Approximate waveforms for
extreme-mass-ratio inspirals in modified gravity spacetimes
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Gair, Jonathan, and Nicolás Yunes. “Approximate waveforms for
extreme-mass-ratio inspirals in modified gravity spacetimes.”
Physical Review D 84.6 (2011): n. pag. Web. 25 Jan. 2012. © 2011
American Physical Society
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http://dx.doi.org/10.1103/PhysRevD.84.064016
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Approximate waveforms for extreme-mass-ratio inspirals in modified gravity spacetimes
Jonathan Gair1and Nicola´s Yunes2
1Institute of Astronomy, Madingley Road, Cambridge, CB30HA, United Kingdom 2MIT and Kavli Institute, Cambridge, Massachusetts 02139, USA
(Received 30 June 2011; published 14 September 2011)
Extreme mass-ratio inspirals, in which a stellar-mass compact object spirals into a supermassive black hole, are prime candidates for detection with space-borne milliHertz gravitational wave detectors, similar to the Laser Interferometer Space Antenna. The gravitational waves generated during such inspirals encode information about the background in which the small object is moving, providing a tracer of the spacetime geometry and a probe of strong-field physics. In this paper, we construct approximate, ‘‘analytic-kludge’’ waveforms for such inspirals with parametrized post-Einsteinian corrections that allow for generic, model-independent deformations of the supermassive black hole background away from the Kerr metric. These approximate waveforms include all of the qualitative features of true waveforms for generic inspirals, including orbital eccentricity and relativistic precession. The deformations of the Kerr metric are modeled using a recently proposed, modified gravity bumpy metric, which parametrically deforms the Kerr spacetime while ensuring that three approximate constants of the motion remain for geodesic orbits: a conserved energy, azimuthal angular momentum and Carter constant. The deformations represent modified gravity effects and have been analytically mapped to several modified gravity black hole solutions in four dimensions. In the analytic kludge waveforms, the conservative motion is modeled by a post-Newtonian expansion of the geodesic equations in the deformed spacetimes, which in turn induce modifications to the radiation-reaction force. These analytic-kludge waveforms serve as a first step toward complete and model-independent tests of general relativity with extreme mass-ratio inspirals.
DOI:10.1103/PhysRevD.84.064016 PACS numbers: 04.30.w, 04.50.Kd, 04.25.g, 04.25.Nx
I. INTRODUCTION
‘‘I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.’’ [1] Isaac Newton’s quote reminds us of the great unknowns that remain in gravitational astrophysics [2]. Our under-standing of this field should soon be revolutionized by the detection of gravitational waves (GWs) with ground [3,4] and space-based interferometers [5–8]. Indeed, the ground-based detectors LIGO [9] and Virgo [10] are currently undergoing upgrades toward a sensitivity at which a first direct detection of gravitational waves is likely. Space-borne detectors are being planned and will hopefully be operational in the next decade.
Detectors in space will be sensitive to low-frequency [ð105–101Þ Hz] GWs. One of the most promising sources in this frequency range are extreme-mass ratio inspirals (EMRIS) [11]. These systems consist of a stellar-mass compact object [ð100–102ÞM], such as a neutron star or black hole (BH), that spirals into a super-massive BH [ð105–107ÞM], typically on an inclined and eccentric orbit. Because of their extreme mass-ratio, these inspirals proceed slowly, generating hundreds of thousands of cycles of gravitational radiation while the smaller object is in the strong-field region close to the central supermas-sive black hole. These GWs encode detailed information about the structure of the spacetime exterior to massive
compact objects and the nonlinear, ‘‘strong-field’’ nature of the gravitational theory that describes the dynamics of the inspiral.
The detection of such GWs requires the construction of accurate waveform templates that allow the extraction of signals from noisy data via matched filtering. EMRI wave-forms, however, are particularly difficult to construct, as one requires these to be accurate over hundreds of thou-sands of cycles. Progress has been made by constructing approximate waveforms that encode all the key features of EMRI waveforms, while being computationally inexpen-sive. There are two families of such ‘‘kludge’’ waveforms that have been used extensively—the ‘‘analytic kludge’’ (AK) developed by Barack and Cutler [12], which we will focus on in this paper, and the numerical kludge [13,14]. In the numerical kludge, the orbital trajectories are based on exact Kerr geodesics, the parameters of which are slowly evolving under the influence of radiation-reaction. Once the orbital trajectories have been obtained, the waveforms are constructed using a quadrupole approximation. The AK model is built around the gravitational waveforms gener-ated by particles describing Keplerian ellipses [15,16], whose semimajor axis, eccentricity and inclination angle evolve according to certain post-Newtonian (PN) equations [15–17]. Relativistic precession of the orbital plane and the perihelion are also included using post-Newtonian approximations.
In the past, attempts have been made to modify EMRI waveforms to study their ability to test general relativity
(GR). Such attempts can be divided into two classes: ‘‘extrinsic’’ or ‘‘intrinsic.’’ The former picks a concrete alternative theory, such as dynamical Chern-Simons (CS) gravity [18,19], constructs waveforms for that particular theory [20–23] and compares these to the predictions of GR. The latter are null-tests of GR, where one assumes the validity of GR a priori and concentrates on internal con-sistency tests, such as measuring the multipolar structure of the metric [24–27], or multimodal spectroscopy of in-spiral and ringdown waveforms [28–30]. Both classes of tests are intrinsically valuable, but they are not ideal to search for departures from GR in a systematic and model-independent way.
