Stability and Quasinormal Modes of Black holes in Tensor-Vector-Scalar theory: Scalar Field Perturbations

1.1 MB·English

by  Paul D. Lasky

Stability and Quasinormal Modes of Black holes in Tensor-Vector-Scalar theory:
Scalar Field Perturbations
Paul D. Lasky∗
Theoretical Astrophysics, Eberhard Karls University of Tu¨bingen, Tu¨bingen 72076, Germany
Daniela D. Doneva†
Department of Astronomy, Faculty of Physics, St. Kliment Ohridski University of Sofia, 1164 Sofia, Bulgaria and
Theoretical Astrophysics, Eberhard Karls University of Tu¨bingen, Tu¨bingen 72076, Germany
(Received January 5, 2011; Published)
The imminent detection of gravitational waves will trigger precision tests of gravity through
observations of quasinormal ringing of black holes. While General Relativity predicts just two po-
larizations of gravitational waves, the so-called plus and cross polarizations, numerous alternative
theories of gravity predict up to six different polarizations which will potentially be observed in
1
current and future generations of gravitational wave detectors. Bekenstein’s Tensor-Vector-Scalar
1
(TeVeS) theory and its generalization fall into one such class of theory that predict the full gamut
0
of six polarizations of gravitational waves. In this paper we begin the study of quasinormal modes
2
(QNMs) in TeVeS by studying perturbations of the scalar field in a spherically symmetric back-

n ground. We show that, at least in the case where superluminal propagation of perturbations is
a not present, black holes are generically stable to this kind of perturbation. We also make a unique
J
predictionthat,asthelimitofthevariouscouplingparametersofthetheorytendtozero,theQNM

4 spectrumtendsto1/√2timestheQNMspectruminducedbyscalarperturbationsofaSchwarzschild
black hole in General Relativity due to the intrinsic presence of the background vector field. We
] furthershowthattheQNMspectrumdoesnotvarysignificantly from thisvalueforsmallvaluesof
c
q thetheory’scouplingparameters,howevercanvarybyasmuchasafewpercentforlarger, butstill
- physically relevant parameters.
r
g
PACSnumbers: 04.50.Kd,04.30.-w,04.70.Bw,04.80.Cc
[

2
v I. INTRODUCTION single blackhole cangive rigorousconstraintsonthe no-
7 hair theorem– one of the major science goalsfor ground
4 based and space based gravitationalwave detectors [e.g.
7 Perturbations of black holes have been intensely stud- 16–19]. An alternative approach to the measurement of
0
ied during the past decades with relation to black hole thefrequenciesofblackholeQNMsbasedonaconnection
.
1 stability,astrophysicalimplicationsofgravitationalwave between the emission of gravitational waves and strong
1 detectionand,morerecently,gauge/gravitydualities(see gravitationallensing hasrecently beenelucidatedinSte-
0
the reviews [1–4]). From an astrophysical perspective, fanov et al. [20].
1
the impending observation of gravitational waves from A recently popular alternative theory of gravity that
:
v black holes will herald new and unprecedented tests of predicts both extra polarizations of gravitational wave
i gravity through two main avenues. The first is through propagationaswellasa violationofthe no-hairtheorem
X
thepossibledetectionofgravitationalwavepolarizations is Bekenstein’s Tensor-Vector-Scalar(TeVeS) theory [21]
r
a that are not predicted by General Relativity (GR) such and its generalization [22–24]. In this paper we take a
as scalar“breathing modes”andextra tensor andvector first step towards exploring the perturbations of black
degreesoffreedom1. Thesecondmethodisadirectprobe holes within TeVeS and its generalization by studying
oftheno-hairtheoremofGRthroughobservationsofthe perturbations of the background scalar field. This en-
quasinormalringingofblackholesknownasquasinormal ablesustolookatboththestabilityofblackholesinthe
modes (QNMs). General Relativity predicts that the si- theories, as well as looking at how the QNM frequencies
multaneous measurement of the frequency and damping and damping times are affected by the various parame-
time of a single QNM is sufficient to determine both the ters of the theories.
mass and angular momentum of a black hole [12–16], TeVeS is a covariant generalization of the Modified
implying the detection of more than one QNM from a Newtonian Dynamics (MoND) paradigm [25], which at-
tempts to explain the discrepancybetweenobservedand
predictedmassdistributionsongalacticscales. Asatool
for interpreting galactic scale observations, MoND has
∗Electronicaddress: [email protected] provenextremelysuccessful(see [26]fora review). How-
†Electronicaddress: [email protected]fia.bg ever, as a non-covariant theory it is ill-equipped to ex-
1 See [5, 6] for possible gravitational modes in metric theories of
plain the full gamut of astrophysical and cosmological
gravity,and[7,8]aswellas[9–11]andreferencesthereinfordis-
observations. To this end, TeVeS has recently been em-
cussions of possible detections with inteferometer and spherical
gravitationalwavedetectors resepctively. ployed in an attempt to explain, amongst other things,
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gravitationallensing[27–31],thecosmicmicrowaveback- thatobservationsofgravitationallensingandgalacticro-
ground power spectrum [32, 33], large-scale cluster sur- tationcurvesprovideaninconsistentparameterspacefor
veys [34] and even type Ia supernova [35, 36], while re- the theory. Although their results were based on the
taining the successful predictions of MoND in the weak original TeVeS theory, their formulation of the problem
accelerationlimit (see [37]for a recentreview of TeVeS). carries over to the generalized theory based on their use
IntheNewtonianregime,TeVeShasbeenshowntore- of a vector field with only a temporal component (the
producethe parametrizedpost-Newtoniancoefficients to proof of this is the same as that given in Lasky [47] in
a level consistent with solar system experiments [21]. In thestrong-fieldregime). However,itisexactlythispoint
the strong field regime however,comparatively little has thatdiminishestherobustnessoftheirclaims. Onemain
been studied. Giannios [38] first solved the field equa- drivingforceforintroducingthevectorfieldintothethe-
tions for static, spherically symmetric, vacuum space- ory is to provide sufficient gravitational lensing without
times within TeVeS. He found two branches of solutions requiring dark matter [21]. It is therefore not surpris-
dependent on the degrees of freedom allowed in the vec- ing that suppressing the degrees of freedom of this vec-
torfield. Giannios’solutionwas,however,plaguedbythe tor field by setting all spatial components to zero also
factthatthescalarfieldwasnecessarilynegativeforsome suppresses the degree to which gravitational lensing ob-
radii,implying superluminalpropagationofscalarwaves servations can be made to be consistent with other ob-
[21]. Sagi and Bekenstein [39] remedied this situation servations. With the extra complexity of the vector field
by showing that another branch of solutions to the field in the generalized theory, combined with the relatively
equations exist that has the same physical metric, but primitivestateofparameterspaceestimation,onewould
where the scalar field is positive throughout the space- expect that gravitational lensing observations could be
time. Sagi and Bekenstein [39] also found the charged induced into conformingwith galacticrotationcurve ob-
Reissner-Nordstro¨msolutionwithin TeVeS andprovided servations with a more robust form of the vector field.
