.41.20)


The variation of £ w ith respect to W*v gives the usual N G T compatabil-

ity equation. (1.7), since the connection is not involved at all in the Lagrange

multiplier terms. This gives the usual NGT ‘auxiliary’ and ‘com patability’

field equations. (1 .8 ) and (1 -1 2 ).

The variation of £ with respect to g^u, using

and.

gives,

S g ^ = - g ^ g ^ S g ^ ,

g a*Ra3 {W ) + - ^ - W ^ S + 16-p o

16-Ai — l) — 16~p,Xo^u^

g ^ g ^ R ^ i W ) + w - V ^ F ^ u 1'.

(.41.21)

(.41.22)

(.41.23)

This is the ‘prim ary’ field equation from which TpU is extracted, once the

Lagrange multipliers Aj and A2 are evaluated in terms of the m a tte r variables.

Exam ining equation (Al.20) in light of equation (A l.12) shows tha t,

or, using equation (A1.2),

■ £ - (— ) = 16trT. dr \p ,_

Contracting equation (A1.1S) w ith u M gives,

(.41.24)

(.41.25)

2AX = - - W ^ S u * .0

(.41.26)


138

de a d n Q

M ultiplying equation (Al.19) by n a and summing over a gives

p«X2 .llu lt = n a -f po(l + e) + po'y ' n' CC “ a

= , + P + ( .4 1 .27)Or

using equation (A l.1 2 ) again. Substituting this into equation (A l.26) then

gives.

2A: = p -t P -r U v .u * . (.41.2S)

Using the model for S7*, through equation (A l.13), then gives

2Ai = p + P

p . A o . ^ = P 4- P + -W MS u ,‘.! (-41.29)

••

6

Insertion of these Lagrange multipliers back into equation (A1.22), m ultipli

cation by g^agpul\J—g and relabelling indices gives,

GllI/(W ) = 8vgllpgcil/(p + P ) u au0 - g^^P. (-41.30)

Then, since the prim ary field equation with unspecified takes the form,

GpU{W ) = 8 7tTm„, this gives,

TpV = g^pgauip -r P )u Qup - g ^ P . (A1.31)

This is the NGT perfect fluid energy-momentum tensor.

C o m m en t on th e M a tte r V ariab les

Note th a t the p term in L gives rise to both the usual 1 6 7x-J—g p0( l + e)

term and a new term , - k y j —g S 2. This new term can be rew ritten as


i

■ | 139

^ f’[ ^ k g ^ J.„)S'/]y/—g u ^ S using the constraint, g ^ ^ u ^ u '" = 1. This term

can then be combined with the W ^ S 1* term in £ , changing to -f

4:-kg(llv)S !'.

If this change were treated as a redefinition of W the ex tra term

from p would be absorbed. Another term would replace it, however. The

\ \ / —g <7^W [,itl/] part of y/—g g ^ R ^ W ) would also have to be shifted,

creating a new term . — ky/—g g >il' \g { liX)Sx) iI/.

Integrating this term by parts, using the secondary field equation (2.3.3)

and using g ^ ^ u ^ u " = 1 again, turns this term into + ~ j~ ■sf —g~kS2, exactly

the original term. The same is true if is redefined by adding any multiple

of 4~g(ill/) S !/. The extra term in p cannot be absorbed in this way. This is

suggestive, however, especially since such a redefinition, equation (2.4.19),

comes naturally out of solving for W 4 in the SSS case.

E u le r ’s E q u a tio n

Vincent goes on to show tha t using the perfect fluid in the NGT

m atter response equation gives a result consistent w ith the m anipulation of

the above equations. This will be shown here in order to get the NGT Euler

equations. If a small linear perturbation of thestr equations about hydro

static equilibrium takes the form of a Sturm-Liouville equation, as happens

in Newtonian theory and GR, then the stability conditions derived in Section

2 . 6 are exact.

M ultiplying equation (A l.lS ) by ux and taking its divergence gives,

0 = jr N/m‘S'u 'V|,a + 2 W-g'^3(^)u‘'uX]iX ~ W~FP* 2, ux] x

W ~ 9 ^3-s«iMua] a . (A1.36)

IReproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Using the conservation of pm and S'rl and equation (A1.24) this becomes.

