Almost Birkhoff Theorem in General Relativity

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by  Rituparno Goswami

Almost Birkhoff Theorem in General Relativity
Rituparno Goswami and George F R Ellis
ACGC and Department of Mathematics and Applied Mathematics,
University of Cape Town, Rondebosch, 7701, South Africa∗
(Dated: January 25, 2011)
We extend Birkhoff’s theorem for almost LRS-IIvacuum spacetimes to show that therigidity of
spherical vacuum solutions of Einstein’s field equations continues even in the perturbedscenario.
PACSnumbers:
I. INTRODUCTION vacuum General Relativity solutions, or solutions with
a trace-free matter tensor. One should also note that
1 (like the EGS result) the result is local: both Birkhoff’s
The core content of Birkhoff’s theorem [1] is that any
1
theorem,andourgeneralizationofit, areindependent of
spherically symmetric solution of the vacuum field equa-
0
boundaryconditionsatinfinity: theyholdinlocalneigh-
2 tions has an extra symmetry: it must be either locally
borhoods U. There are of course crucial differences be-
static or spatially homogeneous. This underlies the cru-
n tweentheexactlysphericalcaseandthatofapproximate
cial importance in astrophysics of the Schwarzschild so-
a
symmetry: specifically, an almost spherically symmetric
J lution,asitmeansthattheexteriormetricofanyexactly
4 sphericalstarmustbe givenby the Schwarzschildmetric pulsating star can emit gravitational waves, but this is
not possible in the exactly spherical case. However we
2 (it cannot be Minkowski spacetime if the mass is non-
show that in such a case if the intensity of the radiation
zero); and this also underlies the uniqueness results for

] non-rotating black holes. is such as to leave the solution almost spherically sym-
c
metric, the space time will remain almost static, which
q Howeveritisanexacttheoremthatisonlyvalidforex-
is defined in a way we will make precise later.
- actsphericalsymmetry;butnorealstarisexactlyspher-
r Weprovetheresultbyusingthe1+1+2covariantper-
g icallysymmetric. So akeyquestionis whetherthe result
turbationformalism[7,8],whichdevelopedfromthe1+3
[ isapproximatelytrueforapproximatelysphericallysym-
covariant perturbation formalism [9]. This enable us to
metric vacuum solutions. We prove an “almost Birkhoff
1 provethe approximateresultas a straightforwardgener-
theorem” in this paper that shows this is indeed the
v alization of the exact result, which we prove first, using
0 case, so those results carry over to astrophysically re-
the 1+1+2 formalism. Actually we prove a small gener-
2 alistic situations (such as the Solar System). There are
alization of the standard Birkhoff result: it holds for all
5 of course many papers discussing perturbations of the
4 Schwarzschildsolution, but none appear to focus on this ClassIILocallyRotationalSymmetric(LRS)spacetimes
. [10] (which include Schwarzschild as a special case).
1 specific issue. It is in a sense an analogue of another
0 important result, the “almost EGS theorem” [2] that
1 generalizes from exact isotropy of radiation to approx-
II. 1+1+2 COVARIANT SPLITTING OF
1 imate isotropy the crucial Ehlers-Geren-Sachs theorem
SPACETIME
:
v [3], proving that isotropic cosmic background radiation
i everywhere in an expanding universe domain U implies
X The 1+3 covariant formalism [9] has proven to be a
a Robertson-Walker geometry in that domain. In both
r cases an exact mathematical result, depending on exact very useful technique in many aspects of relativistic cos-
a
mology. In this approach first we define a timelike con-
symmetry, is generalized to a more physically realistic
gruence by a timelike unit vector ua (uau = 1). Then
result, depending on approximate symmetry. a −
thespacetimeissplitintheformR V whereRdenotes
Thealmost-Birkhoffresultwillnotbetrueifspacetime the timeline along ua and V is the⊗tangent 3-space per-
is not a vacuum (empty) spacetime, for the degrees of
pendicular to ua. Then any vector Xa can be projected
freedomavailablethroughamattersourcegenericallyin-
on the 3-space by the projection tensor ha =ga+uau .
