The Combridge-Janne Solution and the Gravitational $g_{22}$ Metric Function

7/28/2019 The Combridge-Janne Solution and the Gravitational $g_{22}$ Metric Function

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The Combridge-Janne Solution and the

Gravitational g22 Metric Function

R.E. Salvino

# 604

1 South Shamian Street

Guangzhou, China 510133

R.D. Puff

Department of Physics, Box 351560

University Of Washington

Seattle WA 98195

Revision date: 16 Oct 2013

Abstract

The time-independent spherically symmetric general relativistic vac-uum field equations are a system of two independent field equationsin three unknown functions. This system has a solution consistingof a one-function family of solutions. The family of solutions, theCombridge-Janne solution, is characterized by the undetermined g22metric function and the imposed auxiliary condition that is used todetermine that function. From this starting point, we then:

1. establish a rigorous equation that must be satisfied by the g22metric function (Section 3);

2. derive the generalized Schwarzschild solution, a 1-parameter gen-eralization of the Schwarzschild solution that contains both thetextbook solution and Schwarzschilds original solution as specialcases (Section 4);

3. derive an exact relation between the classical potential and themetric functions (Section 5);

4. derive the GR Newtonian solution, a 1-parameter solution basedon the form of the classical potential which also contains solutionsfor wormholes, a point mass, and objects of finite size (Section6); and

5. derive a correction to the classical potential based on the solutionin item (2) and connect to experiment (Section 7).

Current address: 9 Thomson Lane, 15-06 Sky@Eleven, Singapore 297726.

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Our results indicate that, for time-independent and spherically sym-

metric systems, the g22 metric function has a much more pivotal rolethan that of a simple radial coordinate in the description of space-timecurvature.

Keywords: Combridge-Janne solution, generalized Schwarzschild solu-tion, Schwarzschild solution, Hilbert solution, GR Newtonian solution,non-Newtonian gravitational potential

1 Introduction

The Schwarzschild metrics of the time-independent spherically symmet-ric gravitational field equations consist of the conventional textbook solu-tion [1,2]

ds2 =

1 2mG

r

c2dt2

1 2mG

r

1

dr2 r2d2 (1.1)

and Schwarzschilds original solution [35]

ds2 =

1 2mGRs(r)

c2dt2

1 2mG

Rs(r)

1R2s (r)dr

2 R2s(r)d2

(1.2)

Rs(r) =

r3 + (2mG)31/3

where Rs(r) dRs(r)/dr and d2 = d2 + sin2 d2 is the square of thedifferential solid angle. It is straightforward to verify that both of thesesolutions satisfy the field equations. It is also straightrforward to verify thatboth of these solutions have the same asymptotic limit that identifies theparameter 2mG with the mass of the source object, 2mG = 2Gm/c

2 where

m is the mass of the source in conventional mass units. These solutions,however, do not satisfy the same small r boundary conditions.

The field equations consist of a system of differential equations, the high-est order of which are second order differential equations. A second order dif-ferential equation requires two distinct and well-posed boundary conditions

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to uniquely specify a solution. Apart from the asymptotic condition that

permits a connection with Newtonian gravitational theory, Schwarzschildoriginally imposed a second boundary condition that the metric functions befinite and continuous everywhere except at r = 0, the site of the point masswhich he took as his source object. This provided the solution (1.2) withthe specific function Rs(r). It is clear that the metric (1.2) is well-behavedeverywhere except at the origin, r = 0. On the other hand, the textbook ap-proach proceeds to the uniquely specified solution (1.1) by application of thesingle asymptotic boundary condition at infinity. This solution has no abil-ity to satisfy a second boundary condition. However, we show in Section 4that the condition g22 = r2 everywhere actually serves as the requiredsecond boundary condition. In the conventional approach, this condition is

imposed before the field equations are even established and is disguised as achoice of coordinates. Consequently, the role of g22 = r2 as a boundarycondition is not recognized. The solutions (1.1) and (1.2) are distinguishedfrom each by means of the second small r boundary condition: the twodistinct small r boundary conditions are the mathematical statementsreferring to two distinct physical objects that serve as the sources for thetwo solutions. At bottom, these are nothing but basic concepts of potentialtheory and relate to a view of general relativity as a tensor, nonlinear, andrelativistic form of potential theory.

Distinguishing among solutions by means of boundary conditions is animportant facet of potential theory. Nevertheless, in general relativity thisfeature is overshadowed by an even more fundamental issue. Within the

context of a perturbative treatment of the time-independent inhomogeneousand spherically symmetric gravitation problem, de Sitter demonstrated thatthere are only two independent field equations, not three [6]. In the limitof vacuum field conditions, this is an exact and rigorous statement. Asa consequence, it should be expected that the vacuum field equations willproduce a solution for two of the metric functions in terms of the thirdmetric function while the third metric function remains undetermined bythe field equations. This, in fact, was explicitly demonstrated independentlyby Combridge [7] and Janne [8].

The focus of this paper relates to the consequences of the Combridge-Janne solution, concentrating specifically on the undetermined g22 metric

function. In Section 2, we present the Combridge-Janne solution and de-scribe the main aspects of the solution. After presenting a brief backgroundand the general structure of the metric, we then:

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1. Establish a simple but rigorous subsidiary equation that relates the

g22 function with the determinant of the metric (Section 3).

2. Impose a specific condition ong and show how the resulting equa-

tion may be used to establish a direct connection with both the text-book solution and the original Schwarzschild solution by means of aone-parameter family of solutions. We denote this one parameter-family by the name generalized Schwarzshild solution and relate thesolution parameter to the small r boundary condition imposed on thesolution (Section 4).

3. Derive an exact relation between the classical gravitational potentialand the metric functions (Section 5).

4. Demonstrate how the form of the classical potential can then be usedas a basis for imposing a different condition to specify the undeter-mined g22 metric function. Given the null results obtained to date fordeviations of the classical gravitational inverse square force law, weenforce a new auxiliary condition that the classical potential obtainedin Section 5 have the Newtonian form exactly. This provides the basisof another one-parameter family of solutions which we denote by thename GR Newtonian solution (Section 6).

