INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf ·...
Transcript of INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf ·...
![Page 1: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/1.jpg)
^r-iMf^r- ic/85/100
INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS
STATIONARY AXIALLY SYMMETRIC EXTERIOR SOLUTIONS
IN THE FIVE DIMENSIONAL REPRESENTATION
OF THE BRANS-DICKE-JORDAN THEORY OF GRAVITATION
William Bruckman
INTERNATIONALATOMIC ENERGY
AGENCY
UNITED NATIONSEDUCATIONAL,
SCIENTIFICAND CULTURALORGANIZATION 1985 MIRAMARE-TRIESTE
![Page 2: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/2.jpg)
![Page 3: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/3.jpg)
IC/85/1OO
International Atomic Energy Agency
and
United Nations Educational Scientific and Cultural Organization
INTERNATIONAL CEJJTRE FOR THEORETICAL PHYSICS
STATIONARY AXIALLY SYMMETRIC EXTERIOR SOLUTIONS
IN THE FIVE DIMENSIONAL REPRESEIWATION
OF THE BRASS-DICKE-JORDAN THEORY OF GRAVITATION •
William Bruckman **
International Centre for Theoretical Physics, Trieste, Italy.
MIRAMARE - TRIESTE
July
* To be submitted, for pub] i cation.
** Permanent address: Depart amenta de Fisica, Colegio Universitario deHumacao, P.O. Box 10136, Huinacao, Puerto Rico 00661,
![Page 4: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/4.jpg)
![Page 5: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/5.jpg)
ABSTRACT I , INTRODUCTION
The Inverse Scattering Method of Belinsky and Zakharov i s used to
investigate axially symmetric stationary vacuum soliton solutions in the
five-dimensional representation of the Brans-Dicke-Jordan theory of
gravitation, where the scalar field of the theory is an element of a five-
dimensional metric. The resulting equations for the space-time metric are
similar to those of solitons in general re la t iv i ty , while the scalar field
generated is the product of a simple function of the co-ordinates ana an
already known scalar field solution. Kerr-like solutions are considered
that reduce, in the absence of rotation, to the five-dimensional form of
the well-known Weyl-Levi Civita axially symmetric s tat ic vacuum solution.
Hence these metrics can he interpreted as the exterior gravitational
fields of rotating non-spherical configurations in the Brans-Dicke-Jordan
theory. With a proper choice of the parameters in the solutions, the
deviation from spherical symmetry will he due only to the angular momentum
of the system, as i t is with the Kerr metric in general re la t iv i ty . An
explicit exact /solution i s given.
The Brans-Dicke-Jordan (BDJ) scalar tensor theory was f i rs t in-
vestigated in connection with Dirao's large number hypothesis, by Jordan ,and, in connection with Maoh's principle, by Brans and Dicke 2),3) The
theory was originally expressed in a representation in vhich the local
measurable value of the gravitational "constant", G, is a function of a1) 2)
scalar field * . It can also be put in a form which is Einstein's
genera], relativity with the scalar field of the theory acting as an additional
external non-gravitational field . Another way of representing the BDJ
theory is with a five-dimensional field equation ) > where the metric
is independent of the fifth co-ordinate, and the elements g , , where
V = (0,1,2,3} is a space-time index, vanish. In vacuum, this is just a
special case of the Klein-Jordan-Thiry theory, where the g , components
are identified with an "electromagnetic" vector potential A . The Klein-
Jordan-Thiry theory is in turn a generalisation of the original Kaluza-Klein
unified theory of gravity ana eleetromagnetism, In which g, , = constant.
In order to study the physical implications of a theory we must find
solutions of its field equations. Fortunately, it has been possible to find
exact solutions of relativistic gravitational theories, like general
relativity and BDJ, when physically reasonable symmetries have been assumed.
For instance, many astrophysical systems of interest are approximately
stationary and axially symmetric, and can be very well described with metrics7)
having these symmetries. The Kerr solution ' is perhaps the moat important
known eact solution to the vacuum Einstein field equations. It represents
the exterior gravitational field of stationary axially symmetric rotating
systems, which In the absence of rotation reduces to the Schwarzschild
spherically symmetric solution. That is, the Kerr solution does not possess
deformations from spherical symmetry, other than those caused by the angular
velocity.
There are also axially symmetric stationary solutions in the BDJp V
theory representing the exterior metric of rotating configurations , butthey can only become spherically symmetric If the scalar field 1B a constant.On the other hand, there i s a simple extension of the Cosgrove-Tomimatsu-Sato
o\solutions of general relat ivi ty yl to the BDJ theory that allows the con-struction of solutions in which the system i s spherically symmetric In theabsence of rotation. However, the Cosgrove-Tomimatsu-Sato solutions involve,in general, transcendental func^ionc.
-1- -2-
![Page 6: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/6.jpg)
In this paper the inverse scattering meth of Belinsky and Zakharov
' is used to construct axially symmetric stationary vacuum
solutions to the BDJ field equations. In particular, I shall consider
solutions that reduce to the well-known spherically symmetric static EDJ
vacuum metric in the absence of rotation, and therefore can be thought of as
generalizations of the Kerr solution in the presence of a scalar field. This
solution could be a starting point for the study of external gravitational'
fields of rotating stars in the BDJ theory. The analysis is carried out in
a five-dimensional represenLation of the BDJ theory, where the application of
the BZ technique is generalized in a straightforward way. Furthermore, this
representation provides a basis for future possible applications of the BS
formalism to the problem of finding classical solutions of, Kaluza-Klein type,
field theories of more than four dimensions. We point out, however, that the
BZ formalism could be applied directly to a particular four-dimensional
representation of the BDJ theory (the Einstein-scalar theory), since for an
axially symmetric stationary metric in vacuum or with an electromagnetic field
source, the scalar field decouples from the second-order field equations for
the space-time metric, and therefore these equations are identical to those
of general relativity 12'<5J_
In fact, Belinsky studied exact solutions of the Einstein-scalar
theory that describe the evolution of gravitational soliton waves against the
background of Friedmann cosmologieal models. A very interesting feature of
this work is that the energy-momentum tensor of the scalar field is now
representing the matter field of a perfect fluid with an equation of state
pressure = energy density.