a detailed study of black hole thermodynamics. Lasky The study of gravitational wave propagation in Gen-
et al. [40] then found the Tolman-Oppenheimer-Volkoff eralized TeVeS was first specifically broached by Sagi
solution in order to study neutron star structure, work [48], who studied linear perturbations of the field equa-
which has since been extended to include slow rotation tionstolookatthe speedandformofgravitationalwave
[41] and fluid & spacetime perturbations [42, 43]. propagation. Sagi showed that there exist six different
Meanwhile, trouble was brewing with the original for- modes of gravitationalwaves (as opposed to two in GR)
mulation of the TeVeS field equations. Seifert [44] first which propagate at four different speeds, all different to
showed that the Schwarzschild-TeVeS solution is unsta- the speed of light. As expected, the various propaga-
ble to linear perturbations for experimentally and phe- tion speeds of the gravitational wave modes were found
nomenologically valid values of the various coupling pa- to be dependent on the various coupling parameters in
rameters. Contaldi et al. [22] then showed that the vec- the theory. It is interesting to note that this is seem-
torfield is proneto the formationofcaustics inavariety ingly in contrast with the result of Kahya and Woodard
of simple dynamical situations, analogously to Einstein- [49]andDesaietal.[50]who predictgravitationalwaves
Æther theories where the vector field is described by a will propagate at the same speed independently of the
Maxwellian action. Finally, Sagi [24] showed that the coupling parameters. The obvious discrepancy between
vector field is constrained by the cosmological value of these two results implies further investigation is neces-
the scalar field in such a way that it prevents the scalar sary.
field from evolving.
Spherically symmetric vacuum, charged and perfect
Toovercometheaforementionedissues,Contaldietal. fluid solutions of the generalized TeVeS field equations
[22], Skordis [23] and Sagi [24] independently provided a were studied in Lasky [47] where the vector field was as-
generalization of TeVeS by taking a more general action sumed to contain only a temporal component. Under
forthevectorfieldwhichismotivatedbyEinstein-Æther these symmetry assumptionsit wasshownthat these so-
theory. Complicating an already complicated theory is lutions are the same as for the original TeVeS theory by
a complicated process. It has implied that little rigor- Giannios [38], Sagi and Bekenstein [39] and Lasky et al.
ous work has been achieved looking at the structure of [40]respectively,whereonlyarescalingofthevectorfield
the fieldequations. Skordis[23]studiedthe cosmological couplingparametersisrequired. Thatis,giventheafore-
equations of the theory and showed they are identical to mentioned rescaling, the form of the background scalar,
theequationsgoverningtheoriginalTeVeScosmologyup vector and tensor fields, and therefore also the form of
to a rescaling of Hubble’s constant. Sagi [24] looked at the physicalmetric, are exactly the same in the two the-
the parametrized post-Newtonian (PPN) parameters of ories. In Lasky [47] it was further shown that this result
the theory and found them not to be in conflict with so- is not generalizable – solutions with time dependence or
larsystemexperimentsforsuitablevaluesofthecoupling more complicatedgeometry in the vectoror tensor fields
parameters. will be different between the two theories. As we are
It should be mentioned that the generalized TeVeS dealing in the present paper with perturbations of the
theory is also not without its problems. In two papers, spherically symmetric vacuum solution, we are dealing
Mavromatosetal.[45]andFerrerasetal.[46]haveshown with the solution of both the original and generalized
3
versions of TeVeS. Moreover, as the difference between The modified Einstein field equations are
thesetwotheoriesisinthevectorfield,theperturbations
we performofthe scalarfieldalsoholdfor boththeories. G =8πG T˜ + 1 e−4ϕ AαT˜ A +τ +Θ ,
µν µν α(µ ν) µν µν
That is, all work presented in this paper is applicable −
to both the original version and generalized version of h (cid:0) (cid:1) i (2)
TeVeS. As such, unless explicitly mentioned, we herein
whereG istheEinsteintensorassociatedwiththeEin-
refer to TeVeS to mean the all-encompassinggeneralized µν
version of the theory. stein frame, T˜µν is the physical stress-energy tensor and
τ and Θ are the effective stress-energy terms asso-
TeVeS admits the Schwarzschildsolution as a possible µν µν
ciated with the scalar and vector fields respectively. In
geometry[38,47],andmostlikelythe Kerrsolutionas is
particular
the case with a majority of alternative theories of grav-
ity [51](althoughwenote thereexistsvarioustheoriesof
µ 1
gravitythat do notadmitthe Kerrsolution,for example τ := ϕ ϕ gαβ ϕ ϕg
µν µ ν α β µν
Chern-Simons gravity [52]). Therefore, electromagnetic kG ∇ ∇ − 2 ∇ ∇
observations of the spacetime surrounding black holes Ahα ϕ A ϕ 1Aβ ϕg
maynotallowforthedistinctionbetweenGR,TeVeSand − ∇α (µ∇ν) − 2 ∇β µν