0 = - V IT S u XW , , x + [ V = F { p - r P ) g ( MV)Ul' u X],A

V — 9 p*u ^ 2 , ^ a + V —g p - T s . tfl + - ~ V —g a 3 u a s . uxlO T T

= [ v '- F ’O’ t P)^(pV)u "u A] >A 4- £v''z ?"S uAWm,a + y / ^ f p*Ts.'i

- (m ;A 2.AtXA) - 77Z* A2,XuX,H

4" —\ f —g '^ 3 [(•S . ,AUA) , m — S . ,A U A,p ] .

Now using equation (A1.15) and (A1.19) this gives,

(-41.37)

0 = [vc j"(/) -!-P )3(^)U 1'u A] a 4- - V ^ S u H v ^ x + \ f —g~p.Tsm

- v/ = < r X ] nQ

>A

— u ,n ( V —g~P - ^ 2 ,A - f Y g Z ^ - ^ ^ 3 " 5 - , A ) • (.41.3S)

Adding to this u x iti multiplying equation (A1.18) with free index A then

0 = [■/irf ( p Jr P ) 9 ( ^ 1' ^ X] tx + ^ V ~ 9 ~ S (uX w ^,^ + uX,nW x)

<e / l TTr „ 4tt \ dSm “ ( 1 + £ ) 4 + {6W’U + T f c s ) a Z

+ V ^ P - T s . ' H + y / ^ i p ^ r P ) g {\ „ ) U * u x t)t. (-41.39)

Using equation (A1.13) this becomes.

0 = [ y / - T ( p - r P ) 9{ ^ ) u ''uX] x 4- y f F f ^ p 4 - P)g(ax)V-a u x til + y /^ g ~ p .T s . t

~ \ / = i~ (p d -P ),^ + yfT ^Lde 4~

m a ( 1 -r c) ■+■ Pott 4* ~ ~kS~d n Q a,/i

+ ^ v ^ p X . A + 5 ua,mTV’a - S (W „U*)i/t (.41.40)


141

then using,

P.* = 2Or

this simplifies to,

. de 4tr 9 5m a ( l + e) + p o - F —C'T’l q, 0 C/TLq

den a,n + 5*,P>

(.41.41)

0 = [v / = ?’ ( /’ T P ) j M i i ‘'u A] iA + Vc=F '(p + P ) ^ ( £rA)U,ru AiM

- v /-^ - P,» -r ^ 7 = 7 •S’u aT’V[m.a]. (.41.42)

This is the Euler equation for am NGT perfect fluid, in a form which makes

it easy to compare with the m atter response equations.

P u tting the TMJ/ of equation (A1.30) into the m atter response equation

(1.16) yields,

0 = -r { y / ^ f g ^ T ^ ) iV — ^ T g 9VtllT9V -f

= - v/= F ^ P ) ^ - y /z T ( p + P)g<rwii

+ (>/=T(p + - y/^f6"P)iV -i- V^TPg^g^

+ 3 v = g (.41.43)

Using yj—g = \ y / —g g !TUgav,ii and the norm alization of this becomes,

0 = [v/= P~(p + P)£(m*')u ‘/uA] 1a ~ W ~ 9~ p ] tll

+ V ~ F ( P ^ P)g<rxu‘ru x itl + yfPg >flP -f ^ ' / - f W [ , i , u ) S v

= [VZI7 { p +P)g(,nu)y-V^ x] a + +

- v/=FP.M + ^ WM S \ (.41.44)

This is exactly the same as equation (A.1.42). The two methods axe consis

tent.


142

T h e E xp lic it Form o f ^

The Euler equation is most useful when rewritten to give the acceler

ation, of a fluid element. W ith some work equation (A1.42) can be

pu t into this form. Here, v' = u l/ u 4 and, for notational simplicity, u° = 1.

Note th a t ^ and g ^ u)vliv u = (u4)- 2 . Equation (A1.42) can be

rew ritten.