5. Illustrate the connections between a non-Newtonian classical potentialand the g22 metric function R(r). In particular, we derive a general

relativistic correction to the classical potential based on the general-ized Schwarzschild solution of Section 4. This correction permits anexperimental determination of the generalized Schwarzschild param-eter and may also impact investigations of short-range deviations ofthe inverse-square force law (Section 7).

We close with a discussion of the relationships among the Combridge-Jannesolution, the generalized Schwarzshild and GR Newtonian solutions, andthe original and textbook Schwarzschild solutions. We also include a briefdiscussion of the physical implications of these solutions.

2 The Combridge-Janne Solution

Making no assumptions concerning the g22 metric function, Combridgeand Janne each obtained a solution which expressed the g00 and g11 metric

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functions as simple functionals of the undetermined g22 metric function.

Writing g00 = e

, g11 = e

, and g22 = e

r2

= R2

, the Combridge-Janne solution [7,8] may be expressed as

ds2 = ec2dt2 edr2 er2d2 + sin2 d2 (2.1)e = 1 2mG

R(2.2)

e =R2

1 2mGR(2.3)

e/2 r = R(r) = undetermined (2.4)

R(r) r as r (2.5)

where the prime in (2.3) means differention with respect to the radial coordi-nate r, R dR/dr. The asymptotic condition (2.5) is the only requirementon the g22 metric function R(r) and guarantees that the circumference ofa great circle and the surface area of a sphere are given by C = 2r andA = 4r2, respectively, in the asymptotically flat Lorentzian spacetime.

The Combridge-Janne solution provides a family of solutions in the co-ordinate basis (t,r,,) which is characterized by the g22 function R(r). Byusing the chain rule, it is possible to put the metric (2.1) into the textbookSchwarzschild form and use R(r) as the radial coordinate. This does not

mean, however, that the metric is therefore equivalent to the textbook met-ric. This simply restates the family nature of the metric in terms of thecoordinate bases (t,R,,) since (a) there is a one-to-one correspondencebetween the radial coordinate R and the function R(r) which produces thatcoordinate and (b) the dependence of the g22 function on the common ra-dial coordinate r provides the connections between various R-coordinates.For example, suppose we have two Combridge-Janne metrics, one with g22function R1(r) and one with g22 function R2(r). By using the chain rulewith the functions R1(r) and R2(r), we may write the metric (2.1) for thetwo cases as

ds2 =

1 2mG

R1

c2dt2

1 2mG

R1

1

dR21 R21d2 (2.6)

ds2 =

1 2mG

R2

c2dt2

1 2mG

R2

1

dR22 R22d2 (2.7)

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The metrics (2.6) and (2.7) are not identical since the functions R1(r) and

R2(r) provide the relationship R2(R1) between the two radial coordinates R1and R2 in implicit form. The function R2(R1) maps the regions of spacetimefor the metric (2.6) into the regions of spacetime described by the metric(2.7). Note that R2(R1) does not serve as the coordinate transformationthat shows these two forms are equivalent: using R2(R1) in (2.7) does notproduce (2.6). Also note that we are not claiming that the coordinatesR1 and R2 must be used to describe the metrics for the two respectiveproblems, but no other choice of radial coordinates will put the Combridge-Janne metric form into the textbook Schwarzschild form. In Section 4 weprovide an explicit example to illustrate these points.

As weve stated, the Combridge-Janne solution contains an undeter-

mined function and so we may characterize this solution as a one-functionfamily of solutions. Alternatively, it may be defined as the class of solu-tions whose members are characterized by the g22 metric function and bythe auxiliary imposed condition that is used to determine that function (seeFigure 1). Since one auxiliary condition used to determine R(r) need haveno relationship to another alternative condition, the R(r) obtained by oneauxiliary condition will have no necessary relation to the function obtainedby a different auxiliary condition.

The result summarized by eqs. (2.1) - (2.5) is, of course, the underly-ing basis of the distinction between the original and textbook versions of

Combridge-Janne Solution:Undetermined g22 Function, R(r)

Solution 1:Determine R(r)by Condition 1

Solution 2:Determine R(r)by Condition 2

... Solution K:Determine R(r)by Condition K

Figure 1: Diagrammatic representation of the family relationship of solutionsgenerated by different auxiliary imposed conditions that may be used to determinethe g22 metric function, R(r). There are as many distinct solutions as there aredistinct auxiliary conditions for determining R(r).

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Condition ong

Original SchwarzschildR3 = r3 + 8m3G

Condition on coordinatesTextbook Schwarzschild

R = r

Figure 2: Diagrammatic representation of relationships between the Combridge-Janne solution, the original Schwarzschild solution, and the textbook Schwarzschildsolution. Choosing coordinates such that

g = 1 and choosing coordinates suchthat R(r) = r are treated as two specific methods for determining R(r). In Section 4we show these are not two separate conditions but result from a single auxiliarycondition.

the Schwarzschild solution. Although the current distinction compares thetwo solutions as proceeding from different methods for determining the g22function R(r) (see Figure 2), we show in Section 4 that the original andtextbook Schwarzschild solutions are consequences of the same imposed con-dition but are distinguished by means of different boundary conditions onR(r). While the asymptotic condition R(r)

r for r

applies to all

sources, the behavior of R(r) as r 0 reflects the nature of the sourceobject: the condition of spherical symmetry is not sufficient to uniquelyspecify the source. Consequently, a specific R(r) not only describes the de-pendence on the radial coordinate r but it is tied to a specific source ofspacetime curvature. The physical content of the boundary condition R(0)provides the basic physical distinction between the two solutions: the sourcefor the original Schwarzschild solution is a point mass while the source forthe textbook Schwarzschild solution is a more exotic object, a wormhole inspacetime [1, 2]. Thus, there are two aspects of the indeterminacy of R(r):(a) the auxiliary condition used to determine the function R(r) and (b)the boundary condition on R(r) that uniquely specifies R(r) for a specific

given physical configuration. These ideas are very much in line with thoseof classical potential theory.

