Proper Formulation of Birkhoff's Theorem

8/10/2019 Proper Formulation of Birkhoff's Theorem

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Proper Formulation of Birkhoffs Theorem

R. E. Salvino

9 Thomson Lane, 15-06 Sky@Eleven

Singapore 927726

7 December 2014

Abstract

The conventional formulation of Birkhoffs theorem assumes thatthe g22 metric function is both independent of the time-like coordi-nate and has a very specific radial dependence, g22 = r2. To removethese assumptions from the theorem without confusion, it is neces-sary to distinguish between the conventional formulation and a properformulation of the theorem which explicitly incorporates the assump-tions. Proof of the proper formulation demonstrates that the timedependent solution of the spherically symmetric vacuum field equa-

tions is the time independent solution if the metric is diagonal andif and only if the g22 metric function is independent of the time-likecoordinate. Furthermore, the proper formulation of the theorem estab-lishes the unique dependence of the g00 and the g11 metric functionson theg22 metric function, but it does not establish the dependence ofthe g22 metric function on the radial corrdinate r. Since g22|t = 0 isa necessary and sufficient condition for the proper formulation of thetheorem, other time-dependent, spherically symmetric solutions of thetime dependent vacuum field equations exist as long as the g22 metricfunction is time-dependent. In particular, the time dependent vacuumfield equations support gravitational wave solutions.

Keywords: Birkhoffs theorem, Combridge-Janne solution, time de-

pendent gravitational field equations, gravitational waves

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1 Introduction

Birkhoffs theorem [1,2] is a statement that the time-independent, spher-ically symmetric solution of the vacuum general relativistic field equationsis also the one and only solution of the time-dependent, spherically symmet-ric vacuum field equations. As an immediate consequence of the theorem,it is clear that there is no need to investigate the time-dependent vacuumequations at all since no other solutions exist. This means, in particular,that it is not possible for a time-dependent spherical distribution of massto emit gravitational waves into the region exterior to the distribution sincethe solution in that region is necessarily time-independent.

The conventional form of the theorem may be formally stated in thefollowing way:

Birkhoffs Theorem: Conventional Formulation. The so-lution of the time-dependent spherically symmetric vacuum fieldequations is identical to the solution of the time-independentspherically symmetric vacuum field equations. Furthermore, thetime-independent spherically symmetric solution is the textbookor Hilbert version of the Schwarzschild solution.

Proofs of this conventional formulation of Birkhoffs theorem are predicated

on two unstated and unacknowledged assumptions about the g22 metricfunction:

(A1) the g22 metric function is assumed to be independent of the time-likecoordinate, and

(A2) the g22 metric function is assumed to have a specified dependence onthe radial coordinate 1, g22 = r2.

From a logical point of view, these statements should be consequences of thetheorem, not assumptions upon which the theorem is based. This, however,is not the case. In addition, the condition that the metric is diagonal, that

the g0k metric functions are zero for k = 1, 2, 3, is always assumed to besatisfied. Since this diagonal condition is actually a necessary condition forthe theorem, it will be listed as an additional assumption:

1A generalized Birkhoff theorem [3] addresses, to some extent, this second assumptionconcerning the specified dependence on the radial coordinate. It does not, however, addressthe assumption of time-independence.

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(A3) the metric must be diagonalizable, that is, g0k = 0 for k = 1, 2, 3

All three statements, A1, A2, and A3, are essential to prove the conventionalform of the theorem; a proper formulation of the theorem requires thesestatements to be incorporated into the theorem.

It should be clear that a diagonal metric is a necessary condition forthe theorem: if the metric can not be diagonalized, then it can not beput into the form of the time-independent solution. It should also be clearthat a time-independentg22 function is also a necessary condition since thetime independent solution cannot have a time dependent g22 function. Itis shown below in Section 6 that a time-independent g22 metric function isalso a sufficient condition for the time-dependent solution of the vacuum field

equations to be identical to the time-independent solution once a diagonalmetric has been established. Both conditions, the diagonal property of themetric and the time independence of the g22 function, are crucial for thetheorem to hold. In addition, the proper form of the theorem establishesthe unique dependence of the g00 and g11 metric functions on the timeindependent g22 metric function but it makes no statement regarding thefunctional dependence of the time independent g22 function on the radialcoordinate r .

