a r X i v :0710.4564v 2 [a s t r o -p h ] 7 J a n 2008
Kerr Black Holes are Not Unique to General Relativity
Dimitrios Psaltis 1,2,Delphine Perrodin 1,Keith R.Dienes 1,and Irina Mocioiu 3
1
Department of Physics,University of Arizona,Tucson,AZ 857212Department of Astronomy,University of Arizona,Tucson,AZ 85721
3
Department of Physics,Pennsylvania State University,University Park,PA 16802
Considerable attention has recently focud on gravity theories obtained by extending general relativity with additional scalar,vector,or tensor degrees of freedom.In this paper,we show that the black-hole solutions of the theories are esntially indistinguishable from tho of general relativity.Thus,we conclude that a potential obrvational verification of the Kerr metric around an astrophysical black hole cannot,in and of itlf,be ud to distinguish between the theories.On the other hand,it remains
true that detection of deviations from the Kerr metric will signify the need for a major change in our understanding of gravitational physics.
Black holes are among the most extreme astrophysical objects predicted by general relativity.They are vacuum solutions of the Einstein field equations realized astro-physically at the end stages of the collap of massive stars.According to a variety of no-hair theorems,a gen-eral relativistic black hole is characterized only by three parameters identified with its gravitational mass,spin,and charge.Any additional “hair”on the black hole,as-sociated with the properties of the progenitor star or the collap itlf,are radiated away in the form of gravita-tional waves over a finite amount of time.
Black holes might look different if general relativity is only an effective theory of gravity,valid at the curvature scales probed by current terrestrial and astrophysical ex-periments.If the more fundamental gravity theory has additional degrees of freedom,they might appear as addi-tional “hair”to the black hole.This would be important for a number of reasons.First,additional degrees of free-dom appear naturally in all attempts to quantize gravity,either in a perturbative approach [1]or within the context of string theory [2].Detecting obrvational signatures of the additional degrees of freedom in black-hole space-times would rve as a confirmation of quantum gravity effects.Second,black-hole solutions not described by the Kerr-Newman metric may follow a t of th
ermodynamic relations different than tho calculated by Bekenstein [3]and Hawking [4]with important implications for string theory [5].Finally,the external spacetimes of astrophys-ical black hole will soon be mapped with gravitational-wave [6]and high-energy obrvations [7]and the means for arching for black holes with additional degrees of freedom will become readily available.
Introducing additional degrees of freedom to the Einstein-Hilbert action of the gravitational field does not necessarily alter the resulting field equations and hence the black-hole solutions.For example,the addition of a Gauss-Bonnet term to the action leaves the field equa-tion completely unchanged [1].Moreover,a large class of gravity theories in the Palatini formalism for which the action is a general function f (R )of the Ricci scalar curva-ture R ,lead to field equations that are indistinguishable from the general relativistic ones [8].In all the situ-
ations,no astrophysical obrvation of a classical phe-nomenon,such as test particle orbits or gravitational lensing,can distinguish between the theories.Never-theless,this leaves a large number of Lagrangian gravity theories that incorporate general relativity as a limiting ca but are described by more general field equations.The most widely studied such extension of general rel-ativity is the Brans-Dicke gravity,which incorporates a dynamical scalar field in addition to the metric ten-sor.Black hole solutions in this theory were studied by Thorne &Dykla [9].Following a conjecture
by Penro,the authors showed that the Kerr solution of general relativity is also an exact solution of the field equations in Brans-Dicke gravity and offered a number of argu-ments to support the claim that the collap of a star in this gravity theory will produce uniquely a Kerr black hole.Additional analytic [4,10,11]and numerical [12]arguments were offered by other authors providing fur-ther evidence for the uniqueness of the Kerr solution in Brans-Dicke gravity.
