Bumpy black holes in alternative theories of gravity

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(1)Bumpy black holes in alternative theories of gravity. The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters.. Citation. Vigeland, Sarah, Nicolás Yunes, and Leo Stein. “Bumpy black holes in alternative theories of gravity.” Physical Review D 83 (2011). © 2011 American Physical Society.. As Published. http://dx.doi.org/10.1103/PhysRevD.83.104027. Publisher. American Physical Society. Version. Final published version. Citable link. http://hdl.handle.net/1721.1/66116. Terms of Use. Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use..

(2) PHYSICAL REVIEW D 83, 104027 (2011). Bumpy black holes in alternative theories of gravity Sarah Vigeland, Nicola´s Yunes, and Leo C. Stein Department of Physics and MIT Kavli Institute, Cambridge, Massachusetts 02139, USA (Received 17 February 2011; published 16 May 2011) We generalize the bumpy black hole framework to allow for alternative theory deformations. We construct two model-independent parametric deviations from the Kerr metric: one built from a generalization of the quasi-Kerr and bumpy metrics and one built directly from perturbations of the Kerr spacetime in Lewis-Papapetrou form. We find the conditions that these ‘‘bumps’’ must satisfy for there to exist an approximate second-order Killing tensor so that the perturbed spacetime still possesses three constants of the motion (a deformed energy, angular momentum and Carter constant) and the geodesic equations can be written in first-order form. We map these parametrized metrics to each other via a diffeomorphism and to known analytical black hole solutions in alternative theories of gravity. The parametrized metrics presented here serve as frameworks for the systematic calculation of extreme mass-ratio inspiral waveforms in parametrized non-general relativity theories and the investigation of the accuracy to which space-borne gravitational wave detectors can constrain such deviations. DOI: 10.1103/PhysRevD.83.104027. PACS numbers: 04.30.w, 04.50.Kd, 04.70.s. I. INTRODUCTION Gravitational waves (GWs) will be powerful tools for learning about source astrophysics and testing strong field gravity [1,2]. The Laser Interferometer Space Antenna (LISA) [3–6], for example, is expected to be sensitive to roughly a few radians or better of the GW phase in a 1 yr observation. Such levels of precision are achieved through matched filtering, where the data is cross correlated with a family of waveform models (see, e.g., [7]). If a signal is present in the data, this cross correlation acts as a filter that selects the member of the waveform family that most closely resembles the signal. Extreme mass-ratio inspirals (EMRIs) are ideal astrophysical sources to perform precision GW tests of strong field gravity with LISA [1,8]. These sources consist of a small compact object (SCO) that spirals into a supermassive black hole (SMBH) in a generic orbit, producing millions of radians in GW phase inside LISA’s sensitivity band. These waves carry detailed information about the spacetime geometry in which the SCO moves, thus serving as a probe to test general relativity (GR). Tests of strong field gravity cannot rely on waveform families that assume GR is correct a priori. Instead, such tests must employ more generic waveforms that allow for GR deviations. Recently, Yunes and Pretorius [9] proposed the parametrized post-Einsteinian (ppE) framework, in which analytic waveforms that represent comparable mass-ratio coalescences in GR are parametrically deformed. For specific values of the ppE parameters, one recovers GR waveforms; otherwise they describe waveforms in non-GR theories. The ppE scheme has been shown to be sufficiently flexible to map all known alternative theory predictions for comparable mass-ratio coalescences [9].. 1550-7998= 2011=83(10)=104027(16). PpE deformations of EMRI waveforms have not yet been constructed because of the complexities associated with the computation of such waveforms. EMRIs are not amenable to post-Newtonian approximation schemes [10–14], from which many GR templates are today analytically constructed. Instead, to build EMRI waveforms one must solve geodesic-like equations, enhanced with a radiation-reaction force that induces an inspiral. The solution of such differential equations can only be found numerically (see, e.g., [15] for a recent review of the self-force problem). One can parametrically deform EMRI waveforms by introducing non-GR deviations in the numerical scheme used to build such waveforms. Two main ingredients can then be modified: the conservative sector, which controls the shape of nonradiative orbits, and the dissipative sector, which controls the rate of inspiral and the GW generation mechanism. The conservative sector, to leading order in the mass-ratio, depends only on the background spacetime metric, the Kerr metric, on which the SCO moves like a test-particle. The goal of this work is to find a parametric deformation of the Kerr metric that allows for non-GR deviations while retaining a smooth Kerr limit, i.e., as the deformation parameters go to zero, the deformed Kerr metric reduces smoothly and exactly to Kerr. This limit then guarantees that the main properties of the Kerr background survive to the deformation, such as the existence of an event horizon, an ergosphere, and, in particular, three constants of the motion, such that the geodesic equations can be separated into first-order form. Although motion in alternative theories of gravity need not be geodesic, we will here assume that the geodesic equations hold to leading order in the mass ratio. This then allows us to focus only on metric deformations that correct. 104027-1. Ó 2011 American Physical Society.

(3) ´ S YUNES, AND LEO C. STEIN SARAH VIGELAND, NICOLA. the conservative sector of EMRI waveforms. This assumption is justified for theories that derive from certain diffeomorphism-covariant Lagrangians [16], as is the case, for example, in dynamical Chern-Simons (CS) modified gravity [17], in Einstein-Dilaton-Gauss-Bonnet theory [18–22], and in dynamical quadratic gravity [23–25]. We are not aware of any alternative theory where this is not the case to leading order in the mass ratio (and neglecting spins), although it is not hard to imagine that one may exist. A framework already exists to parametrically deform the metric tensor through the construction of so-called bumpy spacetimes [26–28]. Such metrics are deformations of the Kerr metric, where the bumps are required to satisfy the Einstein equations. Because of this last condition, bumpy black holes (BHs) are not sufficiently generic to represent BHs in certain alternative theories of gravity (e.g. solutions that are not Ricci flat). A better interpretation is to think of these bumps as representing exterior matter distributions. The bumpy BH formalism then allows tests of whether compact objects are truly described by the vacuum Kerr metric or by some more general tensor with external matter sources, assuming GR still holds. We here propose two generalizations of the bumpy BH formalism to allow for metric deformations that can represent vacuum BH solutions in alternative theories. The first approach, the generalized bumpy Kerr (BK) scheme, takes the standard bumpy metric and relaxes the requirement that the bumps satisfy the Einstein equations. The second approach, the generalized deformed Kerr (DK) metric, perturbs the most general stationary, axisymmetric metric in Lewis-Papapetrou form and transforms it to Boyer-Lindquist coordinates without assuming the deformations satisfy the Einstein equations. In both cases, the metric is exactly Kerr to zeroth order in the deformation parameter, but it deviates from Kerr at first order through a set of arbitrary functions. The metric deformation is then restricted by requiring that the full metric still possesses three constants of the motion: a conserved energy (associated with time invariance), a conserved angular momentum component (associated with azimuthal rotational invariance), and an approximate second-order Killing tensor that leads to a conserved Carter constant. These conditions restrict the set of arbitrary functions that parametrize the metric deformations to have a certain functional form. It is these conditions that select the appropriate bumps instead of the imposition of the Einstein equations. The restrictions imposed above are not strictly necessary, as there is no guarantee that BHs in alternative theories of gravity will continue to have three conserved quantities. However, all BH solutions in alternative theories of gravity known to date and that are not pathological (i.e. they are stationary, axisymmetric, asymptotically flat, and contain no spacetime regions with closed timelike curves outside the horizon) possess three constants of the motion [29].. PHYSICAL REVIEW D 83, 104027 (2011). Examples include the slowly rotating solution found in dynamical CS gravity [30] and the spherically symmetric solution found in dynamical quadratic gravity [25]. Much effort has gone into finding spacetimes with an exact second-order (or higher-order) generalized ‘‘Carter constant,’’ thus allowing for the separation of the equations of motion [31–35]. That work demonstrates that, for a broad class of spacetimes, such separation can be done provided that the Carter constant is quartic in the orbit’s 4-momentum (i.e., the constant is C ¼ p p p p , where is a 4th-rank Killing tensor and p is the 4-momentum). In this analysis, we show that one can in fact find an approximate Carter constant that is quadratic in the 4-momentum, C ¼ p p , for many relevant spacetimes, provided they differ from Kerr only perturbatively. Finally, we gain insight on the different proposed parametrizations by studying certain key limits. First, we show that the BK and DK deformations are related to each other by a gauge transformation. Second, we show that both the BK and DK metrics can be exactly mapped to specific non-GR BH metrics, the dynamical CS gravity one and the dynamical quadratic one. Third, we study the structure that the deformations must take when only deforming frame dragging or the location of the event horizon. Fourth, we separate the geodesic equations into first-order form in both the BK and DK metrics, and calculate the modified Kepler law. This modification corrects the dissipative dynamics through the conversion of radial quantities to frequency space. The parametrized metrics (BK and DK) proposed in this paper lay the foundations for a systematic construction and study of ppE EMRI waveforms. With these metrics and their associated separated equations of motion, one can now study modified SCO trajectories and see how these impact the GW observable. Numerical implementation and a detailed data analysis study will be presented in a forthcoming publication. This paper is organized as follows. Section II reviews BHs in GR and derives the first-order form of the geodesic equations. Section III introduces the BK formalism, which generalizes the bumpy BH formalism. Section IV presents the DK parametrized metric. Section V compares the parametrizations to each other and to alternative theory predictions. Section VI discusses some general properties of the parametrizations and points to future work. In this paper we primarily follow the notation of Misner, Thorne, and Wheeler [36]. Greek letters stand for spacetime indices, while Latin letters in the middle of the alphabet ði; j; k; . . .Þ stand for spatial indices only. Parenthesis and brackets in index lists stand for symmetrization and antisymmetrization, respectively, i.e., AðabÞ ¼ ðAab þ Aba Þ=2 and A½ab ¼ ðAab Aba Þ=2. Background quantities are denoted with an overhead bar, such that the full metric g ¼ g þ h can be decomposed into a. 104027-2.