The proper formulation of the theorem demonstrates that the conven-tional formulation of the theorem is false. That is, there are solutions to thetime dependent vacuum field equations that are not identical to the time in-

dependent solution. These solutions necessarily require a time dependent g22metric function. An appeal to the source-free limit of the Robertson-Walker(RW) metric [4, 5] or the McVittie metric [6] would provide a sufficient in-dication that the conventional formulation of Birkhoffs theorem is false.Since the assumption of time independence for the g22 metric function wasnever acknowledged, there was no perceived need to make a connection tothe source-free limit of the RW metric, the McVittie metric, or any othermetric.

To establish the proper formulation of Birkhoffs theorem, the time de-pendent, spherically symmetric vacuum field equations are developed ab ini-tio. In Section 2, the form of the line element is established, including the

diagonality of the metric tensor. That line element is then used in Section 3to provide the basic ingredients that are used for deriving the geodesic equa-tions and the non-zero Christoffel symbols of the second kind. These resultsare then used in Section 4 to derive the explicit forms of the time-dependentRicci tensor components and the time-dependent, spherically symmetric vac-uum field equations. In Section 5, some basic symmetry properties of the

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time-dependent Ricci tensor and field equations are examined. Section 6

provides the proof that the time-independence of the g22 metric function isa sufficient condition for the theorem; Section 7 provides the proof that thetime-independence of theg22metric function is also a necessary conditon forthe theorem. Some remarks on the solution presented in Section 7 are givenin Section 8, explicit gravitational wave solutions using linearized equationsare demonstrated in Section 9, and conclusions are presented in Section 10.

2 Time-dependent Metric Functions andg01= g10= 0

The most general line element for a spherically symmetric system thatis neither static nor stationary has the form 2

ds2 =ec2dt2

+ 2 c T dt dr edr2 R2d2 (2.1)

d2 =d2 + sin2 d2 (2.2)

where the , T, , and R are functions of r and t only and d is thedifferential solid angle. This line element is not static (g01 = g10= 0), isnot stationary (, T, , and R are not independent of the time coordinatet), but is rotationally invariant ( and both leave the metricunchanged). The distinguishing feature of equation (2.1) is contained in thefunction R(r, t) which is treated as a function ofr and t and not, as in thestandard approach, as identical to the independent radial coordinate r.

The metric may be diagonalized by a coordinate transformtaion in thefollowing way. First, the metric is rewritten as

ds2 =ec2dt + T

c2

edr2

e

T2

c2

edr2 R2d2 (2.3)

Now a new time marker is introduced either by

2The material in this section follows a line of development that is outlined in ref [2].

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dt= dt + Tc2

edr (2.4)

or by

e/2 dt= e/2 dt + T

c2 e/2dr (2.5)

depending on which form is integrable. In either case, the new time markerprovides a diagonalized metric with new metric functions that utilize thenew time marker defined by (2.4) or (2.5),

ds2 =ec2dt2 edr2 R2d2 (2.6)

e =e Tc2

e (2.7)

Now the time-dependent problem may be approached in a simplified manner,with a diagonalized metric, by using the coordinate set (ct,r,,). Makingthe conversion back to the original coordinate set (ct,r,,), if necessary at

all, may be a much more difficult task. In any case, the analysis can nowbegin with the diagonalized metric (2.6), the coordinate basis (ct,r,,)using the new time marker t, and the new set of metric functions , , andR.

The introduction of an integration factor, Eq. (2.4) or Eq. (2.5), toobtain Eq. (2.6) may not be able guarantee integrability in all cases sincethe integration factor is also subject to the field equations and it is not atall obvious that the integration factor can fulfill both roles simultaneously.However, a non-diagonal metric immediately provides a counterexample tothe conventional form of Birkhoffs theorem which requires the metric tensorto be diagonal. Thus, it is clear that a diagonal metric is a necessary con-

dition for the theorem to hold and, consequently, the focus is placed solelyon metrics that have diagonal form.