In the next section, we will describe the Belinsky and Zakharov method
in i t s essentials, vithout restricting the number of dimensions to four. In
Sec.I l l , the BDJ theory in i t s five-dimensional representation (BDJ-5) Is
discussed in detail . Next, in Se~.IV we apply the BZ technique to the BDJ-5
theory, while in Sec.V the diagonal form of the solutions Is i-onnidered. ri'!;e
BZ technique relies on the knowledge of a seed solution, g , from which a
more general one, g is generated, hence in Sec.VI we •.-• • •§:M.ir.-reu conditions
for the BZ solution, g . , to become equal to g . Sees.VII &nd VIII deal
with specific applications of the formulas developed in previous sections
to the two cases where the background, g , represents a flat space-time, and
a Weyl-Levi Civita static metric. The solution .-unstruoted with the Weyl-
Levi Civita metric may be thought of as a particular generalization of the
Kerr solution to the BDJ theory, since i t is possible to obtain the limit
where, in the absence of rotation, g , approaches the static spherically
symmetric solution of the theory. The paper ends with an explicit exact
solution of this type .-3-
II. THE INVERSE SCATTERING METHOD OF BELINSKY AND ZAKHAROV
This method allows the generation of large classes of new solutions
from old, when the metric tensor depends only on two variables. Belinsky
and Zakharov * developed and employed the method to obtain exterior
solutions in general relativity. In particular, they have obtained the
Kerr solution starting from a flat space-time metric background. The
technique can also be used in relativistic theories of more than fourI1*)
dimensions. This was done by Belinsky and Ruffini , in the framework
of the five-dimensional Jordan-Thiry-Kaluza-Klein theory, to generate
stationary axially symmetric solutions starting from a constant metric.
In what follows we will briefly describe the Belinsky and Zakharov
technique. Their notation will be followed in its essentials. For
the sake of generality, we will not, yet, restrict the number of dimensions
considered, since the generalization only introduces trivial modifications
to the BZ formulas. Hence, let us consider a m+2 dimensional metric that
depends only on two co-ordinates p and Z. The line element can be written
in the Lewis form: ''
dS2 - R
f(p, 2) [dp2 + dZ2] + g (p, Z) dXa dXb;ab
i, b = 1 - m . (2.1)
Furthermore, the source-free Einstein equations in m+2 dimensions
admit the following co-ordinate condition-
(2.2)
det g = det g « -p' ,ab
(2.3)
With the metric (2.1) and the condition (2.3) the field equation (2.2) takes
the form
CP8»
. z]. P
0,
-h-
![Page 7: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/7.jpg)
(In f) = -1/p + l/4p Trace [u - V ] , (2.5)
(In f ) , = l /2p Trace UV, (2.6)
where ve used the conventional notation ( )3X
Eq. (2.M is the compatibility condition for the following system of
linear equations:
D.Oi = — - (2.7)
V- I-'(2.8)
where \ is a complex variable. Moreover, when X= O, Eqs.(2.7) and (2.8)
are just
P*, (2.9)
*PU,
(2.10)
vhich imply that
X - 0) - g (2.11)
The solitonic solution i(i is given as a function of a particular
solution ty , corresponding to a given background metric g , in the following
way:
where
(2.12)
(2.13)
- 5 -
.> -. Jc (2.H)
C. : arbitrary constants.
i(il (k) is the inverse of i|i_ evaluated at
- \ ~ Z ± (2.15)
W : arbitrary constants and T are the elements of the inverse of the
following matrix:
(2.16)
The metric g is then given hy
0) - g -(2.17)
It also follows, from Eqs.(2.5) and (2.6) that
f - cn
Pn n
nnk«l
r , (2.13)
where f is the solution corresponding to g , and C is a constant.
Even though g satisfies Eq.(2.i*), it is not a solution of the field
equations (£.2), since now det g is not equal to det g = -p , but instead
equal to
det g - (-l)n n (P/uk)2 dec g <
(2.19)
where n is taken to be an even number in order to preserve the signature
of g. However, the new metric
• *(2.20)
-6-
![Page 8: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/8.jpg)
is still a solution of Eq.. (2.1+) and, furthermore, satisfies the condition
det gph - V . ( 2 > 2 1 )
The funct ion f i s a l so modified by the t ransformat ion ( 2 . 2 0 ) .
The new func t ion , f , i s
phf ,
where
Const. Pn1 (l
k=!
n - l
\ >
n
k=i
2 n
nk>
(2.22)
(2.23)
The power 1/m is the only explicit reference to the dimensionality
of the geometry. The Belinsky and Zalcharov formulas of Eef.ll are recovered
simply by putting m = 2.
III. THE feDJ THEORY IN FIVE DIMEHSIOHS
The EDJ field equations are conventionally given in the form 2)
WH »
SITE,, ,Qt
L_1 ™ ™tn8ir tf
(3.1)
( 3 .2 )
where g is the metric for the four-dimensional space-time, and u> isuv
the coupling constant for the scalar field + . The comas and the semi-
colon denote par t ia l and covariant derivatives, respectively. The energy-
momentum tensor, T , includes a l l non-gravitational fields and matter,
and sat isf ies the conservation law
TWV = 0 . (3.3)
- 7 -
Units are chosen such that the gravitational constant and the speed of light
are equal to one.
The theory has also "been expressed in the more general form :
G rr—y v ij)
(2u+3-3cQ |Y T
2a $2
- 2 - 8TT a _• C2oo+3)
(3.M
(3.5)
where the new metric g and the scalar field
g by the conformal transformation:
are related to if and
a = constant, (3.6)
8uv (3.7)
The Einstein tensor 3 and the covariant derivatives are built with the
metric g , and the new energy momentum tensor T is given in terms of
T by(IV
T (3.8)
The above transformation, Eqs.(3.6) and (3.7), have been interpreted
by Dicke as a space-time dependent change in the units of measurement.
