Uniqueness of the static spacetimes with a photon sphere in Einstein-scalar field theory

0.1 MB·English

by  Stoytcho S. Yazadjiev

Uniqueness of the static spacetimes with a photon
sphere in Einstein-scalar field theory
Stoytcho Yazadjiev
∗
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1 1Department of Theoretical Physics, Faculty of Physics, Sofia University
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2 5 J. Bourchier Blvd., Sofia 1164, Bulgaria

n 2Theoretical Astrophysics, Eberhard-Karls University of Tübingen,
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J
Tübingen 72076, Germany

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Abstract
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r
g
Inthepresentpaperweproveauniquenesstheoremforthestaticandasymptotically
[
flat solutions to the Einstein-scalar field equations which possess a photon sphere. We

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showthatsuchsolutionsareuniquelyspecifiedbytheirmassM andscalarchargeqand
v
7 thattheyareisometric totheJanis-Newman-Winicour solution withthesamemassand
3 scalarcharge subjecttotheinequality q2 <3.
8 M2
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. 1 Introduction
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General relativity and the generalized gravitational theories predict the existence of space-
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: time regions where light can be confined in closed orbits [1]-[10]. These regions called
v
i photon spheres play a very important role in gravitational lensing which is a standard and
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powerful tool in modern observational astronomy and astrophysics [4],[6]. From the point
r
a ofview of gravitationallensing thephoton sphere can beconsidered as a timelikehypersur-
face on which the bending angle of a light ray is unboundedly large [4],[6]. A more precise
definition will be given below. In the presence of a photon sphere the gravitational lensing
effect gives rise to the so-called relativistic images which are characteristic for relativistic
gravity. It is commonlyexpected that ultracompactobjects such as black holes, neutron and
boson stars, wormholes and naked singularities are surrounded by a photon sphere [1]-[10].
Apart from its significance for gravitational lensing the photon sphere is intimately related
to thequasinormalmodesofultracompact objectsand iscrucially relevant fortheirstability
[9], [11]-[14].
Thephotonspherepossessessomeveryspecificcharacteristicswhichmakeitaveryspe-
cial and unique object. For example, in static spacetimes, as the explicit exact or numerical
solutions show, the lapse function is constant on the photon sphere. Moreover, the photon
[e-mail]
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sphere is a totally umbilic hypersurface with constant mean curvature and constant surface
gravity[5],[7],[15]. Withthesepropertiesthephotonsphereresemblesverymuchtheblack
holehorizon. It is wellknownthatthepresenceofan eventhorizonallowsus toclassifythe
asymptotically flat spacetimes only in terms of their conserved asymptotic charges, such as
the mass and the electric charge in the static case [16]. In analogy with the black hole case,
thenaturalquestionthatarisesis: Doesthepresenceofaphotonsphereuniquelyspecifythe
spacetimes with given asymptotic charges? In other words, we ask whether the spacetimes
possessing a photon sphere can be classified only in terms of asymptotic charges (or other
global physical quantities). The problem set in this way seems to be much more difficult
that the classification of black hole spacetimes since the spectrum of spacetimes possessing
a photon sphere is much richer than the spectrum of black hole spacetimes. Nevertheless,
in some cases the desired classification can be achieved. Recently Cederbaum [15] proved
that the static, asymptotically flat solutions to the vacuum Einstein equations with mass M
possessingaphoton sphere are isometricto the Schwarzschild solutionwith the samemass.
In the present paper we consider the more general case of static Einstein-scalar field equa-
tions. We prove that the static and the asymptotically flat solutions to the Einstein-scalar
fieldequationspossessingaphotonsphereareuniquelyspecifiedbytheirmassM andscalar
charge q. Moreprecisely weprovethatthe staticand the asymptoticallyflat solutionsto the
Einstein-scalar field equations with mass M and scalar charge q possessing a photon sphere
are isometric to the Janis-Newman-Winicour solution [17] with the same mass and scalar
chargeas well as withacertain restrictionontheratioq/M.
