the two schwarzschild solutions: a critical assessment

7/28/2019 The Two Schwarzschild Solutions: A Critical Assessment

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The Two Schwarzschild Solutions: A Critical

Assessment

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

Revised: 29 Sep 2013

Abstract

We present a pedagogically sound derivation of the most general solution of thetime-independent, spherically-symmetric gravitational field equations. We use thatsolution, the Combridge-Janne solution, as a basis for evaluating the original and text-book Schwarzschild solutions. We demonstrate that both versions of the Schwarzschildsolution are valid, are distinct and not equivalent to each other, but are related bymeans of a one-parameter family of solutions. We explicitly show that the originalsolution is the appropriate solution for a point mass source while the textbook so-lution is the appropriate solution for a wormhole source. In addition, the textbooksolution necessarily has a time-dependent aspect while the original solution is truly

time-independent. A number of issues surrounding these two solutions are clarifiedand resolved.

Keywords: Combridge-Janne solution, generalized Schwarzschild solution, Schwarzschildsolution, original Schwarzschild solution, Hilbert solution, textbook Schwarzschild so-lution

1 Introduction

It is now known that the Schwarzschild solution, the first and best known exact solution

to the general relativistic field equations, actually comes in two versions. The first version,which we will temporarily call the textbook solution [1,2] or the Hilbert solution [3,4], is by

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

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far the most familiar, has the well-known coordinate singularity and event horizon at theSchwarzschild radius, was the origin for the concept of a singularity in spacetime, and hasthe preferred interpretation as the gravitational field of a wormhole in spacetime [1, 2, 5].The second version, which we will call the original Schwarzschild solution [68], is regular

everywhere, has a physical singularity and event horizon at the single point r = 0, and hasthe simple interpretation as the gravitational field for a point mass located at r = 0 [69]

The basic distinction between these two Schwarzschild solutions is traced to a singlemetric function, g22. The textbook solution develops from three general relativistic fieldequations in two unknown metric functions, g00 and g11. The g22 function, in effect, is cho-sen to be g22 = r2 by means of an argument based on a suitable choice of coordinatesand is not determined by a field equation [1, 2]. It also satisfies the asymptotic boundarycondition that the gravitational field be described by the Newtonian theory far from thesource. The original Schwarzschild solution develops from four equations in three unknownmetric functions, g00, g11, and g22. The four equations consist of the three general rela-tivistic field equations plus the subsidiary condition

g = 1 where g is the determinant

of the metric [68]. While this additional condition was originally part of the structure ofthe theory of general relativity and later abandoned, Einstein continued to promote thisadditional equation as defining a particularly convenient choice of coordinates that maysimplify resulting equations [10]. Consequently, we view Schwarzschilds approach as uti-lizing a choice of coordinates such that

g = 1 and, having completed the solution inthose coordinates, then expressing the solution in the original contravariant coordinates.In addition, Schwarzschilds solution was designed to satisfy two boundary conditions. Thefirst is the same asymptotic condition that the textbook solution satisifies, that the fieldbe described by Newtonian theory far from the source. The second is the condition thatthe metric functions be finite and continuous everywhere except at r = 0, the presumedsite of the point mass source object. Schwarzschilds original solution then states that

g22 = R2

s(r) = (r3

+ 3

)2/3

where = 2mG and mG = Gm/c2

is the geometric mass ofthe object [68].The difference between the number of boundary conditions imposed to uniquely de-

termine the solutions is important. The field equations consist of a system of differentialequations and the highest order of those equations is two. Consequently, we expect thattwo boundary conditions are needed to uniquely determine the solution. As we pointed outin the above discussion, Schwarzschild did originally impose two boundary conditions whilethe textbook presentation imposes only one. However, as we show below in Section 7, thetextbook condition that g22 = r2 everywhere serves as the necessary second boundarycondition for the textbook solution. It is not recognized as a boundary condition becauseit is imposed before the field equations are even established and its role is obscured by theclaim that the condition is the result of a choice of coordinates.