validatetheresult,asisshownforexamplebythefamily b b b
Thevectorua isusedtodefinethecovariant timederiva-
ofLemaˆıtre-Tolman-Bondi(LTB)models[4]. Howeverit
tive (denoted by a dot) for any tensor Ta..b along the
remainstruefortheelectrovaccase[5]andsolutionswith c..d
observers’worldlines defined by
acosmologicalconstant. Itwillalsonotbe trueforgrav-
itationaltheorieswithascalardegreeoffreedom,suchas T˙a..b =ue Ta..b , (1)
c..d e c..d
scalar-tensor theories [6]. The rigidity embodied in this ∇
property of the Einstein Field Equations is specific to andthetensorh isusedtodefinethefullyorthogonally
ab
projected covariant derivative D for any tensor Ta..b ,
c..d
D Ta..b =ha hp ...hb hq hr Tf..g , (2)
e c..d f c g d e r p..q
∗Electronicaddress: [email protected];Electronicad- ∇
dress: [email protected] with total projection on all the free indices.
2
In the (1+1+2) approach we further split the 3-space The hat-derivative is the derivative along the ea vector-
V,byintroducingthespacelikeunitvectorea orthogonal field in the surfaces orthogonal to ua. The δ -derivative
to ua so that is the projected derivative onto the orthogonal 2-sheet,
with the projection on every free index. We can now
eaua =0, eaea =1. (3) decompose the covariantderivative ofea in the direction
orthogonalto ua into it’s irreducible parts giving
Then the projection tensor
1
N b h b e eb =g b+u ub e eb , Na =2 , (4) D e =e a + φN +ξε +ζ , (12)
a ≡ a − a a a − a a a b a b 2 ab ab ab
projects vectors onto the tangent 2-surfaces orthogonal
where
to ea and ua, which, following [7], we will refer to as
‘sheets’. Hence it is obvious that eaN =0=uaN . In a ecD e =eˆ , (13)
ab ab a ≡ c a a
(1+3)approachanysecondranksymmetric 4-tensorcan φ δ ea , (14)
a
be splitinto ascalaralongua, a 3-vectoranda projected ≡
1
symmetric trace free (PSTF) 3-tensor. In (1+1+2) slic- ξ εabδaeb , (15)
≡ 2
ing,wecantakethissplitfurtherbysplittingthe3-vector
ζ δ e . (16)
andPSTF3-tensorwithrespecttoea. Any3-vector,ψa, ab ≡ {a b}
canbe irreducibly split into a componentalong ea and a We see that along the spatial direction e , φ represents
a
sheet component Ψa, orthogonal to ea i.e. the expansion of the sheet, ζ is the shear of ea (i.e. the
ab
distortion of the sheet) and aa its acceleration. We can
ψa =Ψea+Ψa, Ψ ψae , Ψa Nabψ . (5)
≡ a ≡ b alsointerpretξ asthevorticity associatedwithea sothat
A similar decompositioncan be done for PSTF3-tensor, it is a representationof the “twisting” or rotation of the
ψ , which can be split into scalar (along ea), 2-vector sheet. The other derivative of ea is its change along ua,
ab
and 2-tensor part as follows:
e˙ = u +α , (17)
a a a
A
1
ψab =ψhabi =Ψ(cid:18)eaeb− 2Nab(cid:19)+2Ψ(aeb)+Ψab , (6) where we have A = eau˙a and αa = Nace˙c. Also we can
writethe(1+3)kinematicalvariablesandWeyltensoras
follows
where
Ψ eaebψab = Nabψab , Θ = hba∇bua (18)
Ψ ≡ N becψ ,− u˙a = ea+ a , (19)
a a bc A A
≡ 1 ωa = Ωea+Ωa , (20)
Ψ ψ Nc N d N Ncd ψ , (7)
ab ≡ {ab} ≡(cid:18) (a b) − 2 ab (cid:19) cd
1
and the curly bracketsdenote the PSTFpartof a tensor σab = Σ(cid:18)eaeb− 2Nab(cid:19)+2Σ(aeb)+Σab , (21)
with respect to ea.