alternative theories. For example, observing the dynam- (µ) (cid:18) (cid:19)i
ics of stars orbiting close to Sgr A⋆ will soon be yielding − 2Fk2ℓ2Ggµν, (3)
fruitfulinformationaboutthespacetimestructureofthat
blackhole[53,54]. Thisproblemhowever,iswellapprox- wherekisthescalarfieldcouplingconstant,µ,afunction
imated by the stars acting like zero-mass test particles, of the theory’s coupling parameters and the scalar field,
implyingoneisonlyprobinggeodesicsofthebackground is associated with the MoND acceleration scale, (µ) is
F
spacetime. Therefore, measuring the spacetime to be a relatedtotheinterpolationfunctioninMoNDandhence
Kerrblackholeusingthismethoddoesnotruleoutalter- is not a priori predicted by the theory and ℓ is a fixed
native theories of gravity that admit Kerr as a possible lengthscaleassociatedwiththefreefunction. Forthere-
geometry. The perturbations of such spacetimes (rele- mainder of this article we only consider the strong-field
vant for gravitational waves) however, depends on the regime of the theory, whereby µ = 1 is an excellent ap-
specific field equations of the theory, and hence will dif- proximation (for more details see [21, 22, 38, 39]). In
ferbetweentheories. Thisimplies QNMsofvariousKerr this case,Contaldietal.[22]showedthatthe function
F
(orSchwarzschild)geometriesbehavedifferently andcan logarithmically diverges, but is exactly cancelled in the
be usedtoproviderigoroustestsofthe theoryofgravity. fieldequations,implying this functionhas zerocontribu-
The article is set out as follows; in section II we pro- tion in this limit. Moreover, it has been shown [39] that
vide abriefprimeronthe relevantfieldequationsforthe this limit of the strong-field approximation, whereby we
generalized TeVeS theory, reviewing the structure of the allow µ=1, is correctout to at least a million times the
background spacetime in section III. In section IV we gravitationalradiusoffablackhole. Notonlyisthiswell
derivethe waveequationgoverningscalarfieldperturba- into the asymptotically flat region of the black hole, but
tionsandalsodiscussanalyticresultsinthelimitofsmall for the purpose of QNM calculations this is more than
couplings. In sectionVA, we analysethe stability ofthe sufficient for the external boundary condition. That is,
black hole solutions and then in section VB we compute itissufficientinanastrophysicalsense,thatwhenweare
the QNM spectrum. Throughout the article Greek in- treating QNMs and their boundary conditions at “infin-
dices range from 0...3 and antisymmemetrization and ity”,we arestilldealingwiththe strong-fieldlimitofthe
symmetrization are respectively denoted by square and theory.
roundbrackets: A[µν] :=Aµν Aνµ,A(µν) :=Aµν+Aνµ. One key difference between the original version of
−
TeVeSdevelopedby Bekenstein[21] andthe currentver-
sion with Maxwellian action [22, 23] is the introduction
II. GENERALIZED TEVES EQUATIONS of three extra vector coupling constants which describe
the relative strengths of the individual terms in the vec-
tor field action. The four vector field coupling constants
TeVeS is built upon three dynamical fields; the Ein-
are denoted by K, K , K and K , where the origi-
steinmetric, g ,atime-likenormalizedvectorfield, Aµ, + 2 4
µν
nal theory is regained by setting the last of these three
and a dynamical scalar field, ϕ. A physical metric, g˜ ,
µν
parameters to zero. The vector field contribution to the
inwhichclocksandrulersaremeasured,isrelatedtothe
effectivestress-energyinthemodifiedEinsteinequations,
other three fields according to
Θ , can now be expressed as a sum of terms associated
µν
g˜ =e−2ϕ(g +A A ) e2ϕA A . (1) with each of the vector field coupling constants, plus a
µν µν µ ν µ ν
− term associated with the Lagrange multiplier, λ (which
ensures normalization of the vector field);
Thisimpliesthatvariationsinthetensor,vectororscalar
fields can be measured through their coupling to the
Θ :=ΘK +ΘK+ +ΘK2 +ΘK4 +Θλ , (4)
physical metric. µν µν µν µν µν µν
4
where in TeVeS. Giannios [38] first solved the original TeVeS
equationsofBekensteinforthecaseofasphericallysym-
1
ΘK :=K F Fα g F Fαβ , (5) metric, static, vacuum spacetime. He showed that there
µν αµ ν − 4 µν αβ exist two branches of solutions based on the form of the
(cid:18) (cid:19)
vector field; one branch where the vector field is tempo-
rally aligned and another when the vector field has both
1
ΘK+ :=K S S α g S Sαβ atemporalandradialcomponent. Lasky[47]thenfound
µν + µα ν − 4 µν αβ
the equivalent solution in the case of a purely temporal
h
+ α AαSµν Sα(µAν) , (6) vector field for the more general TeVeS theory. In that
∇ −
work it was shown that the solution in the generalized
(cid:0) (cid:1)i
TeVeS theory is the same as that in the original TeVeS
ΘK2 :=K g Aα Aβ A ( Aα) theory with a simple substitution of the parameters of
µν 2 µν∇α ∇β − (µ∇ν) ∇α the theory
h1g (cid:0)Aα Aβ (cid:1), (7) Following the notation of Giannios [38], we write the
µν α β
− 2 ∇ ∇ physicalline element in isotropic coordinates in terms of
i two constants, r and a;
c
ΘK4 :=K A˙ A˙ +A˙ A Aα A˙αA A 1 r /r a
µν 4 µ ν α (µ∇ν) −∇α µ ν ds˜2 = − c dt2