(4) BUMPY BLACK HOLES IN ALTERNATIVE THEORIES OF . . .. PHYSICAL REVIEW D 83, 104027 (2011). background g plus a small deformation h , where 1 is a bookkeeping parameter. The background spacetime is always taken to be the Kerr metric. We use geometric units with G ¼ c ¼ 1.. four-dimensional Dirac density, defined via R 4 pffiffiffiffiffiffiffi ð4Þ d x g ðxÞ ¼ 1. The divergence of this stress-energy tensor must vanish in GR, which implies that test particles follow geodesics:. II. BLACK HOLES AND TEST-PARTICLE MOTION IN GENERAL RELATIVITY. D dz ¼ 0; d d. In this section, we review some basic material on the Kerr BH solution in GR and the motion of test particles (see, e.g., [36] for more details). The most general stationary, axisymmetric BH solution in GR is the Kerr metric, which in Boyer-Lindquist coordinates ðt; r; ; Þ takes the form 2Mr 2Mar g tt ¼ 1 2 ; g t ¼ 2 sin2 ;. (1) 2. 2 2 g ¼ ; g. ¼ 2 sin ; g rr ¼ ; . where we have introduced the quantities 2 r2 þ a2 cos2 ; r2 f þ a2 ;. f 1. (2) 2M ; r. ðr2 þ a2 Þ2 a2 sin2 :. (3). where D=d is a covariant derivative. The divergence of the stress-energy tensor vanishes due to local energymomentum conservation and the equivalence principle [36]. We have assumed here that the particle is nonspinning, so that geodesic motion is simply described by Eq. (7); otherwise other terms would arise in the action that would lead to spin-dependent modifications. B. Constants of the motion and separability The geodesic equations as written in Eq. (7) are of second-order form, but they can be simplified to first-order form if there exist at least three constants of the motion plus a normalization condition on the 4-momentum. If the metric is sufficiently symmetric, then Killing vectors and Killing tensors exist that satisfy the vectorial and tensorial Killing equations. (4). The overhead bars are to remind us that the Kerr metric will be taken as our background spacetime. This background describes a BH with mass M and spin angular momentum Si ¼ ð0; 0; MaÞ, where a is the Kerr spin parameter with units of length. A. Equations of motion With this background, we can now compute the equations of motion for a test particle of mass moving on a background worldline x ¼ z ð

(5) Þ, where

(6) is an affine parameter. One way to derive these equations is via the action for a nonspinning test particle (see, e.g., [37]) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z SMat ¼ d

(7) g ðzÞz_ z_ ; (5) . (7). rð Þ ¼ 0;. (8). rð Þ ¼ 0:. (9). The Kerr metric possesses two Killing vectors and one Killing tensor. The vectors, t ¼ ½1; 0; 0; 0 and ¼ ½0; 0; 0; 1, are associated with the stationarity and axisymmetry of the metric, respectively. In addition, the Kerr metric also possesses the Killing tensor: ¼ kð lÞ þ r2 g ;. (10). where k and l are principal null directions: 2 2 r þ a2 a r þ a2 a l ¼ ; 1; 0; ; ; 1; 0; : k ¼ (11). where z_ ¼ dz =d

(8) is the tangent to z and g is the determinant of the background metric. The contribution of this action to the field equations can be obtained by varying it with respect to the metric tensor. Doing so, we obtain the stress-energy for the test particle: Z d TMat ðx Þ ¼ pffiffiffiffiffiffiffi u u ð4Þ ½x zð Þ; (6) g. The contraction of these Killing vectors and tensors allows us to define constants of the motion, namely, the energy E, z component of the angular momentum L, and the Carter constant C:. where is proper time, related to

(9) via d ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d

(10) g ðzÞz_ z_ , u ¼ dz =d is the particle’s 4velocity, normalized via g u u ¼ 1 and ð4Þ is the. E t u ;. (12). L u ;. (13). C u u :. (14). It is conventional to define another version of the Carter constant, namely,. 104027-3.

(11) ´ S YUNES, AND LEO C. STEIN SARAH VIGELAND, NICOLA. Q C ðL aEÞ2 :. (15). With these three constants of the motion [either ðE; L; CÞ or ðE; L; QÞ] and the momentum condition p p ¼ 2 , or simply u u ¼ 1, we can separate the geodesic equations and write them in first-order form: P (16) 2 t_ ¼ aðaEsin2 LÞ þ ðr2 þ a2 Þ ; L aP 2 _ ¼ aE 2 þ ; sin . (17). 4 r_ 2 ¼ ½ðr2 þ a2 ÞE aL2 ½Q þ ðaE LÞ2 þ r2 ; (18) 4 _ ¼ Q cot2 L2 a2 cos2 ð1 E2 Þ; 2. (19). where overhead dots stand for differentiation with respect to proper time and P Eðr2 þ a2 Þ aL. Notice that not only have the equations been written in first-order form, but they have also been separated. Once the geodesic equations have been derived, one can compute Kepler’s third law, relating radial separations to orbital frequencies for an object moving in an equatorial, circular orbit. For such an orbit, the equatorial, circular orbit conditions r_ ¼ 0 and dr_ =dr ¼ 0 lead to [38] j = _ t_ j ¼ jj. M1=2 : r3=2 þ aM1=2. 2Mr 4aMr SBK SBK hSBK ¼ 2 c 1 ; 1 þ tt 1 2 4. (20). PHYSICAL REVIEW D 83, 104027 (2011). In the far field limit, M=r 1, this equation reduces to the familiar Kepler law: ðM=r3 Þ1=2 . III. BLACK HOLES AND TEST-PARTICLE MOTION IN ALTERNATIVE THEORIES: GENERALIZED BUMPY FORMALISM We here introduce the standard bumpy BH formalism [26–28] and generalize it to the BK framework. The bumps are constrained by requiring that an approximate, secondorder Killing tensor exists. We then rewrite the geodesic equations in first-order form. A. From standard to generalized bumpy black holes Let us first review the basic concepts associated with the bumpy BH formalism [26–28]. This framework was initially introduced by Collins and Hughes [26] to model deformations of the Schwarzschild metric and expanded by Vigeland and Hughes [27] to describe deformations on a Kerr background. In its standard formulation, generalized to spinning BHs [27], the metric is perturbatively expanded via g ¼ g þ hSBK ;. (21). where is a bookkeeping parameter that reminds us that is the Kerr jhSBK j=jg j 1. The background metric g solution of Eq. (1), while in the standard bumpy formalism, the metric functions hSBK are parametrized via. hSBK ¼ SBK tr 1. 2a2 Mrsin2 ; 2. 2 2 2aMrsin2 SBK ðr þ a Þ2Mr ; 2 þ 1 2 4 2 2 2 4Mr 2 2 2 SBK Þ ; SBK ¼ SBK 1 þ 2Mrðr þ a Þ asin2 ; SBK c SBK Þ 2 ; h ¼ 2ðSBK c hSBK 1 1 1 ¼ 2ð1 1 r. 2 2 2 2 2 4 4aMr 2Mrðr2 þ a2 Þ sin2 SBK 8a M r sin SBK SBK ¼ ðSBK c Þ c ; þ 2 þ 1 1 1 2 2Mr 2 2Mr 2 2Mr 1 2 2. SBK SBK Þ hSBK 1 t ¼ ð2 c 1. hSBK rr hSBK. SBK where ð c SBK ; SBK Þ are functions of ðr; Þ only that 1 1 ; 1 represent the ‘‘standard bumps.’’ A few comments are due at this point. First, notice that this metric contains only three arbitrary functions SBK Þ, instead of four, as one would expect ð c SBK ; SBK 1 1 ; 1 from the most general stationary and axisymmetric metric. This is because of the specific way Eq. (22) is derived (see below), which assumes a Ricci-flat metric in the a ¼ 0 limit. Second, note that many metric components are nonvanishing: apart from the usual ðt; tÞ, ðt; Þ, ðr; rÞ, ð ; Þ, and ð ; Þ metric components that are nonzero in the Kerr metric, Eq. (22) also has nonvanishing ðt; rÞ and ðr; Þ components.. (22). The derivation of this metric is as follows. One begins with the most general stationary and spherically symmetric metric in Lewis-Pappapetrou form that is Ricci-flat. One then perturbs the two arbitrary functions in this metric and maps it to Schwarschild coordinates. Through the NewmanJanis procedure, one then performs a complex rotation of the tetrad associated with this metric to transform it into a deformed Kerr spacetime. At no stage in this procedure is one guaranteed that the resulting spacetime will still possess a Carter constant or that it will remain vacuum. In fact, the metric constructed from Eq. (22) does not have a Carter constant, nor does it satisfy the vacuum Einstein equations, leading to a nonzero effective stress-energy tensor.. 104027-4.