1
We also have
E = e e N +2 e + , (22)
ab E(cid:18) a b− 2 ab(cid:19) E(a b) Eab
2
h{ab} =0 , Nhabi =−ehaebi =Nab− 3hab . (8) H = e e 1N +2 e + . (23)
ab H(cid:18) a b− 2 ab(cid:19) H(a b) Hab
The sheet carries a natural 2-volume element, the alter-
nating Levi-Civita 2-tensor: where E and H are the electric and magnetic partof
ab ab
theWeyltensorrespectively. Thereforethekeyvariables
ε ε ec =η ecud , (9)
ab ≡ abc dabc of the 1+1+2 formalism are
where εabc is the 3-space permutation symbol the vol- [Θ, ,Ω,Σ, , ,φ,ξ, a,Ωa,Σa,αa,aa, a, a,
ume element of the 3-space and η is the space-time A E H A E H
abcd
Σ , , ,ζ ] . (24)
permutator or the 4-volume. With these definitions it ab Eab Hab ab
follows that any 1+3 quantity can be locally split in the These variables (scalars , 2-vectorsand PSTF 2-tensors)
1+1+2 setting into only three types of objects: scalars, formanirreducible setandcompletelydescribeavacuum
2-vectors in the sheet, and PSTF 2-tensors (also defined spacetime. In terms of these variables the full covariant
on the sheet). derivatives of ea and ua are
Nowapartfromthe‘time’(dot)derivativeofanobject
(scalar, vector or tensor) which is the derivative along
the timelike congruence ua, we now introduce two new e = u u u α + Σ+ 1Θ e u
a b a b a b a b
derivatives, which ea defines, for any object ψ c...d: ∇ −A − (cid:18) 3 (cid:19)
a...b
1
ψˆa..bc..d ≡ efDfψa..bc..d , (10) +(Σa−εacΩc)ub+eaab+ 2φNab
δ ψ c..d N f...N gN c..N dN jD ψ i..j .(11) +ξε +ζ , (25)
f a..b a b h i f j f..g ab ab
≡
3
u = u ( e + )+e e 1Θ+Σ ˆ Θ˙ = ( +φ) + 1Θ2+ 3Σ2. (35)
∇a b − a A b Ab a b(cid:18)3 (cid:19) A− − A A 3 2
+ea(Σb+εbcΩc)+(Σa εacΩc)eb Since the vorticity vanishes, the unit vector field ua
−
1 1 is hypersurface-orthogonal to the spacelike 3-surfaces
+N Θ Σ +Ωε +Σ . (26)
ab(cid:18)3 − 2 (cid:19) ab ab whose intrinsic curvature can be calculated from the
Gauss equation for ua that is generally given as [11]:
For the complete set of evolution equations, propaga-
tion equations, mixed equations and constraints for the
(3)R =(R ) K K +K K , (36)
aboveirreduciblesetofvariablespleaseseeequations(48- abcd abcd ⊥− ac bd bc ad
81) of [8]. Also we have the following commutation re-
where (3)R is the 3-curvature tensor, means pro-
lations for the different derivatives of any scalar ψ, abcd ⊥
jection with h on all indices and K is the extrinsic
ab ab
ψˆ˙ ψˆ˙ = ψ˙+(1Θ+Σ)ψˆ+(Σ +ε Ωc α )δaψ , (27) curvature. Also we note that in case of LRS-II space-
a ac a
− −A 3 − times the sheets at each point mesh together to form
2-surfaces. The Gauss equation for ea together with the
δaδbψ δbδaψ =2εab(Ωψ˙ ξψˆ)+2a[aδb]ψ (28) 3-Ricci identities determine the 3-Ricci curvature tensor
− −
From the above two relations it is clear that the 2-sheet of the spacelike 3-surfaces orthogonal to ua to be
is a genuine two surface (rather than just a collection
1 1 1
of tangent planes), in the sense that the commutator of 3R = φˆ+ φ2 e e φˆ+ φ2 K N . (37)
ab a b ab
the time and hat derivative do not depend on any sheet −(cid:20) 2 (cid:21) −(cid:20)2 2 − (cid:21)
component and also the sheet derivatives commute, if
and only if Σ +ε Ωc α =0 and Ω=ξ =aa =0. This gives the 3-Ricci-scalar as
a ac a
−
1 3
3R= 2 φˆ+ φ2 K (38)
III. BIRKHOFF THEOREM FOR VACUUM − (cid:20)2 4 − (cid:21)
LRS-II SPACETIMES
where K is the Gaussian curvature of the sheet, 2R =
ab
KN . Fromthisequationand(29)anexpressionforK
Locally Rotationally Symmetric (LRS) spacetimes ab
is obtained in the form [11]
posses a continuous isotropy group at each point and