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3 Geodesic Equations and Christoffel Symbols

It was shown in Section 2 that the most general line element for time-dependent and spherically symmetric systems may be put in the diagonalizedform

ds2 =e(dx0)2 edr2 R2d2 (3.1)

d2 =d2 + sin2 d2 (3.2)

where the , , and R are functions ofr and t. The line element containsthree unknown metric functions (,, andR), is rotationally invariant, andis static but not stationary (, , and R are functions ofr and t).

Now to set up the Euler-Lagrange equations from this metric, it is con-venient to define the function

F =e(x0)2 e(r)2 R2()2 R2 sin2 ( )2 (3.3)

where x0 = dx0/ds, r = dr/ds, = d/ds, and = d/ds. The Euler-

Lagrange equations

d

ds

F

x

F

x = 0 (3.4)

then provide the equations for the geodesics. Using the vertical slash| todenote standard partial differentiation, f|x f/x, the equations are

x

0

+|t

c 1

2

|t

c

x

0

x

0

+ |rrx

0

+

1

2 e

|t

c (r)

2

+ eRR|t

c ()2 + eR

R|tc

sin2 ( )2 = 0 (3.5)

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r+1

2e|r(x

0)2 +|tc

x0r+1

2|r(r)

2

eRR|r()2 eRR|rsin2 ( )2 = 0 (3.6)

+ 2R|t

Rcx0+ 2

R|r

Rr sin cos ( )2 = 0 (3.7)

+ 2R|tRc

x0 + 2R|rR

r + 2 cot = 0 (3.8)

for = 0, 1, 2, and 3, respectively. Comparing these equations with thegeodesic equations written in the form

x +

xx = 0 (3.9)

then provides the determination of the Christoffel symbols of the secondkind. These results are summarized in Table 1. It should be noted thatthese results revert to the time-independent functions for R(r, t) =R(r) [7]and to the textbook functions for R(r, t) =r [4] as they must.

4 The Ricci Tensor and Field Equations

The Ricci tensor may be written as

R=

lng

||

|

+

ln g| (4.1)For spherically symmetric and time-dependent systems, the only non-zeroderivatives are the derivatives with respect to x0 = ct (for index = 0),x1 =r (for index = 1), and x2 = (for index = 2).

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Table 1: The non-zero Christoffel symbols of second kind obtained from thegeodesic equations, eqs. (3.5), (3.6), (3.7), and (3.8). The determinant ofthe metric g = ||g|| and ln

g are also included in the table.

0

0 0= 12c|t

01 0=

00 1=

12|r

0

1 1

= 12ce

|t

0

2 2

= 1ce

RR|t

0

3 3

= 1ce

RR|tsin2

1

0 0

= 12e

|r

1

0 1

=

1

1 0

= 12c|t

1

1 1

= 12|r

12 2

= eRR|r 1

3 3= e

RR|rsin2

2

0 2

=

2

2 0

=

R|tRc

2

2 1

=

2

1 2

=

R|rR

2

3 3

= sin cos

3

0 3

=

3

3 0

=

R|tRc

3

1 3

=

3

3 1

=

R|rR

3

2 3

=

3

3 2

= cot

g=

e+R4 sin2 ln

g= 1

2

(+ ) + 2ln R+ 1

2

ln(sin2 )

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Computing the Ricci tensor components is a straightforward but tedious

and time-consuming task. The results of the computation are

R00= 1

2c2

|t|t+

2|t2 |t|t

2 2|tR|t

R +

4R|t|tR

e

2

|r|r+

2|r

2 |r|r

2 +

2|rR|r

R

(4.2)

R01= R10=2R|t|r

Rc

|rR|t

Rc

|tR|r

Rc

(4.3)

R11=1

2

|r|r+

2|r

2 |r|r

2 2|rR|r

R +

4R|r|r

R

e

2c2

|t|t+

2|t2 |t|t

2 +

2|tR|tR

(4.4)