It is important to recognize that Birkhoffs theorem [2] does not provethe uniqueness of the textbook Schwarzschild solution since proofs of the

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theorem assume the textbook form for the g22 function. In essence, this

is a point of elementary logic: one cannot assume that which one wishesto prove. Removing the assumption on the g22 metric function so thatg22 = R2(r), the derivation of Birkhoffs theorem then shows that theCombridge-Janne solution is the unique solution of the time-dependent andtime-independent spherically symmetric problem. Birkhoffs theorem doesestablish the uniqueness of the dependence of the g00 and g11 functions onthe g22 function, but it does not and can not determine the dependence ofthe g22 function on the radial coordinate r. Properly generalized, Birkhoffstheorem neither supports nor rejects any particular form for the g22 metricfunction R(r).

We also note that current experimental evidence does not rule out any

particular form for R(r) as long as it has the correct asymptotic behavior.The classic calculation of the perhelion shift of Mercury [1] is not sensitiveto redefinitions of the radial coordinate and the form ofR(r) does not play arole in the calculation. This is also true for the bending of the trajectory of alight ray [1] by a source object but it does lead to the interpretation of R(0)as the distance of closest approach of the light ray to the source object. Weakfield or far field behavior may use R(r) r to lowest order without needingto specify the full functional form of R(r). For example, Schwarzschildsoriginal solution may be expanded in a power series in (2mG/r)

3 and usingonly the lowest order term R(r) = r introduces an error of less than 0.5 % forr 8mG; even at r = 4mG, the error is only 4 %. Detecting differences dueto different forms for R(r) will require very massive objects to provide very

small effects in the asymptotic region or, for more ordinary objects, probesinto the small r regions where R(r) = r is not a good approximation. SinceR(r) contains information about the source object, an individual probe ofnear-field behavior will provide significant information about R(r) for thegiven source, but not for an arbitrary source object.

3 The Metric, Coordinate Basis, and an Equation for R(r)

The tensor equations R = 0 determine the metric tensor g and the cor-

responding geometry of spacetime whether we choose to use a differentialgeometry approach or select a coordinate basis and generate a system ofdifferential equations. We choose a single coordinate basis, a time-like co-ordinate together with the standard spatial spherical coordinate system,and write the basic vacuum equations in coordinate form as R = 0. The

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line element in this basis for a spherically symmetric and time-independent

configuration is

ds2 = ec2dt2 edr2 R2d2 (3.1)

where d is the differential solid angle, d2 = d2+sin2 d2. The sphericalsymmetry is reflected by the imposed conditions that g33 = g22 sin

2 =R2 sin2 and the metric functions are functions of r = |r| only; time-independence is reflected by the imposed stationary (g are independentof the time coordinate t) and static (g0k = 0 for k = 1, 2, 3) conditions.We emphasize that at no time do we ever change the coordinate basis, our

coordinate basis always remains (t,r,,).The metric form (3.1) leads to the Combridge-Janne solution, Eqs. (2.2),(2.3), and (2.4), expressing and in terms of the undetermined functionR. Since the field equations provide no basis for determining the g22 metricfunction, we would like to establish an equation that must be satisfied byR(r). The conventional approach [1, 2] and the postulated relations amongthe metric functions , , and [6] provide no guidance. On the otherhand, Schwarzschilds original solution [35] implemented a condition onthe determinant of the metric and chose coordinates such that

g = 1could be imposed. Consequently, we turn our attention to the determinantof the metric,

g ||g || = e+R4 sin2 (3.2)

Using eqs. (2.2) and (2.3) in (3.2) and taking the positive square root thenprovides a simple but rigorous equation for R(r),

R2R =1

3

dR3

dr=

gsin

(3.3)

The sin factor in (3.3) removes the angular dependence in

g to yield a

function ofr only. Eq. (3.3) clearly shows that a solution for R(r) uniquelydetermines the determinant of the metric tensor. Conversely, if the deter-minant of the metric tensor is not uniquely specified, unless a condition isimposed on the determinant of the metric tensor either by means of thefield equations or by a subsidiary condition, then (3.3) indicates that the

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metric function R(r) must remain undetermined. Although eq. (3.3) is a

rigorous consequence of the spherically symmetric vacuum field equations,we stress that an arbitrary specification of the determinant of the metricstill produces an arbitrarily specified metric function R(r).

4 The Generalized Schwarzschild Solution

Eq. (3.3) may be used as a basis to generalize Schwarzshilds originalsolution and to provide a fundamental relationship between the original andtextbook Schwarzschild solutions. Noting that the determinant of the metrictensor is not a scalar but a scalar density of weight W =

2,

g =XX

1

g (4.1)

where |X/X| is the Jacobean for the coordinate transformation that con-nects the barred coordinate system (with determinant denoted by g) tothe unbarred coordinate system (with determinant denoted by g). If, in aspecific coordinate system, the determinant of the metric tensor has a par-ticularly simple form, then equation (4.1), in conjunction with eq. (3.3), canthen be used to determine the function R(r).