Note also that in vacuum, and if m = -3/2 the field equations (3.^) and
(3.5) are conformally invariant, a result pointed out by Anderson
In the limit a -+ 0 we obtain the field equations
G = 8it T + ( U 1 + , 2 ) f a ; 6; - % g *>'•% ] ,UV v-v ^ L V v y^ ;a J '
(3.9)
(3.10)
where now the bars mean that the tensors are built with the metric
(3.11)
![Page 9: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/9.jpg)
This representation, Eqs.(3.9) and (3.10), is just Einstein's equations with
a scalar field, $, as external source; -the so-called Einstein-scalar theory.
"The case when
2(D+3 (3.12)
is of,particular interest. In this situation, Eqs.(3.M and (3.5) become
_ yvG = B I T -=^
UVg
J 3{2<u+3)
I t also follows from £qs.(3.13) and (3. lit) that
-8TT |1-
-V
V - K. • 3
Let us introduce the following five-dimensional metric:
g = * 2 ( x y ) ,
(3.13)
(.3.Ill)
(3.15)
(3.16)
Note that g is "static" with respect to the additional fifth dimension
X . It is straightfoward to show that
GUV = Guv - f t* • v " 8 v D *] • (3.17)
G* •= %Ry - (3.IS)
V = 5,y - 0 . (3.19)
Therefore Eqs.(3.13), (3.11') ani (3.15) can be put in the form
- 9 -
g - Sir (3.20)
"1] 4 (3.21)
Finally, defining
and
we obtain the concise form
- ^
(3.22)
0.23)
GAB ° TAB (3.21*)
The field equations (3.2U) are the five-dimensional representation of the
BDJ theory that is equivalent to Einstein's equations for a five-dimensional
metric which is "static" with repsect to the additional non-space-tine
dimension (BDJ-5 theory).
For a specific application of the five-dimensional BDJ theory in
vacuum, where the effect of a scalar field on the cosmological singularityh)
is studied, see Belinsky and Khalatnikov .
IV. APPLICATION OF THE BZ METHOD TO THE BDJ-5 THEORY
We can apply the BZ technique, described in Sec.II, to a five-
dimensional vacuum metric g , when thi3 is a function of only two
variables. Thus let us assume that the metric g , Eq.{3.l6) have the
form (2.1):
f[dp2 + dZz]+g dXadXb; a, b - 1, 2, 3, (U.I)
where we made the identification
-10-
![Page 10: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/10.jpg)
i f a = 3
5 , - 0, if a 4 3) '
(1*.?)
Then i t f o l l o w s t h a t t h e m e t r i c (2 .SO) and ( ? . ? ? ) , w i t h m = 3 :
f p h
is a solution of the BDJ-5 theory in vacuum, 5 = 0, if the -background
metric, g , is also a solution, and, furthermore,
? - * M d g d a M c 8C J
t.5)
We now consider a sufficient condition for Eq,(lt.5) to be valid.
We see that if
M - 0, k > q; a 4 3,
(It.6)
(It.7)
then the matrix
takes the block form
vi o
Ct.8)
where
-11-
r n r l 2 . . . r,(It.9)
Hence
r r rV q j ' • ' qq
q+i " • •rq+m
r E (it.10}
(it. I D
vhich implies that
or, equivalently^
g - g33 a
0 ; a i i 3 . CW13)
Therefore, we will assume that M^ satisfies Eqs.(U.6) and (I*.7).
We now explore further consequences of this assumption. We knov that
*ocb
Then the conditions (I4.6) and Ct.7) can be implemented if we choose
- 0; C 4 3, (lt.15)
and
-12-
![Page 11: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/11.jpg)
C- = 0; k < q,
<T = 0; k > q; a + 3, l % i | )
Note tha t the assumption in Eq.Ct.15) i s consistent with £qs.(2.7) and (2 .8 ) ,
i f the matrix g sa t i s f i e s Eq.Ct,13). Hence, we wi l l assume the va l id i ty
of Eqs.Cul5), U.l6) and C*.17).
The metric (U.3) takes the form
= 21 "z
0 0
With the metric in the form Ct.l8), the field equations (2.M tecome
(PS 1 l) + (fig,, ¥ "' ) z " 0.
[tatBp^.J.p^ = 0 . (It.20)
Eq.(lt.2O) is equivalent to Laplace's equation with cylindrical symmetry.
Eq.Ct.19) has the same form that the second-order equation for a metric
of the type Eq.(2.1) in general relativity. However, we see, from Eqs.(2.2l)
and (!*.l8) , that
(h.21)
while a oorres.ponding 2x2 metric in general relativity satisfies
det gg
ggr
(It. 22)
Therefore, a solution g in general relativity is not a solution in the
BDJ-5 theory, nevertheless, we can use g to Duild a solution g, in
the following way:
-13-
Cfc.23)
Then we see that indeed g, as given 'by the expression ft,23), is & solution
of Eq.(l4.19)» if g is also, since
<Pg 8'P >P ,z
B;1>
0 ,
where we used Eq.Ci.20). Furthermore, Eq.Ct.23) implies that
det g
(!t.25)
and therefore
det g p h = (g p h) 3 3 det g - - P% ^ - ^
as required by Eq.(2.3).
Summarizing, we have the following result . Given a solution, g ,
of the general relat ivi ty axially symmetric and stationary vacuum field
equations, we get a corresponding solution in the BDJ-5 theory
v»
where Hn(g )^T is a solution of Laplace's equation Ct.20). The equivalent
result in the four-dimensional representation has teen given 'by Sned ion
and Mclntosh . It is also possible to generalize this conclusion to
the case when electromagnetic field sources are present ' .
- l i t -
![Page 12: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/12.jpg)
Let us return to Eq.Ct.3) and make use of the implications of
conditions (4.15)-(U.17). We obtain
C,Jl=l
Thus ve have
(4.28)
' a b = n ab
q v k » ..*.