2 General definitions and equations
In thepresent paperweconsiderEinstein-scalarfield theorydescribed bytheaction
1
S = 16p Z d4x g(4) R(4)−2g(4)µn (cid:209) (µ4)(cid:209)j (n 4)j , (1)
q (cid:16) (cid:17)
where j is the scalar field, (cid:209) (4) and R(4) are the Levi-Civitaconnection and theRicci scalar
µ
curvature with respect to the spacetime metric gµn . This action gives rise to the following
field equationson thespacetimemanifoldM(4)
Ric(µ4n) =2(cid:209) (µ4)(cid:209)j (n 4)j , (2)
(cid:209) (4)(cid:209) (4)µj =0,
µ
(4)
with Ricµn being the Ricci tensor. It is important to note the following. Although the
Einstein-scalarfield equationscan be considered in theirownrightthey are in fact the"vac-
uum" field equations of scalar-tensor theories presented in the so-called Einstein frame. In
this way all results for (2) obtained in the present paper can be easily transformed as results
foran arbitrary classofscalar-tensortheories.
In the present work we focus on static and asymptotically flat spacetimes. A spacetime
iscalledstaticifthereexistsasmoothRiemannianmanifold(M(3),g(3))and asmoothlapse
functionN :M(3) R+ such that
→
2
M(4) =R M(3), g= N2dt2+g(3). (3)
× −
In addition to the metric staticity we have to defined the scalar field staticity. The scalar
field is called static if Lx j = 0 where Lx is the Lie derivative along the Killing field x =
¶ . We note that both notions of staticity are consistent since the Ricci 1-form Ric(4)[x ] =
¶ t
x µRic(µ4n)dxn is zero duetothefield equationsandthefact that x µ(cid:209) (µ4)j =Lx j =0.
We will adopt the following notion of asymptotic flatness. The spacetime is called
asymptoticallyflatifthereexistsacompactsetK M(3) suchthatM(3) K isdiffeomorphic
toR3 B¯ whereB¯ is theclosed unitballcentered a⊂t theorigininR3,and−such that
\
M q
g(3) =d +O(r−1), N =1 +O(r−2), j =j ¥ +O(r−2), (4)
− r − r
with respect to the standard radial coordinate r of R3. Here M, j ¥ and q are constants with
M and q being themass and thescalarcharge. We willconsiderspacetimes withM >0 and
withoutlossofgeneralitywewillset j ¥ =0.
Thedimensionallyreduced staticEinstein-scalarfield equationsarethefollowing
Ric(3) =N 1(cid:209) (3)(cid:209) (3)N+2(cid:209) (3)(cid:209)j (3)j ,
ij − i j i j
(cid:209) (3)(cid:209) (3)iN =D (3)N =0, (5)
i
(cid:209) (3) N(cid:209) (3)ij =0,
i
(cid:16) (cid:17)
where(cid:209) (3) andRic(3) aretheLevi-CivitaconnectionandtheRiccitensorwithrespecttothe
i ij
(3)
metricg .
ij
Now we can define the notion of photon sphere. First we give the definition of photon
surface[5],[15].
Definition An embedded timelike hypersurface P ֒ M(4) is called a photon surface if
→
andonlyifanynullgeodesicinitiallytangenttoP remainstangenttoP aslongasitexists.
Thedefinitionofphotonsphereisanaturalextensionofthedefinitioninthepurevacuum
case[15].
Definition Let P ֒ M(4) be a photon surface. Then P is called a photon sphere if the
lapsefunctionN andt→hescalarfieldj areconstantalongP.
It is worth noting that the definition of the photon sphere is very well justified by the
known exact and numerical solutions in general relativity and the scalar-tensor theories as
wellas thestudiesofthegravitationallensingin thesetheories.
As an additional technical assumption we shall assume that the lapse function regu-
(3)
larly foliates the region of spacetime M exterior to the photon sphere. This means that
ext
g(3)((cid:209) (3)N,(cid:209) (3)N) = 0 everywhere on M(3). In what follows we will use the function
ext
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r :M(3) R+ defined by
ext
→
3
1/2
r = g(3)((cid:209) (3)N,(cid:209) (3)N) − . (6)
h i
Nowletusconsiderthe2-dimensionalintersectionS ofthephotonsphereP andthetime
slice M(3). By the definition of the photon sphere, S which is the inner boundary of M(3),
ext
is given by N =N for some N R+. The metric induced on S will be denoted by s . It is
0 0
∈
notdifficulttoseethatourassumptionsrestrictourconsiderationstothecaseofaconnected
photon sphere. Moreover, all the level sets N =const, including S , are topological spheres
whichisa directconsequencefrom ourassumptions.
