Gravitational spectrum of black holes in the Einstein-Aether theory

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by  R. A. Konoplya

Gravitational spectrum of black holes in the Einstein-Aether theory.
R.A. Konoplya∗ and A. Zhidenko†
Instituto de F´ısica, Universidade de S˜ao Paulo
C.P. 66318, 05315-970, S˜ao Paulo-SP, Brazil
Evolutionofgravitationalperturbations,bothintimeandfrequencydomains,isconsideredfora
spherically symmetric black hole in the non-reduced Einstein-Aether theory. It is shown that real
oscillation frequency and damping rate are larger for the Einstein-Aether black hole than for the
Schwarzschild black hole. This may provide an opportunity to observe aether in the forthcoming
experimentswith new generation of gravitational antennas.
PACSnumbers: 04.30.Nk,04.50.+h
7 One of the most intriguing issues of modern physics time domain.
0 consists in attempts to go beyond local Lorentz symme- We shallstartfromthe lagrangianofthe fullEinstein-
0 try [1]. In theory of gravity, breaking of local Lorentz Aethertheoryformsthemostgeneraldiffeomorphismin-
2
invariance leads to a general relativity coupled to a dy- variantactionofthespace-timemetricg andtheaether
ab
n namical time-like vector field ua, called “aether”. More field ua involving no more than two derivatives given by
a exactly,uabreakslocalboostinvariance,whilerotational
J L=−R−Kab ∇ um∇ un−λ(g uaub−1), (1)
symmetry in a preferredframe is preserved[2]. Thereby, mn a b ab

5 aether is a kind of locally preferred state of rest at each here R is the Ricci scalar, λ is a Lagrange multiplier
1
point of space-time due-to some unknown physics. Re- which provides the unit time-like constraint,

centlyobservableconsequencesofEinstein-Aethertheory
2
v attracted considerable interest [3]. Gravitational conse- Kabmn =c1gabgmn+c2δmaδnb +c3δnaδmb +c4uaubgmn,
6 quences of Local Lorentz symmetry violation must show
where the c are dimensionless constants.
2 themselves in radiative processes around black holes. It i
Spherically symmetry allows to fix c = 0. In this
2 is known that gravitational radiation damping of binary 4
1 pulsarsorbitsreproducestheweakfieldgeneralrelativity letter, following [6], we shall consider the so-called non-
1 at lowest post- Newtonian order [4]. Yet, the significant reducedEinstein-Aether theory,forwhichc3 =0,andwe
6 can use the field redefinition that fixes the coefficient c
differencebetweenEinsteinandEinstein-Aethertheories 2
0 [6]:
/ should be seen in the regime of strong field, for instance
h in observing of the characteristic quasi-normalspectrum c3
-t of black holes. Thus, existence of aether could be tested c2 =−2−4c1+3c2,
p in the forthcoming experiments with new generation of 1 1
e
h gravitationalantennas. Motivated by the above reasons, so that c1 is the free parameter.
: inapreviousletter[5]wedevelopedamethodforfinding The metric for a spherically symmetric static black
v of the quasinormal modes for the perturbations of met- holes in Eddington-Finkelstein coordinates can be writ-
i
X rics which are not known analytically, but instead are ten in the form [6]:
given only numerically in some region near black holes.
r ds2 =N(r)dv2−2B(r)dvdr−r2dΩ2, (2)
a Thatisthe caseofthe Einstein-Aetherblackholesfound
in[6]. In[5],therewerefoundthequasinormalmodesfor where the functions N(r) and B(r) are givenby numeri-
testscalarandelectromagneticfieldsinthevicinityofthe calintegrationneartheblackholeeventhorizon[6]. One
Einstein-Aetherblackholes. Itwasshownthatthescalar can re-write this metric in a Schwarzschildlike form:
and electromagnetic quasinormal modes in the Einstein-
B2(r)
Aether theory, have larger real oscillation frequency and ds2 =−N(r)dt2+ dr2+r2dΩ2. (3)
dampingratethanthoseoftheSchwarzschildblackholes N(r)
in the Einstein theory. As quasinormal spectrum does
Since the background value of aether coupling, de-
not depend on the spin of the field in eikonal regime,
termined by constants c is small in comparison with
i
qualitatively the same QN behavior was suggested in [5]
the background characteristics of large black hole, de-
forthe gravitationalperturbationsasforscalarandelec-
termined by the mass of the black hole M. There-
tromagnetic ones. In the present work we show that it
fore, the background black hole metric is, in fact, the
is indeed true, and analyze gravitational perturbations
Schwarzschild metric slightly corrected by the aether.