R22= e

c2 RR|t|t+ R2|t

RR|t|t2

+RR|t|t

2 +

e

RR|r|r+ R

2|r+

RR|r|r2

RR|r|r2

1

(4.5)

R33= e

c2 sin2

RR|t|t+ R

2|t

RR|t|t

2 +

RR|t|t

2

+ sin2

e

RR|r|r+ R

2|r+

RR|r|r2

RR|r|r2

1

(4.6)

with all other R identically zero. As in the time-independent case, R33is proportional to R22, R33 = sin2 R22, and so it does not provide an ad-ditional vacuum field equation. It is easily verified that the Ricci tensorcomponents reduce to the time-independent spherically symmetric compo-nents for R(r, t) = R(r) [7]. And, of course, the equations reduce to thestandard time-independent textbook equations forR(r, t) =r, R|r = 1 andR|r|r = 0 [4].

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The vacuum equations for the time-dependent case, R = 0, may be

conveniently written as

1

2

|r|r+

2|r2 |r|r

2 +

2|rR|rR

=e

2c2

|t|t+

2|t2 |t|t

2 2|tR|t

R +

4R|t|tR

(4.7)

2R|t|rRc

|rR|tRc

|tR|rRc

= 0 (4.8)

1

2

|r|r+

2|r2 |r|r

2 2|rR|r

R +

4R|r|rR

=e

2c2

|t|t+

2|t2 |t|t

2 +

2|tR|t

R

(4.9)

(eRR|r)|r 1 + (eRR|r)

|r+ |r2

= 1c2

(eRR|t)|t+ (e

RR|t)

|t+ |t2

(4.10)

which are the 00, 01, 11, and 22 equations, respectively. Since the R33equation duplicates the R22 equation, this is a system of four equations inthree unknowns. Presumably at least one of these equations will duplicateanother so that the system is not overdetermined.

5 Symmetry Properties of the Ricci Tensor

Before proceeding with the proof of the proper formulation of Birkhoffstheorem, it is worth noting some symmetry properties of the Ricci tensorcomponents. Interchanging the coordinates r and ct and the functions and and comparing the results with the original Ricci tensor components,Eqs. (4.2) - (4.5), shows that

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R00= R11 for r ct and (5.1)

R01= R01 for r ct and (5.2)

R22+ 1 = (R22+ 1) for r ct and (5.3)

R33+ sin2 = (R33+ sin2 ) for r ct and (5.4)

The interchange of the R00 and R11 components of the Ricci tensor reflectsthe fact that interchanging r and ct corresponds to an interchange of the0 (ct) and 1 (r) indices. SinceR is symmetric in its indices, interchang-ing the 0 and 1 indices leaves the R01 = R10 components unchanged. Thebehavior of the R22 and R33 components is ultimately due to the changeof signature of the metric under the interchange of the 0 and 1 indices, aswapping of the time-like and radial space-like coordinates. For the inter-change of , the metric functions g00 = e and g11 =e becomeg00 e =g11 and g11 =e =g00 which introduces the change inthe signature of the metric. In the R00 and R11 components, these factorsappear in product form so that the interchange does not introduce an overall

sign change; in the R22 and R33 components, the factors e

and e

appearseparately and so a sign change is introduced. This can also be seen inthe = 0 and = 1 geodesic equations and the corresponding Christoffelsymbols (0, 2, 2 1, 2, 2 provides an example).

However, the interchange ct r and g00 = e g11 =e providesboth the interchange of the functions and and the requisite sign changeto keep the signature of the metric unchanged. Under this interchange, it isfound that both R22 and R33 are invariant

R22= R22 for r ct and g00 g11 (5.5)

R33= R33 for r ct and g00 g11 (5.6)

Thus, under the composite interchange x0 x1 and g00 g11, the R00and R11 components interchange, R00 R11, while the other Ricci ten-sor components are unchanged. Since the vacuum field equations require

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R = 0, this means the field equations are invariant under this composite

interchange.