In both versions of the solution, only two field equations are necessary in the determi-nation of the solution, but the third field equation is satisfied by that solution. Textbookpresentations often state that a third field equation is not needed and it is a common

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exercise to verify that the textbook solution does indeed satisfy the third field equation.Schwarzschild had pointed out in his original paper that only two of the three field equa-tions, with the addition of the auxiliary condition, were needed, but he verified that thesolution did satisfy the third equation. The fact that the third field equation is not needed

in either case and yet is satisfied by both solutions is never explored and is simply acceptedas a pleasant fact. It is not, however, just a matter of luck. In 1916, within the context of aperturbative treatment of the time-independent inhomogeneous and spherically symmetricgravitation problem, de Sitter showed that there are only two independent field equations,not three [11]. This is an exact and rigorous statement in the limit of vacuum field condi-tions. This means that the third field equation is necessarily satisifed by the solution sinceit is a consequence of the other two equations. Furthermore, it should be expected that thevacuum field equations will produce a solution for two of the metric functions in terms ofthe third metric function while the third metric function remains undetermined by the fieldequations. This, in fact, was explicitly demonstrated independently by Combridge [12] andJanne [13] in 1923.

The solution provided by Combridge and Janne provides the basis for understanding thethe relationship between the two Schwarzschild solutions. Only two equations are neededfor the solution because the third field equation is not independent of the other two. Thus,imposing the condition

g = 1 provides Schwarzschild with the basis for determiningthe third metric function; stating that g22 = r2 everywhere appears to provide the thirdmetric function simply by fiat. We will show below in Section 7 that both the original andtextbook Schwarzschild solutions come from the same imposed condition,

g = 1, butcorrespond to two different boundary conditions imposed on the solution.

While work has been done on the original Schwarzschild solution, much of the emphasishas been on attempting to discredit the textbook solution as a solution [1421] On the otherhand, the proponents of the textbook solution view the original solution as misguided at

worst and nothing new and equivalent to the textbook solution at best [2224]; most often,it is simply ignored. In addition, the use of gauge function terminology [25], distinguishingbetween the original Schwarzschild solution and the textbook solution, or Hilberts solution,as different gauges, provides a false sense of understanding and obscures the underlyingrelationship between these two solutions.

To clarify the issues surrounding and underlying the two Schwarzschild solutions, webegin by simply removing all assumptions and agendas in the derivation of the solution andexamine the rigorous exact solution without bias. To facilitate connections with historicalpapers that have worked with the differential equation approach and to reach the widestpossible audience, we choose to work in a standard coordinate basis and with the resultingdifferential equations. In Section 2 we define our coordinate basis and derive the geodesicequations, Christoffel symbols of the second kind, the components of the Ricci tensor, andthe vacuum field equations for the spherically symmetric system without any assumptionsor imposed conditons on the coordinates. In Section 3 we provide our derivation of the mostgeneral time-independent, spherically symmetric solution of the vacuum field equations.

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This is the solution mentioned above that was derived independently by Combridge andJanne. In the process of doing so, we explicitly demonstrate de Sitters discovery relating tothe number of independent field equations which underlies the Combridge-Janne solution.While we use the Combridge-Janne solution as the basis for our analyses, we focus almost

exclusively on understanding the relationship between the original Schwarzschild solutionand Hilberts version of the Schwarzschild solution. The importance of the Combridge-Janne solution, however, goes well-beyond understanding the connections between thesetwo Schwarzschild solutions [26].

We show that Hilberts version of the Schwarzschild solution and the original Schwarzschildsolution are distinct solutions and can not be shown to be equivalent by means of a coordi-nate transformation (Section 4). In Section 5, we then discuss the issues involved in usingthe g22 metric function as a radial coordinate. In Section 6 we derive a very simple butrigorous equation relating the g22 metric function and the determinant of the metric.

1 Wethen show that that equation is the groundwork for a generalization of the Schwarzschildsolution into a one-parameter family of solutions that includes both Hilberts version of

the solution and the original solution on an equal footing as special cases (Section 7). Itis explicitly shown that boundary conditions provide the basis for the individual solutionswhich establishes them as separate and distinct solutions. We summarize and discuss ourresults in Section 8.