While we could use (4.1) together with Schwarzschilds transformation tocoordinates such that

g = 1 [35], it is simpler and more straightforwardto note that this is equivalent to imposing the condition

g = r2 sin inthe original spherical coordinate basis. Using this form in eq (3.3) producesthe simple differential equation

dR3

dr= 3r2 (4.2)

Eq. (4.2) can be easily integrated to give

Rr0 = (r3 + r30)

1/3 (4.3)

where r0 is an integration constant. The subscript on the function Rr0(r)is necessary since r0 parametrizes the solution, providing different functions

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for different values of r0. Eq. (4.3) provides the formal connection between

the original Schwarzschild solution and the textbook Schwarzschild solution:they are distinctly different instances of a one-parameter family of solutionsdefined by eq. (4.3). Just as the asymptotic boundary condition on g00 iden-tifies the parameter mG as the geometric mass, the integration constant, r0,must be determined by the specification of a boundary condition, the valueof Rr0(0) which characterizes the nature of the source. The conventionalsolution imposes the condition that Rr0(r) = r everywhere, which demandsr0 = 0. The explication of this solution led to the interpretation that thesource of the field is a wormhole, an object with no classical analogue. Thepoint mass solution imposes the condition that the solution is regular ev-erywhere except at the location of the point mass, r = 0, which demands

r0 = 2mG. These are the results referenced in the discussion in Section 1.As we mentioned in Section 2, we may write the Combridge-Janne metricin the conventional textbook form by using Rr0(r) as the radial coordinate.But as we also pointed out, this involves the introduction of a family ofradial coordinates with each family member providing a distinct metric inthe textbook Schwarzschild form. If we examine two metrics obtained fromEq. (4.2), one with the parameter r0 = a and one with the parameter r0 = bwe have

ds2 = 1 2mG

Ra(r)c2dt2 1

2mG

Ra(r)1

R2a (r)dr2

R2a(r)d

2

(4.4)

Ra(r) = (r3 + a3)1/3

ds2 =

1 2mG

Rb(r)

c2dt2

1 2mG

Rb(r)

1

R2b (r)dr2 R2b(r)d2

(4.5)

Rb(r) = (r3 + b3)1/3

where a and b are determined by two unspecified but distinct boundaryconditions. If we now choose to write these metrics in the conventionaltextbook form, we have

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ds2 =

1 2mG

Ra

c2dt2

1 2mG

Ra

1

dR2a R2ad2 (4.6)

ds2 =

1 2mG

Rb

c2dt2

1 2mG

Rb

1

dR2b R2bd2 (4.7)

R3a a3 = R3b b3 (4.8)

The relation between Ra and Rb given in Eq. (4.8) is a necessary consequence

of the functions Ra(r) and Rb(r) given in Eqs. (4.4) and (4.5), respectively.While we may choose to use any coordinates we wish, if we want to obtain themetrics in (4.6) and (4.7), then we must use the functions in (4.4) and (4.5),no other choice for the radial coordinates will convert the Combridge-Janneform of the metrics into the conventional textbook form. This necessarilyrequires the relationship between the new radial coordinates given in (4.8).

To make a very explicit comparision, we may now specialize to the case ofthe two Schwarzschild solutions by setting a = 0 so that Ra(r) R0(r) = rand b = 2mG so that Rb(r) R2mG(r) = [r3 + (2mG)3]1/3 Rs(r), wherethe symbol Rs has been introduced for notational simplicity and to connectto the discussion in Section 1. Consequently, Eqs. (4.6), (4.7), and (4.8)

become

ds2 =

1 2mG

r

c2dt2

1 2mG

r

1

dr2 r2d2 (4.9)

ds2 =

1 2mG

Rs

c2dt2

1 2mG

Rs

1

dR2s R2sd2 (4.10)

R3s = r3 + (2mG)

3 (4.11)

Since r is in the range [0, ), the relation (4.11) between Rs and r showsthat Rs is in the range [2mG, ). But this does not mean that the metricin (4.10) is just the metric (4.9) outside of the event horizon at r = 2mG.

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Rs = 2mG corresponds to r = 0 and the location r = 2mG corresponds

to Rs = (2)1/3

2mG, so that the region outside of the textbook horizon isrepresented by

2mG r(2)1/3 2mG Rs

and the region inside of the event horizon is represented by

0 r < 2mG2mG Rs < (2)1/3 2mG

The metric (4.10) covers the entire spacetime, corresponding to both theregion inside and outside of the textbook event horizon. Furthermore, (4.10)does not have an event horizon at Rs = (2)

1/3 2mG so there is no eventhorizon corresponding to the textbook event horizon in the metric (4.10).Thus, we see that the metric (4.9) has an event horizon at r = 2mG and atime-like singularity at r = 0 (t and r swap roles inside the event horizon)while the metric (4.10) has no event horizon corresponding to the textbook

horizon and a space-like singularity with a coinciding event horizon at Rs =2mG. The appearance of the spherical surface singularity at Rs = 2mGis an artifact of using the g22 function as the radial coordinate. In theoriginal Combridge-Janne form, Rs = 2mG corresponds to Rs(0) = 2mGwhich identifies it as a point singularity at r = 0.

This exercise demonstrates that there is nothing to be gained by writingthe Combridge-Janne metric in the conventional textbook form. Comparingtwo metrics that have been put into this form requires careful delineationsof the various corresponding regions of spacetime and produces spuriousproblems of interpretation (for example, the spherical surface singularitymentioned in the previous paragraph). Comparing two metrics in the same

coordinate basis is much more straightforward and unambiguous than com-paring two metrics that have been forced into the same form.

The generalized Schwarzschild solution provides three types of solutionscorresponding to three distinct ranges of values for the parameter r0. Forr0 > 2mG, the solution is regular everywhere with no singularities or event

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horizons of any kind and may be considered to represent solutions for ob-

jects of finite size. Such solutions may be used to specify the potential2mG/Rr0(r) on a surface (Dirichlet boundary conditions) or the normalgradient 2mGRr0(r)/R2r0(r) on a surface (Neumann boundary conditions)without the complication of introducing step functions. For r0 < 2mGthe solution is of the wormhole type, with the event horizon located atRr0(reh) = [r

3eh + r

30]1/3 = 2mG. The conventional solution, as weve

shown, is obtained by setting r0 = 0. Note that if r0 is negative, thenRr0 = [r

3 |r0|3]1/3 means that Rr0(r) < 0 for r < |r0| which may precludemeaningful physical interpretation. The point mass solution r0 = 2mG pro-vides the boundary between wormholes and objects of finite size: it is afinite size object in the limit of zero size and it is a wormhole in the limit of

zero wormhole radius.Figure 3 provides a comparison of the generalized Schwarzschild function,eq. (4.3), to the textbook function R0(r) = r. It is clear that differencesbetween the two solutions for Rr0(r) are confined to the small r/r0 region.As mentioned in Section 1, for the point mass case, r0 = 2mG, differencesare noticeable for r 4mG but are completely negligible for r0 8mG.For the case of a finite object such as the sun, r0 = solar radius. Sincethe distance from the sun to Mercurys perihelion is approximately 66 solarradii, Figure 3 shows that deviations of (4.3) from R0(r) = r are completelyundetectable at that distance. Measurable deviations would seem to requireprobing distances well within Mercurys orbit, r 2 solar radii.