hence a multi-transitive isometry group acting on the
2
spacetime manifold [10]. These spacetimes exhibit lo- K = + 1φ2 1Θ 1Σ (39)
cally(ateachpoint)aunique preferredspatialdirection, −E 4 −(cid:18)3 − 2 (cid:19)
covariantlydefined. Since LRSspacetimesaredefined to
be isotropic about a preferred direction, this allows for From (29-34), the evolution and propagation equations
the vanishing of all orthogonal 1+1+2 vectors and ten- of K can be determined as
sors, such that there are no preferred directions in the
2 2
sheet. Then,allthenon-zero1+1+2variablesarecovari- K˙ = Θ Σ K, (40)
antly defined scalars. A subclass of the LRS spacetimes, −3(cid:18)3 − (cid:19)
called LRS-II, contains all the LRS spacetimes that are Kˆ = φK. (41)
rotation free. As consequence in LRS-II spacetimes the −
variables Ω, ξ and are identically zero and the vari- Fromequations(31), (34), (40)and(41)wegetaninter-
H
ables ,Θ,φ,Σ, fully characterize the kinematics of esting geometrical result for vacuum LRS-II spacetime
{A E}
the vacuum spacetime. The propagation and evolution
equations of these variables are: =CK3/2. (42)
E
1 1 2
φˆ = φ2+ Θ+Σ Θ Σ (,29) Thatis,the1+1+2scalaroftheelectricpartoftheWeyl
− 2 (cid:18)3 (cid:19)(cid:18)3 − (cid:19)−E
tensor is alwaysproportionalto a power of the Gaussian
Σˆ 2Θˆ = 3φΣ, (30) curvatureofthe2-sheet. TheproportionalityconstantC
− 3 − 2 sets up a scale in the problem. We can immediately see
3 that for Minkowski spacetime C =0.
ˆ = φ . (31)
E − 2 E Tocovariantlyinvestigatethe geometryofthe vacuum
LRS-IIspacetime,letustrytosolvethe Killing equation
φ˙ = Σ 2Θ 1φ , (32) for a Killing vector of the form ξa = Ψua+Φea, where
− (cid:18) − 3 (cid:19)(cid:18)A− 2 (cid:19) Ψ and Φ are scalars. The Killing equation gives
2
2 1 1
Σ˙ Θ˙ = φ+2 Θ Σ , (33) (Ψu +Φe )+ (Ψu +Φe )=0. (43)
a b b b a a
− 3 − A (cid:18)3 − 2 (cid:19) −E ∇ ∇
3 Using equations (25) and (26), and multiplying the
˙ = Σ Θ . (34)
E (cid:18)2 − (cid:19)E Killing equation by uaub, uaeb, eaeb and Nab we get the
4
following differential equations and constraints: solution
Ψ˙ + Φ = 0, (44) φ= 2 1 2m , = m 1 2m −12 (49)
A rr − r A r2 (cid:20) − r (cid:21)
1
Ψˆ Φ˙ Ψ +Φ(Σ+ Θ) = 0;, (45) 2m 1
− − A 3 = , K = (50)
1 E r3 r2
Φˆ +Ψ( Θ+Σ) = 0, (46)
3 Herethe constantmis the constantofintegration. Solv-
2
ing for the metric components using the definition of
Ψ( Θ Σ)+Φφ = 0. (47)
3 − these geometrical quantities we get [11]
Nowwe knowξ ξa = Ψ2+Φ2. Ifξa istimelike (that is 2m dr2
a − ds2 = 1 dt2+ +r2dΩ2, (51)
ξthaξeav<ect0o)r,utha,enwebeccaanusaelwoafytshme aakrbeitΦra=rin0e.sOs inntchheoootshinegr −(cid:18) − r (cid:19) (1− 2rm)
hand, if ξa is spacelike (that is ξ ξa > 0), we can make which is the metric of a static Schwarzschild exterior.
a
Ψ=0. In the second case, when φ = = 0, we can choose
A
Let us first assume that ξa is timelike and Φ = 0. In ua = 2m 1δa wheremisaconstantandthensolving
that case we know that the solution of equations (44) q t − t
(29)- (35), we get the unique solution
and (45) always exists while the constraints (46) and
(47) together imply that in general, (for a non trivial 3m 2t 2 3m t
Θ= − , Σ= − , (52)
Ψ),Θ=Σ=0. Thusthe expansionandshearofanunit t t(2m t) −3t t(2m t)
vectorfield alongthe timelike Killingvectorvanishes. In − −
p 2m 1 p
this case the spacetime is static. Now if ξa is spacelike = , K = (53)
E − t3 t2
and Ψ = 0, solution of equations (45) and (46) always
exists and the constraints (44) and (47) together imply Again solving for the metric components we get
that in general, (for a non trivial Φ), φ = = 0.