2 Derivation of the Field Equations

We choose our coordinate basis to be a time-like coordinate and the standard sphericalspatial coordinates, (x0 = ct,r,,). In all that follows, the symbols t,r,, and alwaysrefer to this coordinate basis and any change of coordinate basis is explicitly stated and

utilizes typographically distinct symbols. It is well-known that the most general line el-ement for a time-independent and spherically symmetric system in this coordinate basishas the form [1,2]

ds2 = A(r)(dx0)2 B(r)dr2 C(r)d2 (2.1)

d2 = d2 + sin2 d2 (2.2)

where the three metric functions A, B, and C are functions of r only and d is thedifferential solid angle. This line element is stationary (A, B, and C are independent oft),

1The Schwarzschild condition on the determinant of the metric is not the only imposed condition thatmay be used to determine the g22 metric function. A wide variety of imposed conditions may be used inits place [26].

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static (g0k = 0 for k = 1, 2, 3), and rotationally invariant (g33 = g22 sin2 and A, B, and C

are functions of |r| only). Since the limiting form of the line element for sufficiently largedistances from the source of spacetime curvature is expected to be Lorentzian

ds2 (dx0)2 dr2 r2d2 (2.3)

then A(r) 1, B(r) 1, and C(r) r2 in the asymptotic region. The last conditionensures the circumference of a great circle is given by 2r and the surface area of a sphereis given by 4r2 in the asymptotically flat spacetime.

We now introduce the radial function 2 R(r) =

C(r) rather than choose C(r) = r2.The choice C(r) = r2 obscures the underlying structure of the equations and provides nosignificant simplification in the solution process. In addition, it is unnecessarily restrictive.The conventional approach is to caste such a choice in the language of a coordinate trans-formation, r = C(r). However, such an approach requires the new radial coordinate toobey

C(0) r < . To maintain that 0 r < means either the radial coordinate

r was not obtained from C(r), effectively resulting in C(r) = r2 as a choice for C(r), orC(0) = 0 was tacitly imposed. The condition

C(0) = 0, however, is not a general

condition but is a specific boundary condition on the function

C(r) and is appropriatefor a specific source object, the source now known as a wormhole. This has been shown tobe a specific case of a more general solution [26] which will also be demonstrated below inSection 7.

To display obvious parallels with the standard results and field equations, we will adoptthe customary exponential forms for A(r) and B(r) [1,2] so that we write the metric as

ds

2

= e

(dx

0

)

2

e

dr

2

R2

d

2

(2.4)

where , , and R are functions of r only. The Lorentzian character of the metric in theasymptotic region now means that both and vanish sufficiently far from the sourcewhile R(r) r.

We follow the standard approach to calculating the Christoffel symbols of the secondkind and note that the equations of the geodesic lines

x +

xx = 0 (2.5)

result from the Euler-Lagrange equations obtained from the variational problem

2The symbols R and R(r) will always refer to the g22 metric function. To avoid confusion with the Ricciscalar, any references to the Ricci scalar will use the contracted tensor symbol R.

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F ds = 0 (2.6)

where the function F is given by

F = e(x0)2 e(r)2 R2()2 + sin2 ( )2 (2.7)In the above equations, the overhead dot denotes differentiation with respect to the lineelement variable s, x0 = dx0/ds, r = dr/ds, = d/ds, and = d/ds. The Euler-Lagrange equations resulting from this variational problem are

d

ds

F

x

F

x

= 0 (2.8)

Calculating the partial derivatives and simplifying the resulting equations, we obtain thegeodesic equations for = 0, 1, 2, 3, respectively,

2e

x0 + rx0

= 0 (2.9)

r +1

2e(x0)2 +

1

2(r)2 eRR()2 eRR sin2 ( )2 = 0 (2.10)

+ 2R

R r sin cos ( )2

= 0 (2.11)

+ 2R

Rr + 2 cot = 0 (2.12)

where the prime denotes differentiation with respect to the radial coordinate r. Comparingthese equations to the geodesic equations (2.5) for = 0, 1, 2, and 3 in turn provides all ofthe non-zero Christoffel symbols which we list in Table 1. In addition, the determinant ofthe metric tensor, g, and the relevant term that appears in the Ricci tensor, ln

g, arealso given in Table 1. These equations are the generalization of the standard results forthe geodesic equations, Christoffel symbols, and determinant of the metric by including

the g22 function R(r). It is very easy to verify that these equations revert to the standardtextbook results for R(r) = r.