A characteristic feature that distinguishes the classes of solutions in the

generalized Schwarzschild solution is time-dependence. It is clear that thegeneralized Schwarzschild solution for r0 2mG provides a time-independentsolution: it satisfies the stationary and static conditions that were initiallyimposed on the form of the metric. This is not, however, the case for worm-hole solutions, the solutions for which r0 < 2mG. These solutions have acoordinate singularity and event horizon located at r3eh = (2mG)

3 r30 andare time-independent outside the event horizon but display time-dependenceinside the event horizon. This is in agreement with analyses based onKruskal-Szekeres coordinates [2, 9], isotropic coordinates [10], and the con-tinued use of the original (t,r,,) coordinate basis inside the event horizonat r = 2mG for which the radial coordinate is time-like and the time coor-

dinate is space-like. The solutions which have an event horizon are time-independent outside the event horizon but display time-dependence insidethe event horizon. The coordinate singularity of the textbook solution atr = 2mG does have physical significance: while it does not indicate a physi-cal singularity in the spacetime manifold, it does indicate the radius at which

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0 2 4 6 8 10

0

2

4

6

8

10

rr0

Rr0

(r

)

r0

r0 > 0r0 = 0

Figure 3: Comparison of the generalized Schwarzschild solution, eq. (4.3) forabitrary r0, with R0(r) = r. The value r0 = 2mG provides the comparison of theoriginal Schwarzschild solution with the textbook solution. Although differencesare visible for r 4r0, significant deviations appear for r 2r0.

the metric abruptly changes symmetry from time-independence (r > 2mG)to spatial-independence (r < 2mG).1

5 The Classical Gravitational Potential

Since the metric functions are connected to the classical gravitationalpotential in the weak-field limit, and since the metric functions depend onthe g22 function R(r), we seek to determine the connection between the clas-sical potential and the function R(r). This will provide the basis for anothercondition to determine R(r) and will naturally lead to a connection between

the solution of the general relativistic field equations and the experimental1This is the basic significance of the acceleration invariant based on the time-like Killing

vector [11], aa = m2

G/[R3(R 2mG)]: if an object has an event horizon in one set of

coordinates, then it has an event horizon in any set of coordinates. An event horizon cannot be removed by a coordinate transformation.

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activity that tests the Newtonian inverse-square force law [12]. To this end,

corrections to the classical potential from the general relativistic frameworkmust be identified. Consequently, we return to the general relativistic equa-tions of motion, the geodesic equations, to develop an exact relation betweenthe metric tensor components and the classical gravitational potential.

Our goal is to make an identification of a static classical gravitationalfield from the general relativistic equations of motion and we will begin byestablishing the general relativistic radial geodesic equation

d2r

ds2+

1

dx

ds

dx

ds= 0 (5.1)

By direct calculation, we find

d2r

ds2= d

dr

L20

2g11g00 1

2g11+

L22g11g22

(5.2)

where L0 and L are the constants of motion for the x0 and x2 equations,

respectively. In obtaining (5.2), we have used the fact that the constant solution, = 0, is a consistent solution with L = 0 and, by choosing0 = /2, the motion is then confined to the x y plane as for the classicalcase. The solution for then produces the equation for , g22 =

L, where

we have explicitly introduced a minus sign for convenience since g22 = R2.In addition, directly from the line element we have

dr

ds

2= 1 L

20

g11g00 L

2

g11g22(5.3)

The correct asymptotic behavior for g00 requires L0 = 1. Formally, theseresults are in strict analogy to the classical gravitational results

d2

rdt2

= ddr

VG(r) +

2

2r2

(5.4)

dr

dt

2= 2(e VG(r))

2r2

(5.5)

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where we use the constant polar angle solution = /2 to restrict motion

to the x y plane, the constants of the motion are the angular momentum and the energy per unit mass e = E/m0, and m0 is the mass of the testparticle located at the radial position r.

Now we write the general relativistic radial equation in terms of coor-dinate time t rather than s and identify the terms that correspond to theclassical gravitational potential. Alternatively, we may write the classicalradial equation in terms of s rather than the coordinate time t and identifythe classical gravitational potential with the corresponding terms in the gen-eral relativistic radial equation. In either case, this procedure correspondsto the demand that the classical gravitational potential VG(r) be such thatthe classical radial equation and the general relativistic radial equation be

identical. We then find that

d

dr

1

c2

(e VG(r))

g200

2

2r2g200

= d

dr

(1 g00)2g11g00

L2

2g11R2

(5.6)

This equation is easily integrated to produce

(e VG(r))

c2g200

2

2r2c2g200

=

(1 g00)2g11g00

2

2c2g11R2

+

e

c2(5.7)

where the constant of integration e/c2 and the identification L = /c havebeen obtained from the asymptotic limit far from the source.

Rather than solve this equation for the classical gravitational potential,we will now specialize to the static case, that is, to the case for which thetest particle is fixed in space and there is no particle motion. In this case,eq. (5.7) provides the exact relationship between the metric tensor and theclassical gravitational potential by setting e = VG(r) = constant and = 0,

VG(r) = c2

(1 g00)2g11g00

(5.8)

For the most general solution, g11g00 = R2 and since g00 = 1 2Gm/Rc2the classical gravitational potential is

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VG(r) = Gm

RR2 (5.9)

This is the exact relation between the classical gravitational potential andthe undetermined g22 metric function R(r).