I r1 ^ . a> w 3i(U.29)
k,£>q
(U.30)
*.31)
The component (g h> can be re-expressed in a simpler form. In order to dot h i s , we note that
(It. 32)
which implies that
3 * 3 3 - 3
Consequently, Eq.(4.3O) becomes
C.33)
where
- 1 5 -
( U . 3 5 )
On the other hand, we know the following general result for a non-singular
matrix H, „:
det [1 + Ny] - det N^ + (det N (k.36)
from which we obtain
"k j
c-Dn"q nn>q (It. 37)
Therefore, substituting Eq. (4.37) into Eq^lt.S1*), we get
k>q \ J 3 3 k>q [P j k>q U(U.38)
where n and q have been chosen as even numbers, in order to preserve
the signature of the metric. Thus, we see that the expression for the
scalar field (g u)-3-i involves in a very simple way the background S -it
the poles u. , and is independent of the matrix +^.
V. THE DIAGONAL FORM OF THE SOLUTION
In order to study the non-rotational limit of the solutions,
we will consider in this section the case where g = g = 0. We can
diagonalize the metric g using the same procedure that led to the
"block form (U.18). That is, let us assume that § and ty^ are diagonal
and furthermore,
0 ; i f k >
C = 0 ; i f k < s o r k > q .
(5-1)
(5.2)
-16-
![Page 13: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/13.jpg)
vhere s < q, is a positive integer. Then it follows that
= ck i/r1 = ck ir1 •= o, if k > s,C DCl 1 O H
Mk = C k If "' - 0, if k < s, or k > q,
and therefore
(5.3)
(5.1*)
8 i l
= l U ^ t5.ll)
(5.12)
(5.5)and also, from Eq.(1*.38),
where
and
'kj.
0, if k or Ji > s
r i u ' i f k a t l d ^ -
'•22Jk£
0, if (k or I) < s , or, (k or i.) > q ]
F „ , i f s < (k and t ) < q IikS. ~ '
(5.6)
(5.T)
( 8 . ) , , - 1 I " 4
where the expressions at the extreme right in Eqs.(5.1l) and (5.12) follow
in the same way we obtained Eq.U.38), in Sec.IV, and we have chosen 's1 an
even number to preserve the signature of g b.
To get an explicit expression for the metric component f h ,
corresponding to the above diagonal metric, we will calculate the determinant
of r k r We find,
Hence, T takes the block form
r n o o
o o rwhere we put
(5-S)
(5.9)
det T = det rtl det V22 det r3J,
dec T]E = det
det T_ = det | —•q
nk>s
det
+ p2)
(5.13)
(
(5.15)
We find, from Eq.(lt.29), that g h simplifies todet r 3 3 = det | ^ - ; — j !1 (Kk) I33| det
k
-i • (5.16)
(5-10) Using the following resul t , shown in Appendix A
-17--18-
![Page 14: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/14.jpg)
d e t
1 + V (5.17)
We can see t h a t , for m poles \i ,
Therefore
(5.18)
det r - n (Mk) | _k - i
s(s-l)
• «. ( V * +(5.19)
r « • i i CM*)2 '•n (p u
,£ > s k H(5.20)
(5.21)
Sutistitutior, of Eqs.(5.19)-{5.21) in Eq.(5-13), and th i s in Eq.(it.li), gives
fk" 1
P£
1+
3-1)
3 2 )IT
Bn
(uk-u
(P U1 k ^
+ P 2 ) ~n~7wf+p2) nkk>s
p(n-qXn-q-l) I (
(5.22)
-19-
VI. THE LIHTT Cj • g
I t i s important to studj under wlii1 ri roLonstanccs ttic solution
g reduces to the "background metric g . For example, i t i s useful
to invest igate the new solut ion, g , "near" the background when the
physical in terpreta t ion of g i s wel.l known.
We can easily show that the diagonal element,, Eqs.(U.38), (5.1l)
and (5-12), reduce to g if ei ther one of 1 >>e following two conditions
are sa t i s f i ed :
= g-B = n-q and ^ = u s + k = U ? s + k ; k < s = .
i i ) In each group of poles , l < k < s , s < k < q , q < k < n ,
the poles come in pairs u < U, t , such that
- z (6.D
We see that the condition (i) implies the following results:
(6.2)
n pn - 2
k>q k
•5 2•= I! £j-
k>s uk
s= n
k=l
o ^ l
k-l yk n(6.3)
Also from the assumption (ii) it follows that
k k'
and therefore, using Eq.(6,10,
(6.5)
n 4k=l \
' 4k>S Uk
" 4k>q Uk
(6.6)
Hence, in both cases (i) and (ii), g » | , since the coefficients of
eab l n Eqs.(lt.36), (5.11) and (5.12) are unity.
-20-
![Page 15: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/15.jpg)
We also expect that, when 5 p hthen f
ph = V In
Appendix B, it is shown explicitly that, when g is diagonal, then for
each one of the conditions (i) and (ii), indeed we have f = fph 0
VII. SOLUTIONS WITH PSEUDO-EUCLIDEAN BACKGROUND
The simplest application of the Belinsky and Zakharov method to
the EDJ-5 theory is the generation of solutions using a flat space-time
metric with a constant scalar field as background:
' P2
0
0
f
0
-1
0
= 1.
0
0
1
(7.1)
(7.2)
where the constant scalar field, g , has been normalised to unity. A
particular solution iK, corresponding to the metric (7.1) is
-XV
0
0
0
- 1
0
0
0
1
XW = (7.3)
j
We can get new soliton solutions g , f . in the BDJ-5 theory in aph ph
straightforward way, using Eqs.(T.l)-(7.3) in Eqs.(lt.U), (14.29) and (It.38).
The problem of creating new soliton solutions in general relativity
from a flat space-time metric has been discussed in detail by Belinsky
and Zakharov ' . In particular, they considered the two soliton
case to build the Kerr metric. These solutions can readily be transformed
into BDJ-5 solutions using the prescription in Eq.Ct.27). Hence an
alternative way of generating solutions from flat space-time is to construct
them as in general relativity and then use Eq.C4.27), with the scalar field
33 by
k>q \= 1,
-21-
or any other solution of Laplace's equation (it. 20). For instance, the
Kerr solution can be transformed to get a rotating solution in the BDJ theory.