0 = [v/= 5r (p -r -P)(u4)2ff(/lv)u‘/uA] _A + %/^g~(p + P )u Ag(<Tx ^ u ^ v

+ V - j ’b v P y . j u 4 - v'-iT-P./i +

= ^ [ V ^ i p + p ) ^ ) 29(»v)vU] - S T P *

-r y / ~ 9 (P 4- P ) (u 4) ' + u 4^3 9(<t\ ) v v ,h

4 ..A

Contracting this w ith u** gives,

, / • 1 du4 . ,,o u dvl+ a/ w TCp + -P) (^\>- + ^ r " ^ r + (u ‘ )‘sr(>»u

. dP

= | [ ^ F ( , + P)1 -

+ V =FC p + -P) (« '.i + ^ l f ) - (A1'46)

M ultiplication by u4 then gives,

[V=5"(p + (A1-47)


143

This can be pu t into an interesting form by use of equation (A l.2), which

says,

so tha t,d (p /p .) = _ p d ( l / p . )

• 1 dpm p . dr ’

(.41.48)

(.41.49)dr dr

Comparing this to the first law of thermodynamics, equation (A l.l) ,

and using constraint equation (A1.15) gives,

d{nQ/p . )o =

E dna d-r

dr

Pa ~ (.41.50)

The conservation of p. enforces this constraint on the various particle num ber

densities.

M ultiplying equation (A1.45) by where 7 is the inverse of g(pv),

and using equation (A l.46) gives,

dv" d - M ^dt

v ^ (uff) or i~ ~ g^ ) v v

du4

dt + y <" > - )

_r 3(p + P)(u-*)27

Combining the a = i and a = 4 parts of this then gives,

(.41.51)

dv'dt

= ( 7 ( o ,V 7 ( .» ) dg(ttv) „ , _ o; 7

( tP )3 ‘r 9{aj)V v ,/xd t

{P + P)(W ( y + i 5^ ) (.41.52)


144

In a static, spherically symmetric system the radial acceleration reduces

to,

dvr 7( P ' 4- ^ z iP + P ) - (.41.53)dt a(p + P )

In hydrostatic equilibrium the acceleration is zero and this equation reduces

to the P ' equation (2.3.23).


145

A P P E N D IX 2

C onservation Laws and th e T otal E n ergy

V ariational M eth od for F ind ing th e C onservation Laws

In this appendix the Bianchi identities, the m atte r response equations,

the conserved canonical energy-momentum pseudo-tensor and its related

super-potential are examined for a static, spherically sym m etric (SSS) s ta r in

NGT. The total conserved energy and mom entum for a spherical s ta r is calcu

lated in an asymptotically Minkowskian variation of the N G T Schwarzchild-

like coordinates.

In GR, the Christoffel connection and the Ricci curvature can be thought

of simply as useful functions of <7(M„) and its derivatives which appear in the

Lagrangian and in the field equations. The Bianchi identities are then truly

identities, needing only algebraic m anipulation to prove them true, totally

independent of any field equation. This is the point of view taken when

performing the Euler-Lagrange variations of the Lagrangian to obtain the

field equations.

The same Bianchi identities can be derived by using the invariance of

the gravitational Lagrangian and the known tensor natu re of the m etric un

der coordinate transform ations, in short, using general covariance. For in

finitesimal coordinate transform ations, which vanish at spatial infinity but

are otherwise arbitrary, general covariance enforces conditions which are the

Bianchi identities.


146

If the change in the coordinates is instead restricted to be constant

in space and time, a different equation is found, one which guarantees the

vanishing of the divergence of a quantity, which is called the energy-

m om entum pseudo-tensor. It is this quantity which is conserved in the usual

sense of the word (t£„ = 0). Its volume integrals over all space are indepen

dent of time and can be used to define the total energy and mom entum of

the field solutions being considered,

p n = J Tt • U 2-1)

Finally, if the change of coordinates is restricted so th a t its second deriva

tives "vanish, although the first derivatives do not, then an equation is derived

which shows th a t the pseudo-tensor is itself the divergence of a quantity, A£a ,

which is called the super-potential. Using this fact, the conserved volume in

tegrals can all be rew ritten as surface integrals at spatial infinity and their

calculation simplified. Only the values of the fields at spatial infinity, far out

side m atter, need to be known, not the complicated structure of spacetime

when m atter is present.