(12) BUMPY BLACK HOLES IN ALTERNATIVE THEORIES OF . . .. Such features make the standard bumpy formalism not ideal for null-tests of GR. One would prefer to have a framework that is generic enough to allow for non-GR tests while still possessing a smooth GR limit, such that as the deformations are taken to zero, one recovers exactly the Kerr metric. This limit implies that the deformed metric will retain many of the nice properties of the Kerr background, such as the existence of an event horizon, an ergosphere, and a Carter constant. This can be achieved by generalizing the bumpy BH formalism through the relaxation of the initial assumption that the metric be Ricci-flat prior to the Newman-Janis procedure. Inspired by [26–28], we therefore promote the nonvanishing components of the metric perturbation, i.e., BK BK BK BK BK BK ðhBK tt ; htr ; ht ; hrr ; hr ; h ; h. Þ, to arbitrary functions of ðr; Þ. These functions are restricted only by requiring that an approximate second-order Killing tensor still exists. This restriction is not strictly necessary, but it is appealing on several fronts. First, the few analytic non-GR BH solutions that are known happen to have a Carter constant, at least perturbatively as an expansion in the non-Kerr deviation. Second, a Carter constant allows for the separation of the equations of motion into first-order form, which then renders the system easily integrable with already developed numerical techniques.. PHYSICAL REVIEW D 83, 104027 (2011). Notice that depends here on the full metric g and on null vectors k and l that are not required to be the Kerr ones or the principal congruences of the full spacetime: k k and l l . We decompose these vectors via k ¼ k þ k ;. l ¼ l þ l ;. (25). where k and l are given in Eq. (11), and the tensor into ¼ þ ;. (26). where is given in Eq. (10), while 2 BK ½kð lÞ þ lð kÞ þ 2hBK ð kÞ l þ 3r h :. (27) All lowering and raising of indices is carried out with the background metric. Although this particular ansatz only allows 6 independent degrees of freedom (as the vectors are assumed to be null), we will see it suffices to find a Carter constant. With this ansatz, the tensor Killing equation [Eq. (9)] becomes. B. Existence conditions for the Carter constant Let us now investigate what conditions must be enforced on the metric perturbation hBK so that a Killing tensor and its associated Carter constant exist, at least perturbatively to OðÞ. Killing’s equations can be used to infer some fairly general properties that such a tensor must have. First, this tensor must be nonvanishing in the same components as the metric perturbation. For the generalized bumpy metric, this means that ðt ; r ; Þ must all vanish. Furthermore, if we expand the Killing tensor as. ¼ þ ;. (23). we then find that must have the same parity as for the full Killing tensor to have a definite parity. For the generalized bumpy metric, this means that ðtt ; rr ; ; . ; t Þ must be even under reflection: ! . These conditions imply that of the 10 independent degrees of freedom in , only 7 are truly necessary. With this in mind, we parametrize the Killing tensor as in Eq. (10), namely,. ¼ kð lÞ þ r2 g :. (24). @ð Þ 2 ð Þ ¼ 2ð Þ :. (28). The system of equations one must solve is truly formidable and in fact overconstrained. Equation (28) is a set of 20 partial differential equations, while the normalization conditions l l ¼ 0 ¼ k k add two additional algebraic equations. This means there are a total of 13 degrees of freedom (7 in hBK and 6 in l and k after imposing the normalization condition) but 20 partial differential equations to solve. In spite of these difficulties, we have solved this system of equations with MAPLE and the GRTENSORII package [39] and found that the perturbation to the null vectors must satisfy 2 r þ a2 r a r r kBK þ 1 ;kBK ;0; kBK þ 2 ; 2 2 r þa a krBK þ 3 ;krBK þ 4 ;0; krBK þ 5 ; lBK ¼ (29) kBK ¼. where i i ðrÞ are arbitrary functions of r, generated upon solving the differential system. These functions are fully determined by the metric perturbation, which is given by. 104027-5.

(13) ´ S YUNES, AND LEO C. STEIN SARAH VIGELAND, NICOLA. hBK tt. PHYSICAL REVIEW D 83, 104027 (2011). a 4 @hBK a PBK a2 sin2 ð 2 4MrÞ2 @1 @3 a 2 2 PBK @2 @5 t. 2 3 BK sin BK þ þ ¼ ht 2M PBK M PBK 4M 4M @r @r @r @r @r PBK P1 1 1 1 PBK a PBK ðr2 þ a2 Þ 2 4 5 ð1 þ 3 Þ ð2 þ 5 Þ þ 2 1 ; BK 2 PBK 2 2M P1 PBK 1. 1 1 2Mr aMrsin2 1 1 2 þ 4 ; ¼ 1 2 ð1 3 Þ þ ð2 5 Þ 4 ; hBK rr ¼ 2 2 2 . þ. hBK tr. hBK r ¼. aMrsin2 sin2 BK a ð Þ ðP3 2a2 Mrsin2 Þð2 5 Þ þ sin2 4 ; 1 3 2 2 2. 2. 2 hBK ¼ 1 þ 2 ;. ðr2 þ a2 Þ2 BK 2ðr2 þ a2 Þ BK 2 2 sin2 h ð2 þ 5 Þ; h þ þ ð þ Þ tt 1 1 3 t. a a a2 a2 a2 2 2 @ 2 @2 5 BK @ 1 @2 3 BK @2 @5 PBK @1 @3 PBK 8;n 9;n ¼ þ þ þ þ þ þ þ P P BK @r @r PBK @r @r P @r2 @r2 6 @r2 @r2 7 8;d 9;d. hBK. ¼ @2 hBK t. @r2. BK @hBK PBK PBK PBK t P12;n 10;n 11;n 13;n þ ð2 þ 5 Þ BK þ ð1 þ 3 Þ BK þ þ hBK þ 1 PBK 14 ; t. BK @r P12;d P10;d P11;d PBK 13;d. where 2 r2 a2 cos2 , 1;2 1;2 ð Þ are functions of and PBK are polynomials in r and cos , which are given i explicitly in Appendix A. Notice that krBK does not appear in the metric perturbation at all, so we are free to set it to zero, i.e., krBK ¼ 0. With this at hand, given some metric perturbation hBK that satisfies Eq. (30), one can construct the arbitrary functions i and i and thus build the null directions of the perturbed spacetime, such that a Killing tensor and a Carter constant exist perturbatively to OðÞ. We have verified that the conditions described above satisfy the generic Killing tensor properties, described at the beginning of this subsection.. where we recall that f 1 2M=r, force the metric components to take the form BK hBK tt ¼ htt;0 ðrÞ;. Let us now investigate the nonrotating limit a ! 0. To do so, we expand all arbitrary functions via. BK BK 2 hBK ðr; Þ ¼ h;0 ðr; Þ þ ah;1 ðr; Þ þ Oða Þ:. (32). The following choices: 1;0 ¼ 0 ¼ 2;n ; 2;0 ¼ 0 ¼ 5;0 ; 4;0 ¼ fhBK rr;0 ;. hBK hBK rr;0 þ tt;02 ; 2 2f r 4M BK ¼ 5;1 ¼ h ; (33) 2r3 f2 tt;0. 1;0 ¼ 2;1. 3;0 ¼ . hBK hBK rr;0 þ tt;02 ; 2 2f. kBK ¼. . lBK. (31). where F i is any of i ðrÞ or i ð Þ. Let us also expand the metric components in a similar fashion, i.e.,. BK hBK rr ¼ hrr;0 ðrÞ;. (34). where we have set all integration constants to zero. All other components of the metric vanish to this order. To obtain this result, it is crucial to use the slow-rotation expansion postulated above, as some of the conditions in Eq. (30) have seemingly divergent pieces. We see then that in the nonrotating limit, one can always choose free functions i and i such that the only two independent metric BK perturbations are hBK tt and hrr . The Killing tensor is then given by Eq. (24), with the perturbed null vector components. Nonrotating limit. F i ¼ F i;0 þ aF i;1 þ Oða2 Þ;. (30). hBK hBK rr;0 þ tt;02 ; 0; 0; OðaÞ ; 2 2f. hBK hBK ¼ rr;0 þ tt;02 ; fhBK ; 0; OðaÞ ; rr;0 2 2f. (35). where we have set krBK ¼ 0. Composing the Killing tensor, we find that all OðÞ terms vanish in the a ! 0 limit. This result makes perfect sense, considering that the Schwarzschild Killing tensor, ¼ r2 with diagð0; 0; r2 ; r2 sin2 Þ, is also a Killing tensor for the most generic static and spherically symmetric spacetime with arbitrary ðt; tÞ and ðr; rÞ components. C. Equations of motion and geodesics in first-order form The equations of motion for test particles in alternative theories of gravity need not be geodesic. But as we review in Appendix B, they can be approximated as geodesics in. 104027-6.