h1g A˙ A˙α , (cid:16) (cid:17)(8) −(cid:18)1+rc/r(cid:19)
µν α r 2+a r 2−a
− 2 + 1+ c 1 c dr2+r2dΩ2 , (12)
i r − r
(cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1)
Θλ := λA A . (9) where dΩ2 := dθ2 + sin2θdφ2. The two constants in
µν − µ ν
the physical metric are related to the various coupling
Here, F := A , S := A and A˙ := parameters of the theory, including the “scalar mass”,
µν [ν µ] µν (ν µ) µ
∇ ∇
Aα αAµ. mϕ, aswell asthe characteristicgravitationalradius, rg,
∇
Variationofthetotalactionwithrespecttothe vector according to
field gives the vector field equation
2
r k Gm
K∇αFµα+K+∇αSαµ+K2∇µ(∇αA8απ)µ+λAµ rc = 4gs1+ π (cid:18) rgϕ(cid:19) − K2, (13)
+K A˙µAα A˙α µA + Aα ϕgµβ ϕ r kGm
4 ∇α − ∇ α k ∇α ∇β a= g + ϕ, (14)
=8πG 1h e−(cid:16)4ϕ gµα(cid:17)T˜ Aβ. i (10) 2rc 4πrc
αβ
−
where := K +K K . Note that all quantities in
Contra(cid:0)ctionofth(cid:1)isequationwiththevectorfieldisolates equatioKn(14)aregre+a−terth4anorequalto zero,implying
the Lagrange multiplier. This subsequent equation can a 0. Moreover, there are only two situations where a
then be used in the modified Einstein equation, in par- ca≥n be identically zero. Firstly, if both the gravitational
ticular in the term expressed in equation (9), such that radius and the scalar field coupling constant, r and k,
g
the system is fully determined. both vanish. Notonly does this reduce the theory to the
The final field equation is that of the scalar field Einstein-Æthertheory,butitalsoimpliesthatthemetric
becomes Minkowski. The second possibility is that r
µ gαβ AαAβ ϕ c
β α diverges,whichisunphysical. Itisthereforeonlypossible
∇ − ∇
(cid:2)=k(cid:0)G gαβ + 1(cid:1)+e−4ϕ(cid:3) AαAβ T˜αβ. (11) that a is strictly greater than zero.
Equation (12) describes more than just a black hole
Itisofinterestfo(cid:2)rthecu(cid:0)rrentwor(cid:1)khowt(cid:3)heTeVeSfield spacetime. Giannios [38] showed that r is a black hole
c
equationslimittothe GRequations. This isachievedby event horizon if and only if a = 2. This is based on
continuously limiting k and all of the K’s to zero, as the requirement that the surface at r = r must have a
c
well as taking ℓ . This was shown in detail for the finite surface area, and also the singularity must be re-
→ ∞
originalTeVeStheorybyBekenstein[21],his sectionsIII movable. EvaluatingthephysicalRicciscalarshowsthat
C and D. it divergesfor values of a<2 and also2<a<4. Mean-
while, as the surface area is proportional to the g˜ (r )
rr c
component of the metric, one finds that this diverges for
III. SPHERICALLY SYMMETRIC, STATIC, a>2. Onethereforehasablackholesolutiononlywhen
VACUUM SOLUTIONS a = 2. When 0 < a < 2 and 2 < a < 4 one has a
naked singularity at r = r , and a 4, r represents a
c c
≥
In order to discuss perturbations of black holes, it is removable singularity with a divergent surface area.
worth spending some time discussing the current sta- As mentioned, the only solution given by the metric
tus of spherically symmetric, static, vacuum solutions (12) thatrepresentsa black hole is thatwith a=2. One
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immediately notes that this is exactly equivalent to the tion VB that perturbations in TeVeS have a different
Schwarzschildmetric in isotropic coordinates. spectrum.
A. Equivalence with Brans-Dicke Theory B. Vector and Scalar Field Parameters
Itisfurtherinterestingtonotethatthemetricofequa- The above metric (12) is that found by Giannios [38].
tion(12)isexactlyequivalenttotheBrans‘typeI’metric He displayed trepidation towards this solution because
[55]inBrans-Dickescalartensortheory[56](withtheas- the scalar field becomes negative close to the horizon,
sociation a = 2Q and χ = 0 from the notation of Scheel which therefore allows for superluminal propagation of
etal.[57])[58]. NotethatintheBransIsolution,setting scalarwaves[21]. SagiandBekenstein[39]overcamethis
χ =0 necessarily implies that the parameter Q is unity, issue by showing that Giannios [38] overlookeda branch
and hence the Schwarzschild solution is recovered. This of solutions where the scalar field was everywhere posi-
is because the Brans I solution has an extra algebraic tive,althoughtheydidthisonlyforthecasewherea=2.
equation which links Q and χ. Using equations(13)and(14)we canre-derivethe extra
A popular representation of the Brans type I metric solution of Sagi and Bekenstein [39] with the generaliza-
can be found by taking the coordinate transformation tion that a remains arbitrary. In this case,one canshow
given by thatthescalarfieldthroughoutthespacetimeisgivenby
R(r)=r(1+rc/r)2, (15) 1 rc/r
ϕ(r)=ϕ +δ ln −
c ±
1+r /r
from the metric (12) and defining M :=2r , one gets (cid:18) c (cid:19)
c
2M
ds˜2 = 1 2M a/2dt2+ 1 2M −a/2dR2 =ϕc+δ±ln r1− R !, (17)
− − R − R
(cid:18) (cid:19) (cid:18) (cid:19)
2M 1−a/2 where ϕc is the cosmological value of the scalar field,
+R2 1 dΩ2. (16) which can be determined through cosmological observa-
− R
(cid:18) (cid:19) tions, and δ± can be found to be
In particular, this is the line element expressed as equa-
a(2 )k/2 2k[(k a2π)(2 )+8π]
tion (7) of Campanelli and Lousto [59] with the associa- δ := −K ± − −K .