However, this solution does not become spherically symmetric in the absence
of rotation, if the scalar field is non-constant, as we shall see below.
The Brans-Dicke-Jordan spherically symmetric static vacuum solution,
t, tai
(Appendix C)
g , f , takes the following form in the five-dimensional representation
s p
-ai1-*-" [i-f *ij
where fi and v are constants related by
(7.5)
(7.6)
(7.7>
(7.8)
(7.9)
and 6(5 must be identified with the gravitational mass of the system
(Appendix C). The spherical co-ordinates r , 8 , are related to the
cylindrical co-ordinates p, z in the following way:
Z = (r - S)cos 9 .
We can re-express g in terms of p and z:
T- -, S-v
(7.11)
(7.12)
(7.13)
(7.Ht)
where the functions p and u are just the poles, with W = -S and +S,
-22-
![Page 16: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/16.jpg)
u = -(6 + z) + + p2 - (r-2g) [l- cos e ] ,
V = (3 - z) - /(@-z)2 + p2 --(r-28) [l+ cos 9 ] ,
(7.15)
(7.16)
Conversely, if the scalar field (&^),-, is not a constant the solution (7.20)
does not represent a spherically symmetric configuration.
More generally, if half of the poles, w (k odd) are chosen equal2 - 2 -
to M (or -p /ii) and the other half equal to u (or -p /v) then the
diagonal solution, Eqs.(U.38), (5.U), (5.12), takes the form
On the other hand, according to Eq.(lt.27) the F, , BDJ-5 solution generatedpr:
from the Kerr solution background, g , j s given t>-
-It
(7.17)
(7.22)
(7.23)
ana since In the absence of rotation the Kerr solution 'becomes the Schwarzschild
solution
r2 sin 2 6 0
0 -fl- •?
then we have that, without rotation.
Ph(7.19)
To compare the metric (7.5)-(7.7) with Eqs.(T.19) we must identify (g )
with [1-2 p/r] . Thus we get
gph rJ
o (7-20)
which is Just like g but with 6 = 1 . Therefore the condition i2 + 3v2 = 1,
can only be satisfied if v = 0, or equivalently when
V " = constant. ( T_ 2 l )
which is similar to the functional form of g . Nevertheless, in order for
g to be equal to g we must identify
Ks — ) ,
6 - v - ± ( H - q + s) ,
2v - ± (-|n+ q ) ,
or, eq^uivalently,
5 ' ± (s - * q) ,
V - ± ftq-jn),2 2
but again, 6 + 3v" cannot be unity un.esr. v vani shes.
(7.25)
(7.26)
Thus it seems that to obtain the spherically symmetric solution as
a diagonal limit of a soliton metric we must start with a non-flat metric
g . In particular, if we put § = g , it ia possible to obtain the reduction
g _> g when either of the conditions (i) or (ii), discussed in Sec.VI,
are satisfied. Or, another possibility is to start with a static axially
symmetric but not necessarily spherical background solution, and, instead of
imposing the conditions (i) or (ii), require that the diagonal metric
generated is equal to g . The study of solutions of this type, where g
is given by the BDJ-5 generalization of the well-known Weyl - Levi Civita
solution of general relativity, is discussed in the next section.
-23-
fm
![Page 17: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/17.jpg)
VIII. SOLUTIONS WITH WEfL-LEVI CIVITA BACKGROUND
A solution of the BDJ-5 field equations that describe the exterior
gravitational field of a static axially symmetric configuration is given by
(Appx .C )
(8.1)
(8.2)
(8.3)
where
[1 - -2a 3
[l- f |JSin
o 2 = <52 + 3v2 (8.5)
When V = 0, this solution is the well-known Weyl-Levi Civita s ta t ic metric.
We can verify that a particular solution of ECLS.(2.7) a nd (2.8)
when g = g is
= - Awl (-6-v
(8.6)
r ex -u) ( A - u) i 2 v
* . , » • L xw J 'The evaluation of <J* at A = u gives
U K
2W, M,k k
-S-\j
(8.7)
(8.8)
(8.9)
(8.10)
-23-
- U)(8.11)
Hote that
T4J0 CX> •> g ,L i m X -> 0
as required by Eq.(2.1l). An alternative form for the solution can te
obtained using the expression
(8.12)
XW = X2 + 2X z - pz = (X - u )<X - p0) , (8.13)
where the functions yQ and jj are poles with Wfc = 0:
" o =z 2 + P J (8.lit)
(8.15)
It is worth noticing that, as follows from Eq_s.(8.l)-(8,3)_,
Lim
pz 0 00 - 1 00 0 1 (8.16)
ana furthermore, using Eqs. (8.6)-(8.8) and (7.15), (7.1
(X)l im
-AW00
0-I
0
001
(8.17)
Thus, when £ = 0, the seed solutions g , ipQ are the same aa for the pseudo-
Euclidean case (Sec.VII).
Let us consider the diagonal form, Eqa.(5.1l)) (5.12) and (It.33).
We see that a Weyl-Levi Civita type solution can "be obtained when the poles
are chosen in the following way:
- 2 6 -
![Page 18: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/18.jpg)
- — =- r [ l - cos 6] , k odd I
- 4r = - r [ l + cos 6] , k evenV >
(6.18)
since, in th is case,
(8.19)
(8.20)
(8.21)
where
• = a + f - » , (6.22)
(8.23)
In other words, if we s tar t with g(6,\i) as background, the new diagonalmetric will be g(J ,v) . Furthermore, if the parameters 6 and v are
2 "2 2 o - -chosen such that 5 = & +3v = 1 , we will obtain gU.u) = g and*M) -t .
spFor these types of solutions, the functions iji (k), Eqs.(8.9)-(8.1l)
take the form, for k odd
*C)^ = 2Br(l - cos 6)[2{1 - - -(1 + cos 9)}]"S 'V , ,„ , .mi r to. 24)
"5"V
• J I J - - c « i - ! (8.25)
(8.26)
and for k even,
-27-
M t SS * . , « • . ; ! '::
= 2Br(l + cos 6)[2{1 - | (1 - cos
- 7 (i - cos e))] S-v
e)} ] / v .