Note th a t bo th the conservation of the energy-momentum pseudo-tensor

and its relationship to the super-potential can be found directly as identities

in the same way tha t the Bianchi identities themselves can. The method

of infinitesimal coordinate variations only serves to isolate and select these

identities. It is not essential for their derivation.

The Palatini approach to obtaining the field equations from the La

grangian leads to another picture. Here, the metric and the connection are


147

independent fields. The variation of the Lagrangian with respect to the con

nection produces the compatibility of the m etric and the connection at the

level of a field equation. Here the Bianchi identities are not identities as

such, bu t ra ther a consequence of the compatibility field equation. (They

do rem ain independent of the prim ary field equation, however.) After the

com patibility field equation has removed the independence of g ^ u) and

this approach becomes identical to the Euler-Lagrange approach.

In NGT, the connections cannot be expressed in term s of g in any

simple way, thus the Palatini approach is the only avenue available. The

variation of the NGT Lagrangian,

L = g - S- g ^ % v + Y W ftS^ ('42’2)

with respect to W ^u produces the NGT ‘comp at ability’ field equation, which

can be w ritten as,

s ^ . a = - g ' n . + (-4 2 .3 )

Here the W connection has been decomposed into the vector-torsion-free

connection, r£ „ , and the vector field, WM, which does not enter into this

equation.

Equation (A2.3) gives the 64 components of g ^ A in term s of the 60

components of and the 4 S ^ . It can be inverted to give S * =■ ^ g ^

and sixty equations (which cannot easily be put in closed form) for r* „ as a

functional of the remaining 60 components of g^.A-


Using equation (A2.3) all the derivatives of in the Lagrangian can

be clustered together into a single to tal divergence term so that.

9C = £ 4- U \ p + p [ H U W ] + M atter, (A2A)

where,

r p F a — F a F p1 tMT1- p v 1 p.uL (crp) (.42.5)£ = f "

and.

l £ = g ' T * , - g ^ r f ^ . (.42.6)

The final two terms of Lagrangian (A2.2) contain all the m atter fields and

have been grouped together into £ matter- The rest of the Lagrangian is

present even in vacuum and will be called the gravitational Lagrangian. C e

lt is the variation of C c which allows the identification of the Bianchi iden

tities, the canonical pseudo-tensor and the canonical super-potential. Note

tha t, as in the Palatin i approach to GR, once the ‘com patibility1 field equa

tion is specified the rest can be found by algebraic m anipulation alone. The

variational m ethod is a useful guide but is not essential.

Under the infinitesimal transform ation x>* —► + £M(r) the Eulerian

variation of a quantity Q(x) is,

SQ = Q'(x) - Q(x)

= (Q '( x ' ) - Q ( x ) \ - (Q '( x ' ) -Q '( x ) )

= A Q - Q , ^ , (.42.7)

to first order in f**. The first term , A Q, is called the Lagrangian variation

of Q and is determ ined by the tensor character of Q, by its transform ation


149

law under coordinate transformations. In particular, since

dxAg'11' = dx'

dx * dx u a3 d x a d x @2

— J . a>l a f v — o ^ P *— & *> .» 1 o ? ,a o *» ,ai (.42.8)

to first order in and thus,

st u = - g ^ r . a - g ^ . a r * ^ 2 .9 )

Similarly, because of its co-vector nature, A = —Wa £“ ^ and so,

s w M = - w Qc , » -

= - 2i'7[#i,a]r . (.42.10)

A vector density, such as transforms as,

<55" = S Q^ , a - ~ (.42.11)

Any scalar density, such as L q . transform s as

8Cg = - L c C . a - = ~(jO'Gn a - (.42.12)

Note tha t this is a to tal divergence and thus the variation of the action

J C c d*X over all spacetime is zero, for any f “ which vanishes a t the boundary

of spacetime.

Finally, it is also useful to note tha t in general = (8Q) . The

Eulerian variation always commutes with the partial derivative operator.