By the maximum principle for harmonic functions and by the asymptoticbehavior of N
forr ¥ weobtainthatthevaluesofN onM(3) satisfy
ext
→
N N <1. (7)
0
≤
3 Functional dependence between the lapse function and
the scalar field
Here we show that there is a functional dependence between the lapse function N and the
scalarfield j . Morepreciselywehavethefollowing
Lemma. The lapsefunctionand thescalarfield aresubjecttotherelation
q
j = ln(N), (8)
M
withM andq beingthemassand thescalarchargerespectively.
Proof. LetusfirstusethefactthatN isharmoniconM(3). IntegratingD (3)N =0onM(3)
ext ext
andapplyingtheGausstheoremwe have
0= D (3)N g(3)d3x= (cid:209) (3)Nd2S i (cid:209) (3)Nd2S i, (9)
ZMe(x3t) q IS¥2 i −IS i
where S2 is the 2-dimensional sphere at infinity. This result and the asymptoticbehavior of
¥
N give
1 1
M = (cid:209) (3)Nd2S i = (cid:209) (3)Nd2S i. (10)
4p IS¥2 i 4p IS i
Inthesamewaymakinguseofthefieldequationforj ,namely(cid:209) (3) N(cid:209) (3)ij =0,and
i
theasymptoticbehaviorofN and j , wefind (cid:16) (cid:17)
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1 1
q= (cid:209) (3)j d2S i = N (cid:209) (3)j d2S i. (11)
4p IS¥2 i 4p 0IS i
The next step is to consider J =(cid:209)j (3)N Nln(N)(cid:209) (3)j . As a consequence of the field
i i i
−
equations for N and j , it is not difficult one to show that (cid:209) (3)Ji =0. Integrating (cid:209) (3)Ji =0
i i
(3)
on M , applying the Gauss theorem and taking into account the asymptotic behavior of N
ext
andj , weobtain
(cid:209)j (3)Nd2S i = Nln(N)(cid:209) (3)j d2S i, (12)
IS i IS i
whichinview of(11)gives
q
j = ln(N ). (13)
0 0
M
Thefinal stepis toconsiderthedivergenceidentity
q
N 1 w w i =(cid:209) (3) j ln(N) w i , (14)
− i i −M
h(cid:16) (cid:17) i
(cid:0) (cid:1)
where
q
w =N(cid:209) (3)j (cid:209) (3)N. (15)
i i −M i
As one can verify this identity is a consequence of the field equations for the lapse function
andthescalarfield. By applyingtheGausstheoremto theaboveidentityweobtain
q q
N 1 w w i g(3)d3x= j ln(N) w d2S i j ln(N) w d2S i =0,(16)
− i i i
ZMe(x3t) q IS¥2 (cid:16) −M (cid:17) −IS (cid:16) −M (cid:17)
(cid:0) (cid:1)
where we have taken into account the asymptoticbehavior of N and j as well as (13) in the
evaluation of the surface integrals. Since N > 0 on M(3) we conclude that w = N(cid:209) (3)j
ext i i
−
q(cid:209) (3)N =0 on M(3). Therefore we obtain j = q ln(N)+C withC being a constant. From
M i ext M
theasymptoticbehaviorofN and j orfromeq.(13)wefind thatC=0 whichproves(8).
4 Some relations for the photon sphere and inequality for
the scalar charge to mass ratio
In thissectionweshallderivesomekey relationsforthephotonsphereandan importantin-
equalityforthescalarchargetomassratio. Alltheseresultswillbedirectlyorindirectlyused
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in the proof of the main theorem. We first recall the important result of Claudel-Virbhadra-
Ellis[5]:
Theorem Let P ֒ M(4) be an embedded timelike hypersurface. Then P is a photon
→
surfaceifand onlyifit istotallyumbilic(iff itssecondfundamentalformis puretrace).