(14) BUMPY BLACK HOLES IN ALTERNATIVE THEORIES OF . . .. the test-particle limit for a wide class of alternative theories, provided we neglect the spin of the small body. Here, we restrict attention to theories where the modified equations of motion remain geodesic, but of the generalized bumpy Kerr background constructed above. The geodesic equations can be rewritten in first-order form provided that three constants of the motion exist. In Sec. III B, we showed that provided the generalized bumpy functions satisfy certain conditions, then an approximate, second-order Killing tensor and a conserved Carter constant exist. Furthermore, it is easy to see that the metric in Eqs. (21) and (22) remains stationary and axisymmetric, BK BK BK BK BK BK since ðhBK tt ; htr ; ht ; hrr ; hr ; h ; h. Þ are functions of only ðr; Þ. The constants of the motion then become. PHYSICAL REVIEW D 83, 104027 (2011). Rðr; Þ ¼ ½Aðr; Þr2 þ Bðr; Þ;. (43). ðr; Þ ¼ Aðr; Þa2 cos2 Bðr; Þ;. (44). where the functions Aðr; Þ and Bðr; Þ are proportional to the perturbation: _ u h u u ; Aðr; Þ ¼ 2½ht t_ þ h. _ t_ þ ð t_ þ Þ _ _ Bðr; Þ ¼ 2½ðtt t_ þ t. Þ t. þ u u :. (36). L ¼ L þ ðh u þ g u Þ;. (37). C ¼ C þ ð u u þ 2 u ð uÞ Þ;. (38). M jj ¼ jj. (39). þ. while the normalization condition becomes. since by definition 1 ¼ g u u . In these equations, _ is the unperturbed, Kerr 4-velocity, while _ u ¼ ½t_ ; r_ ; ; u is a perturbation. Equations (36)–(39) can be decoupled once we make a choice for ðE; L; QÞ. This choice affects whether the turning points in the orbit are kept the same as in GR, or whether the constants of the motion are kept the same. We here choose the latter for simplicity by setting E ¼ E, L ¼ L, and C ¼ C, which then implies that u must be such that all terms in parentheses in Eqs. (36)–(38) vanish. Using this condition and Eq. (39), we then find that 2 t_ ¼ TK ðr; Þ þ Tðr; Þ; 4 r_ 2 ¼ RK ðrÞ þ Rðr; Þ; 2 _ ¼ K ðr; Þ þ ðr; Þ;. 1=2. ðr1=2 ðr 3MÞ þ 2aM1=2 Þ TðrÞ r5=4 ðr3=2 þ aM1=2 Þ2. ðr1=2 ðr 3MÞ þ 2aM1=2 Þ1=2 ðrÞ; r5=4 ðr3=2 þ aM1=2 Þ. (47). was already given in Eq. (20), T where we recall that jj and are given by Eq. (41) and (42), and we have used In the far field limit (M=r 1) that E ¼ E and L ¼ L. this becomes þ 1 : jj jj (48) r2 where we have assumed that and T are both of order unity for simplicity. Given any particular metric perturbation, one can easily recalculate such a correction to Kepler’s law from Eq. (47). Clearly, a modification of this type in the Kepler relation automatically modifies the dissipative dynamics when converting quantities that depend on the orbital frequency to radius and vice-versa.. (40). IV. BLACK HOLES AND TEST-PARTICLE MOTION IN ALTERNATIVE THEORIES: DEFORMED KERR FORMALISM. 4 _ 2 ¼ K ð Þ þ ðr; Þ; where ðTK ; RK ; K ; K Þ are given by the right-hand sides of Eqs. (16), (18), (19), and (17), respectively. The perturbations to the potential functions ðT; R; ; Þ are given by 2 ðr þ a2 Þ2 2aMr 2 2 Tðr; Þ ¼ h u ; a sin ht u þ (41). (46). As before, we have dropped the superscript BK here. Interestingly, notice that the perturbation automatically couples the ðr; Þ sector, so that the first-order equations are not necessarily separable. One can choose, however, the i functions in such a way so that the equations remain separable, as will be shown elsewhere. The perturbations to the potential functions result in a modification of Kepler’s law [Eq. (20)]:. E ¼ E þ ðh t u þ g t u Þ;. 0 ¼ h u u þ 2g u u ;. (45). In the previous section, we generalized the bumpy BH framework at the cost of introducing a large number of arbitrary functions, later constrained by the requirement of the existence of a Carter constant. In this section, we investigate a different parameterization (DK) that isolates the physically independent degrees of freedom from the start so as to minimize the introduction of arbitrary functions. A. Deformed Kerr geometry. 2aMr 2 2Mr ht u ðr; Þ ¼ h u ; sin2 . (42). Let us first consider the most general spacetime metric that one can construct for a stationary and axisymmetric. 104027-7.

(15) ´ S YUNES, AND LEO C. STEIN SARAH VIGELAND, NICOLA. spacetime. In such a geometry, there will exist two Killing vectors, ta and a , that represent invariance under a time and azimuthal coordinate transformation. Because these Killing vectors are independent, they will commute satisfying t½a b rc td ¼ 0 ¼ t½a b rc d . Let us further assume the integrability condition ½b c. ½b c. t Ra t ¼ 0 ¼ Ra t : a. d. a. d. where ðV; w; ; Þ are arbitrary functions of ð ; zÞ. In GR, by imposing Ricci flatness, one can eliminate one of these functions via a coordinate transformation, and we can thus set ¼ 1. The Einstein equations then further restrict the form of these arbitrary functions [40] 2. ðrþ þ r þ 2MÞ2 þ M2aa2 ðrþ r Þ2 2. ðrþ r Þ 2aMðM þ rþ þr 2 Þð1 4ðM2 a2 ÞÞ. ;. 2. w ¼. 1 4 ðrþ. ðrþ r Þ þ r Þ2 M2 þ a2 4ðM 2 a2 Þ 2. ;. (51). ¼ 1; 2. a 2 2 2 ¼ V ðrþ þ r Þ 4M þ M2 a2 ðrþ r Þ ; 4rþ r. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 where r ¼ þ ðz M2 a2 Þ2 . One can map from this coordinate system to Boyer-Lindquist pffiffiffiffi coordinates ðt; r; ; Þ via the transformation ¼ sin and z ¼ ðr MÞ sin to find the metric in Eq. (1). We will not impose Ricci flatness here, as we wish to model alternative theory deviations from Kerr. We thus work directly with Eq. (50), keeping ðV; w; ; Þ as arbitrary functions, but transforming this to Boyer-Lindiquist coordinates. Let us then reparametrize 2 ,

(16) , and q Vw and perturb the metric functions via V ¼ V þ V;

(17) ¼

(18) þ

(19) ;. q ¼ q þ q; ¼ þ :. g ¼ g þ hDK ;. (53). where the perturbation is given by. (49). ds2 ¼ Vðdt wd Þ2 þ V 1 2 d 2 þ 2 ðd 2 þ dz2 Þ; (50). ðrþ þ r Þ2 4M2 þ M2aa2 ðrþ r Þ2. The metric then becomes. hDK tt ¼ V;. The condition in Eq. (49) guarantees that the 2-planes orthogonal to the Killing vectors ta and a are integrable. Generic stationary and axisymmetric solutions to modified field equations do not need to satisfy Eq. (49). However, all known analytic solutions in GR and in alternative theories do happen to satisfy this condition. We will thus assume it holds here as well. Given these conditions, the most general stationary and axisymmetric line element can be written in LewisPapapetrou canonical form, using cylindrical coordinates ðt; ; ; zÞ:. V ¼. PHYSICAL REVIEW D 83, 104027 (2011). (52). hDK t ¼ q;. hDK r. ðr MÞ2 sin2 ; ¼ ðr MÞ cos sin ð

(20) Þ;. hDK . ¼. 2 hDK rr ¼

(21) cos þ . hDK. ¼.

(22) sin2 ðr. . MÞ2. þ. (54). cos2 ;. 2. sin f4aMrq 2Mr 2. ½ðr2 þ a2 Þ2 a2 sin2 Vg; and all other components vanish. As is clear from the above expressions, the most general perturbation to a stationary, axisymmetric metric yields five nonvanishing metric components that depend still only on four arbitrary functions ðV; q; ;

(23) Þ of two coordinates ðr; Þ. This is unlike the standard bumpy formalism that introduces six nonvanishing metric components that depend on four arbitrary functions (courtesy of the Newman-Janis algorithm applied to a non-Kerr metric). Furthermore, this analysis shows that two of the six arbitrary functions introduced in the generalized bumpy scheme of the previous section must not be independent, as argued earlier. Following the insight from the previous section, we will henceforth allow these 5 metric components DK DK DK DK ðhDK tt ; hrr ; hr ; h ; h. Þ to be arbitrary functions of ðr; Þ, although technically there are only four independent degrees of freedom. We do this because it eases the analytic calculations to come when one investigates which conditions hDK must satisfy for there to exist an approximate second-order Killing tensor. Moreover, it allows us to relax the undesirable requirement, implicit in Eq. (54), that when V 0, then both htt and h. must be nonzero in the a ! 0 limit. B. Existence conditions for the Carter constant Let us now follow the same methodology of Sec. III B to determine what conditions the arbitrary functions must satisfy in order for there to be a second-order Killing tensor. We begin by parametrizing the Killing tensor just as in Eq. (24), with the expansion of the null vectors as in Eq. (25) and the replacement of BK ! DK everywhere. With this at hand, the Killing equation acquires the same structure as Eq. (28). We have solved the Killing equations with MAPLE and the GRTENSORII package [39] to find that the null vectors must satisfy. 104027-8.