150

To consider the variation of £ ', first use the variation of equation (A2.3),

S |" ,A = - f K * ~ r ST^ + s r ^ i - - r L « s "

+ - Y ^ s s “ - 6^ s ^ ’ (--12-13)

to find tha t,

+ s i r ^ , ) - r « s " .» =

Thus,

where,

and.

s c + s g " ( r ' 7r ; „ - r ; „ r f , f)) . (.4 2 .1 4 )

« £ ' = + i j . i g " , a , (.42.15)

d Cr = . = _r? ■p0’ 4. r (r^PSf — O ____ “*- llff-1- ou T A /II/-*- (

X/A =pi/ — = " rh + § + «X~>) . (42.1C4)

(.42.16a)

o g ^ A

From these definitions it can be shown th a t . R ^ r ) = — L ^v

C = —g ^ L p v = 2gli‘/ 1\ Z A„ and ZYA = — Usi ng these relations and

equation (A2.S), equation (A2.15) can be rew ritten els,

S C = R ^6 g > lv - L x 6*‘tv

= AaZa + + [cAr + ©2Ar./*]iA. (-42.17)

wnere,

•Aa — —Rfivg*1 ,o

B* = R „„g* + Ra, - Rr, g “ 6’

c f = - l L sT *

(.42.18a)

(.42.1S6)

(A2.13c)

and,


151

V 3X = L x0f 3 + L x a g3tr + Ux 8 l (A2.1M)

Now consider the variation of U * itr. Like C \ lAa can be thought of a a

functional of g”u and g'1" t\ alone so,

m ° = u% 8 f " A U °x 8 g \ x , (A2.19)

with,

V ’ = — and ET* = A H - M , , m” d ^ u ~~ dg— g 1 ' •

and, for the same reason,

Now, using equations (A2.9), (A2.19) and (A2.21),

GU\„ = {8U °)a

= [-W,aa r + V 'V , * + V ^ - 0 . 7] |(r, (A2.22)

v ?* ee A C/Q% g ^ - x

+ ^ V ",X - ~ u * j g \ a (.42.23a)

ee L ^ g 1"3 A U H g v - (-42.236)

where.

and.

The forms of these V functions can be shown to be quite simple by-

com paring equation (A2.22) to the results of pu tting equation (A2.9) and

the equivalent transform ation law for a connection into definition (A2.6).

r -\ ' = - d x ° dx? d2x' A ( 4 9 041d x f d x A d x '” °-3 d x » d x ” d x ° d x ? ’ K -}


so, to first order in f Q,

= r “^ \ Q - r U a,^ - r £ tte % + - £ \„ ,„ . (-4 2 .2 5 )

This gives,

SU9 = U aZ9,a - U9 ,a - U * e , a + - g{9fl)C , ^ - (-42.26)

Comparison of this to equation (A2.22) then shows tha t,

- Ux8?, (.42.27a)

and,

yM-r = gP-t)6* _ g ( ^ ) 57. (.42.276)

The th ird and final term in Cg is fg^ 1 W[/i,j/], which is itself a tensor

density. From equation (A2.12) the variation of this term is,

= - f ( g M w W ]{ “) • M 2-28)I®

P utting all three of these contributions together, and using equation

(A2.12),

8£-G = - £G ,a tQ - ,?

= ( • * . + c i , - u ’ , . , . - | { i ^ w {ll. ^ j r

+ ( b> + cl + v t \ x - w * . + v f , „ - | s3

+ W + V ? + v : * \ r ) C , 0 , v + VW-'C,0n.6- (A2.29)

This equation m ust hold true for arb itrary transformations, £Q.


153

Three types of coordinate variations will be considered here. The first

type are variations which vanish on the boundary of spacetime bu t are oth

erwise arbitrary. These generate the Bianchi identities. Only term s in SCg

which can cannot be w ritten as to tal divergences will contribute here so,

elim inating all such terms equation (A2.29) becomes,

0 = ( a R- I cCt (-42.30)

Since this must be true for arbitrary it can be rew ritten using definitions

(A2.1S) us,

0 = Raag*3 A R aa g' — R o-'"' (A2.31)

which are the NGT Bianchi identities.