±
wtioilnldmea=lw−itnh=thea/c2as−eo1f.aFo=r2a,mfoarjworhiitcyhotfhtehleinaeretliecmleewnet (p2−K)k+8π (18)
(16) becomes exactly the Schwarzschild line element.
The association with the Brans-Dicke theory is inter- Note here that when a = 2, equation (18) reduces to
esting given the vast complexity of the field equations in exactly equation (67) of Ref. [39] (where K as
K ≡
TeVeS as compared with Brans-Dicke theory. However, theywereworkingintheoriginalTeVeSwhichhasK+ =
the associationisdue tothe simplifying assumptionthat K2 =K4 =0).
the vector field is purely timelike. It was shown for the Given that 0 < < 2 and k > 0, one can trivially
K
original version of TeVeS that black hole solutions with show that δ+ > 0. Moreover, after some extra work a
a non-zeroradialcomponent for the vector field produce conditions is derived for δ− that is
significantly more complicated geometries than the case
8
wheretheradialcomponentvanishes[38]. Thiscomplex- a2 < = δ <0 (19)
−
ity will only increase when one analyses the black hole 2 ⇒
−K
solutions with non-zero radial vector fields in the fully
which further implies that δ < 0 for all a 2. A
−
general theory. ≤
majority of the results of this paper are for the cases
It is clear that, while the background spacetime in
where a = 2, which is the only black hole solution of
TeVeS and Brans-Dicke theories are equivalent, pertur-
the field equations. In this case it is clear that δ is
−
bations of these spacetimes will differ as the scalar field
always negative, a fact that will be important for the
equation of TeVeS is not a Klein-Gordon equation but
perturbation analysis in section V.
includes contributions from the background vector field.
It is interesting to note that there exists a specialcase
Infact,whilstscalarfieldperturbationsofSchwarzschild-
of the above functions whereby δ vanishes identically,
Brans-Dicke black holes [equation (16) with a = 2] have −
implying that ϕ(r)=ϕ throughout the spacetime. It is
the same QNM spectrum as that of scalar perturbations c
of Schwarzshildblack holes in GR2 we shall show in sec-
the stability of such perturbations, however the wave equation
theyderiveisequivalenttotheRegge-Wheelerwaveequationfor
2 Thiswas not explicitlyshowninKwonet al.[60] whoanalyzed scalarperturbationsofSchwarzschildblackholesinGR.
6
apparentthat this is only the case where a2 =8/(2 ) can be expressed component-wise as
−K
(or when = 0 which is not of interest for the present
work). WKhilst this case exhibits a constant scalar field δg˜tt =2δϕ˚g˜tt and δg˜ii = 2δϕ˚g˜ii. (23)
−
throughout the spacetime, there still exists an irremov-
able naked singularity at r = r . Although the stability From the above form of the perturbed physical met-
c
of such objects is of great interest, we shall see below ric it is obvious that a perturbation of the scalar field
that the wave equation governing the perturbations for only affects the diagonal components of the line ele-
this case is no simpler than the generic case when there ment. Thesearetheso-calledbreathingmodes ofgravita-
is a naked singularity at the horizon. As such, we leave tionalwaveswhichare not presentin GR. The detection
this for further exploration in a latter paper. of these modes is possible using current interferometric
The generic behaviour of the scalar field throughout techniques, although requires multiple suitably oriented
the spacetime is of great importance to the present dis- detectors to be operating simultaneously (see Ref. [8]
cussion due to the presence of superluminal propaga- and references therein), spherical resonant-mass detec-
tion of scalar waves. Indeed, Bekenstein [21] showed tors (for example [9–11, 61, 62]), or through the differ-
that, in the eikonal approximation, scalar perturbations ence inexpected energiesbetweenthe standardplus and
in the Newtonian limit of the theory travel with veloc- crosspolarizationsinstandardinterferometricdetection.
ity v = exp( 2ϕ)/√2. This implies that scalar per- The detection of such modes would instantly falsify GR,
turbations tra−vel superluminally if and only if ϕ < 0. howevernotdetectingthesemodeswillsimplyacttocon-
Equation (17) shows that the scalar field necessarily di- strainboth sources andparameter spacesin variousthe-
verges as r r . When δ <0, the scalar field diverges ories.
c ±
to positive →infinity, and indeed the scalar field is every- Thescalarfieldequation(11)issimpletoperturb,and
where positive (given ϕ > 0). However, when δ > 0, turns out to be
c ±
the scalar field diverges to negative infinity as one ap-
gαβ AαAβ ∂ δϕ =0. (24)
proachesthe critical radius, rc, implying for some values ∇β − α
of the radial coordinate the scalar field is negative.
Forthe remain(cid:2)d(cid:0)erofthe artic(cid:1)lewe(cid:3)workonly withthe
Finally, for the remaining sections we will require the
black hole case in which the metric parameter a=2. In
form of the vector field in the prescribed background
theAppendixweshowthemoregeneralformofthewave
spacetime, which can be shown to be
equationwith arbitrarya, howeverwe leavethe study of
−a/2 the stability of such spacetimes to future work.
1 r /r
At =eϕ − c
1+r /r
(cid:18) c (cid:19)
2M −a/4 A. The wave equation
=eϕ 1 . (20)
− R
(cid:18) (cid:19)
Performing the standard seperation of variables in
Wenotethatthevariouscouplingparametersofthethe- Schwarzschildcoordinates
oryaremainlyexpressedhereintheeϕterm[seeequation
(17)]. Φ(t,R) ∞ ℓ
δϕ(t,R,θ,φ)= Y , (25)
ℓm
R
ℓ=0m=−ℓ
X X
IV. PERTURBATION EQUATIONS
where the Y are the standard spherical harmonics,
ℓm
leads to the wave equation
We express the perturbed scalar field as
∂2Φ ∂2Φ 2M ℓ(ℓ+1) 2M
δϕ 2e4ϕ˚ + 1 + Φ=0,
ϕ=ϕ˚+δϕ, where <<1, (21) − ∂t2 ∂R2 − − R R2 R3
ϕ˚ ⋆ (cid:18) (cid:19)(cid:20) (cid:21)
(26)
where ϕ˚ and δϕ are respectively the background and
perturbed scalar fields. Since the scalar field pertur- wherethe standardtortoisecoordinate, R⋆(R), has been
bations decouple from perturbations of the vector and defined according to
tensor fields [48], we are free to let perturbations of the
dR 2M
vector and tensor fields vanish, i.e. δAµ = δgµν = 0, =1 . (27)
throughoutthearticle,andweleavetheanalysisofthese dR⋆ − R
perturbations to a future article.