(8.27)
(8.28)
(8.29)
Thus we can construct generalizations of the Weyl-Levi Civita metric by-
substituting the above matrix tyQ and the metric {8.1)-(8.U) in Eqs.(U.£9),
(1*.38) and (k,k). The case ti = q = s = 2, i s given explicitly below
& ~ * sia , v s v - f ,
(gp h)1 2 - ^ - [A(a - b cos 9 ) - a sin2 e ( a2 - b1 - MR)] ,
fP h - ^ ^ - , 6 E « . i ,
where
_ B -2
Mo (B +
2Mo)
1
1 -
r
r
4- =y Sin! 9
M = i o B )126
(8.
( 8 .
(B.
( 8 .
(8.
30)
31)
32)
33)
34)
(8.35)
C8.36)
3.37)
-28-
![Page 19: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/19.jpg)
- 4rI"26 (bo+ a.)
8) + - — —
R = (r - &) + M,
E R2 + (h - a cos 6)2
-261
(8.. 39)
(8.40}
A E 1- [r(r _ 20)] = (R - H)' - y' .fl2
(8.U1)
The constants aQ, b Q, H , and 6 have been chosen such that M? - 8 = a -b 2,
in order to obtain the Kerr-IJut solution when fi = v = 0, as we shall see
below.
The study of the physical interpretation of the above solution is
under way, but we can already state the following results:
0, then Y » S , M * MQ, a = aQ, ~o = hQt and thei} If S«
solution becomes equivalent to the Kerr-Nut metric, since one can. verify that
the time co-ordinate transformation
T - t + 2aQ^
leads to the Boyer-Lindquist Kerr-Nut line element
cos 6 - a sin2 e[MR + bz])dT
S Jn2
If b = 0, we get the Kerr solution with mass M and geometric angular
momentum Ma .
ii) If a » b = 0, we have (g 1 1 ) - , 2 = 0, and therefore xero angular
momentum. The metric (8.3O)-(8.3't) becomes a Weyl-Levi Civita solution
2S-.1-S-V! (8.U)
(8.1*4)
-29-
. [! . 1 | ] 2V i (3.1*5)
{8.1*6)
where 5 : J U , depending on the choice MQ = +_ B. If, furthermore,
o = S + 3v = 1, we have spherical symmetry.
ACKNOWLEDGMENTS
The author would like to thank Professor Emil Kazes from the
Pennsylvania State University for his continuous encouragement and advice
in this work. Thanks are also due to Professor Belinsky for useful
discussions. He is grateful to Professor Atdus Salam, the International
Atomic Energy Agency and UNESCO for hospitality at the International
Centre for Theoretical Physics, Trieste. This work was supported in part
by a grant from the Resource Centre for Science and Engineering of the
University of Puerto Pico, and by the Humacao University College.
-30-
![Page 20: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/20.jpg)
SUMMARY OF THE APPENDICES OF THIS WORK APPENDIX A
In Appendices A-F we show the following:
1Appendix A: Determinant
1 + L *t ( w <vv] [^.T1
In this appendix it is shovn that
det
(A.I)
Appendix B: If g is diagonal and any one of the conditions (i) or (ii),
defined in Sec.VI, are satisfied then f = f .ph 0
Appendix C: The tfeyl Levl Civita metric in the BDJ-5 theory is derived.
Appendix D: Determinant iJi(A) = \0{v), where G(w) i s a function only of
W = [X2 - p * ] A " 1 .
Appendix E:
Appendix F: If g and i|i are diagonal, then Ji{)i) g 1 IJJ(—p2/X) = F(Vf),
where F is an arbitrary function of W.
First note that
det H?i ^ m
{1 + det
where we used the fact that det
equivalent to
J =[H bj det
det R {1 + A ^ ) - K <Ak - AjXBt - Bk)
(A.2)
. Hence Eq_. (A.l) is
U.3)
The proof of Eq.(A.3) will be by induction. Hence let us assume that Eq.(A,3)
is valid for a. (n-l) x (n-l) matrix:
n-l - v-(A.5)
Then we have
n
k>£(Ak" V ( V
n-l n-l
"n-l
5
where we used Eq_. (A. 5). We know that
n_!C (A.6)
-31--32-
![Page 21: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/21.jpg)
det C = f cn-l 1... cn
n-1
SI
(A,7)
Also, it is easy to show that
" " V - Vn-1
it (A.8)
for any particular choice of n-1, i . Consequently, using Eqs.(A.T) and
(A.8),
" n-I
n (AJ^-A^(BJJ-BJ^) = E. , n c
" (l+A.B ](1+U. )ij £ n * f
- K,t •
n-1
n
(A.9)
The terms
E (I+AnBn)(HAs,Bi^c"^ = (HAnBn) H (1 + A ^ J , (A..10)
are independent of 1 . Hence any term on the right-hand side of Eq.(A.9)
that contains a factor like K.K ...K is nymmet. '• i i a i under pernrutationp
of i., i ,.,.,i . Therefore these term;; do siot .•o::tribut!> t.n the sum inx. m s
Eq.(A.9). Thus, the only surviving terms are either independent of K
or linear in K . Consequently, Eq..(A,9) becomes
-33-
n-1 n-1
(A.U)
Using Eq.(A.10), we obtain after some rearrangement
n - 1 ''mn^in,n J I D c " - - — • -irm n i m j , H j , criCL SLi£
(A.12)
•where we use the definitions
C- 5n-1
(A.13)
ni
n-1
n-1
(A.15)
Substituting Eq.U.12) and using again the definitions (A.13)-(A.15) in
Eq.(A.ll) we get
n-1 C ^ (? n-1 (A.16)
On the other hand, we know that
![Page 22: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/22.jpg)
(A.IT)
Eq.(A,17) can te re-expressed in the following form:
n-1det H C
+ £. ki k snn
•'• n , i 2 , . . . i o k kifc i
?n- l
We see that
c". cn » eCkiv
CniCsn
uSi«,
Therefore, applying Eq.(A,19) to Eq. (A.l8),ve obtain
det < ^ - e. . V <£ cn - e
ni
(A.19)
<u
V TT
{A.20)
- 3 5 -
Comparing Eq.(A.2O) with Eq,(A,l6) we see that
d e t C° = det n (1 +k > J l
- Aj)(B r B ) ,
(A.21)
or, equivalently, from (A.2)
det(A.22)
Since A.22 is valid for a 2 * 2 matrix, then we have shown that is
valid for all n >_ 2. Q.E.D. In particular, if A = B = y , we have
det1 +
(A.23)
-36-
![Page 23: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/23.jpg)
APPENDIX B
It is shown below that when g is diagonal, then for each one ofph
the conditions (i) or (ii), in Sec,VI, we have
f . " const. foph (B.I)
First consider the case (i). In this situation Eq.(5,22) becomes
fph3 Q ' fo P
k-1
!! k k k 2 •> •> •
n ( M IM 2M 3 ) g l l g 2 2g 3 3
k-1
P(P-l)
k=l
We know that
(uv+p2) nk J i
(B.2)
11 It 33det g - - Q2 , (B.3)
and
k Mk Mk ck ck ck - (k ck ck(B.U)
In Appendix D it is shown that
det i> (X) - XG(w(B.5)
where G(W) is only a function of
(B.6)
Since, using E<i.(2.15).