ofthe Einstein-Aetherblack holesbothinfrequency and
Thus, one can neglect small perturbations of aether,
keeping only linear perturbations of Ricci tensor. Then
theperturbationequationswithunperturbedaetherhave
the form:
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] δR =0. (4)
αβ
2
The general form of the perturbed metric, according to Einstein-Aether theory, we are in position to apply the
Chandrasekhar designations, is method developed in our previous paper [5]. Here we
shall give only a brief summary of the whole procedure
ds2 =e2νdt2−e2ψ(dφ2−ωdt−q2dr−q3dθ)2− of [5].
We approximate the numerical data for the metric by
e−2µ2dr2−e−2µ3dθ2. (5) a fit of the form
NN NB
Here a(N)ri a(B)ri
i i
e2ν =N(r), e2µ2 =N(r)/B2(r), N(r)= Xi=0 , B(r)= Xi=0 .
NN NB
1+ b(N)ri 1+ b(B)ri
i i
e2µ3 =r2, e2ψ =r2sin2θ. (6) Xi=1 Xi=1
which are substituted into equations (11) and (12). The
Let us introduce new variables
numbers N and N determine the number of terms in
N B
Q =q −q , Q =q −ω , i,k =2,3. (7) the polynomials and are chosen in order to provide best
ik i,k k,i i0 i,0 ,i
convergence of the WKB series. Coefficients a(N), b(N),
i i
Here we used 0,1,2,3 for t, φ, r and θ coordinates re-
a(B), b(B) are determined by the fitting procedure. The
spectively. Hereweshallconsidertheaxialtypeofgravi- i i
WKB expansion has the form
tationalperturbations. The equations(11),(12)onp.143
of [7] can be reduced to a single equation
6
ıQ 1
0 − Λ =n+ , (14)
r4 N ∂Q r2∂2Q ∂ 1 ∂Q 2Q′′ i 2
− +sin3θ =0 0 Xi=2
B (cid:18)Br2 ∂r (cid:19) B ∂t2 ∂θ (cid:18)sin3θ ∂θ (cid:19) p
,r
wherethecorrectiontermsofthei-thWKBorderΛ can
(8) i
be found in [10, 11] and [12], Q=V −ω2 and Qi means
where we used 0
the i-th derivative of Q at its maximum.
Q=e3ψ+ν−µ2−µ3Q . (9) Alternatively, we shall use the above mentioned fits
23
ofthe metric functions, andconsequentlyof the effective
The following representation of the function Q(t,r,θ) potential,inthetime-domainanalysis: usingtheintegra-
tion scheme described for instance in [13]. In detail, we
Q(t,r,θ)=rRℓ(t,r)Cℓ−+32/2(θ) (10) usedanumericalcharacteristicintegrationscheme,based
in the light-cone variables u = t−r and v = t+r . In
⋆ ⋆
leads to separation of angular variable θ.
the characteristic initial value problem, initial data are
Finally we have the wave-like equation for the radial specified on the two null surfaces u = u and v = v .
0 0
coordinate
The discretization scheme applied, is
d2Ψ B(r)
+(ω2−V(r))Ψ=0, dr = dr. (11) Ψ(N)=Ψ(W)+Ψ(E)−Ψ(S)
dr2 ∗ N(r)
∗ Ψ(W)+Ψ(E)
−∆2V(S) +O(∆4) , (15)
with the effective potential 8
(ℓ+2)(ℓ−1) 2N2(r) 1d(N(r)/B(r)) where we have used the definitions for the points: N =
V(r)=N(r) + − . (u+∆,v +∆), W = (u +∆,v), E = (u,v +∆) and
r2 B2(r)r2 r dr∗
S =(u,v).