(24) BUMPY BLACK HOLES IN ALTERNATIVE THEORIES OF . . .. kDK ¼. . r2 þ a2 r hDK a ðlDK þ 1 Þ þ 4 ;lrDK þ 1 ; r 2 ; ðlrDK þ 1 Þ þ 3 ;. 2 r þ a2 r hDK a lDK þ 4 ;lrDK ; r 2 ; lrDK þ 3 ; lDK ¼ . (55). hDK tt ¼ þ. PHYSICAL REVIEW D 83, 104027 (2011). where i i ðrÞ are arbitrary functions of radius, while lrDK and hDK r are arbitrary functions of both ðr; Þ. As before with krBK , we find below that the function lrDK does not enter the metric perturbation, so we can set it to zero. The functions i are completely determined by the metric perturbation:. a PDK a 4 @hDK 2a2 rðr2 þ a2 Þ sin cos DK ðr2 þ a2 Þ 2 2a2 r2 sin2 t. 2 DK h h þ I þ 1 r t. M PDK 2M PDK @r PDK 2 PDK 2 PDK 1 1 1 1 1 2 ðr2 þ a2 Þ a sin2 PDK 2 PDK a2 2 2 sin2 d1 3 4 þ 3 3 4 M 2 PDK 2M dr PDK 2 PDK 2 PDK 1 1 1 1. a 2 ð þ 2a2 Mrsin2 Þsin2 d3 a2 2 ð 2 4MrÞsin2 d4 ; DK 2M dr 2M dr P1 PDK 1 1 1 I 3 ; hDK rr ¼ 2 2 2 ðr þ a Þ DK 2ðr2 þ a2 Þ DK 22 sin2 22 h ¼ h þ I þ þ 4 ; hDK 3 3 tt. t. a a a2 a2 a2 a2 @hDK 2r 2a2 sin cos DK @hDK 2r 2r r ¼ 2 hDK þ 2 I 2r1 þ 2 3 ; þ h þ 2 r 2 @r @. . @2 hDK t. @r2. ¼. 2 2 DK 2 2 DK DK 8aM sin cos PDK 5 DK 4aMrðr þ a Þ sin cos @hr þ 2a sin P6 hDK 2r P7 hDK h r t. t. @r PDK PDK 8 4 2 PDK 6 1 1 1. þ. 4aMrsin2 PDK 4aMrsin2 PDK 4aMr PDK 2sin2 PDK 16aMsin2 PDK 8 9 10 11 15 I þ þ 4 1 3 3 PDK PDK PDK 4 4 4 PDK 4 PDK 4 1 1 1 1 16. . 2a PDK 2sin2 PDK 2asin2 PDK asin2 d2 1 sin2 d2 12 d1 13 d3 14 d4 ð þ 2a2 Mrsin2 Þ 23 DK dr DK dr DK dr 4 4 4 2 2 4 P1 P1 P1. dr dr. . að 2 4MrÞsin2 d2 4 ; 4 dr2. (56). where we recall that 2 r2 a2 cos2 , r ¼ r ð Þ is an arbitrary function of polar angle, while PDK are polynoi mials in r and cos , given explicitly in Appendix A. Notice that many of these relations are integro-differential, as I is defined as. I ¼. Z. 2 2a sin cos DK 2 d1 dr : (57) hr þ 2r1 þ dr 2. Notice also that the component hDK r is free and thus there are here truly only four independent metric components. We see then that any metric perturbation hDK that satisfies Eq. (56) will possess a Carter constant. If given a specific non-Kerr metric, one can then use these equations to reconstruct the i and 2 functions to automatically. obtain the perturbative second-order Killing tensor associated with this spacetime. With this at hand, one can now construct the constants of the motion. The metric in Eqs. (53) and (54) remains stationary and axisymmetric since all metric components hDK depend only on ðr; Þ. Therefore, the constants of the motion can be expanded in exactly the same way as in Eqs. (36)–(38). With this at hand, the geodesic equations can be rewritten in first-order form in exactly the same way as in the BK case, except that here different metric components are nonvanishing. Of course, all perturbed quantities must here have a DK superscript. Nonrotating limit Let us now take the nonrotating limit, i.e., a ! 0, of the DK Carter conditions. As before, we expand all arbitrary functions as in Eqs. (31) and (32). Imposing the constraints. 104027-9.

(25) ´ S YUNES, AND LEO C. STEIN SARAH VIGELAND, NICOLA. 3;0 ¼ 0;. 1;0 ¼ hDK rr;0 f;. 3;0 ¼ 0; 4;0 ¼. PHYSICAL REVIEW D 83, 104027 (2011). 3;1. hDK tt;0 2f2. þ. hDK rr;0 2. 1 ¼ F1 ðrÞ . hDK r 4M DK ¼ rr;0 h ; 2r3 f2 tt;0 2r2 ;. (58). forces the metric components to take the form DK hDK tt ¼ htt;0 ;. DK hDK rr ¼ hrr;0 ;. hDK. ¼ 0:. (59). All other pieces of Oða0 Þ in the remaining metric compoDK DK nents can be set to zero by setting ðhDK r ; h ; ht Þ to zero. With these choices, the differential conditions are automatically satisfied, where we have set all integration constants to zero. The Killing tensor is then given by Eq. (24), with the perturbed null vector components hDK hDK rr;0 tt;0 DK kDK ¼ þ 2 ; hrr;0 f; 0; OðaÞ ; 2 2f DK (60) DK h h lDK ¼ rr;0 þ tt;02 ; 0; 0; OðaÞ ; 2 2f where here we have set lrDK ¼ 0. As before, the Killing tensor reduces exactly to that of the Schwarzschild spacetime, with the only nonzero components being ð ; . Þ ¼ r4 ð1; sin2 Þ. V. RELATING PARAMETRIZATIONS A. To each other In the previous sections, we proposed two different parametrizations of deformed spacetimes suitable for modeling alternative theory predictions. These parametrizations are related via a gauge transformation. Under a general diffeomorphism, the metric transforms according DK0 DK to hDK ! h h þ rð Þ , where we parametrize the generating vectors as ¼ ½0 ðrÞ; 1 ðr; Þ; 2 ðr; Þ; 3 ðrÞ:. (61). @r. þ. 2hDK r d ; 2. (63). where F1 ðrÞ is a free integration function and 2 ðr; Þ is free. The generating vector of Eq. (61) with the condition in 0 Eq. (63) not only guarantees that hDK r ¼ 0, but also ensures that the only nonvanishing components of the gaugetransformed metrics are the ðt; tÞ, ðt; rÞ, ðt; Þ, ðr; rÞ, ðr; Þ, ð ; Þ, and ð ; Þ, exactly the same nonzero components as in the BK metric. The new components are M 2 2a2 Mrsin cos 0 DK hDK 1 þ 2 ; tt ¼ htt 2 2 1 2Mr d0 aMrsin2 d3 0 1 hDK ; ¼ tr 2 dr dr 2 sin2 2aMrðr2 þ a2 Þcos sin 0 DK þ aM ¼ h 2 ; hDK 1 t. t. 2 2 a2 r a2 sin cos @1 0 DK 2 þ hDK ; ¼ h þ rr rr 1 2 @r sin2 d0 0 2 2 2 2 2 d3 2aMr hDK þ ½ðr ; ¼ þ a Þ þ a sin r. 2 dr dr @2 0 DK 2 hDK ; ¼ h þ r1 a sin cos 2 þ @ 2 0 DK þ sin fr5 a2 Mr2 þ a2 cos2 ½2rðr2 þ M2 Þ hDK ¼ h. 2 þ M þ a4 cos4 ðr MÞg1 2Mrðr2 þ a2 Þ2 2 ; þ sin cos þ (64) 2 where we recall that 2 r2 a2 cos2 and 1 is given by Eq. (63). The above result can be simplified somewhat by setting 2 ¼ 0 and F1 ðrÞ ¼ 0 in Eq. (63). Notice that the ðt; rÞ and ðr; Þ components are not modified by these vector components. The modified components then become M 2 1 ; 2 2 sin2 0 DK þ aM hDK ¼ h 1 ; t. t. 2 a2 r @1 0 DK hDK ; rr ¼ hrr þ 2 1 þ @r 0 DK hDK ¼ h þ r1 ; 0. The question is whether a generating vector exists that could take a generic hDK metric perturbation to one of hBK form. The DK parametrization has an ðr; Þ component that is absent in the BK one, while the BK one has ðt; rÞ and ðr; Þ components that are absent in the DK parametrization. Let us assume that we have a certain metric in DK form. The first task is to remove the ðr; Þ component, i.e. to find 0 a diffeomorphism such that hDK r ¼ 0. This can be achieved by requiring that 1 1 @1 @2 þ ¼ 0; (62) þ hDK r 2 @ @r whose solution is. Z @2. DK hDK tt ¼ htt . 0. sin2 5 fr a2 Mr2 þ a2 cos2 ½2rðr2 þ M2 Þ 2 (65) þ M þ a4 cos4 ðr MÞg1 :. DK hDK. ¼ h. þ. We have thus found a generic diffeomorphism that maps a DK metric to one that has the same nonzero components. 104027-10.

(26) BUMPY BLACK HOLES IN ALTERNATIVE THEORIES OF . . .. as a BK metric. Notice that 0 and 3 only enter to generate ðt; rÞ and ðr; Þ components. If we know the form of the components that we are trying to map to, then we could solve for these two vector components. For example, let us BK assume that hBK tr and hr are given and we wish to find 0;3 0 BK DK0 BK such that hDK tr ¼ htr and hr ¼ hr . This implies 1 2Mr d0 aMrsin2 d3 BK ; (66) 0 ¼ htr þ 1 2 dr dr sin2 0 ¼ hBK þ r. 2 d0 d3 2 2 2 2 2 2aMr þ ½ðr þ a Þ þ a sin : (67) dr dr. PHYSICAL REVIEW D 83, 104027 (2011). (expanded in the slow-rotation limit), while the only nonvanishing component of the deformation tensor in BoyerLindquist coordinates is 5 a M5 2 12M 27M2 hCS þ ¼ sin. 1 þ ; (72) t. 8 CS M r4 7r 10r2 where a is the Kerr spin parameter, M is the BH mass and CS is a dimensionless constant that depends on the CS couplings and is assumed to be small. Such a non-GR BH solution can be mapped to the generalized bumpy parametrization of Eq. (21) and (30) via 2 ¼ gCS. ¼ 5 , and all other i and i vanish. The CS quantity g is given by [41]. Let us, for one moment, allow 0;3 to be arbitrary functions of ðr; Þ. The solution to the above system is then 0 ¼ F2 ð Þ þ. 2 Z dr BK 2 ½h cos ð2a2 Mr a2 r2 a4 Þ 2 tr. þa r þa Þ 2 Z dr BK 2 ½h ð 2MrÞ 3 ¼ F3 ð Þ þ 2 r. . 2 hBK tr ð2a Mr. 2 2. 4. 2MrahBK r ;. 2aMrsin2 hBK tr :. hBK r ¼. 2Mrasin2 F2 ðrÞ F3 ðrÞ 4 2 ½ þ a4 ð 2 þ 2MrÞsin2 : 2. (70). BK Provided hBK tr and hr components can be written in the above form, the generating function becomes. 0 ¼ 2. Z. F2 ðrÞdr;. 3 ¼ 2. Z. F3 ðrÞdr:. (71). . r2 þ a2 a CS ; 1; 0; þ g ; ¼ 2 2 r þa a ; 1; 0; þ gCS l ¼. ; . 2. Dynamical, quadratic gravity We can similarly map the BK and DK parametrizations to the BH solution found in dynamical quadratic gravity [25]. Treating GR corrections as small deformations (i.e. working in the small-coupling limit), Yunes and Stein [25] found that the unique nonspinning BH solution is is the Schwarzschild metric, g ¼ g þ hQG , where g QG while h in Schwarzschild coordinates is given by [25] QG M3 3 r3 26M 66 M2 96 M3 80M4 þ 1þ þ ; r 5 r2 5 r3 r4 QG M2 ¼ 2 2 f r M 52 M2 2M3 16 M4 368 M5 1þ þ þ þ ; r 3 r2 5 r4 3 r5 r3 (75). hQG tt ¼ . The BK and DK parametrizations can also be mapped to known alternative theory BH metrics. In dynamical CS gravity [17], one can employ the slow-rotation approximation, where one assumes the BH’s spin angular momentum is small, jSj=M2 1, and the small-coupling approximation, where one assumes the theory’s corrections are small deformations away from GR to find an analytical BH solution. Yunes and Pretorius [30] found that this solution is the Kerr metric is simply g ¼ g þ hCS , where g. (74). which agrees with Eq. (35) of [41]. Moreover, with these choices all the metric components vanish except for the ðt; Þ one, which must be equal to Eq. (72) for the differential constraint in Eq. (30) to be satisfied. Similarly, we can also map it to the DK parameterization of Eq. (53) and (54) via 3 ¼ gCS. , and all other arbitrary functions vanish. With this, the perturbation to the principal null directions are equal to Eq. (74), which agrees with [41]. Moreover, we have checked that Eq. (72) satisfies the differential constraint found in Sec. IV.. B. To alternative theories 1. Dynamical Chern-Simons modified gravity. (73). k. (69). F2 ðrÞð 2 2MrÞ þ 2Mrasin2 F3 ðrÞ ; 2. 70r2 þ 120Mr þ 189M2 : 112r8 ð1 2M=rÞ. With these choices, the full null vectors become. (68). But of course, to avoid introducing other spurious components to the DK0 metric, such as hDK t , we must require that 0;3 be functions of r only. This implies that the integrands must be themselves also functions of r only. Setting the integrands equal to F2 ðrÞ and F3 ðrÞ, we find the conditions hBK tr ¼. 4 gCS. ¼ CS aM. hQG rr. 104027-11.