6 R|t= 0: The Sufficient Condition

It will now be shown that imposing the condition R|t = 0 immediatelyleads to the time-independent solution. This is actually an important re-stricted case of the full time-dependent field equations since the textbookchoice R(r, t) =r falls into this category. The field equations for R|t= 0 are

1

2

|r|r+

2|r

2 |r|r

2 +

2|rR|r

R

=

e

2c2

|t|t+

2|t

2 |t|t

2

(6.1)

|tR|rRc

= 0 (6.2)

1

2|r|r+

2|r

2

|r|r

2

2|rR|r

R

+4R|r|r

R

=e

2c2

|t|t+

2|t

2 |t|t

2

(6.3)

e

RR|r|r+ R

2|r+

RR|r|r

2 RR|r|r

2

1

= 0 (6.4)

First, it should be noted that the condition R|t = 0 immediately reducestheR22= 0 equation, Eq. (6.4), to the time-independent equation. Second,the form of R00 = 0 and R11 = 0 equations, Eq. (6.1) and Eq. (6.3),show that the spatial derivative parts, although not separately equal to zero

as in the time-independent case, are equal to each other. These two factsare sufficient to show that the time-dependent solutions for and are, towithin an overall scaling factor, the time-independent solutions.

Proceeding with the direct solution, the second equation requires eitherR|r = 0 or |t = 0. Subtstituting R|r = 0 into Eq. (6.4), the R22 equation,shows that R|r = 0 produces a contradiction and so is not a permissable

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condition. Consequently,|t = 0. Thus, as a result ofR|t= 0, the only pos-

sible time-dependence in the metric is contained in the g00 function . Thelast two field equations for this R|t = 0 case are the time-independent R22andR33 equations and, since the R01 equation requires to be independentof t, the R00 and R11 equations reduce to the time-independent equationsas well,

e

2

|r|r+

2|r

2 |r|r

2 +

2|rR|r

R

= 0 (6.5)

1

2|r|r+2|r

2 |r|r

2 2|rR|r

R +

4R|r|r

R = 0 (6.6)From the initial four time-dependent field equations, the single constraintR|t = 0 produces the three time-independent field equations with the addi-tionalR01= 0 condition which requires |t= 0. The three time-independentfield equations are reducible to two independent equations [79] which pro-duce the time-independent solution for in terms ofR. SinceR is indepen-dent oft by supposition, |t = 0 is satisfied automatically.

Although the metric function is, in principle, a function of r and t,its t-dependence is not a factor in the field equations and so will remainundetermined. Thist-dependence must take the form of an overall scalingfactor on the g00 function which translates to an additive function of t tothe time-independent solution 0(r), (r, t) =0(r) +f(t), so that all spa-tial derivatives contain the function 0(r) only. Consequently, the singlerestriction thatR is a function ofr only leads to the time-independent so-lution to the field equations with an arbitrary function oft multiplying thetime-independentg00,

e =e0(r)+f(t) =

1

R

ef(t) (6.7)

e =R2

|r1

R

1

(6.8)

R(r) = undetermined (6.9)

f(t) = undetermined (6.10)

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The function f(t) may be asbsorbed into a new time coordinate defined by

dt =ef(t)dt. Alternatively, the far-field behavior may be used to show that,as r ,

e

1 r

ef(t) (6.11)

which indicates the possibility of a time-dependent classical gravitationalpotential. It has been previously shown [9] that c2(1 g00)/(2g00g11) is theclassical potential, so that

VG(r) =c2(ef(t) 1)

2R2 Gm

RR2 (6.12)

The second term is the static gravitational potential that tends to the New-tonian form for R r in the asymptotically flat spacetime and identifies as twice the geometric mass of the source, = 2mG = 2Gm/c

2. Thefirst term is a time-dependent non-Newtonian potential which tends to aspatially-independent function of time in the asymptotically flat spacetime,where R 1 as R r. If no such time-dependent non-Newtonian term isto be expected, then the requirement that f(t) = 0 follows. This, of course,

is consistent with simply absorbing the exponential function into the time co-ordinate. Thus, it has been established that R|t= 0 is a sufficient conditionthat the time-independent solution is also the time-dependent solution of thefield equations. Furthermore, it was explicitly shown that the relevant time-independent solution is the solution found independently by Combridge [10]and Janne [11], denoted by the name Combridge-Janne solution [9], and notthe textbook or Hilbert version of the Schwarzschild solution. Note that forR(r)< 2mG,t becomes space-like and must be identified as the radial coor-dinate whiler becomes the time-like coordinate. ConsequentlyR(r) 2mGmust hold since the g22 metric function was required to be independent ofthe time-like coordinate in the derivation by supposition.