(12)
The application of the above two methods shows ex-
Quasi-normalmodesofasymptoticallyAdSblackholes
cellent agreement: for instance the fundamental mode
have been studies recent years extensively, because of
0.7686−0.1887iintimedomainisveryclosetotheWKB
their interpretation in Conformal Field Theory [8] with
value0.769470−0.187783iforc =0.4,ascanbeseenin
some specific boundary conditions. In astrophysically 1
Fig. 2. Fromthe obtainednumericaldatainTableIand
relevant problem, one should require natural boundary
time domain pictures in Fig. 1-2, one can see that when
conditions for QN modes of purely in-going wavesat the
increasing c , both real oscillation frequency and damp-
event horizon and purely out-going waves at spatial in- 1
ing rate are increasing. Evenfor a smallaether c ∼0.1,
finity 1
the increase in Reω and Imω is of about half percent,
Ψ∼e±ir∗ω r →±∞. (13) and could, in principle, be detected by new generation
∗
of gravitational antennas. For larger c , the difference
1
Under these boundary conditions, the quasinormal between, Schwarzschild and Einstein-Aether QNMs can
modes were studied in a great number of papers [9], yet be very significant and reach six-seven percents. From
in those cases the background metric and the effective the Table I, it is evident that both real and imaginary
potential were known in analytical form. For the case of parts of ω grows when increasing the multipole number
3
Table I: Axialgravitational perturbations for thenon-reduced Einstein-Aethertheory: The fundamental mode.
c1 ℓ=2 ℓ=3 ℓ=4 ℓ=5
0.1 0.751958−0.179578i 1.206905−0.187387i 1.629395−0.190378i 2.038526−0.191824i
0.2 0.757144−0.180786i 1.215669−0.188850i 1.641492−0.191932i 2.053816−0.193422i
0.3 0.762871−0.184443i 1.225598−0.192601i 1.655131−0.195746i 2.071021−0.197268i
0.4 0.769470−0.187783i 1.236966−0.196172i 1.670792−0.199412i 2.090803−0.200981i
0.5 0.777059−0.192176i 1.250474−0.200794i 1.689421−0.204151i 2.114345−0.205777i
0.6 0.786784−0.198087i 1.267215−0.207061i 1.712545−0.210574i 2.143595−0.212276i
0.7 0.799391−0.206961i 1.289844−0.216264i 1.743878−0.220000i 2.183271−0.221813i
0.77 0.811083−0.216302i 1.311449−0.225807i 1.773893−0.229784i 2.221342−0.231720i
ℓ. Here we considered only axial gravitational perturba- perturbationsofthemetricofalargeastrophysicalblack
tions,whichareiso-spectralwithpolargravitationalper- hole. Therefore this difference between axial and polar
turbationsfor Schwarzschildblackholes. For blackholes QN spectra was neglected as well.
inEinstein-Aethertheorythisiso-spectralitywillbebro-
ken, and QNMs for polar perturbations should slightly Note that we used here the method based on the sup-
differfromaxial,whenconsideringthefullperturbations positionthatQNfrequenciesaredeterminedmainlynear
of Einstein-Aether equations. The same breaking of iso- the peak of the potential barrier, while behavior of the
spectrality happens, for instance, when perturbing dila- potential barrier far from black hole is not significant.
ton black holes or black holes in higher than four di- Even despite this idea was inspired by WKB approach,
mensional space-times [14]. In our approach, the per- it is not dependent on WKB technique, as was shown
turbations of aether were neglected in comparison with here by computations in time domain.
Figure 1: Evolution of axial gravitational perturbations in Figure 2: Evolution of axial gravitational perturbations in
timedomainc1 =0.4, ℓ=3,non-reducedtheory(redline)in time domain c1 = 0.1 (green line), c1 = 0.4 (red line), ℓ =
comparison with theSchwarzschild case (blueline). 2, non-reduced theory in comparison with the Schwarzschild
case (blue line).
ÈYÈ
ÈYÈ
1
0.1
0.01
0.001
0.0001
0.00001
1.´10-6
1.´10-7
t
0 10 20 30 40 50 60
t
0 20 40 60 80
Acknowledgments
This work was supported by Fundac¸a˜o de Amparo a`
Pesquisa do Estado de Sa˜o Paulo (FAPESP), Brazil.
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