(27) ´ S YUNES, AND LEO C. STEIN SARAH VIGELAND, NICOLA. where M is the mass of the Schwarzschild BH and QG is a dimensionless constant that depends on the couplings of the theory and is assumed to be small. For a spherically symmetry background, one can easily show that the Schwarzschild Killing tensor (i.e. one whose only nonvanishing components are ¼ r4 and . ¼ r4 sin2 ) satisfies the Killing equation, regardless of the functional form of the ðt; tÞ and ðr; rÞ components of the metric. This means that the perturbed vectors k and l must adjust so that the above is true. We find that the following vector components do just that: QG M2 r4 8 M 14M2 128 M3 48M4 þ 2 þ k ¼ þ 4 ; 1þ 3 r 5 r3 r r 2fr6. PHYSICAL REVIEW D 83, 104027 (2011). htt ¼ f1 ðrÞ þ 2f2 4 ðrÞ; hrr ¼ f1 1 ðrÞ; 4 r d1 r3 ðr 4MÞ d4 h. ¼ f2 sin2 þ 2r2 4 ðrÞ þ : 2M 2M dr dr (78) We then see that in the DK parametrization 3 controls modifications to frame-dragging (and thus the ergosphere), while 1 and 4 control modifications to the location of the event-horizon and the innermost-stable circular orbit. VI. DISCUSSION. t. QG M2 r4 kr ¼ 6. fr M 52 M2 2M3 16 M4 368 M5 1þ þ þ þ ; r 3 r2 5 r4 3 r5 r3 QG M2 r4 lt ¼ 2f2 r6 4 M 26M2 32 M3 48 M4 448 M5 þ 2 þ 1þ þ ; 3r 5 r3 5 r4 3 r5 r (76). and all other components vanish. That perturbed null vectors exist to reduce the Killing tensor to its Schwarzschild counterpart must surely be the case, as we have already shown in Secs. III B and IV Bthat in the nonrotating limit both the BK and DK parametrizations allow for generic ðt; tÞ and ðr; rÞ deformations. 3. Arbitrary perturbations: Perturbation to 3 , perturbations to 1 and 4 We can understand the functions that parametrize the metric perturbation by considering perturbations where we allow only a few of the parametrizing functions to be nonzero. Consider a perturbation in the deformed Kerr parametrization of Eqs. (53) and (54) described by 1 ¼ 4 ¼ 3 ¼ 0 and hr ¼ h ¼ 0. In the small a limit, the only nonzero component of the metric is the ðt; Þ component:. ht ¼ fðr2 sin2 3 ðrÞ þ CÞ;. (77). where C is a constant. Now consider a perturbation in the DK parametrization whose only nonzero parameters are 1 and 4 . In the small a limit, this produces perturbations to the metric given by. The detection of GWs from EMRIs should allow for the detailed mapping of the spacetime metric on which the SCOs move. Such a mapping in turn allows for null tests of GR, as one would in principle be able to constrain deviations of the background spacetime from the Kerr metric. To carry out such tests, however, one requires a parametrization of the metric tensor that can handle modelindependent, non-GR deviations. This was the purpose of this paper. We have constructed two such parametrizations: one as a generalization of the bumpy BH formalism (the BK scheme) and one as a generalized deformation of the Kerr metric (the DK scheme). The former promotes the nonvanishing components of the metric perturbation that contain bumps to arbitrary functions of radius and polar angle. The latter takes the most general stationary, axisymmetric line element without assuming Ricci-flatness and constructs a generic metric deformation from it. In both cases, the arbitrary functions introduced are constrained by requiring that there exists an approximate second-order Killing tensor. These schemes differ from each other in the metric components that are assumed not to vanish. We have found a gauge transformation, however, that relates them. That is, we have shown that a generating vector exists such that the BK metric can be mapped to the DK one. We have also mapped both parametrizations to known analytical non-GR solutions, thus automatically finding their respective Killing tensors and Carter constants. The perturbative nature of our approach puts some limitation on the generality of the deformation. All throughout we restricted the metric deformation to be a small deviation away from the GR solution. But due to the structure of the perturbation, there can be regions in spacetime where this deformation dominates over the GR background. For example, the full metric tensor might lose a Lorentzian signature if the metric perturbation dominates over the Kerr background and it possesses a certain definite sign, i.e., the metric will no longer have detðgÞ < 0. Clearly, such regions are unphysical, and the coupling constants that control the magnitude of the perturbation should be. 104027-12.

(28) BUMPY BLACK HOLES IN ALTERNATIVE THEORIES OF . . .. adjusted appropriately to ensure that if they exist, they are hidden inside the horizon. We should emphasize that we were guided by two principles in the construction of the parameterized schemes proposed here. First, we wanted to ensure that the parametrizations would easily map to known analytic solutions. Second, we wished to retain a smooth Kerr limit, such that when the deformation parameters are taken to zero, the deformed metric goes smoothly to Kerr. This in turn guaranteed that certain properties of the Kerr background were retained, such as the existence of a horizon. We additionally required that the metric tensors allowed a second-order perturbative Killing tensor, so that there exists a perturbative Carter constant. This latter requirement is not strictly necessary to perform null tests, as one could build GWs from the evolution of the full second-order equations of motion without rewriting them in first-order form. An interesting avenue of future research would be to investigate the Petrov type [42] of the BK and DK metrics. Brink has shown that GR spacetimes that admit a Carter constant must also be of Type D [33]. In alternative theories of gravity, however, a formal mathematical proof of the previous statement is lacking. We have here found Ricci-curved metrics that admit a Carter constant. One could now attempt to prove that the perturbative Carter existence conditions found in this paper also automatically imply that the spacetime is approximately Petrov Type D. This is a purely mathematical result that is beyond the scope of this paper.. PHYSICAL REVIEW D 83, 104027 (2011). An interesting physical question that could now be answered with either the DK or BK frameworks is whether LISA could place interesting constraints on non-GR Kerr deformations. To answer this question, one would have to evolve geodesics in either framework and construct the associated GWs. One could then carry out a Fisher analysis to determine the magnitude of the non-GR deviation from Kerr that could be measured or constrained by LISA. We will investigate this problem in a forthcoming publication. ACKNOWLEDGMENTS We would like to thank Scott Hughes and Eric Poisson for useful discussions. N. Y. acknowledges support from NASA through the Einstein Postdoctoral Fellowship Award No. PF0-110080 issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under Contract No. NAS8-03060. S. V. and L. S. acknowledge support from NSF Grant No. PHY0449884 and NASA Grant No. NNX08AL42G. L. S. acknowledges support from MIT’s Solomon Buchsbaum fund. APPENDIX A: POLYNOMIAL FUNCTIONS We here provide explicit expressions for the polynomials appearing in the conditions that guarantee the existence of a Carter constant. In Eq. (30), the polynomials PDK are given by i. 6 2 4 4 2 2 2 4 2 2 4 4 4 2 2 PBK 1 ¼ r þ 5a r þ 2a r 2a cos ð2r þ a r þ a Þ a cos ðr a Þ; 4 2 2 2 2 2 2 4 4 PBK 2 ¼ r M þ a r ðr þ 2MÞ a cos ðr ðr þ 2MÞ a ðr 2MÞÞ a cos ðr MÞ; 3 2 2 2 2 2 PBK 3 ¼ rðr þ a ðr þ 4MÞÞ þ a cos ðr 4Mr þ a Þ; 3 2 2 2 2 2 2 2 4 4 2 2 PBK 4 ¼ r ðr ðr 2MÞ þ a ð3r 4MÞÞ 2a r cos ðr 2Mr a Þ a cos ð3r 2Mr þ a Þ; 3 2 2 3 2 4 2 2 2 2 2 4 PBK 5 ¼ r ðr 2MÞðr ðr þ 2MÞ þ a ðr þ 4MÞÞ r cos ðr 4M r a ðr 6Mr þ 16M Þ 2a Þ. a2 cos4 ð2r3 ðr2 2Mr þ 4M2 Þ þ a2 rðr2 þ 4Mr 4M2 Þ a4 ðr 2MÞÞ a4 cos6 ðr 2MÞ; PBK 6 ¼ PBK 7 ¼. 1 ð 2 2Mr 2 þ 4Mra2 þ 4Mr3 Þsin2 ; 2 4. 1 asin2 ð 2 þ 4MrÞ ; 2 4. 11 7 4 9 2 9 2 6 2 10 4 4 2 4 2 3 2 6 2 2 PBK 8;n ¼ ð8r 4r þ 16r M þ 2r 44a M 8a M 68a Mr þ 128a M r 56a rM . 12a4 r5 2 16r3 a6 2 6a8 r 2 36a2 Mr6 2 þ 40a2 Mr4 4 þ 32r9 a2 3r5 6 8r10 M 20a2 rM2 6 16r8 Ma2 16r7 a2 M2 48r3 a6 M2 þ 16r4 Ma6 56a2 M2 r3 4 þ 64a2 M2 r5 2 þ 72a4 rM2 4 þ 48r7 a4 þ 32r5 a6 þ 8r3 a8 þ 60a4 M 4 18a2 M 6 80a4 M2 r5 þ 8a8 M þ 16a8 M2 r þ 36a8 M 2 32r7 M2 2 4r8 M 2 12r3 a4 4 12a2 r5 4 4Mr6 4 4a6 r 4 44a6 M 4 þ 12Mr4 6 8M2 r3 6 þ 24M2 r5 4 þ 2r3 a2 6 þ 5a4 r 6 þ 12a4 M 6 Þsin2 ;. 104027-13.