U2. + 2ukz - p2
(B.T)
we have
det <KUk) - Const. (B.8)
-31-
Furthermore, we will also use the following relation, which is shown in
Appendix E:
(M - p )k p2) j k (B.9)
Subs t i tu t ing Eqs . (B .3 ) , (B.I*), (B.8) and (B.9) in Eq.(B.2) we obtain
Const.
11
k.Jt(B.10)
On the other hand, from Eq.(2.23), we have
Q » Const. p
k=l j k.A Va (B.ll)
where we used Eq.(B.9). The last factor of Eq.(B.ll) can be re-expressed
in the following form:
(B.12)
where we use the condition (i). Thus we have, putting n = 3s.
Q'1 - Const, p(3s+l)2 f s
II Uk
k=ln (u u +p2)
(B.13)
Substitution of Eq..(B.13) into Eq.(B.lO) gives the desired result:
f - Const,ph (B.lU)
-38-
![Page 24: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/24.jpg)
(6.5)
He now consider the condition (ii). Let us recall Eqs.C6.l4) and
Some consequences of Eqs.C6.lt) and (6.5) will be derived below. First
consider
it
(B.15)
but
" z> =
where we used Eqs.(6.1t), (6.5) and (B.7). Consequently,
(B.16)
)K -Const. II —i— lip, .
CB.17)
Also it can easily be shown that
II(u5 + p2) - H(y2 + p2)(u= + p2) = n p2(y5 - y.,)2
(B.18)
T h e r e f o r e v e h a v e , f o r m p o l e s u , o r e q u i v a l e n t l y ~ p a i r s u D » U a i
ji (u«2+ P 2 )Const. s
m
Const. Const, p
(E.19)
where we used Eq,(6.!*)
-39-
We can now simplify the last three factors of Eq.(5.£2) using
Eq.CB.19), since we have, for m poles y ,
pjnCm-1) TJ (yk-yj,) Const, p
2 (m-1)
k k
m(m-1)-3m
Const. P ,
CB.20)
where «e used Eqs.(B.9), (B.19) and Eq.(6.1t). On the other hand, with
-1EIJS.(B.19) and C6.it), the expression for Q~ also simplies to
Q T - Const, p ' .
Using the results (B.20) and CB.21) in Eq.(5.22), we get
CB.21)
f . =ph
Const. f o n(M ) .gq
n (k>s
k 2 •IT (Mj) g
k>q CB.22)
Finally, it is shown in Appendix F that, if g and ip are diagonal,
then the matrix
(B.23)
is only a function of W(JO, and therefore
= Const. ,CB.2U)
or, equivalently, for a = 1,2,3
C' * l ia °aa a a "i
(No s u m o v e r a r ) .
'aa «.' °aaConst.
(B.25)
-1*0-
![Page 25: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/25.jpg)
This implies that APPEBDTX C
k>s k>qConst. (B.26)
A solution of the Einstein scalar field enunLionr. (3.9) and (3.1Q)
that represents the exterior gravitational field of a stat ic axially
symmetry configuration is
Hence again
gK.S l! ?l -o
sen^O ,(C.I)
f = Const, fph (C.2)
Q.E.D.
1 -
; G2 ? S2 + 3\J=, (C3)
(C.I+)
where B and 6 are constants, and the co-ordinate? r, S are related to
P and .z ty Eq.s.(7.1O) and (7.11). If '" = 0, the above equations transform
into the well-known Weyl-Levi Civita meLri'r- * of general re la t ivi ty .