In this Letter,we show that black-hole solutions of the general relativistic field equations are indistinguishable from solutions of a wide variety of gravity theories that ari by adding dynamical vector and tensor degrees of freedom to the Einstein-Hilbert action.Although we do not prove that the general relativistic vacuum solution is the unique solution of the extended Lagrangian theories,we u our results to argue that an obrvational verifica-tion of the Kerr solution for an astrophysical object can-not be ud in distinguishing between general relativity and other Lagrangian theories such as tho considered here.Note that we are only considering four-dimensional theories that obey the equivalence principle,and hence we are not studying theories with prior geometry [13],that are Lorentz violating [14],or braneworld gravity theories [15].Although veral of the extensions lead to predictions of an unstable quantum vacuum and of ghosts,we are focusing here on their classical black-hole solutions.
In general relativity,the external spacetimes of black holes that are astrophysically ,with zero
2 charge,are completely specified by the relation
Rµν=
R
16πG d4x√
2
g kl f(R)=0,(3) where primes denote differentiation with
choo的用法
respect to
R and
we have ud the sign convention of Ref.[16].
A general relativistic black-hole ,one that satisfies equation(1)with R,µ=0,will also be a solution of thefield equation(3)if
1
2
R+
1
2
a n R n+...=0.(6)
There are three cas to consider:(i)If a0=0,then the Kerr solution,which corresponds to R=0,will al-ways be a solution of thefield equations of a general f(R) theory.Thus,in the abnce of a cosmological constant, we conclude that the Kerr solution of general relativity remains an exact solution to all f(R)theories as long as f(R)has a Taylor expansion of the form in Eq.(5). (ii)Moreover,independent of the value of a0,all of the constant-curvature solutions of General Relativity in vacuum–including the Kerr solution–remain exact so-lutions of the f(R)theory,if the Taylor ries for f(R) terminates after the quadratic ,if a n≥3=0). Indeed,this statement remains true independently of the value of a0,and thus holds for both vanishing and non-vanishing cosmological constants.
(iii)Finally,if a0=0and the Taylor expansion ex-tends beyond the quadratic term,then Kerr-like black-h
ole solution will always be possible.The only change is that the value of its constant curvature will be shifted rel-ative to the value predicted in General Relativity.Since terrestrial and solar-system tests require any extra non-linear terms in the gravity action to be perturbative,this shift in the curvature will also be correspondingly small. However,even in this ca,it is straightforward to show that the corrections to the curvature are actually sup-presd by additional powers of the cosmological constant relative to what might naively have been expected on the
3 basis of dimensional analysis.For example,given the ex-
pansion for f(R)in Eq.(5),we would have expected the
curvature term to have a leading correction term which
scales as R=−2a0[1+O(a0a2)+...].However,explicitly
solving Eq.(6),wefind that the true leading correction
is actually given by
R=−2a0 1+4a20a3+... .(7)
Thus the deviations of the vacuum curvature solutions of
f(R)gravity from tho of General Relativity are partic-
ularly suppresd.
f(R)Gravity in the Palatini Formalism.—In deriving
thefield equation(3),we extremized the action of the
gravitationalfield with respect only to variations in the
metric.In the so-called Palatini formalism,field equa-
tions of lower order can be derived from the same action
of the gravitationalfield,by extremizing it over both the
metric and the connection[22].A large class of f(R)
theories in the Palatini formalism are known to result in
the samefield equations as general relativity[8].
Applying this procedure for a gravitational action that
is a general function f(R)of the Ricci scalar curvature,
we obtain the well-known t of equations[22]
R kl f′(R)−
1
−gf′(R)gµν =0.(9)
In order to look for constant curvature solutions in
vacuum for this theory,wefirst take the trace of equa-
tion(8).The result is simply the algebraic equation(4),
which we can solve for the value of the constant curva-
ture(7)as before.For a solution with constant curva-
ture,the factor f′(R)in equation(9)is a constant,and
the solutions to this equation are simply the Christoffel
symbols of general relativity.As a result,any general
杭州考会计证relativistic solution of constant curvature,such as the
black-hole solutions with cosmological constant,will also
be solutions(with the same or slightly different value of
the cosmological constant)to thefield equations of an
f(R)gravity in the Palatini formalism.