(29) ´ S YUNES, AND LEO C. STEIN SARAH VIGELAND, NICOLA. PHYSICAL REVIEW D 83, 104027 (2011). 6 4 2 2 2 2 2 4 2 4 2 4 4 4 2 4 PBK 8;d ¼ ð4r þ 2 a 4a 8a Mr þ 2a þ 8r a 2 Mr 2r þ 4a . þ 8Mra4 4a6 Þ 4 ; 2 2 5 2 2 3 4 4 4 6 2 2 2 5 8 4 5 8 9 PBK 9;n ¼ asin ð16M r þ 8M r 4Mr 20Mr 32a M r þ 24r M 24a r 8a M 8r. þ 56a2 Mr6 þ 40a4 Mr4 24r7 a2 8r3 a6 þ 8a6 M þ 16a6 M2 r 12a4 4 M þ 28a6 M 2 36a4 M 2 þ 12a2 4 M 48a4 M2 r3 þ 8ra2 M2 4 3a2 r 6 þ 16r7 M2 þ 2a2 M 6 þ 18a2 r5 2 þ 6r7 2 þ 18r3 a4 2 þ 6a6 r 2 þ r3 6 þ 48a2 M2 r3 2 40a4 M2 r 2 48a2 Mr4 2 Þ; 4 6 4 2 2 2 2 2 4 2 4 2 4 4 4 2 4 PBK 9;d ¼ ð4r þ 2 a 4a 8a Mr þ 2a þ 8r a 2 Mr 2r þ 4a . þ 8Mra4 4a6 Þ; 2 4 4 2 8 4 2 4 4 6 6 2 6 3 6 2 2 2 6 10 PBK 10;n ¼ sin ð12rMa 48a r þ 6r a 48a r þ 16a M þ 32a M r þ 8a 8a M 16r. 16M2 r8 þ 6a2 6 Mr þ 24r4 a4 2 þ 32r9 M 32a2 4 Mr3 8a8 2 þ 8a6 M2 2 þ 40M2 r6 2 þ 16rMa6 2 32r5 Ma2 2 þ 8r8 2 16a8 M2 þ 96a2 M2 r4 2 16r4 a6 þ 32r5 Ma4 80r6 a2 M2 48r4 a4 M2 þ 16r3 Ma4 2 þ 24a2 r6 2 6Mr3 6 þ 3r4 6 þ 2r6 4 þ 5a4 6 þ 64r7 Ma2 þ 24Mr5 4 24M2 r4 4 PBK 10;d. 48Mr7 2 24a6 M2 2 sin2 6a2 6 þ 6a4 4 4a6 4 Þ; ¼ 4 ð4r6 þ 2 4 a2 4a2 2 8a2 2 Mr þ 2a4 2 þ 8r4 a2 4 2 4 Mr 2r4 2 þ 4a4 þ 8Mra4 4a6 Þ;. 2 3 4 2 5 7 2 4 2 3 2 2 4 2 2 2 2 2 PBK 11;n ¼ 8aMsin ðr 5 r þ 4r 3ra þ 6a r þ a M þ a M þ 2a M r. 3a4 M 2 8a2 Mr4 þ 3a4 r 2 2Mr6 4a4 r3 þ 3Mr4 2 þ 2a4 M þ 4a4 M2 r 2a6 MÞ; 4 6 4 2 2 2 2 2 4 2 4 2 4 4 4 2 4 PBK 11;d ¼ ð4r þ 2 a 4a 8a Mr þ 2a þ 8r a 2 Mr 2r þ 4a . þ 8Mra4 4a6 Þ; 2 2 2 2 4 2 6 4 4 4 2 4 2 6 PBK 12;n ¼ 2rð28a 56a Mr þ 18a þ þ 2 þ 4 Mr þ 4 a 18r þ 16r. þ 32r4 a2 þ 16a4 þ 32Mra4 16a6 Þ; 2 6 4 2 2 2 2 2 4 2 4 2 4 4 4 2 4 4 6 PBK 12;d ¼ ð4r þ 2 a 4a 8a Mr þ 2a þ 8r a 2 Mr 2r þ 4a þ 8Mra 4a Þ; 2 2 2 2 2 2 4 2 6 4 4 4 2 4 2 6 4 2 PBK 13;n ¼ 2a sin ð8a þ 16a Mr 8a 6 12 Mr þ 8 a þ 24r 16r 16r a Þ; 4 6 4 2 2 2 2 2 4 2 4 2 4 4 4 2 4 4 6 PBK 13;d ¼ ð4r þ 2 a 4a 8a Mr þ 2a þ 8r a 2 Mr 2r þ 4a þ 8Mra 4a Þ;. PBK 14 ¼. 4 ð4r6. þ. 2 4 a2. . 4a2 2 . 4aMrsin2 ð4r4 þ 2 þ 2 2 Mr 4a2 2 þ 4r2 a2 Þ : 8a2 2 Mr þ 2a4 2 þ 8r4 a2 4 2 4 Mr 2r4 2 þ 4a4 þ 8Mra4 4a6 Þ (A1). Similarly, the polynomial functions that appear in the Carter conditions for the DK metric of Eq. (56) are. 2 4 2 2 4 2 2 4 2 2 4 4 4 2 2 PDK 1 ¼ r ðr þ 5a þ r þ 2a Þ 2a cos ð2r þ a r þ a Þ a cos ðr a Þ; 2 2 2 2 2 2 2 2 4 4 PDK 2 ¼ r ðr M þ a r þ 2a MÞ a cos ðr ðr þ 2MÞ a ðr 2MÞÞ a cos ðr MÞ; 3 2 2 3 2 2 2 2 2 4 4 PDK 3 ¼ r ðr 2MÞðr ðr þ 2MÞ þ a ðr þ 4MÞÞ þ 2r a cos ðr 2Mr þ 4M þ a Þ þ a cos ðr 2MÞ; 3 2 2 2 2 2 2 2 4 4 2 2 PDK 4 ¼ r ðr ðr 2MÞ þ a ð3r 4MÞÞ 2r a cos ðr 2Mr a Þ a cos ð3r 2Mr þ a Þ; 4 6 2 4 4 2 6 4 4 2 2 2 4 4 6 2 4 4 2 6 PDK 5 ¼ r ðr þ 3a r þ 8a r þ 2a Þ 3a r cos ð3r a Þ þ a cos ð5r 3a r þ 6a r þ 2a Þ. þ a6 cos6 ð2r4 3a2 r2 a4 Þ; 4 2 2 2 2 2 2 2 4 2 2 2 6 6 PDK 6 ¼ r ðr 6a Þ þ 3a r cos ð3r þ 4a Þ a cos ð9r 2a Þ a cos ; 6 2 4 4 2 2 2 4 2 2 4 4 4 2 2 6 6 PDK 7 ¼ r þ 10a r þ 6a r a cos ð11r þ 16a r þ 10a Þ þ a cos ð5r þ 6a Þ þ a cos ;. 104027-14.