A
From the LRS-II equations (29)- (35), we then imme- dt2 2m
ds2 = + 1 dr2+t2dΩ2, (54)
diately see that the spatial derivatives of all quantity −(2m 1) (cid:18) t − (cid:19)
vanishandhencethespacetimeisspatiallyhomogeneous. t −
which is a part of the Schwarzschild solution inside the
In other words, we can say: There always exists a Schwarzschildradius.
Killing vector in the local [u,e] plane for a vacuum
LRS-II spacetime. If the Killing vector is timelike then Thus we have provedthe (local) Birkhoff Theorem:
the spacetime is locally static, and if the Killing vector is Any C2 solution of Einstein’s equations in empty space,
spacelike the spacetime is locally spatially homogeneous. which is of the class LRS-II in an open set , is locally
S
equivalent to part of maximally extended Schwarzschild
solution is . Also it is interesting to note that the
In fact in the first case, when Θ = Σ = 0, we have
S
K˙ = 0. Furthermore if choose coordinates to make the modulusoftheproportionalityconstantinequation(42),
which sets a scale in the problem, is exactly equal to the
Gaussian curvature ‘K’ of the spherical sheets propor-
Schwarzschildradius.
tional to the inverse square of the radius co-ordinate ‘r’,
(suchthatthiscoordinatebecomesthearea radiusofthe
sheets), then this geometrically relates the ‘hat’ deriva-
IV. ALMOST BIRKHOFF THEOREM
tive with the radial co-ordinate ‘r’. As we have already
seen, Kˆ = φK, where the hat derivative, defined in
terms of the−derivative with respect to the co-ordinate As we have already seen, for a spherically symmetric
‘r’, depends on the specific choice of ea (orthogonal to spacetime, the 1+1+2 scalar of the electric part of
ua and the sheet). If we choose the ’radial’ co-ordinate the Weyl tensor is always proportional to the (3/2)th
as the area radiusof the sphericalsheets, then from (10) power of the Gaussian curvature of the 2-sheet, with the
and (14) the hat derivative of any scalar M becomes proportionalityconstant defining a scale in the problem.
Let us now perturb this geometry and define the notion
of an almost spherically symmetric spacetime in the
1 dM
Mˆ = rφ . (48) following way:
2 dr
Any C2 spacetime that admits a local 1+1+2 splitting
fora static spacetime. Ifthe spacetime is notstatic then at every point such that the magnitude of all 2-vectors
thereisalsoadotderivativeintheRHSofequation(48). on the sheet, the sheet gradient of scalars (defined by
Details are given in [11]. √ψ ψa), the magnitude of all PSTF 2-tensors on the
a
Now solving equations (29)- (35), we get the unique sheet, and the sheet derivative of 2-vectors (defined by
5
ψ ψab) at any given point are either zero or much Evolution equation for eˆ :
ab a
psmaller than the scale defined by the modulus of the
proportionality constant in equation (42), is called an αˆa−a˙a = − 12φ+A αa+ 31θ+Σ (Aa−aa)
almost spherically symmetric spacetime. +(cid:0)1φ (cid:1) Σ +(cid:0)ε Ωb (cid:1) ε b. (63)
2 −A a ab − abH
(cid:0) (cid:1)(cid:0) (cid:1)
We would like to emphasize here that though Electric Weyl evolution:
Minkowskispacetimebelongstothe setofLRS-II,inthe
above definition of the perturbed spacetime we exclude ˙ + 12ε ˆb = 3ε δb + 1ε δb c 3 Σ
Ea abH 4 ab H 2 bc H a− 4E a
the Minkowski background, as in that case the scale is
+3 ε Ωb 3 α + 3Σ θ
identically zero. As we have seen from equations (27) 4E ab − 2E a 4 − Ea
and (28), the sheet will be a genuine two surface if and − 14φ+A εabHb , (cid:0) (cid:1) (64)