The total gravitational mass of the syatem in (SB , whicti follows from the
requirement that the equations of no1 innsi become Mewt.nn Lan in the limit
The case a ' = i i« of pat IJi.ul ai interest , since we get a
spherically symnintrir1 gravitationa] field. The line element takes the form
d s " dr' + r ' + sin 0
(C5)
1 - 6(C.6)
A s o l u t i o n of t.hfi BDJ-S f i e l d e q u a t i o r j , g , i s " b u i l t , from t h e m e t r i c
(C.I. ' - ( C . 3 ) and t h e sca.la:r f i e l d [C,h] u s i n g t h e t r a n s f o r r a a t i o t i s ( 3 . 1 6 ) ,
( 3 . 6 ) , ( 3 . 7 ) , t o g e t h e r wi th E q . ( 3 . 1 1 ) :
- 4 1 -
![Page 26: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/26.jpg)
* aV,
*>•>• • - I1 - T J
or, in the canonical form {k,1 },
E.S
(C.7)
(C.8)
(C.9)
( CIO)
(C.ll)
APPENDIX D
2Let us show that if det. g = -P , then
det *(*) - AG(H),(D.I)
vhere G(W) is a function only of W, subjected to the following condition:
(D.2)
We know, from Eq.(2.8) that
X2 + p2
Consider the trace of the above equation
(D.3)
Trace_! pTrace U+ X Trace V P2 T r a c e 8 g' 1 + PX Trace g > z g'
X2 + p2 X2 + p2
(D.I*)
Using the Hell-known resul t , valid for any non-singular matrix H,
Trace
we obtain, from Eq.(
D,(det ['
det N(D.5)
g) >z] (det g)"2 p
det
where we used
X2 +
(0.6)
-1*3-
![Page 27: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/27.jpg)
det g = - p .
Similarly, from Eq.(2.T) it follows that
det V
On the other hand
D2(ln X) = \4-
+ p4) det g
-2\
, , -2XIn X =
2 2
(D.7)
(D.8)
(D.9)
(D.1Q)
particular solution is
G(W) = W,
det l)J( X ) = XW.
(D.16)
(D.IT)
Therefore
D In <i^Ji) - 0, (D.ll)
The solution to Eqa.(D.ll) and (D.12) Is
Furthermore, since ip(X » 0) = g, we must have
(D.12)
(D.13)
det Y - det g - - Plim X-fO
and consequently
XG(W)--p2
lim(D.15)
-1*6-
![Page 28: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/28.jpg)
APPENDIX E
We w i l l see lielow tha t
2(Wk - (E. I )
where n is the number of poles, u^. Thus consider two poles u , \i .
They satisfy Eq.(£.15), which in turn implies
U2 - 2u.(W0 - z) - p2 = 0
(E.2)
(E.3)
The difference of these two equations gives, after some re-arrangement
Substitution of Eq.(E.lt) in Eq.(E.2) yields
CE.lt)
(E.5)
or, equiveilently,
(E.6)
Thus we have
(E.7J
-1*7-
APPENDIX F
We will prove that if lfi is diagonal then
* (X)g l U -y- ) - F{W),(F.I)
where FCW) is an arbitrary function of W. First i t i s shown that D^ and
D are invariant under the following transformation:
P - P
(F.2)
(P.3)
Hence ve have
_3_ _ _3_ + ZX _3_3p " 3p p j X "
(F.5)
3 _ 3 (F.T)
Then it follows that
3z ' 3X " Yl~ I2 + p2 (F.8)
: ) 3X 3 ^* 2 1T 1 J si
(x2+u2)D2(X) . (F.9)
The field equations(2.7) and (2.8) imply that
-U8-
![Page 29: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/29.jpg)
T % I / T V I - i T , pv -= r ^
- [Di(X)i|)(X)]g , z g
(F.10)
and similarly,
[p U
\ z + p 2 'P
Us ing E t i s . ( F . a ) and ( 7 . 9 ) , i n E q s . ( F . l O ) and ( F . l l ) , we ofctain
l (X) - [ D j g j g " 1 - 0 , (F.1E)
)]i(j-I(X) - [D2g] g"1 = 0. CF.13)
If g and iji are diagonal, then Eqs.(F.12) and (F.13) simplify to
g"1) - 0,Dj(X)
D2(X) ln(((i(X)i|)(X) g"!) - 0 .
The solution to Eqs.tF.lM and (F.15) is
ili(A)i|J(X) g"' =' ( —^— ) ^(X) g"1 " F(W) ,X
where F{w) is an arbitrary function of
X2 + 2Xz - p2
(F.llt)
(F.16)
-1*9-
REFEBEMCES
1) P. Jordan, Z. Physik 151, 11? (1959).
2) G. Brans and R.H. Dicke, Phys. Rev. 2}i_, So.3, 925 (l96l).
3) R.H. Dicke, The Theoretical Significance of Experimental Relativity
(Gordan and Breach ig6U).
It) V.A.Belinsky. and I. Khalatnikov, Sov. Phys.-JETP 36, 591 (1973).
5) U, Eruckman, Thesis Diseertation, Dept. of Astronomy, The Pennsylvania
State University (1979).
6) A. Liehnnerowicz, Theories Relativlstes de la Gravitation et de
1'electromagnetistae (Mason Paris, 1955).
7) E. Kerr, Phys. Rev. Lett. 11_, 237 (1963).
8) C. Mclntosh, Commun. Math. Phys. 3J_, 335 (19T1*).
9) C M . Cosgrove, J. Phys. A 10, lWl (19T7).
10) V.A. Belinsky and V.E. Zakharov, Sov. Phys.-JETP U5_, 985 (1973).
11) V.A. Belinsky and V.E. Zakharov, Sov. Phys.-JETP ^0, 1 (1979).
12) A. Eris and M. Gurses, J. Hath. Phys.l8_ , 1303 (1977).
13) V.A. Belinsky, Sov. Phys.-JETP 5£, 623 (1979).
14) V.A. Belinsky and R. Ruffini, Phys. Letts. B 89, No.2, 195 (1980).
15) T. Lewis, Proc. Roy. Soc. (London) A136, 176 (1932).
16) J.L. Anderson, Phys. Rev. D3, 1689 (1971).
17) G. Sneddon and C. Mclntosh, Int. J. Theor. Phys. £J_, 1*11 (1971*).
IS) H. Weyl, Ann, Phys. Leipzig, 5}*., 117 (1917).
19) T. Levi Givita, Atti Acad. Haz. Lincei, %, 26 (1917).
-50-
![Page 30: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/85/100.pdf · 2005-02-23 · simply by putting m = 2. III. THE feDJ THEORY I N FIVE DIMEHSIOHS The EDJ field](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facb13bbdd4a71fb1179e78/html5/thumbnails/30.jpg)