If we now specialize to the generalized Schwarzschild solution for whichR = (r3 + r30)

1/3 and R2 = r2/R2, then the classical gravitational potentialis

VGS(r) = Gmr

1 +

r30r3

(5.10)

where the subscript GSdenotes the generalized Schwarzschild solution. Thisis the exact expression for the classical gravitational potential as derivedfrom the general relativistic framework specifying the g22 function by meansof the generalized Schwarzschild solution.

6 The GR Newtonian Solution

The discussion on the Schwarzschild solution has kept the focus on oneparticular auxiliary condition, the condition imposed on the square root ofthe determinant of the metric. We now propose a different auxiliary con-dition to be satisfied by the g22 metric function. We note that our resultfor the static classical gravitational potential, eq. (5.9), may be used as amethod to specify the undetermined g22 function R(r) by requiring R(r)to be such that the classical potential have a specific dependence on theradial coordinate r. As one example, suppose we demand that the classi-cal potential have the Newtonian form exactly. This would mean that allexperimental searches for deviations from the inverse square force law, nomatter how sophisticated, must always return a null result. Such a directand total connection between Einsteinian gravity and Newtonian gravity isintellectually pleasing and would be more in keeping with the spirit of clas-

sical general relativity. This can be accomplished by imposing the condition

RR2 = r (6.1)

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This may be integrated to obtain

RrN = (r3/2 + r

3/2N )

2/3 (6.2)

where rN is a constant of integration. As for the generalized Schwarzschildsolution, the constant of integration parametrizes the solution and is usedas a subscript to identify the solutions associated with a specific value ofrN. If we demand that RrN(r) be real, then rN 0. From the point of viewof the differential equation established in Section 3, (6.2) requires specify-ing the determinant of the metric by

g =

R3r sin . Since eq. (6.1)imposes the condition that the classical potential obtained from the general

relativistic equations have the Newtonian form exactly, we designate thesolution eq. (6.2) as the GR Newtonian solution. A primary physical dis-tinction between this one-parameter family of solutions and the generalizedSchwarzschild solution (4.3) is that the generalized Schwarzschild solutionprovides corrections to the classical gravitational potential in the asymptoticregion while the GR Newtonian solution, by definition, does not.

The GR Newtonian solution, characterized by eq. (6.2), is a one param-eter family of solutions that is analogous to the generalized Schwarzschildfamily. As for the generalized Schwarzschild parameter r0, the parameterrN should be determined by applying a boundary condition on RrN(r), thevalue of RrN(0), which is specified by the nature of the source object. Wenote that the textbook Schwarzschild solution may also obtained from this

family of solutions by setting rN = 0. The textbook solution is the onlymember of the generalized Schwarzschild family that does not provide acorrection to the classical potential, consequently, it is a member of the GRNewtonian family as well.

The singularity in g11 defined by RrN = 2Gm/c2 = 2mG is located at

r3/2eh = (2mG)

3/2 r3/2N (6.3)

Setting rN = 2mG places that singularity at the origin, r = 0, indicatingthat (6.2) with rN = 2mG is the solution appropriate for a point mass source,

but with asymptotic behavior that agrees with the original Schwarzschildsolution to lowest order only. The same small r boundary condition, re-quiring the solution to be regular everywhere except at the site of the pointmass r = 0, applied to the generalized Schwarzschild and GR Newtonian so-lutions produces two distinct point-mass solutions, since they are solutions

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to the two distinct differential equations (4.2) and (6.1). This underlines

the importance of identifying both the imposed auxiliary condition and theimposed boundary condition for determining R(r).


Generalized Schwarzschild:Determine R(r) by Eq. (4.2)

(Boundary condition determines r0)

OriginalSchwarzschild

(r0 = 2mG)

TextbookSchwarzschild

(r0 = 0)

GR Newtonian:Determine R(r) by Eq. (6.1)

(Boundary condition determines rN)

Point-massGR Newtonian

(rN = 2mG)

TextbookSchwarzschild

(rN = 0)

Figure 4: Diagrammatic representation of relationships among the various solu-tions presented in the text. The middle level of the tree identifies the auxiliarycondition used to determine R(r). The lowest level of the tree identifies the valuesofr0 and rN obtained from the boundary condition that uniquely determines R(r)for the imposed auxiliary condition.

The GR Newtonian solution defined by (6.2) consists of two well-definedclasses of solutions, depending on the value of rN, just as the generalizedSchwarzschild solution contains two well-defined classes according to thevalue of the parameter r0. For this family, the classes are: (1) rN 2mG,providing regular solutions of finite-sized objects with the point-mass solu-tion obtained in the limit of zero size, or rN = 2mG; (2) 0 rN < 2mG,providing wormhole solutions with event horizons of various wormhole di-ameters, the textbook solution corresponding to rN = 0. This is analogousto the discussion on the generalized Schwarzschild solution at the end of

Section 4. The parallels between the generalized Schwarzschild and GRNewtonian solutions are depicted in Figure 4.

Figure 5 compares the GR Newtonian, generalized Schwarzschild, andtextbook solution R0(r) = r for r0 = rN. The functional form of (6.2) re-quires the deviation from the linear R0(r) = r behavior to persist for larger

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0 2 4 6 8 10

0

0.5

1

1.5

rr0

R(r

)r

r

GR NewtonianOriginal Schwarzschild

R(r) = r

Figure 5: Comparison of GR Newtonian, generalized Schwarzschild, and textbooksolution R0(r) = r with rN = r0. Although differences among the three solutionsare restricted to the region of small r, the deviations of the GR Newtonian solutionfrom R0(r) = r persist much farther into the mid-r region compared to those ofthe generalized Schwarzschild solution.

values ofr than the corresponding behavior for the generalized Schwarzschildsolution. Nevertheless, the differences are still confined to the small r re-gion. For example, for r0 = rN = 2mG, the point mass solutions, GR Newto-nian shows deviations beyond 20mG while at those distances the generalizedSchwarzschild shows no detectable difference from R0(r) = r. However, adistance of 20mG is still very small for ordinary objects. For the case of anobject of finite extent such as the sun, r0 = rN = solar radius, there arestill no detectable deviations at distances corresponding to the perihelion ofMercury. But there does seem to be an opportunity to distinguish betweenthe GR Newtonian RrN(r) and R0(r) = r within Mercurys orbital distance,at a distance of several solar radii.