onlyifthe commutatorofthetime andhatderivativedo (cid:0) (cid:1)
not depend on any sheet component and also the sheet
derivatives commute. In the perturbed scenario we will E˙{ab}−εc{aHˆb}c = −εc{aδcHb}− 23EΣab− θ+ 23Σ Eab
requirethesheettobeanalmostgenuine2-surfaceinthe + 1φ+2 ε c . (cid:0) (cid:1)(65)
sensethatthecommutatorofthetimeandhatderivative 2 A c{aHb}
(cid:0) (cid:1)
almost do not depend on any sheet component and the Magnetic Weyl evolution:
sheet derivatives almost commute; this will follow from
our definition of almost spherically symmetric. In that
case the scalars Ω and ξ would be of the same order of ˙ = ε δa b 3ξ + θ+ 3Σ , (66)
H − ab E − E 2 H
smallnessastheothervectorsandPSTF2-tensorsonthe (cid:0) (cid:1)
sheet. Also using the constraint equation
δ Ωa+ε δaΣb =(2 φ)Ω 3ξΣ+ε ζacΣb+ , (55) ˙ 1ε ˆb = 3Σ θ 3 ε b
a ab A− − ab c H Ha− 2 abE 4 − Ha− 2E abA
+(cid:0) 3 ε a(cid:1)b 1ε δb c
weseethescalar alsoisofthesameorderofsmallness. 4E ab − 2 bc E a
Hence the set of 1H+1+2 variables + 1φ+ ε b 3ε δb , (67)
4 A abE − 4 ab E
(cid:0) (cid:1)
[Ω, ,ξ, a,Ωa,Σa,αa,aa,
H A
a, a,Σ , , ,ζ ] . (56)
E H ab Eab Hab ab ˙ +ε ˆ c = +3 ε ζ c 1φ+2 ε c
H{ab} c{aEb} 2E c{a b} − 2 A c{aEb}
are all of (ǫ) with respect to the invariant scale. Using θ+ 3Σ (cid:0)+ε δc(cid:1) . (68)
equationsO(48-81)of [8],we cangetthe propagationand − 2 Hab c{a Eb}
(cid:0) (cid:1)
evolution equations of these small quantities. We now In the above equation all the zeroth order quantities
list all the equations up to the first order (that is up to are background quantities. If the background is static
order ‘ǫ’). The time evolution equations of ξ and ζ{ab} with Θ = Σ = 0 and the time derivative all the back-
are as follows: ground quantities are zero, we can easily see that the
time derivatives of the first order quantities at a given
ξ˙= 1Σ 1θ ξ+ 1φ Ω+ 1ε δaαb+ 1 , (57)
2 − 3 A− 2 2 ab 2H point is of the same order of smallness as themselves.
(cid:0) (cid:1) (cid:0) (cid:1) Hence the first order quantities still remains “small” as
the time evolves.
ζ˙ = 1Σ 1θ ζ + 1φ Σ
{ab} 2 − 3 ab A− 2 ab Similarly we can write the spatial propagation equa-
(cid:0)+δ{aαb}−(cid:1)εc{aH(cid:0)b}c . (cid:1) (58) tion of all the first order quantities up to O(ǫ). The
propagationequations of ξ and ζ are:
{ab}
The Vorticity evolution equations:
Ω˙ = 21εabδaAb+Aξ+Ω Σ− 32θ , (59) ξˆ = −φξ+(cid:0)13θ+Σ(cid:1)Ω+ 21εabδaab , (69)
(cid:0) (cid:1)
Ω˙a+ 12εabAˆb = − 23θ+ 12Σ Ωa ζˆ{ab} = −φζab+δ{aab}+(cid:0)13θ+Σ(cid:1)Σab−Eab .(70)
+21(cid:0)εab −Aa(cid:1)b+δbA− 12φAb .(60) Shear divergence:
(cid:2) (cid:3)
Shear evolution equations: Σˆ ε Ωˆb = 1δ Σ+ 2δ θ ε δbΩ 3φΣ 3Σa
a− ab 2 a 3 a − ab − 2 a− 2 a
Σ˙{ab} = δ{aAb}+Aζab− 23θ+ 12Σ Σab−Eab ,(61) + 21φ+2A εabΩb−δbΣab , (71)
(cid:0) (cid:1) (cid:0) (cid:1)
Σ˙a− 21Aˆa = 12δaA+ A− 41φ Aa− 32θ+ 12Σ Σa Σˆ{ab} = δ{aΣb}−εc{aδcΩb}− 12φΣab
+12Aaa−(cid:0) 23Σαa−(cid:1) Ea (cid:0) (cid:1) (62) +32Σζab−εc{aHb}c . (72)
6
Vorticity divergence equation: Therefore we can say:
Ωˆ = δaΩa+( φ)Ω. (73) For an almost spherically symmetric vacuum space-
− A−
time there always exists a vector in the local [u,e] plane
Electric Weyl Divergence:
which almost solves the Killing equations. If this vector
is timelike then the spacetime is locally almost static,
ˆ = 1δ δb 3 a 3φ . (74)
Ea 2 aE − Eab− 2E a− 2 Ea and if the Killing vector is spacelike the spacetime is
locally almost spatially homogeneous.
