Perturbed Kerr Black Holes Can Probe Deviationsfrom General Relativity

It has been claimed [1] that the existence of theKerr(–de Sitter) solution in many modified gravity theoriesimplies that its verification around an astrophysical blackhole (BH) cannot probe deviations from general relativity(GR). This would have implications for current attempts totest GR using BHs [1]. While this is true for electromag-netic observations, we argue that gravitational wave (GW)experiments can in principle distinguish Kerr BHs in GRand in modified gravity theories.

Reference [1] considered fðRÞ gravity, quadratic gravity,and vector-tensor gravity. These theories are built so as tohave maximally symmetric solutions (Minkowski orde Sitter) in vacuum, and this is done by imposing thatthe field equations reduce to GR if R;� ¼ 0, R�� ¼Rg��=4, and T�� ¼ 0. As such, it had long been known

that Kerr–de Sitter, which satisfies these conditions, is asolution in these theories. Moreover, these theories admitvacuum solutions different from Kerr, as noted also inRef. [1]. [The only exception is Palatini fðRÞ gravity,which reduces to GR in vacuum.] Astrophysical BHsform by gravitational collapse, and there is yet no guaran-tee that this will lead to a Kerr BH if this solution is notunique.

However, we will set this (potentially important) issueaside, as done in Ref. [1], and show that even a Kerr BHpermits probing deviations from GR. Indeed, the (vacuum)metric perturbations over a Kerr BH in the theoriesconsidered in Ref. [1] behave differently from GR.Again, the exception is Palatini fðRÞ gravity, as this re-duces to GR in vacuum. We use metric fðRÞ gravity as anexample, and assume fð0Þ ¼ 0: this is required to have aKerr (and not Kerr–de Sitter) BH [1], which has R ¼R�� ¼ 0 thus simplifying the calculation. With h �r�r� and 0 � @=@R, the vacuum field equations are

f0ðRÞR�� � 12 fðRÞg�� � ðr�r� � g��hÞf0ðRÞ ¼ 0:

(1)

We denote the metric perturbation by h�� and use the

Lorenz gauge, defined by r��h�� ¼ 0, with �h�� � h�� �

hg��=2. This is always possible: �h�� transforms as

�̂h �� ¼ �h�� �r��� �r��� þ g��r���; (2)

and we only need to imposeh�� ¼ r��h�� [2]. Perturbing

Eq. (1) over a Kerr background in this gauge, we have

h �h�� þ 2R�����h�� ¼ ��ðr�r� � g��hÞh �h; (3)

where � � f00ð0Þ=f0ð0Þ. In GR f0ð0Þ ¼ 1 and f00ð0Þ ¼ 0,and thus � ¼ 0. Also, in GR it is possible to set �h ¼ 0,although only in a globally vacuum spacetime, using theresidual freedom of the Lorenz gauge (one can perform atransformation with �� satisfying h�� ¼ 0) [2]. From

Eq. (2) it follows that we need r��� þ �h=2 ¼ 0 in order

to have �̂h ¼ 0. Taking the ‘‘box’’ of r��� þ �h=2, and

using the trace of Eq. (3) and h�� ¼ R ¼ R�� ¼ 0, one

can show that r��� þ �h=2 satisfies the homogeneous

wave equation only if f00ð0Þ ¼ 0 (as in GR): it is thenpossible to choose initial data for �� on a Cauchy hyper-surface such that r��

� þ �h=2 and its derivative normal to

the hypersurface vanish, thus ensuring r��� þ �h=2 ¼ 0

everywhere. This is not possible if f00ð0Þ � 0, and �h cannotthen be zeroed (even in globally vacuum spacetimes).Thus, Eq. (3) differs from its GR analog. For example,over a Minkowski background, besides the propagationmodes of GR, Eq. (3) also has a plane wave solution �h�� /ð��� þ k�k�=m

2Þ expðik�x�Þ, where k � !ð�Þ@t þ �n

(n being the propagation direction) and !ð�Þ2 ��2 þm2 [with m2 � ð3�Þ�1 > 0: if � < 0 the gravity the-ory is nonviable [3]]. These GWs cannot be zeroed in theLorenz gauge, correspond to massive gravitons with ve-locity d!=d� < 1 (detectable by LISA [4]), and theirpolarization differs from GR. Note that this mode corre-sponds to scalar field excitations [cf. equivalent Brans-Dicke theory with a potential [5]]. It is also present inBrans-Dicke theory with no potential [6] and it affects theorbital evolution of binary systems [4,6].In conclusion, while the Kerr solution is common to

many gravity theories, its perturbations are not. BecauseGWs from perturbed Kerr BHs behave differently in differ-ent theories, their detection can be used to discriminatethem, and the concerns of Ref. [1] seem unjustified.We acknowledge discussions with L. Barack, J. Gair,

T. Jacobson, J. C. Miller, D. Psaltis, and L. Rezzolla.

Enrico Barausse1 and Thomas P. Sotiriou21SISSAInternational School for Advanced Studiesand INFNVia Beirut 2, 34014 Trieste, Italy2Center for Fundamental PhysicsUniversity of MarylandCollege Park, Maryland 20742-4111, USA

Received 24 March 2008; published 29 August 2008DOI: 10.1103/PhysRevLett.101.099001PACS numbers: 04.70.Bw, 04.50.�h, 97.60.Lf

[1] D. Psaltis et al., Phys. Rev. Lett. 100, 091101 (2008).[2] E. E. Flanagan and S. A. Hughes, New J. Phys. 7, 204

(2005).[3] A. D. Dolgov and M. Kawasaki, Phys. Lett. B 573, 1

(2003).[4] C.M. Will and N. Yunes, Classical Quantum Gravity 21,

4367 (2004).[5] D. Wands, Classical Quantum Gravity 11, 269 (1994).[6] C.M. Will and H.W. Zaglauer, Astrophys. J. 346, 366

(1989).

PRL 101, 099001 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending

29 AUGUST 2008

0031-9007=08=101(9)=099001(1) 099001-1 � 2008 The American Physical Society