Use of the field equations (2.3.1) and (2.3.3) then turns this into the

m atter response equations,

0 = A T ^ \ q a (.-12.32)

The second type of variation is global or constant, with = 0. Such

variations generate the conservation of the pseudo-tensor. From equation

(A2.29),

= M a A c i f - r , (-42.33)

which mtist be true for all constant £Q. From definitions (A2.IS) and equa

tions (A2.30) this gives,

0 = \ ( g A g^ R va - r / uR , X ) - + C 5Z I , (-42.34)L V~ ~ J ~ , tf}


154

winch shows th a t the pseudo-tensor denned by,

le r r r f = g^ R QtT + g^ R aa - g ^ R ^ S * - L % f \ a + C'S* (.42.35)

• 3is conserved in the usual fashion: Ta $ = 0. Application of the field equations

to equation (A2.35) then produces,

3 -*-ocr i o -L o ,or 12 t t 2 16 t t ~

+ si 1 4 - c -16 -(.42.36)

The th ird type of variation considered here is one for which = 0.

P u tting this into equation (A2.29) and using equation (A2.33) gives,

-CaSir. ) = ( + Ct 4- v i \ , - U\, V U)3

(.42.37)

This m ust be true for any j so, using definitions (A2.1S) and (A2.23) and

equations (A2.27) and (A2.35) this gives,

r a = a *A, a,

where A^a is the super-potential, defined by,

- 1

(.42.3S)

A£A =16 -

<,*0 n_ l x <rPa a . UXS&^v/rn 4*'1 /v (.42.39)

E n e rg y in th e S ta t ic , S p h e ric a lly S y m m e tr ic C ase

So far, all of this is quite general, not referring to any specific sys

tem. Now it will be specialized to the case of static, spherical stars made


155

of a perfect fluid. It will prove necessary to work in coordinates which are

asym ptotically Minkowskian at spatial infinity to assure the convergence of

some integrals and the vanishing of others. For example, in order th a t the

energy integrals. P^, be time independent,

= 0. (.42.40)

In Schwarzchild coordinates, however, this surface integral does not vanish.

A nother example is tha t in Schwarzchild coordinates f C d3x diverges, so

P 4 itself is not finite.

A coordinate transform ation must therefore be performed between the

usual Schwarzchild-like coordinates in which the rest of this work is done

and coordinates which give a Minkowskian m etric at spatial infinity. This is

accomplished simply by the usual transform ation from spherical to Cartesian

coordinates, x1 = x = r sin 9 cos p, x~ = y = r sin 9 sin 6 and x3 = x = r cos 9.

In the new coordinate system, the m etric has components,

x 'x 29(ij) = ( 1 - a ')- )3 S'i ’ 9[ij] = 0,

x 1*<7[i4] = 5 9(H) =

5-44 = 7 , v ' - F = F q t(1 - e) , (.42.41)

with a , 7 , u.’ and e being the SSS metric components used in C hapter 2.

They remain functions of the Schwarzchild r which is now defined by r =

v /x 2 A tr + ~ 2-


156

The components of g^v are then,

g (ij) =

<744 =

1 -

a ( l — e)_X X

r-- 6 tJ [if] -= 0 .

u0 7 ( 1 — e) r

17 ( 1 - e ) '

,(« '* ) _= 0 ,

(.42.42)

Calculating the connection coefficients then gives,

r *4 = Tfu] = r ; i4) = r 4, = r fo = 0

r* — r 144 — 1 44r

it*) = - n~ 1 (14), 3 x lx J 1 • i ,

[f4] = ( 9 ~ 0 6j ) 1141

x 'x ^ x^ ~ (»f) = TT~

a' t f f2 a ' a r r

(.42.43)

+ *,-f— ( - _ — j + :Z - l 6 ? - + 6 t - - ' r a x / 3 r \ r r ,

x k ( 1 t \ s / . ..x -1

Here F ]4, T414 and r | 14j axe the connection coefficients of equation (2.3.16)

and all others are the coefficients in the new coordinate system.

The perfect fluid energy-momentum tensor in this coordinate system has

com ponents,

T(if)X X

r-x l

T[i4] II

■*1 e

T(i4) ii Cl

[ae(p + P) 4 - (1 - a ) P ] 4 - <5,; P,

) ^44 = IP,

(.42.44)

The to tal energy of the system will now be calculated in two different

ways, first the volume integral over r 4 and then the surface integral over A4‘.