Animportantaspectofthis waveequation(26)isthatit
The perturbation of equation (1) implies
is almost identicalto the waveequation governingscalar
δg˜ = 2δϕ e−2ϕ˚(g +A A )+e2ϕ˚A A . (22) perturbations of a Schwarzschild black hole in GR, the
µν µν µ ν µ ν
− difference being the factor of 2exp(4ϕ˚) preceding the
Giventhisequat(cid:2)ion,togetherwiththefactthatt(cid:3)hephys- second-ordertime derivative. In the weak field limit (i.e.
ical metric of the backgroundspacetime is diagonal,this takinglargeR)equation(26)reducestoawaveequation
7
wherethevelocityoftheperturbationsisexp( 2ϕ˚)/√2, parameters tend to zero. However, this expectation
−
inagreementwiththevelocityofscalarperturbationsde- breaks down as the perturbation equation being consid-
rived by Bekenstein [21] and Sagi [48]. eredherein does not exist in GR because of the lack of a
Consider now the limit of equation (26) as the back- vector field, and hence the derivation of the wave equa-
ground scalar field goes to zero. This limit is not well tion is not applicable.
definedinthe sensethatthebackgroundscalarfieldnec-
essarily diverges as R 2M for δ =0. However, if we
±
→ 6
take the limit as , k and ϕ tend towards zero, we find
c
the backgroundscKalarfield tends to zero for allvalues of B. Wave equation in Isotropic Coordinates
R = 2M. In this limit we see that the coefficient of the
6
∂2/∂t2 term tends to two, which remains distinct from Equation (26) is not appropriate for studying details
the generalrelativisticlimit. Considernowarescalingof of QNMs due to the non-trivial coefficient preceding the
the temporal coordinate t = √2t′, which acts to reduce time derivative. Interestingly, performing a coordinate
the wave equation to exactly the Regge-Wheeler equa- transformationfromtheSchwarzschildcoordinatestothe
tionforscalarperturbationsofSchwarzschildblackholes originalisotropic coordinates defined by the transforma-
in GR. If we now followstandardmethods andassume a tion (15), together with the substitution
harmonic time dependence for the perturbations
2
Φ(t,R) eiωt =eiωst′, (28) Ψ(t,r):=eϕ˚ 1+ M Φ(t,r), (30)
∼ 2r
(cid:18) (cid:19)
where ω and ω are the QNMs for our TeVeS black hole
s
and the GR Schwarzschild black hole respectively, then leads to the standard form of the wave equation
we find the following simple relation
∂2 ∂2
ω
s +V Ψ=0. (31)
lim ω = (29) ∂t2 − ∂r2
K,k,ϕc→0 √2 (cid:18) ⋆ (cid:19)
Here, we have introduced a new coordinate defined by
Thatis,asthetheory’svariouscouplingparameterstend
to zero,the QNMfrequencies anddamping times arere- dr M 3−2δ± M −1+2δ±
lated to the Schwarzschild QNM frequencies and damp- ⋆ =√2e2ϕc 1+ 1 , (32)
ing times through a factor of √2. We note again that dr (cid:18) 2r(cid:19) (cid:18) − 2r(cid:19)
the GR Schwarzschild black hole spectrum is equivalent
to the QNM spectrum of Brans-Dicke black holes [60], which is a “tortoise-like”coordinate, in the sense that it
and hence the TeVeS QNMs also deviate by the same maps r (M/2, ) to r⋆ ( , ), providing 1+
∈ ∞ ∈ −∞ ∞ −
relation to the Brans-Dicke QNMs. We have confirmed 2δ± < 0. Note that in isotropic coordinates the horizon
the result of equation (29) numerically in section VB. occurs at r =M/2.
Na¨ıvely, one may expect QNMs in TeVeS to tend ex- The potential, V(r ), is now givenby the complicated
⋆
actly to the QNMs of GR in the limit as the theory’s expression
e−4ϕc M −2(3−2δ±) M 2(1−2δ±)
V = 1+ 1 ℓ(ℓ+1)
2r2 (cid:18) 2r(cid:19) (cid:18) − 2r(cid:19) (
M M M M −2 M −2
+ 2 (1 δ ) 2(1 δ ) 1+ 1 . (33)
± ±
2r (cid:20) − − r (cid:21)(cid:20) − − r (cid:21)(cid:18) 2r(cid:19) (cid:18) − 2r(cid:19) )
Throughout section V we will look for QNMs of the comes
above system of equations. For these purposes it is con-
venient to assume a harmonic time dependence d2Ψ˜
+ ω2 V Ψ˜ =0. (35)
dr2 −
⋆
(cid:0) (cid:1)
Ψ(t,r)=Ψ˜(r)eiωt, (34) In the above definition, ω represents the frequency of
Re
theQNMwhileω isitsdampingtime. Anymodewith
Im
ω < 0 will grow exponentially in time, and hence rep-
Im
where ω = ω +iω , which implies equation (31) be- resents an unstable mode.
Re Im
8
C. Behaviour of the potential
0.03
Determining stability and finding the QNM spectrum
is now a one-dimensional scattering problem that is de-
pendentsolelyontheformofthepotential,V,definedin 0.02
V
equation (33). It is therefore worth spending some time
establishing various properties of this function.