7 R|t= 0: The Necessary Condition

It now remains to show that R|t = 0 is a necessary condition for thetheorem. For this part, it is only necessary to present a time-dependent

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solution of the field equations that is distinct from the Combridge-Janne

solution. As noted in Section 1, an appeal could be made to the source-freelimit of the Robertson-Walker metric [4, 5] or the McVittie metric [6] assuch a time-dependent solution. However, for the purpose of establishingR|t = 0 as a necessary condition for the theorem, it may be of interest toexamine the symmetry properties presented in Section 5 and recognize thata solution exists for the spatially homogeneous case, R|r = 0, so that theg22 function is a function of t only, R(r, t) = R(t). It is straightforward toverify that this solution has the form

ds2 =ec2dt2

edr2

R2d2 (7.1)

e =R2|t

c2AcR 1

(7.2)e =

Ac

R 1

eC0 (7.3)

R= R(t) = undetermined (7.4)

where A and C0 are constants. The undetermined nature of R(t) is thetime-dependent analogue of the undetermined nature of R(r) in the time-independent Combridge-Janne solution and is due to an insufficient numberof independent field equations [711].

The solution given by Eq. (7.1) - Eq. (7.4) is clearly distinguishablefrom the time-independent Combridge-Janne solution and thus establishesthat R|t = 0 is a necessary condition for the time-dependent solution tobe identical to the time-independent solution of the vacuum field equations.This completes the proof that, given a diagonal metric, R|t= 0 is a necessaryand sufficient condition that the solution of the time-dependent vacuum fieldequations is given by the solution of the time-independent field equations.

8 Remarks On The R|r= 0 Solution

The purpose of the spatially uniform R|r = 0 solution is to presenta time-dependent vacuum solution that can not be reduced to the time-independent Combridge-Janne solution. However, since this solution has

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properties which relate it to wormhole solutions [9] that are presented as

time-independent solutions, it is worth making a few observations and com-ments on the relations between a wormhole solution and the R|r = 0 solution.

Three distinct cases need to be considered. First, if R(t) < Ac for allt, then the signature of the metric is constant showing that this solutiondoes not undergo a transition to a time-independent metric for any valueof t. The solution remains time-dependent and spatially homogeneous forall times: R(t) continues to evolve in time but is bounded from above byAc. This is the time-dependent analogue of the strictly time-independentCombridge-Janne solution for which R(r) is bounded from below by a valuegreater than or equal to 2mG.

On the other hand, ifR(t)> Acfor allt, the signature of the metric then

shows that t is space-like and r is time-like. This means that the solutionis actually time-independent. This can most easily be seen by noting thatr= ct assumes the role of the radial coordinate and ct = r assumes the roleof the time-like coordinate. Due to the symmetry of the field equations, thissolution is identical to the time-independent Combridge-Janne solution forwhich R(r)> 2mG for all r .

Finally, for the third case, if there is a time tH for which R(tH) = Ac,then the coordinates t and r swap roles for t > tHvery much as they do inthe textbook wormhole solution for r < 2mG. Then, for t > tH the radialcoordinate is r = ct and the time coordinate is ct = r so that the metrictakes the form

ds2 =ec2dt2 edr2 R2d2 (8.1)

e =

1 Ac

R

eC0 (8.2)

e =R

2|r

(1 AcR)(8.3)

R= R(r) = undetermined (8.4)

This is just the Combridge-Janne form of the wormhole solution outsider = 2mG with Ac = 2mG and C0 = 0. Although this solution is time-independent for timest > tH, it is clearly time-dependent for t < tH.