The Ricci tensor may be written as

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Table 1: The non-zero Christoffel symbols of the second kind obtained from the geodesic equations,

eqs. (2.9) - (2.12). We also include entries for the determinant of the metric g and ln g.

0

1 0

=

0

0 1

1

2

1

0 0

1

2e

1

1 1

1

2

1

2 2

eRR

1

3 3

eRR sin2

2

2 1

=

2

1 2

R

R

2

3 3

sin cos

3

2 3

=

33 2

cot

3

1 3

=

3

3 1

R

R

g = ||g || e+R4 sin2

lng 1

2(+ ) + 2 ln R + 1

2ln(sin2 )

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R =

lng

||

|

+

ln

g|

(2.13)

For static and spherically symmetric systems, the only non-zero derivatives are those withrespect to x1 = r (contravariant coordinate index = 1) and x2 = (contravariantcoordinate index = 2). By direct calculation, the non-zero components of the Riccitensor are

R00 = e

2 +

2

2

2+

2R

R (2.14)

R11 =1

2

+

2

2

2 2

R

R+

4R

R

(2.15)

R22 = (eRR) 1 + (eRR)

+ 2

(2.16)

R33 = sin2

eRR 1 + (eRR) +

2

= sin2 R22 (2.17)

All other R are identically zero and provide no additional information. Eqs. (2.14) (2.17) provide the most general Ricci tensor components for the time-independent spheri-

cally symmetric problem.The vacuum equations, R = 0, now yield the three general relativistic field equations

+2

2

2+

2R

R= 0 (2.18)

+2

2

2 2

R

R+

4R

R= 0 (2.19)

(eRR)

1 + (eRR)

+

2 = 0 (2.20)

since the R33 = 0 equation duplicates the R22 = 0 equation. These are the time-independent spherically symmetric vacuum field equations which explicitly contain the

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g22 metric function R(r). Although it is often stated that, for the time-independent spher-ically symmetric case, R = 0 is a system of three independent equations, we wish tore-state and emphasize that de Sitter has shown that only two of the three field equationsare independent [11]. Consequently, the field equations are not sufficient to fully determine

the solution.

3 Derivation of the Combridge-Janne Solution

We will now provide the solution of the vacuum field equations without any additionalassumptions or imposed conditions, such as Hilberts choice of coordinates such thatR = r or Schwarzschilds choice of coordinates such that

g = 1. It will eventuallybecome clear that a choice of coordinates has nothing to do with such imposed conditions.It will also become evident that the solution is quite straightforward and is not in need of

any purported simplifications.Subtracting (2.19) from (2.18) gives

+ 2R

R= 0 (3.1)

Substituting eq. (3.1) into (2.18) yields

+ 2 R

R+

2R

R= 0 (3.2)

Substituting (3.1) into (2.19) also produces (3.2). Substituting (3.1) into eq. (2.20) andsimplifying yields

d

dr(eRR2) = R (3.3)

So now our three independent equations are (3.1), (3.2), and (3.3). This last equation,(3.3), may be integrated to yield

eR2

= 1

R (3.4)

where is an integration constant. Eq. (3.1) may also be easily integrated to obtain

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e+ = eC0R2 (3.5)

where C0 is another integration constant. Now, comparing (3.5) with (3.4), we find that

e = eC0

1 R

(3.6)

Our three equations are now (3.6) (an algebraic equation relating and R), equation (3.4)(a differential equation relating and R), and equation (3.2) (another differential equationrelating and R).

Using eq. (3.6) to eliminate and from eq. (3.2) does not produce a differentialequation for the function R(r), it produces an identity with no conditions on the functionR(r). To see why this is so, we note that (3.2) contains only and R and their derivatives.

By direct calculation of the derivatives of (3.6), we find

e = eC0

R

R2

(3.7)

e

= eC0

R

R2

(3.8)

Taking the ratio of these equations removes the constant factor eC0 and produces

e

e

=

R/R2

R/R2 (3.9)

Performing the differentiations in (3.9) and simplifying yields

+ 2 R

R+

2R

R= 0 (3.10)

which we immediately recognize as eq. (3.2). Thus, the third field equation (3.2) isultimately a consequence of eqs. (3.1) and (3.3); there are only two independent fieldequations, not three. This provides an explicit demonstration of the result obtained by deSitter [11].