Magnetic Weyl divergence:
ˆ = δ a 3φ 3 Ω, (75) Also as we have seen that in this case the resultant
H − aH − 2 H− E set of equations are the perturbed LRS-II equations,
(that is equations (29)- (35) with (ǫ) terms added to
O
ˆ = 12δ δb 3 ε Σb+ 3 Ω each), and the perturbations locally remain small both
Ha aH− Hab− 2E ab 2E a in space and time, a part of the maximally extended
+3Σε b 3φ . (76)
2 abE − 2 Ha almost-Schwarzschild solution will then solve the field
equations locally.
Hence we see that if the background is spatially homo-
geneous with φ = A = 0 and the ‘hat’ derivative all the Thus we have proved the (local) Almost Birkhoff
backgroundquantitiesarezero,wecaneasilyseethatthe Theorem: Any C2 solution of Einstein’s equations in
‘hat’ derivatives of the first order quantities at a given
empty space, which is almost spherically symmetric in
point are of the same order of smallness as themselves.
an open set , is locally almost equivalent to part of a
Hencethefirstorderquantitiesstillremain“small”along S
maximally extended Schwarzschild solution in .
the spatial direction. In both the cases of a static back- S
ground and a spatially homogeneous backgroundthe re-
Note that we do not consider perturbations acrossthe
sultant set of equations are the perturbed LRS-II equa-
horizon: our result holds for any open set that does
tions,(thatisequations(29)-(35)with (ǫ)termsadded S
O not intersect the horizon in the background spacetime.
to each).
The resultalmostcertainlyholds true acrossthe horizon
Again trying to solve the Killing equation (43) for a
also, but that case needs separate consideration.
Killing vector of the form ξ = Ψu +Φe , using equa-
a a a
tion (26),(25) and multiplying the Killing equation by
Theaboveresultcanbeimmediatelygeneralizedinthe
uaub, uaeb, eaeb, Nab, Naub, Naeb and NaNb, we get
c c c d presence of a cosmological constant. In that case an ‘al-
the following differential equations and constraints:
most’sphericallysymmteric solutionin anopenset , is
S
Ψ˙ + Φ = 0, (77) locallyalmostequivalenttopartofamaximallyextended
A Schwarzschild deSitter/anti-deSitter solution in . Also
Ψˆ Φ˙ Ψ +Φ(Σ+ 1Θ) = 0, (78) theresultholdsforanalmostsphericallysymmetSricelec-
− − A 3
tric charge distribution with no spin or magnetic dipole.
1
Φˆ +Ψ( Θ+Σ) = 0, (79) Inthis casewe haveto use the energymomentumtensor
3 oftheelectromagneticfieldinvacuumwiththemagnetic
2
partbeing equalto zero. Proceedingexactly in a similar
Ψ( Θ Σ)+Φφ = 0, (80)
3 − manner one can then show that the solution of the per-
δ Ψ+Ψ +Φ ε Ωd+α +Σ = 0, (81) turbedfieldequationswillbealmostequivalenttoapart
c c cd c c
− A
δ(cid:0) Φ+Φa +2ΨΣ(cid:1) = 0, (82) of maximally extended Reissner-Nordstro¨m spacetime.
c c c
ΨΣ +Φζ = 0. (83)
cd cd
V. DISCUSSION
Now we see that for both timelike (Φ = 0) or spacelike
(Ψ = 0) vectors, all the above equations are not com-
pletely solved in general unless the first order quantities Therigidityofsphericalvacuumsolutionsofthe EFE,
appearing in the equations above are exactly equal as enshrined in Birkhoff’s theorem, is maintained in the
to zero. This special case corresponds to static but perturbed case: almost spherical symmetry implies al-
distorted black holes in the presence of matter outside moststatic. Thisis animportantreasonfor thestability
black hole, for example when an accretion disk occurs of the solar system, and of black hole spacetimes.