General Quadratic Gravity.—We shall now consider a
gravitational action that incorporates all combinations
of the Ricci curvature,Ricci tensor,and Riemann tensor,a d
up to cond ,
S=1
−g −2Λ+R+αR2+βRστRστ
+γRαβγδRαβγδ .(10)
寒战的意思withα,β,andγthe parameters of the theory.Such terms appear naturally as radiative corrections to the Einstein-Hilbert action in perturbative approaches to quantum gravity[1]or in string theory[2].Note,however,that in general such theories are not free of the Ostrogradski instability[21].
Becau of the Gauss-Bonnet identity,the predictions of the theory described by the action(10)in calculat-ing classical properties of astrophysical black holes are identical to tho of the action[23]
S=
1
−g −2Λ+R+α′R2+β′RστRστ ,(11)
whereα′=α−γandβ′=β+4γ.
Thefield equation for this action in the metric formal-ism is
百知教育Rµν−1
2
R2gµν+2RRµν,(13) Lµν≡−2Rσµ;σν+2Rµν+1
2
gµνRστRστ+2RαµRαν.(14)
It is trivial to show that,for any black-hole solution satisfying equation(1),Kµν=Lµν=0and thefield equation of quadratic gravity reduces to that of general relativity.As a result,the Kerr solution is also a solution of the general quadratic theory considered here.
Vector-Tensor Gravity.—Wefinally consider a gravita-tional theory that incorporates a dynamical vect
orfield in addition to the metric tensor.A priori,such an ad-dition to the Einstein-Hilbert action appears to have the highest probability of requiring black-hole solutions that are not described by the Kerr metric.This is becau the vectorfield has the same spin as photons,the geodesics of which are ud to define the event horizon of a black hole.We restrict our attention to Lagrangian theories that are linear and at most of cond-order in the vector field.The most general action for such a theory is[17]
S=
1
−g(−2Λ+R+ωRKµKµ
+ηKµKνRµν−ǫFµνFµν+τKν;µKµ;ν),(15) with
Fµν=Kν;µ−Kµ;ν.(16) The vectorfield Kµat large distances from an object is meant to asymptote smoothly to a background value determined by a cosmological solution.Note that the values of the model parametersω,η,ǫ,andτare not independent[17].
As in the ca of previous investigations of scalar-tensor gravity[9],we will be eking vacuum soluti词
ons that are characterized by constant curvature,as well as by a constant vector Kµ.In this ca,thefield equations
4
that are derived from the action (15)
are [17]
R
µν−
1
2
g µνK 2R ,
(19)
Θ(η)µν
=2K αK µR να−2K αK νR µα
−
rf是什么意思
14
1+ωK 2
g µν−ηR
2K 2g µν
=0.(21)
Contracting equation (21)with g µν,we obtain for the constant curvature
R =16Λ
4
K 2 .
(22)
As in the previous cas,a black-hole solution that dif-fers only in the value of the constant curvature from the general relativistic one is possible for the vector-tensor gravity theory that we have considered.
Discussion.—Our results have important implications for current attempts to test general relativity in the strong-field regime using astrophysical black holes.On the one hand,we appear to be lacking a parametric the-oretical framework with which to interpret obrvational data and quantify possible deviations from the general relativistic predictions for astrophysical black holes.On the other hand,the detection of deviations from the Kerr metric in the spacetime of an astrophysical black hole will be a very strong indication for the need of a major change in our understanding of gravitation.
2017年6月英语四级
DP is supported in part by the Astrophysics Theory Program of NASA under Grant NAG 513374.KR
D is supported in part by the U.S.National Science Founda-tion under Grant PHY/0301998,by the U.S.Department of Energy under Grant DE-FG02-04ER-41298,and by a
Rearch Innovation Award from the Rearch Corpora-tion.IM is supported by NSF grant PHY-0555368.We are happy to thank Z.Chacko,C.-K.Chan,K.Sigurdson,and U.van Kolck for discussions.
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