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The asymptotic far-field limit determines the constant C0 to be zero and the constant to be twice the geometric mass of the source, = 2mG = 2Gm/c

2, now known as theSchwarzschild radius. We now have determined the constants of integration, but the metricfunctions (r) and (r) remain as functionals of the g22 metric function R(r). Consequently,

having used all the available field equations, the function R(r) remains undetermined andthe solution to the vacuum field equations is not unique. It may be characterized as aone-function family of solutions and can be summarized as

e = 1 2mGR(r)

(3.11)

e =

dR(r)

dr

21 2mG

R(r)

1(3.12)

R(r) = undetermined (3.13)

We do know, however, that R(r) must obey the Lorentzian asymptotic condition R(r) ras r . As we mentioned in Section 2, this will guarantee the circumference of agreat circle in the asymptotic flat spacetime is 2r and the surface area of a sphere is4r2. Although R(r) has the interpretation of the radius of a circle, C = 2R(r), or ofa sphere, A = 4R2(r), in curved spacetime, the flat space conditions must be fulfilled inthe asymptotically Lorentzian spacetime only.

To cast these results in the notation utilized by de Sitter, Combridge, and Janne, wedefine the function (r) by

R = re/2 (3.14)

The metric may now be written as

ds2 = ec2dt2 edr2 er2d2 (3.15)

This is the form of the metric originally established by de Sitter [11] and used by Com-bridge [12] and Janne [13]. The undetermined nature of R(r) means that the function (r)is undetermined.

Previously, the standard approach to removing the indeterminancy of the solution in-volving , , and has been to assume some relation among the functions , , and [11, 18]. For instance, choosing = 0 is equivalent to the symmetry condition thatR(r) = re/2 which produces the isotropic metric

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ds2 = ec2dt2 edr2 + r2d2 (3.16)

e

= e

=R2

r2 =

1 +mG2r

4(3.17)

e =

1 2mGr

2

1 + 2mGr2 (3.18)

We note that this metric has not been obtained by means of a coordinate transformation toisotropic coordinates [1,27] or any other set of coordinates, the coordinate basis remainsthe original spherical spatial coordinate basis introduced in Section 2. It has been obtainedby simply exploiting the extra degree of freedom provided by the indeterminate nature ofR(r) to produce the metric with the desired symmetry. Such an ab initio approach forthe isotropic metric was advocated nearly 100 years ago by Eddington [28]. We should

point out, however, that the condition R(r) = re/2 is valid only for e/2 0: it must besupplemented by another condition if e/2 < 0 is allowed [27].

Another example is provided by imposing the condition + 2 + = 0. Although deSitter identified this condition as corresponding to Einsteins condition of choice [11], itmore accurately produces the condition R2R4 = r4 which may be used as the basis for de-riving the original Schwarzschild solution. This provides an indication that Schwarzschildsoriginal solution and the textbook solution do arise from the same condition. As a finalexample, Hilberts version of the Schwarzschild solution also immediately results from sim-ply choosing = 0 so that R(r) = r. While other choices are admissable, it must bestressed that any such condition is arbitrary and amounts to nothing more than choosinga metric function by decree; such a procedure provides no fundamental determination of

the function R(r).

4 The Two Schwarzschild Solutions

The Combridge-Janne solution reduces to the original Schwarzshild solution and Hilbertsversion of the Schwarzschild solution for Rs(r) = (r

3 + (2mG)3)1/3 and RH(r) = r, respec-

tively. We will now demonstrate that these two versions of the Schwarzschild solution arephysically distinct and are not reducible to each other by means of a coordinate transfor-mation. First, we note that it is not possible to write the two metrics in terms of differentcoordinate bases. For example, if we write

ds2 =

1 2mG

r

c2dt2

1 2mG

r

1dr2 r2d2 (4.1)

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ds2 =

1 2mG

Rs

c2dt2

1 2mG

Rs

1dr2 R2sd2

(4.2)

Rs(r) =

r3 + (2mG)31/3

where Rs = Rs(r), the two spherical coordinate bases (r,,) and (r,,) are related totwo corresponding rectangular coordinate bases (x,y ,z) and (x,y ,z). Because the rangeof the two sets of spherical coordinates are identical, the corresponding ranges of the rect-angular coordinate bases are also identical. At most, the two coordinate systems can bedistinguished by a translation and a rotation, but there can be no deforming transfor-mations that may alter the range of the coordinates. Consequently, there is no way todistinguish between the coordinate bases (r,,) and (r,,) so that r = r without loss

of generality. In fact, we show explicitly in Section 7 that the two Schwarzshild solutionsare obtained in the same coordinate system, (t,r,,), the basis that was introduced inSection 3.