Denotingthemetricinducedon P by p theaboveresult can bewrittenintheform
P
H
KP = p, (17)
3
where KP is the second fundamental form and HP is the mean curvature of P. It is easy
to show that HP is constant on P. Using the contracted Codazzi equation for (P,p) ֒
→
(M(4),g(4))wehave
2
Ric(4)(X,n)= X(HP), (18)
3
wheren is theunitnormal toP and X is a vectorfield tangentto P, i.e. X G (TP). Taking
intoaccountthefieldequationswefindRic(4)(X,n)=2X(j )n(j )=0sinc∈ej isconstanton
P. HenceweconcludethatX(HP)=0which meansthat HP isconstanton P.
The same can be proven for the Ricci scalar curvature RP of P. Applying the Gauss
equationfor(P,p)֒ (M(4),g(4))
→
2 2
R(4) 2Ric(4)(n,n)=RP TrKP +Tr KP , (19)
− −
(cid:16) (cid:17) (cid:16) (cid:17)
andmakinguseofthefield equationsand (17), wefind
2
RP = (HP)2 2(n(j ))2. (20)
3 −
Since HP is constant on P we have to show that n(j ) is constant on P. In view of the
relation (8), it is enough to show that n(N) is constanton P. Since thespacetimeis static, it
issufficientto provethat n(N)is constantonS . Thiswillbedonebelow.
Forthesecond fundamental form KS of(S ,s )֒ (M(3),g(3))(with a unitnormal n)we
→
have
P P
H H
KS (X,Y)=g(3)((cid:209) (3)n,Y)=g(4)((cid:209) n,Y)=KP(X,Y)= p(X,Y)= p(X,Y), (21)
X X 3 3
where X,Y G (TS ). Therefore we find KS = HPs which also gives a simple relation
∈ 3
between themean curvatures HP and HS , namelyHS = 2HP. Usingthisinformationinthe
3
contracted Codazzi equationweeasilyfind that Ric(3)(X,n)=0.
In order to show that n(N) is constant on S we follow [15]. For an arbitrary X G (TS )
∈
wehave
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X(n(N))=X(n(N)) ((cid:209) (3)n)(N)=((cid:209) (3)(cid:209) (3)N)(X,n)=
X
−
N Ric(3)(X,n) 2X(j )n(j ) =0, (22)
−
h i
wherewehavetaken intoaccountthedimensionallyreduced field equations(5)andthefact
thatN and j are constantonS . Therefore n(N)isindeed constantonS .
FromtheGaussequationfor(S ,s )֒ (P,p),andtakingintoaccountthatthespacetime
isstatic,itis easy toshowthattheRicci→scalarcurvatureRS ofS isgivenby
2
RS =RP = (HP)2 2(n(j ))2. (23)
3 −
TheGaussequation
R(3) 2Ric(3)(n,n)=RS TrKS 2+Tr KS 2 (24)
− −
for(S ,s )֒ (M(3),g(3))gives (cid:0) (cid:1) (cid:0) (cid:1)
→
1
R(3) 2Ric(3)(n,n)=RS (HS )2. (25)
− −2
InordertofindRic(3)(n,n)wecanusethedimensionallyreduced fieldequations(5)and
D (3)N =D (2)N+(cid:209) (3)(cid:209) (3)N(n,n)+HS n(N), (26)
whichleads to
N Ric(3)(n,n)= HS n(N)+2N (n(j ))2. (27)
0 0
−
Using again the dimensionally reduced field equations one can easily show that R(3) =
2(n(j ))2 which combinedwith(25) and(27)gives
1
N RS =2HS n(N)+ N (HS )2 2N (n(j ))2. (28)
0 0 0
2 −
WecaneliminateRS from(23)and(28)byintegratingoverS andusingtheGauss-Bonnet
theorem RS √s d2x = 8p for the topological sphere S . The integration reduces (23) and
S
(28) tothRefollowingrelations
3 1
1= (HS )2AS (n(j ))2AS , (29)
16p −4p
1 1 1
N0 = 4p HS n(N)AS +16p N0(HS )2AS −4p N0(n(j ))2AS .
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From theseequationsitis easy toshowthatthefollowingimportantrelationis satisfied
S
2n(N)=N H , (30)
0
orequivalently
N HS r =2, (31)
0 0
bytakingintoaccount thatn(N)=r 1.
−0
Thepresenceofaphotonsphereimposesaveryimportantrestrictiononthescalarcharge
to mass ratio. It is convenient to express this restriction in terms of the parameter n defined
by
q2 −1/2
n = 1+ . (32)
M2
(cid:18) (cid:19)
Forthispurposewefirst rewrite(10)and (11)intheform
1 AS
M = n(N)AS = , (33)
4p 4pr
0
1
q= 4p N0n(j )AS . (34)
Substitutingrelations(33)and (34)in (29)and after somealgebraweobtain
4n 2 1 r
− =N2 0 >0. (35)
n 2 0 M
Thisinequalitycombinedwiththedefinitionofn givesthedesired inequalityforn , namely
1
<n 1, (36)
2 ≤
q2
whichisequivalentto <3.