(30) BUMPY BLACK HOLES IN ALTERNATIVE THEORIES OF . . .. PHYSICAL REVIEW D 83, 104027 (2011). 2 2 2 2 2 2 2 PDK 8 ¼ r ð3r a Þ a cos ðr 3a Þ; 2 4 2 2 4 2 2 2 2 2 2 4 4 2 2 PDK 9 ¼ r ð3r þ 5a r 2a Þ 2a cos ðr þ a Þð2r 3a Þ þ a cos ðr 3a Þ; 2 2 2 2 4 4 2 4 2 2 PDK 10 ¼ r ð3r a Þ 3cos ðr a Þ þ a cos ðr 3a Þ; 4 6 5 2 4 2 3 2 2 2 4 2 4 4 2 6 PDK 11 ¼ r ð3r 2Mr þ 24a r 18a Mr 8a M r þ 19a r 24a Mr þ 48a M þ 6a Þ. þ a2 cos2 ð21r8 18Mr7 8M2 r6 þ 18a2 r6 42a2 Mr5 þ 72a2 M2 r4 þ 33a4 r4 32a4 Mr3 þ 24a4 M2 r2 þ 12a6 r2 16a6 M2 Þ þ a4 cos4 ð11r6 þ 6Mr5 24M2 r4 12a2 r4 þ 22a2 Mr3 24a2 M2 r2 þ 3a4 r2 24a4 Mr þ 24a4 M2 þ 2a6 Þ þ a6 cos6 ð3r4 6Mr3 6a2 r2 þ 18a2 Mr 8a2 M2 a4 Þ; 3 3 2 2 2 4 2 2 5 4 2 2 4 4 PDK 12 ¼ r ðr M þ 3a r 6a Mr a Þ a cos ð3r 3Mr 3a Mr 3a r þ 2a MÞ. PDK 13. þ a4 cos4 ðr3 3a2 r þ a2 MÞ; ¼ r3 ðr6 þ 9a2 r4 10a2 Mr3 þ 6a4 r2 8a4 Mr þ 2a6 Þ cos2 rðr8 þ 18a2 r6 22a2 Mr5 þ 15a4 r4 26a4 Mr3 þ 20a6 r2 12a6 Mr þ 6a8 Þ þ a2 cos4 ð9r7 12Mr6 þ 6a2 r5 22a2 Mr4 þ 27a4 r3 18a4 Mr2 þ 6a6 r þ 4a6 MÞ þ a4 cos6 ð3r5 þ 4Mr4 10a2 r3 þ 6a2 Mr2 þ 3a4 r 6a4 MÞ þ a6 cos8 ðr3 3a2 r þ 2a2 MÞ;. 3 8 7 2 6 2 5 2 2 4 4 4 4 3 6 2 6 6 2 8 PDK 14 ¼ r ð3r 4Mr þ 22a r 30a Mr þ 8a M r þ 29a r 14a Mr þ 12a r 12a Mr 16a M þ 2a Þ. a2 cos2 ð15r9 24Mr8 þ 8M2 r7 þ 10a2 Mr6 þ 21a4 r5 34a4 Mr4 32a4 M2 r3 þ 24a6 r3 þ 12a6 Mr2 32a6 M2 r þ 6a8 r þ 8a8 MÞ a4 cos4 ð13r7 32Mr6 þ 6a2 r5 þ 14a2 Mr4 þ 8a2 M2 r3 3a4 r3 42a4 Mr2 þ 48a4 M2 r þ 4a6 r 16a6 MÞ a6 cos6 ð3r5 12Mr4 þ 8M2 r3 2a2 r3 þ 18a2 Mr2 16a2 M2 r 5a4 r þ 6a4 MÞ; 3 6 2 4 2 4 2 3 4 2 4 4 2 6 PDK 15 ¼ r ðr 8M r þ 9a r 22a Mr þ 6a r 4a Mr þ 16a M þ 2a Þ. a2 cos2 ð9r7 28Mr6 þ 9a2 r5 18a2 Mr4 þ 16a2 M2 r3 þ 16a2 M2 r3 þ 18a4 r3 36a4 Mr2 þ 6a6 r 8a6 MÞ a4 cos4 rð3r4 4Mr3 þ r2 ð8M2 9a2 Þ þ 18a2 Mr 16a2 M2 Þ a6 cos6 ðr3 3a2 r þ 2a2 MÞ:. (A2). APPENDIX B: EQUATIONS OF MOTION IN ALTERNATIVE THEORIES The derivation of the equations of motion in Sec. II A is fairly general. In particular, this derivation is independent of the metric used; in no place did we use that the spacetime was Kerr. We required the divergence of the testparticle stress-energy tensor vanishes; this condition is always true in GR, because of a combination of local stress-energy conservation and the equivalence principle. One might wonder whether the divergence-free condition of the stress-energy tensor holds in more general theories. In any metric GR deformation, the field equations will take the form Mat H G þ H ¼ T þ T ;. (B1). where G is the Einstein tensor, H is a tensorial H is a posdeformation of the Einstein equations, and T sible stress-energy modification, associated with additional fields. The divergence of this equation then leads to Mat H r H ¼ r T þ r T ;. (B2). since the Bianchi identities force the divergence of the Einstein tensor to vanish. The Bianchi identities hold in alternative theories, as this is a geometric constraint and not one that derives from the action. We see then that the divergence of the matter stress-energy tensor vanishes independently provided H r H ¼ r T :. (B3). Whether this condition [Eq. (B3)] is satisfied depends somewhat on the theory of interest. Theories that include additional degrees of freedom that couple both to the geometry and have their own dynamics usually satisfy Eq. (B3). This is because additional equations of motion arise upon variation of the action with respect to these additional degrees of freedom, and these additional equations reduce to Eq. (B3). Such is the case, for example, in dynamical CS modified gravity [41]. If no additional degrees of freedom are introduced, the satisfaction of Eq. (B3) depends on whether the divergence of the new tensor H vanishes, which need not in general be the case. Recently, it was shown that the equations of motion are geodesic to leading order in the mass-ratio for any classical field theory that satisfies the following constraints [16]:. 104027-15.

(31) ´ S YUNES, AND LEO C. STEIN SARAH VIGELAND, NICOLA. PHYSICAL REVIEW D 83, 104027 (2011). (i) It derives from a diffeomorphism-covariant Lagrangian, ensuring a Bianchi identity; (ii) It leads to second-order field equations. The second condition seems somewhat too stringent, as we know of examples where third-order field equations still lead to geodesic motion, i.e., dynamical CS gravity [41].. Therefore, it seems reasonable to assume that this condition could be relaxed in the future. Based on this, we take the viewpoint that the equations of motion are geodesic even in the class of alternative theories we consider here.. [1] B. F. Schutz, J. Centrella, C. Cutler, and S. A. Hughes, Astro2010: The Astronomy and Astrophysics Decadal Survey, Science White Papers, No. 265, (2009) http:// adsabs.harvard.edu/abs/2009astro2010S.265S. [2] C. F. Sopuerta, GW Notes 4, 3 (2010). [3] P. Bender et al., LISA Pre-Phase A Report, Max-PlanckInstitut fu¨r Quantenoptik, Garching, mPQ 233, 1998. [4] K. Danzmann and A. Rudiger, Classical Quantum Gravity 20, S1 (2003). [5] K. Danzmann, Adv. Space Res. 32, 1233 (2003). [6] T. Prince, American Astronomical Society Meeting 202, 3701 (2003). [7] A. K. Piotr Jaranowski, Living Rev. Relativity 8, 3 (2005), http://www.livingreviews.org/lrr-2005-3. [8] P. Amaro-Seoane et al., Classical Quantum Gravity 24, R113 (2007). [9] N. Yunes and F. Pretorius, Phys. Rev. D 80, 122003 (2009). [10] L. Blanchet, Living Rev. Relativity 9, 4 (2006). [11] N. Yunes and E. Berti, Phys. Rev. D 77, 124006 (2008). [12] N. Yunes, GW Notes 2, 3 (2009). [13] N. Yunes, A. Buonanno, S. A. Hughes, M. Coleman Miller, and Y. Pan, Phys. Rev. Lett. 104, 091102 (2010). [14] N. Yunes, A. Buonanno, S. A. Hughes, Y. Pan, E. Barausse et al., Phys. Rev. D 83, 044044 (2011). [15] L. Barack, Classical Quantum Gravity 26, 213001 (2009). [16] S. E. Gralla, Phys. Rev. D 81, 084060 (2010). [17] S. Alexander and N. Yunes, Phys. Rep. 480, 1 (2009). [18] P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley, Phys. Rev. D 54, 5049 (1996). [19] T. Torii, H. Yajima, and K.-i. Maeda, Phys. Rev. D 55, 739 (1997). [20] P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley, Phys. Rev. D 57, 6255 (1998). [21] M. Pomazanov, V. Kolubasova, and S. Alexeyev, arXiv:gr-qc/0301029.. [22] P. Pani and V. Cardoso, Phys. Rev. D 79, 084031 (2009). [23] B. A. Campbell, N. Kaloper, and K. A. Olive, Phys. Lett. B 285, 199 (1992). [24] S. Mignemi and N. R. Stewart, Phys. Lett. B 298, 299 (1993). [25] N. Yunes and L. C. Stein, arXiv:1101.2921 [Phys. Rev. D (to be published)]. [26] N. A. Collins and S. A. Hughes, Phys. Rev. D 69, 124022 (2004). [27] S. J. Vigeland and S. A. Hughes, Phys. Rev. D 81, 024030 (2010). [28] S. J. Vigeland, Phys. Rev. D 82, 104041 (2010). [29] D. A. Brown et al., Phys. Rev. Lett. 99, 201102 (2007). [30] N. Yunes and F. Pretorius, Phys. Rev. D 79, 084043 (2009). [31] J. Brink, Phys. Rev. D 78, 102001 (2008). [32] J. Brink, Phys. Rev. D 78, 102002 (2008). [33] J. Brink, Phys. Rev. D 81, 022001 (2010). [34] J. Brink, Phys. Rev. D 81, 022002 (2010). [35] J. Brink, arXiv:0911.4161. [36] C. W. Misner, K. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman & Co., San Francisco, 1973). [37] E. Poisson, Living Rev. Relativity 7, 6 (2004), http:// www.livingreviews.org/lrr-2004-6. [38] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Astrophys. J. 178, 347 (1972). [39] GRTENSORII, this is a package which runs within MAPLE but distinct from packages distributed with MAPLE. It is distributed freely on the World-Wide-Web from the address: http://grtensor.org. [40] G. C. Jones and J. E. Wang, Phys. Rev. D 71, 124019 (2005). [41] C. F. Sopuerta and N. Yunes, Phys. Rev. D 80, 064006 (2009). [42] A. Z. Petrov, Einstein Spaces (Pergamon Press, Oxford, 1969).. 104027-16.

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