We firstlynotethatthepotentialvanishesatspacelike 0.01
infinity, independent of the value of ϕ . The behaviour
c
approaching the horizon, r = M/2, is more difficult as
this is dependent on the value of δ . Bearing in mind
± 0
that, for the black hole case, δ <0 and δ >0, we find 1 2 3 4 5 6 7 8 9 10 11
− + r / 2M
0 for δ FIG. 1: Potential, V, for the scalar perturbation wave equa-
r→limM2+V(r)=(cid:26) −∞ for δ−+ . (36) tainodnϕfocr=a0b.0la0c3k. Nhooltee(tih.ea.t rais=an2)iswotirtohpKic r=ad0ia.1l,cokor=din0a.0t1e
implyingthehorizon of thespacetime is at r=M/2. This is
As discussed, the case of δ necessarily has regions of
+ thecaseofδ−,althoughatthisrangeplotsofδ+ looksimilar
the spacetime that allows for superluminal propagation (theeffect ofthesign choicein δ± becomes morerelevantfor
of scalar field perturbations. While it has been heavily higherℓ’s). Here,thethickblack,dashedblueanddottedred
debatedwhetherthisisphysicallyallowedtooccurinna- lines represent ℓ=0, 1 and 2 respectively.
ture (see for example Ellis et al. [63], Bruneton [64] and
references therein), it is clear that this induces a pathol-
ogyonthe waveequationgoverningscalarperturbations
2×10–7
in the form of a divergent potential function. When we
are discussing such superluminal perturbations we must 1×10–7 δ−
be careful with our definition of “horizon”. Indeed in
0
this case the surface at r = M/2 is a photon horizon
associatedwiththenullconesofthephysicalmetric. Su- –1×10–7
perluminal propagation of scalar field perturbations im-
plies that a perturbation at some r <M/2 could escape V –2×10–7
through the horizon at r = M/2 out to infinity. We 0
believe that this induces the pathological behaviour of
δ
the potential for the δ case, howeverwe reserve further –2×10–7 +
+
exploration of the effect of this to a future article.
–4×10–7
It is also of interest to determine when the potential
functionisnegativeassuchregionsofapotentialcanrep-
–6×10–7
resent bound states which imply growing modes. Con-
1 1.002 1.004 1.006 1.008
sider, for the moment, the ℓ = 0 case. From equation
r / 2M
(33), one can show that for δ the potential is negative
−
in the region
FIG. 2: A zoomed in view of figure 1 for δ− (top panel)
M M and δ+ (bottom panel). As stated in the text, as r → M/2
r< , (37) the potential goes to zero for all ℓ’s for the δ− case, however
2 ≤ 2(1−δ−) divergestonegativeinfinityforthecaseofδ+. Thisdivergent
behaviourisassociated withthesuperluminalpropagation of
and for δ the potential is negative for
+ thescalar perturbations.
M M(1 δ )
+
<r < − . (38)
2 2
V. RESULTS
It is trivial to show that for all values of ℓ there exists
a negative portion of the potential, however the largest
A. Stability analysis
negative region is realized for the ℓ = 0 case described
above. A generic plot of the potential is shown in figure
1, for the δ case with ℓ = 0, 1 and 2. We note that at The presence of a single unstable mode can do severe
−
thisscalethe differenceinbehaviourbetweenthe δ and damagetoatheoryasaviablealternativetheoryofgrav-
−
δ casescannotbeseen. However,whenonezoomsinto ity. The stability analysisofsphericallysymmetric black
+
the regionclose to the horizon,figure 2, the difference in holesisgenerallyrenderedrathersimpleduetotheprop-
the respective functions is apparent. erty that any growing, unstable mode, must necessarily
9
be purely imaginary (KonoplyaandZhidenko [for exam-
ple see 4]). That is, if an unstable mode exists (i.e. such
that ω <0), it must have zero real part, and therefore
Im 10–2
will not be oscillatory in nature.
|Φ|
We have performed time evolutions of equation (26)
10–4
for variousvalues of the sphericalharmonicℓ. The ℓ=0
modehasthelargestnegativeregioninthepotential,and
hence is the most likely to be unstable. In figure 3 we 10–6
plotthetimeevolutionofatypicalℓ=0mode(thefigure 200 400 600 800 1000
uses the example where = k = 0.1 and ϕ = 0.003). t / M
c
K
Consistently with the case with scalar perturbations of
theSchwarzschildblackholeinGR,theℓ=0modehasa FIG. 3: Typical temporal evolution of scalar field for ℓ = 0
large imaginary part (see section VB), implying the sig- with = k = 0.1 and ϕc = 0.003. The presence of oscilla-
K
nal is significantly damped after only a few oscillations. tionsin theevolution implies non-zeroωRe which,duetothe
spherically symmetric nature of the background spacetime,
Whilst this makes it difficult to extract robust measure-
implies the mode is stable. Due to the strong damping of
ments of the frequencies and damping times for these
the ℓ = 0 mode, only a few oscillations can be seen in the
modes, one canclearly see from figure 3 that oscillations
evolution.
are present, implying these modes are stable. As dis-
cussedabove,largervaluesforthe vectorandscalarfield
coupling parameters act to increase the size of the neg-
ative region in the potential, which implies these cases
are more likely to be unstable. We have extended our 10–2
numerical analysis for extreme values of the parameter
space,including variationsofϕc, andfind no evidenceof 10–4
any unstable modes.
10–6
In figure 4 we plot the temporal evolution for the ℓ =
1, 2 and 3 modes for = k = 0.1 and ϕ = 0.003.
K c 10–2
As expected there is no evidence of instabilities present
|Φ|
for these, or any other values of the parameter space. 10–4
We have further calculated evolutions for higher ℓ’s and 10–6
againfindnoevidenceofanyinstabilitiesforthis typeof
10–8
perturbation.