Second, we note that by using the chain rule, we may convert the line element (4.2)from the Combridge-Janne form to the conventional textbook form

ds2 =

1 2mG

Rs

c2dt2

1 2mG

Rs

1dR2s R2sd2 (4.3)

While the chain rule has converted the g22 function Rs(r) into the radial coordinate, the

only function that can be used in conjunction with the chain rule to produce the lineelement (4.3) for the original Schwarzschild solution is the function defined in (4.2). Thefunction in (4.2) no longer provides the g22 function as a function of the radial coordinater, but it does provide the necessary relationship between the two radial coordinates r andRs in metrics (4.1) and (4.3), respectively:

Rs =

r3 + (2mG)31/3

(4.4)

We point out that eq. (4.4) does not function as the coordinate transformation that putsone metric in the form of the other: using (4.4) in the textbook metric (4.1) does not

produce the original metric (4.3) and using (4.4) in the original metric (4.3) does notproduce the textbook metric (4.1).It is clear that (4.3) has the form of the textbook Schwarzschild metric. It is also clear

that Rs 2mG since eq. (4.4) shows that Rs(0) = 2mG. Thus, it is easy to misinterpret

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(4.3) as corresponding to the textbook solution outside of the event horizon. To clarifythe issue, we note that the textbook solution has an event horizon at r = 2mG whichcorresponds to Rs = (2)

1/3(2mG). The line element (4.3) has no event horizon at thatradius, it is perfectly well-behaved there. In fact, it is perfectly well-behaved everywhere

except for a singularity at Rs = 2mG. But Rs = 2mG corresponds to r = 0 not to r = 2mG.So, delineating the regions relative to the site of the textbook event horizon, the regionexterior to the textbook event horizon corresponds to the ranges

r > 2mG

(4.5)

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

and the region interior to the textbook event horizon corresponds to the ranges

0 r < 2mG(4.6)

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

Consequently, the metric (4.3) covers the entire spacetime, both inside and outside of thetextbook event horizon with no complications or subtleties of any kind. The textbook

metric (4.1) has an event horizon at r = 2mG and a temporal singularity at r = 0 sincethe radial coordinate r is timelike inside the event horizon. The original metric (4.3) hasno event horizon corresponding to the textbook event horizon and has a spatial singularityand coinciding event horizon at Rs = 2mG or r = 0. These are two completely differentmetrics describing two completely different physical configurations.

We note in passing that the singularity of the metric (4.3) at Rs = 2mG appears to beat a spherical surface. This, however, is an artifact of using the g22 function Rs(r) as theradial coordinate. This is a problem of interpretation that does not appear when using theoriginal coordinate basis (t,r,,). The minimum value of Rs(r) simply means that theg22 metric function has a non-zero minimum at r = 0, it does not imply a minimum valuefor the radial coordinate r different from r = 0. The singularity at Rs = 2mG does notindicates a surface singularity, it corresponds to a point singularity at r = 0. Other issuesof interpretation and meaning related to using the general R(r) as the radial coordinateare discussed below in Section 5.

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Finally, we offer one last and perhaps more straightforward approach to demonstratingthe inequivalence of the textbook metric and the original metric. Given Hilberts versionof Schwarzschilds solution [1, 2],

ds2 =

1 2mG

r

c2dt2

1 2mG

r

1dr2 r2d2 (4.7)

where 0 r < , and the original Schwarzschild solution

ds2 =

1 2mG

Rs

c2dt2

1 2mG

Rs

1dR2s R2sd2 (4.8)

where Rs(0) = 2mG

Rs

the two schwarzschild solutions: a critical assessment

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