[12]. If the distribution of the matter outside black Though there are many discussions in the literatures
hole is axisymmetric, then the vacuum metric outside onthestabilityofSchwarzschildsolutioninGeneralRela-
the matter is described by Weyl Solution. However as tivity,forexample[13];mostofthemdealwithaspecific
we proved that these first order quantities generically sector of the maximally extended Schwarzschild mani-
remain (ǫ) both in space and time, we can see that a fold, namely the static exterior part. In this paper we
O
timelike vector with (Θ = Σ = 0) or a spacelike vector have established in a compact and completely different
with (φ = = 0) almost solves the Killing equations. way, an aspect of the stability of the static sector of
A
7
the complete manifold: as long as the solution remains vacuum solutions of Einstein’s field equations.
almost spherically symmetric, it remains almost static, This rigidity has interesting implications for the issue
with a similar result for the spatially homogeneous sec- ofhowauniversemadeupoflocallysphericallysymmet-
tor. Furthermoreourresult,beinglocal,doesnotdepend ricobjectsimbeddedinvacuumregionsisabletoexpand,
onspecific boundaryconditions used for solvingthe per- giventhatBirkhoff’stheoremtellsusthelocalspacetime
turbation equations. Hence it does not depend on the domains have to be static. A two-mass exact solution il-
global topology of the spacetime, and brings out covari- lustratingthissituationisgivenin[14];thepresentpaper
antly the rigidity and uniqueness of the almost-spherical suggests the results given there are stable.
[1] Birkhoff, G. D. (1923). Relativity and Modern Physics. School, ed . R. K, Sachs (New York Academic Press,
(Cambridge,MA:HarvardUniversityPress).SWHawk- 1971);G.F.R.EllisandH.vanElstCosmologicalmodels
ingandGFREllis(1973),The Large Scale Structure of (Carg`eselectures1998),inTheoreticalandObservational
Spacetime (Cambridge UniversityPress), AppendixB. Cosmology, edited by M. Lachi`eze-Rey, p. 1 (Kluwer,
[2] W Stoeger, R Maartens and G F R Ellis: “Proving Dordrecht, 1999).
almost-homogeneity of the universe: an almost-Ehlers, [10] G F R Ellis: “The dynamics of pressure-free matter
Geren and Sachstheorem”. Ap J 443, 1 – 5 (1995). in general relativity”. Journ Math Phys 8, 1171 – 1194
[3] Ehlers, J., Geren, P., Sachs, R.K.: “Isotropic solutions (1967). H. van Elst and G. F. R. Ellis, Class. Quantum
of Einstein-Liouville equations”. J. Math. Phys. 9, 1344 Grav. 13, 1099 (1996), [gr-qc/9510044].
(1968). [11] G.BetschartandC.A.Clarkson,Class.QuantumGrav.
[4] HBondi, “Spherically symmetricmodelsingeneral rela- 21 5587 (2005).
tivity”.Mon. Not. Roy. Astr. Soc. 107, 410 (1947). [12] Valeri P. Frolov, Andrey A. Shoom, Phys. Rev. D 76,
[5] D’Inverno, R. (1992). Introducing Einstein’s Relativity. 064037 (2007) [arXiv:0705.1570 (gr-qc)]
(Oxford: Clarendon Press), Section 18.1. [13] C. V. Vishveshwara (1970) “Stability of the
[6] Valerio Faraoni Cosmology in Scalar-Tensor Gravity Schwarzschild Metric” Phys. Rev. D 1, 2870 - 2879;
(FundamentalTheories of Physics) (Springer: 2004) S Chandrasekhar “On the Equations Governing the
[7] C. A. Clarkson and R. K. Barrett, Class. Quant. Grav. Perturbations of the Schwarzschild Black Hole”, Proc.
20, 3855 (2003) [gr-qc/0209051]. R. Soc. Lon. A 343, 289 (1975).
[8] C. Clarkson, Phys. Rev. D 76, 104034 (2007) [14] GFREllis,J-PUzanandJLarena(2010)“Atwo-mass
[arXiv:0708.1398(gr-qc)]. expanding exact space time solution” GRG 43: 191 –
[9] J. Ehlers Abh. Mainz Akad. Wiss. u. Litt. (Math. Nat. 205 [arXiv:1005.1809].
kl) 11 (1961); G. F. R. Ellis, in General Relativity and
Cosmology, Proceedings ofXLVIIEnricoFermiSummer