15

From equation (A2.35),

r t = y / 0:7(1 - e) P + (-42.45)

a n d ,

r / = ^ 7 ( 1 - e ) ( -3 R ) + + i - g ^ l pF[l-4]. (.42.46)

R ather than calculate £ it is possible to eliminate it by using the fact

that, for a static star r* . = 0 and therefore,

J t- d3x = J (x 'r* ) ,* d3x

= J x ' t? d2crk

= / x ’i — o-^TFr-.i - ~^—L k ^ ■J \ 1 2 t t 2 I1’4! 1 6 t t " " s -■

+ <H ^ £ ' ■ d2° i

= 0. (.42.47)

using equation (A2.36). The gM" lj term is evaluated outside m atter by

using equations (A2.42) and (A2.43) w ith a, 7 and w given by equation

(2.1.3). The terms cancel each other. The result then follows by using

^ rd ra k j = r2 s in ddddS and taking the limit as r —* 0 0 .

Thus, using equations (A2.45) and (A2.46),

P* = J ( d - r j ) d

1 X= J v W l - <0 (p + 3P + - - g M W i ' ) d3x : (.42.48)

which is exactly the same as in Schwarzchild coordinates, the non-tensor

pieces having cancelled out.


158

Inserting W4' from equation (2.3.23) and using equations (2.3.12) and

(2.3.13) gives,

P4 = jr°°47rr2y ^ ^ ( p + 3 P ) d r - J ~ i 2R [14] dr, (.42.49)

and using equation (2.3.IS) for P [i4],

* “ - P

~ l ^ \ P 3+s)(2eiP ) iT-

All factors of s' can be eliminated from the second integral by splitting it up,

integrating by parts and using equations (2.3.11) and (2.3.14):

P ( 2 e t - l s )

dr

(.42.50)

€-a e

dr- rJo

- rJo

- 1 r' 3 Jo

I J ' A ( 2 e _ i ±r V e V 3 t

- V i e dr

dr

r 2v - £o V ae r o

Zrs1 - e

— - dr. (.42.51) ae r

Now £2(0) = 0, p(0) = 0 and s(co) = 0 so, using equations (2.3.49),

r R

Pa ( 1 - e )- / * "Jor e _ [ T

Jo r i j a e

(p -f 3P) dr

et2 . S s22T (o + 5 — -) + dr

r R a ~ f+ 3P + 2 n ( p - P - dr

rR ,_________= / 4 - r 2 — e)

JoM L 4

£ 4 / - u

^ + 3P + ^7 [3P + P ~ ^ ~ 2 dr

P 4 ‘(.42.52)


159

Looking at the surface integral approach, equation (A2.39) gives,

A4/ = LkM g' 4 + L k ° + Uk6l

= [ 2 i 4V 4 + 2 i f 4]s [i4] + W*) <54, (.42.53)

using components (A2.42) and (A2.43) in definition (A2.16b). Outside m at

ter a , 7 , and w are given by equation (2.1.3) so tha t,

Xr* r 3M _ , 9 AfL 4

r* r 5 "h r 6

a n d .

\ 4 * _ 1 c4Am -

M _ , 5 M L 4

r* 2 r5 2 r®

(.42.54)

(.42.55)

Using equation (A2.39), the surface integral for P M is,

P» = J d2 a k

' & J=

M7-2

3 L42 r 5

5 M L2 r 6

-d2a k

(.42.56)

in the limit as r —+ cc.

Thus, P 4 = M and P,- = 0. Every static, spherical solution has zero

m om entum and to tal energy M , as expected.

Finally, relating this to the volume integral (.4.2.52),

■ R

u ( i + w ) = L - e)p + 3P

dr. (.42.57)


160

This relation has been confirmed by the use of the com puter program to

generate numerical solutions. It is the NGT equivalent of the GR expression,

M = J * 4 ' r 2(p -f 3P ) dr.

Note th a t the to tal energy is expressible most simply in. term s of integrals

over p and P , rather than p and P. This lends support to the assertion that

it is the shifted m atter variables which are the physical ones.


I 161

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