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experiments searching for small distance deviations from the Newtonian po-

tential by comparing power law modifications of the inverse square law ineq. (7.1) to the exact classical gravitational potential obtained from theSchwarzschild solution (5.10)

VG(r)Gm/r

=

1 +r30r3

(7.3)

This indicates that experiments should look for a value ofi = 4 to correspondto the correction found in the Schwarzschild solution with

4r34 = 4r34 + r

30 (7.4)

where the barred parameters are due to other mechanisms that also providei = 4 correction terms.

The discussion of the two classes of the generalization of the Schwarzschildsolution in Section 4 suggests the following scheme for interpreting experi-mental or observational determinations of the Schwarzschild parameter r0.First, it is important to recognize that the parameter r0 is tied to the sourceof the spacetime curvature and so there can be no single unique determina-tion of the parameter. For a gravitational field due to an extended objectsuch as the sun, a determination of r0 should find a value that is greaterthan twice the geometric mass of the object, r0 > 2mG, approximating thephysical size of the object, and a value of r0 2mG should be obtainedonly for an object that is a good approximation for a point mass. For anobject that may be a candidate for a wormhole, a value for r0 that is in therange r0 < 2mG should be obtained. However, unless the mass parameteris exorbitantly large, this correction should be detectable only for objects offinite extent that have a sufficiently large radius or that can be probed tosufficiently small r. As Figure 6 shows, significant deviations from the clas-sical potential based upon the generalized Schwarzschild solution eq. (7.3)appear for r 2r0, but are completely negligible for r 8r0. For example,for an object of solar dimensions, at a distance corresponding to Mercurys

perihelion, the correction (r0/r)3 is 3.5 104 %, at a distance of 4 solarradii the correction is 1.5 %, and at a distance of 2 solar radii the correctionis 12.5 %. Such corrections to the classical potential should be detectablewithin 2 solar radii from the surface of the sun. For ordinary objects of

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0 2 4 6 8 100

5

10

15

rr0

VG

(r

)

Gm/r

Original SchwarzschildClassical Potential

Figure 6: Corrections to the classical gravitational potential provided by the gen-eralized Schwarzschild solution, eq. (7.3). Significant deviations from the classicalinverse-distance potential appear for r 2r0.

finite size, it may be possible to isolate this correction in the laboratory by

a Cavendish torsion-style experiment.

8 Conclusion

The Combridge-Jannes solution is the exact solution for the time inde-pendent and spherically symmetric gravitational problem. It is a class ofsolutions that consists of a one-function family of solutions. This familyof solutions is conveniently characterized by the g22 metric function R(r)since the other metric functions, (r) and (r), are simple functionals ofR(r). The tensor field equation, R = 0, establishes the conditions to deter-

mine the metric tensor functions for the vacuum case. Since the principlesof the theory provide no apparent reasons to expect R(r) to be undeter-mined by the field equations, the inescapable conclusion is that the vacuumfield equations provide an incomplete description. The Combridge-Jannesolution demonstrates that any R(r) that approaches r in the asymptotic

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Lorentzian spacetime provides a valid solution. The original Schwarzschild

solution, the textbook Schwarzschild solution, the generalized Schwarzschildsolution, and the GR Newtonian solution all satisfy this condition and allare members of the Combridge-Janne class of solutions (see Figure 4).

We must emphasize that the corresponding forms for R(r) in these solu-tions have not been generated by changes to different coordinate bases. Theyare different functions expressed in a single coordinate basis, (t,r,,), thebasis that was introduced in Section 3. These functions have been uniquelydetermined by a specific imposed auxiliary condition in conjunction withthe appropriate boundary condition corresponding to a desired source ob-ject. The boundary condition, the specification of R(0), provides informa-tion about the nature of the source object and determines the behavior of

R(r) as the source is approached. This is an important and key feature ofthe function R(r). Multiple forms for R(r) are permitted, however, becausethere is no uniquely specifed condition by which to determine the g22 metricfunction. We have examined two specific conditions for determining R(r),imposing a condition on the determinant of the metric and imposing a con-dition on the form of the classical potential, but an unambiguously uniquemethod for determining the function R(r) is needed to proceed beyond theCombridge-Janne class of solutions (see Figure 1).

Within the current formalism of general relativity, we have establisheda deceptively simple but rigorous differential equation,

R2R = 13

dR3

dr= g

sin (3.3)

that links the function R(r) to the determinant of the metric. While thisequation shows the function R(r) is specified by the determinant of the met-ric tensor, R(r) will remain undetermined unless a condition is imposed onthe determinant. Consequently, we emphasize that an arbitrary specificationof the determinant of the metric still leads to an arbitrary specification ofthe function R(r), it simply follows from (3.3) rather than as a direct choicefor R(r) or on some other basis. For example, following Schwarzschild andimposing the appropriate condition on

g, we have provided a simplegeneralization of Schwarzschilds original solution and have shown how boththe original and the textbook solutions are obtained from this generalizedSchwarzschild solution. But we stress that there is no compelling reasonfor demanding that

g = r2 sin . Apart from the historical aspect, the

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strongest justification for applying this condition is thatg = r2 sin

provides the familiar spatial 3-volume element, dV3 = r2

sin dr d d.Both the generalized Schwarzschild solution and the GR Newtonian so-

lution contain two classes of solutions: (a) those for objects of finite sizewhich are regular everywhere and have no event horizon and (b) those withan event horizon. Each family of solutions contains a point mass solution,an object of finite size in the limit of zero size and a wormhole in the limit ofzero wormhole radius. Consequently, the point mass solutions have a physi-cal singularity and coincident event horizon at r = 0. The family parametersof those solutions, r0 and rN, respectively, vary from source to source and donot have a single value for all sources. Such solutions provide a considerablywider range of descriptions than the single textbook wormhole solution.