This solution provides a means for understanding the physical origin ofthe wormhole solution: the time independent state of the solution evolves in

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time out of the spatially homogeneous solution. The metric originates at t =

0, say, as spatially homogeneous as described by Eq. (7.1) with R(0) definingthe original radial extent of spacetime. IfR(0) = 0, then the behavior of theg00 function depends on the behavior ofR|t while theg11 function diverges.As R(t) increases with t, the radial extent of spacetime increases until thetime tH for which R(tH) = Ac. At this time, the signature of the metricabruptly changes indicating that the time-like and radial coordinates swaproles, ct= r and r= ct. Then, for all times t > tHthe metric assumes theform of the time independent generalized wormhole solution.

This time evolution can not be traced in a backward sense by a freelyinfalling observer since the infall terminates at the event horizon. It must beremembered that the point r= 0 does not correspond to the spatial origin

r= 0, it corresponds to the time origin t = 0. But since the solution insidethe event horizon is spatially homogeneous, all spatial points are equivalentwithin the event horizon. Consequently, reaching the event horizon is equiv-alent to reaching the spatial origin. But the time evolution of the metricwithin the event horizon is not retraced in a time-reversed sense.

9 Linearized Equations and Gravitational Waves

The solution presented in Section 7 establishes that g22|t = 0 is a nec-

essary condition for the proper formulation of Birkhoffs theorem. Anothersolution that accomplishes the same task can be obtained from the linearizedfield equations. In addition, this solution to the linearized equations alsodemonstrates that gravitational wave solutions exist for the time dependentvacuum field equations.

It is simplest to provide a perturbation around flat Lorentzian spacetime.In other words, weak field conditions are assumed such that e 1 +,e 1 +, and R = r +R. Thus, to first order in , , and R, themetric is given by

ds

2

(1 + )c2

dt

2

(1 + )dr2

(r2

+ 2rR)d

2

(9.1)

and the field equations (4.7) - (4.10), to first order in , , and R, are

|r|r+2|r

r =

1

c2

|t|t+

4R|t|tr

(9.2)

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2R|t|r |t= 0 (9.3)

|r|r 2|r

r +

4R|r|r

r =

1

c2|t|t (9.4)

+ 2R|r+ rR|r|r+ r

|r |r2

=

1

c2rR|t|t (9.5)

The field equations can be written in a more suggestive form by noting thatfor an arbitrary function of the magnitude of the radial coordinate f(r), theLaplacian contains only the r-derivatives

2f=f|r|r+2f|r

r (9.6)

Consequently, the field equations may then be written as

2= 1c2

|t|t+

4R|t|t

r

(9.7)

2R|t|r

|t= 0 (9.8)

2 2(|r+ |r)r

+4R|r|r

r =

1

c2|t|t (9.9)

2R 1

c2R|t|t

+

|r |r2

r

= 0 (9.10)

Clearly, Eq. (9.10) has the form of an inhomogeneous wave equation for Rwith the other metric functions appearing to act as source functions.

The solutions of Eq. (9.7) - Eq. (9.10) are given by

|r = g0r2

+ 2

c2R|t|t (9.11)

= 2R|r g0

r (9.12)

R = undetermined (9.13)

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Rk =

A(k)j0(kr) + B(k)n0(kr)

ei(k)t (9.18)

(k) =kc (9.19)

where j0(x) and n0(x) are the zero-order spherical Bessel functions. Sinceboth j0(kr) and n0(kr) vanish as r , the boundary condition in theasypmptotic region is satisfied. The boundary condition near the source,for example as r 0, and the initial conditions will provide the necessaryinformation to completely specify the solution.

Note that if the gravitational wave condition, Eq. (9.15), is not im-posed and some other condition is imposed instead, then a different type

of solution will be obtained for R(r, t) that may have nothing to do withgravitational waves. In other words, gravitational waves are possible solu-tions to the linearized equations but they do not follow from the equationsnecessarily. Nevertheless, this solution clearly demonstrates the the time-dependent vacuum field equations do support gravitational wave solutionsas long as the g22 metric function is time-dependent and the gravitationalwave condition is imposed. Consequently, a radially collapsing, expanding,or pulsating spherical distribution of mass can indeed emit gravitationalwaves into the vacuum region outside of the mass distribution.