M2
5 Uniqueness theorem
Themainresultofthepresent paperis thefollowing
Theorem There can be only one static and asymptoticallyflat spacetime (M(4),g(4),j ),
satisfying the static Einstein-scalar field equations, possessing a photon sphere P ֒ M(4)
→
asaninnerboundaryofM(4),withlapsefunctionN regularlyfoliatingM(4) andgivenmass
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M and scalar charge q. Moreover, the solution is isometric to the Janis-Newman-Winicour
solutionwith 1 <n (M,q) 1.
2 ≤
Proof: Thestrategyoftheproofisthefollowing. Thefirstandthemostdifficultstepisto
provethatthespacetimeissphericallysymmetric. Thenweconstructthesolutionexplicitly.
(3)
Let usconsiderthe3-metrichon M defined by
ext
h =N2g(3). (37)
ij ij
In termsofthenewmetricthedimensionallyreduced equationsbecome
R(h) =2D ln(N)D ln(N)+2D j D j ,
ij i j i j
D Diln(N)=0, (38)
i
D Dij =0,
i
whereD andR(h) aretheLevi-ChivitaconnectionandtheRiccitensorwithrespecttoh ,
i ij ij
respectively. Takingintoaccount thefunctionaldependancej = q ln(N) theequations(38)
M
can becast intheform
R(h) =2D ln(N˜)D ln(N˜), (39)
ij i j
D Diln(N˜)=0,
i
wherewehaveintroducedanewfunction
N˜ =N1/n . (40)
Proceeding furtherweconsidertheBach tensor
1
R(h) =2D R(h) + h D R(h). (41)
ijk [i j]k k[i j]
2
Usingeqs. (39)aftera longcalculationwefind
1
Di W 1D c = c 7W 3R(h) R(h)ijk, (42)
− i − ijk
16
(cid:0) (cid:1)
where
1 1 N˜ 4N˜
c = hijD HD H 4, H = − , W = . (43)
i j 1+N˜ 1+N˜ 2
(cid:0) (cid:1)
WecancombinetheequationforN˜ from(39)witheq. (4(cid:0)2)toob(cid:1)tainanotherdivergence
identity
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1
Di W 1(HDc c D H) = Hc 7W 3R(h) R(h)ijk. (44)
− i i − ijk
− 16
(cid:2) (cid:3)
SinceH >0 weobtaintheinequality
Di W 1(HD c c DH) √hd3x 0. (45)
− i i
ZM(3) − ≥
ext
(cid:2) (cid:3)
Anotherinequalitycan beobtainedbytakingintoaccount thatH <1, namely
Di W 1D c √hd3x Di W 1(HD c c DH) √hd3x. (46)
− i − i i
ZM(3) ≥ZM(3) −
ext ext
(cid:0) (cid:1) (cid:2) (cid:3)
In bothcases (45)and (46)theequalityholdsifandonly ifR(h) =0.
ijk
Calculating theintegralin (45) by usingthe Gausstheorem and takinginto account (31)
wegettheinequality
2 2n 1
N˜2 =Nn − . (47)
0 ≤ 2n +1
In thesameway (46)gives
2 2n 1
N˜2 =Nn − . (48)
0 ≥ 2n +1
Hence we conclude that N˜2 = Nn2 = 2n 1 and therefore R(h) = 0. This means that the
0 2n +−1 ijk
metric h is conformally flat. As an immediate consequence we get that g is conformally
ij ij
(3)
flat too(i.e. R(g) =R =0).
ijk ijk
(3)
SinceN regularlyfoliatesM wecanintroduceadaptedcoordinatesinwhichthemetric
ext
(3)
g takes theform
ij
g(3) =r 2dN2+s , (49)
where s is the 2-dimensional metric on the 2-dimensional intersections S of the level
AB N
(3)
setsN =const withthetimesliceM . Usingtheformula
ext
R(3)R(3)ijk = 4 KS N 1HS Ns KS NAB 1HS Ns AB + 1 s AB¶ r¶ r , (50)
ijk N4r 4 AB −2 AB −2 r 2 A B
(cid:20)(cid:18) (cid:19)(cid:18) (cid:19) (cid:21)
where KS N is the extrinsic curvature of (S ,s ) ֒ (M(3),g(3)) and HS N is its trace, we
AB N →
concludethat
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