The exact relation between the classical gravitational potential and themetric functions derived in Section 5 permits a determination of the g22function based upon the form of the classical potential. Imposing the condi-tion that the classical potential have the Newtonian form exactly generatesthe GR Newtonian solution, characterized by eq. (6.2). This solution isclearly different from the generalized Schwarzschild solution, although theydo share the textbook Schwarzschild solution as a special case since thetextbook Schwarzschild solution satisfies both imposed conditions, the con-dition on the determinant of the metric and the condition on the form ofthe classical potential. The GR Newtonian solution also produces a pointmass solution that is inequivalent to the original Schwarzschild solution. Al-though these point mass solutions obey the same boundary conditions for

r 0, they satisfy different differential equations. This exemplifies thetwo separate aspects related to the determination of the g22 metric functionR(r): (1) imposed conditions are used to determine R(r) (see Figure 1) and(2), for a given imposed condition, the boundary condition appropriate tothe given physical configuration provides the unique solution (see Figure 4).Consequently, it is important to identify both the imposed auxiliary condi-tion and the boundary condition that is used to complete the determinationof R(r).

As we noted in Section 6, should there be no convincing evidence fordeviations of the classical inverse square force law, then the GR Newtoniansolution defined by eq. (6.2) is validated. Such null results must also in-

clude tests of the generalized Schwarzschild solution and the parameter r0.Although a single unique determination of the generalized Schwarzschild pa-rameter r0 is not possible (Section 7), if all experimental or observationalevidence provide convincing null results in determining corrections to theclassical gravitational potential, then that, in toto, may be taken as evi-

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dence that all generalized Schwarzschild solutions are invalid except for the

textbook solution, r0 = 0. This, of course, is simply due to the validationof the GR Newtonian solution, (6.2). However, if such deviations from theclassical potential are found, then the GR Newtonian solution would haveto be declared invalid and the function R(r) could then be determined byeq. (7.2), which must include corrections such as those from the generalizedSchwarzschild solution, or by some other method. Since R(r) may be inter-preted as a measure of spatial distance and so may be related to distanceson other length scales, the metric function R(r), as suggested by (7.2), maybe effectively used as a back door through which quantum mechanics couldbe incorporated into the classical general relativistic framework.

In the course of this work, we have examined only two imposed auxiliary

conditions that determined the g22 metric function R(r). The first conditionon the determinant of the metric produces the generalized Schwarzschild so-lution and has a long associated history due to the original and textbookSchwarzschild solutions. The second condition is based on the form of theclassical potential, stimulated by a what if assumption that is consistentwith the current lack of experimental evidence of deviations from the New-tonian inverse-square law, and provides the GR Newtonian solution. Butwe stress that there is no compelling basis for choosing either one of theseimposed conditions. As long as the correct asymptotic behavior R(r) r isretained far from the source, anything is equally acceptable until somethingin the theoretical basis of general relativity requires a particular conditionthat determines R(r).

Acknowledgment

We would like to thank Arthur Kosowsky of the Pittsburgh ParticlePhysics, Astrophysics, and Cosmology Center at the University of Pitts-burgh for an extended email dialogue concerning an earlier version of thispaper. In particular, his comments and observations were instrumental indeveloping the material concerning the use of the g22 metric function, R(r),as a radial coordinate.

References

[1] R. Adler, M. Bazin, and M. Schiffer, Introduction to General Relativity,Second Edition (McGraw-Hill, New York, 1975).

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[2] C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (Freeman,

San Francisco, 1973).

[3] K. Schwarzschild, Uber das Gravitationsfeld eines Massenpunktesnach der Einsteinschen Theorie, Sitzungsber. Preuss. Akad. Wiss.,Phys. Math. Kl., 189, (1916).

[4] K. Schwarzschild, On the Gravitational Field of a Mass Point Accord-ing to Einsteins Theory, trans. by S. Antoci and A. Loinger, Gen.Relativ. Gravit. 35, 951 (2003). Also arXiv:physics/9905030v1 (1999).

[5] K. Schwarzschild On the Gravitational Field of a Point-Mass, Accord-ing to Einsteins Theory, trans. by L. Borissova and D. Rabounski,

Abra. Zel. Jour.1

(2008).[6] W. de Sitter, On Einsteins Theory of Gravitation, and Its Astronom-

ical Consequences (First Paper), Month. Not. R. Astr. Soc., 76, 699(1916). In particular, see Sections 10, pg. 711, and 11, pg. 714.

[7] J.T. Combridge, Phil. Mag., 45, 726 (1923).

[8] H. Janne, Bull. Acad. R. Belg., 9, 484 (1923).

[9] M.D. Kruskal, Maximal Extension of Schwarzschild Metric, Phys.Rev., 119, 1743 (1960).

[10] H.A. Buchdahl, Isotropic Coordinates and Schwarzschild Metric, Int.J. Theor. Phys., 24, 731 (1985).

[11] S. Antoci and D-E Liebscher, Reconsidering Schwarzschilds OriginalSolution, Astron. Nachr., 322, 137 (2001). Also arXiv:gr-qc/0102084v2.

[12] E.G. Adelberger, B.R. Heckel, and A.E. Nelson, Tests of the Gravita-tional Inverse-Square Law, Ann. Rev. Nucl. Part. Sci. 53 (2003) 77.Also arXiv:hep-ph/0307284v1.

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The Combridge-Janne Solution and the Gravitational $g_{22}$ Metric Function

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