10 Conclusion

As stated in Section 1, it is necessary to distinguish between the con-ventional and proper formulations of Birkhoffs theorem. The conventionalformulation of Birkhoffs theorem states that the only solution of the time-dependent spherically symmetric vacuum gravitational field equations is aparticular time-independent solution. The conventional formulation is falsesince it is predicated on the assumptions that (1) the g22 function is inde-pendent of the time-like coordinate and (2) theg22 function has a particulardependence on the radial coordinater,g22=

r2. It is the time-independent

assumption that is the primary reason that the conventional formulation isinvalid.

A proper formulation of Birkhoffs theorem must explicitly state the cru-cial but unacknowledged assumptions of the original theorem. This includesthe condition that the metric tensor must be diagonal. The critical natureof the time-independent assumption,g22|t= 0, is demonstrated in Section 6

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and Section 7: a time-independent g22 function is both a necessary and

sufficient condition that the solution of the time-dependent vacuum fieldequations is the solution of the time-independent field equations. In ad-dition, removing the unnecessary second coordinate dependence assump-tion, that the g22 function is given by g22 =r2, shows that the relevanttime-independent solution is the Combridge-Janne solution. Consequently,as outlined in Section 1, Birkhoffs theorem must be formulated in the fol-lowing way:

Birkhoffs Theorem: Proper Formulation. The solutionof the time-dependent spherically symmetric vacuum field equa-

tions is identical to the solution of the time independent spheri-cally symmetric vacuum field equations if the metric is diagonal(g0k = 0 for k = 1, 2, 3) and if and only if the g22 metric func-tion is independent of the time-like coordinate. Furthermore,the time-independent spherically symmetric solution is the time-independent Combridge-Janne solution.

An immediate and very important consequence of the proper formulationof Birkhoffs theorem is that an entire class of uninvestigated solutions ofthe time-dependent spherically symmetric vacuum field equations exists for

whichR|t= 0. This may be stated with the force of a corollary to the properformulation of Birkhoffs theorem:

Corollary of Proper Formulation. Solutions of the time-dependent spherically symmetric vacuum field equations existand are inequivalent to the time-independent solution if and onlyif theg22metric function is a function of the time-like coordinate.

The time dependent solutions presented in Section 7 and Section 9 providetwo distinct illustrations of this corollary. Of particular interest, the results

of Section 9 demonstrate the existence of spherical gravitational wave solu-tions of the vacuum equations, thereby permitting a collapsing, expanding,or pulsating distribution of mass to emit such waves into the region exteriorto that mass distribution. Since this is not possible for systems for whichR|t = 0, this illustrates the crucial dependence of the gravitational wavesolution on the time dependence of the g22 metric function R(r, t).

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References

[1] G.D. Birkhoff, Relativity and Modern Physics (Harvard UniversityPress, Cambridge MA, 1923).

[2] C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (Freeman,San Francisco, 1973).

[3] A.H. Abbassi, General Birkhoffs Theorem, arXiv:gr-qc/0103103v1(2001).

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

[5] H.P. Roberson, Kinematics and World Structure, Astrophys J., 82,284 (1935).

[6] G.C. McVittie, Mon. Not. R. Astron. Soc.,93, 325 (1933).

[7] R.E. Salvino and R.D. Puff, The Two Schwarzschild Solutions: ACritica Appraisal, preprint (2013). Available on academia.edu.

[8] W. de Sitter, On Einsteins Theory of Gravitation, and Its Astronom-ical Consquences (First Paper), Month. Not. R. Astr. Soc., 76, 699(1916). In particular, see sections 10 on pg. 711 and 11 on pg. 714.

[9] R.E. Salvino and R.D. Puff, The Combridge-Janne Solution and theg22 Metric Function, preprint (2013). Available on academia.edu.

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

[11] H.A. Jannes, Bull. Acad. R. Belg., 9, 484 (1923).

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Proper Formulation of Birkhoff's Theorem