Overlaidonfigure4aredampedsinusoidaloscillations 10–2
where the frequency and damping times are those found 10–4
using the WKB method in the following section. The 10–6
WKB method is sensitive to the peak of the potential, 10–8
whereas the time evolution accounts for the entire po- 10–10
tential. One can see from these figures that the WKB 800 1600 2400
t / M
method gives extremely accurate results, implying the
negative regionofthe potential contributes to the evolu-
FIG. 4: Temporal evolution of scalar field for ℓ = 1 (top
tion of the perturbations on a level commensurate with
panel), ℓ=2 (middle) and ℓ=3 (bottom) with =k =0.1
the overallaccuracy of the various schemes. K
and ϕc = 0.003. The numerical time evolution is shown in
We have further verified the stability of the solutions
blackwhichincludestheregionofquasinormalringingaswell
using the time independent form of the wave equation as the late-time tail. Overlaid in red is the damped sinu-
(35). The method we used is based on the fact that, if soidal oscillations of the fundamental mode calculated using
unstablemodesexist(ω <0),theboundaryconditions theWKB method (see section VB).
Im
fortheradialperturbationfunctionΨ˜(r)becomezeroon
both boundaries and the corresponding boundary value
problem defined by equation (35) is self-adjoint [65, 66].
This implies that, for the unstable modes, ω2 is real and istforallofthe studiedvaluesofthe parametersandthe
negative and ω2 should be greater than the minimum of black holes are therefore stable against the considered
perturbations.
the potential, V , for all of the bound states. Thus,
min
in order to find unstable modes, we integrate equation
(35) with initial condition Ψ˜ = 0 for test values
of ω2, starting from ω2 = V |r⋆→to−∞ω2 = 0. An eigenfre- B. Frequency domain calculations
min
quencyisthenfoundwhentherightboundarycondition,
Ψ˜ =0,isfulfilled[67]. Whenapplyingthismethod Giventhatnounstablemodesexist,thenextstepisto
|r∗→∞
to our problem, it turns out that no unstable modes ex- calculatetheQNMmodesgovernedbyequation(35). We
10

0.340 0.34
0.336
0.33
0.332
Re =0.01 Re
K
=0.05
0.328 K
=0.10
K =0.20 0.32
K
0.068
=0.01, k=0.01
0.068 K
=0.10, k=0.10
K
0.067 K =0.20, k=0.20
0.066
Im Im
0.066
0.064
0.065
0.0 0.1 0.2 0.3 0.4 0.5 0.000 0.005 0.010 0.015 0.020
k
c
FIG. 5: The real (top) and the imaginary (bottom) part of
the fundamental ℓ = 2 QNM frequencies as a function of FIG. 6: The real (top) and the imaginary (bottom) part of
thescalar fieldcouplingparameter k for severalvaluesof the the fundamental ℓ=2 QNM frequencies as a function of the
vector field coupling parameter := K +K+ K4. The cosmological value of the scalar field ϕc for several values of
cosmological value of the scalar fiKeld is ϕc = 0.0−03 and ωRe the parameters K and k. Here, ωRe and ωIm are shown in
and ωIm are shown in units of M. unitsof M
ℓ ω ωs/√2
have used two methods – a direct integration (shooting) 0 0.0797+i0.0749 0.0781+i0.0742
method and also the WKB method. 1 0.2071+i0.0691 0.2071+i0.0691
The first method we used is the shooting method in- 2 0.3422+i0.0684 0.3420+i0.0684
troduced by Chandrasekhar and Detweiler [68] (see also 3 0.4779+i0.0682 0.4776+i0.0682
[69] and [66]). A strong aspect of this method is that it
TABLEI:Fundamentalquasinormalmodesforthefirstthree
takesintoaccountthepresenceofthenegativeminimum
spherical harmonics in the limit as the theory’s various cou-
of the potential, however it is also prone to suffer from
plingparameterstendtozero,ωandforaSchwarzschildblack
numericalinstabilities due to this minimum and alsothe
hole in GR, ωs. The Schwarzschild QNMs in GR are calcu-
fact that the first derivative of the potential (33) is di-
lated using the continued-fraction method, while ω is calcu-
vergent at the left boundary, r = M/2. For this reason lated using the shooting method which has an error in this
it was important to use an alternative method to con- limit of a few percent for the ℓ = 0 mode and significantly
firm our results. As can be seen in figure 1 and the top smaller for the higher modes. These numerical results con-
panel of figure 2, the negative minimum of the potential firm therelation in equation (29).
is extremely small compared to the positive maximum
implying itis reasonablealsotoapply the WKB method
[69–72].
rors for the ℓ = 0 and ℓ = 1 modes are comparable to
In figures 5 and 6 we show the fundamental (n = 0) thechangeinfrequenciesassociatedwiththevariationof
ℓ=2QNMfrequenciesanddampingtimesobtainedfrom theparameterspace. Nevertheless,ourcalculationshave
the WKB method for various values of the theory’s cou- shown that the qualitative behaviour of the ℓ = 0 and
pling parameters3. As can be seen for relevant values of ℓ=1 modes are similar to the shown ℓ=2 case.
theparametersthefrequenciesonlyvarybyafewpercent
Finally, from our calculations we have identified the
throughout the range of the physically relevant param-
limiting behaviour as the theory’s various parametersgo
eter space. We have displayed this in figures 5 and 6
tozerowhichisshownintableI. Thefinalcolumnofta-
only for the ℓ = 2 mode however, as the associated er-
ble I showsthe correspondingQNMs foraSchwarzschild
black hole in GR. One can see that the errors produced
are extremely small, even for the difficult to calculate
ℓ = 0 case. These numerical results confirm the discus-
3 For physically relevant values of the theory’s parameters, the
sionofsectionIVAandparticularlytherelationbetween
resultsobtained fromtheWKB andthe shooting method differ
GR and the limiting behaviour of TeVeS elucidated in
byupto1%andthedifferencebecomessmallerforsmallervalues
ofthecouplingparameters. equation (29).