Recently, there has been a focused effort to develop such a systematic and model-independent approach. Yunes and Pretorius [31–33] proposed the parametrized post-Einsteinian (ppE) framework, a model-independent template family for the quasicircular inspiral of compa-rable mass, nonspinning, compact binaries. In this ap-proach, deformations of the conservative Hamiltonian and of the radiation-reaction force are mapped onto the waveform observable, i.e. the frequency-domain GW re-sponse function. The meta-template family allows for ge-neric deformations away from GR, that have been shown to reproduce waveform predictions from all known alterna-tive theories proposed to date. However, this approach is ill-suited to EMRI waveforms, as the orbits are likely to be inclined and eccentric and the mass ratios are extreme.
To address this shortcoming, Vigeland, Yunes and Stein [34] took the first steps toward extending the ppE framework to EMRIs. They concentrated on conservative corrections to the orbit, realizing that these could be pa-rametrized by deformations of the metric tensor. Previous attempts to construct such ‘‘bumpy spacetimes’’ [25–27] had focused on intrinsic or null tests, and had therefore used metrics that satisfied the Einstein equations, but with potentially unphysical matter distributions, such as naked singularities, or unphysical spacetime regions, e.g., closed timelike curves. The approach of [34] was to only allow for the subclass of metric deviations that would ensure the existence of an approximately conserved energy, (z-component of ) angular momentum, a second-order Killing tensor and a Carter constant, without requiring that the Einstein equations be satisfied. Such an approach led to a parametrically deformed metric that was shown to map to metrics in known alternative theories in four di-mensions [20,35].
In this paper, we construct corrections to the AK wave-forms by considering geodesics in the modified gravity, bumpy metric of [34]. We begin by providing explicit expressions for all components of the metric deformation as an expansion in r M, where r is the field-point distance to the supermassive BH with mass M; we require that the metric remain asymptotically flat, with the same scaling at spatial infinity as that predicted by the GR
peeling theorems, and disallowing pure angular defor-mations. We show that even with these restrictions, all known metrics in alternative theories are still recovered by an appropriate choice of the parameters characterizing the metric functions [20,35]. Arbitrary choices of these parameters lead to parametric, metric deformations at Oð1=r2Þ, Oð1=r3Þ, Oð1=r4Þ and Oð1=r5Þ relative to the Kerr metric.
We then calculate the geodesics equations associated with this background. We parametrize the orbits via the location of their turning points and will relate various quantities to the Kerr geodesic orbit with the same turning points. The existence of three approximately conserved quantities allows us to explicitly separate the geodesic equations into first-order form. From these, we calculate the three orbital frequencies associated with the orbital motion. Although we do not introduce explicit corrections to the radiation-reaction force, modifications to the fluxes of energy, angular momentum and Carter constant will be introduced due to modifications to the orbit itself. We calculate these implicit modifications with a quadru-polar approximation for the fluxes of energy and angular momentum. The Carter constant flux is calculated by as-suming that the inclination angle remains approximately constant and by requiring that exactly circular orbits re-main circular under radiation reaction. Finally, we collect all pieces of the calculation, providing an explicit prescrip-tion to build AK waveforms in these families of parametri-cally deformed spacetimes.
The study performed here is similar, yet more generic than others already carried out in the literature. For ex-ample, Barack and Cutler [28] considered modifications to AK waveforms induced by a perturbation to the quadru-pole moment of a Kerr BH. Using results in [36], they introduced modifications to the precession frequencies and rate of change of orbital frequency that would be induced by a quadrupole moment deformation; they then searched for the accuracy with which LISA could measure the size of such a deformation. This study, however, neglected modifications to the eccentricity evolution. Glampedakis and Babak [37] also considered a class of bumpy space-times which differed from Kerr in the quadrupole moment only. Unfortunately these metrics are of Petrov Type I, and thus do not allow for the existence of a Carter constant or for the separability of the geodesic equations [38]. Moreover, both studies neglected the possibility that the metric could be modified at multipole orders higher than quadrupole. This is a critical disadvantage, as strong-field modifications of GR are likely to introduce corrections at higher than quadrupole order, e.g., dynamical CS gravity modifies the metric at hexadecapole order, leaving the quadrupole and octopole unchanged.
Although the waveforms presented here will be use-ful for studies that determine the accuracy to which instruments like LISA could constrain model-independent
deviations from GR, they are not unique or complete. To demand uniqueness is futile in a ppE scheme, as it is also in the parametrized Newtonian or parametrized post-Keplerian schemes that are used to search for deviations from GR in the Solar System and in binary pulsar obser-vations, respectively. An infinite number of theories predict an infinite number of possible deviations in the observables of interest, while the schemes above can only parametrize a subset of them, i.e., the subset that can reproduce all the predictions of alternative theories that are known to date [20,35].
These waveforms are also not complete because here we will modify only one of the three main ingredients that go into waveform construction, i.e., the metric tensor, and will neglect modifications to the first-order equations of motion (that prescribe how GWs are sourced by matter distributions) and the second-order equations of motion (that describe the self-force and radiation-reaction). How-ever, a large subclass of quadratic gravity theories exist in which the corrections to the metric are dominant over modifications to the wave generation and radiation-reaction [39]. Nonetheless, a more complete analysis should also investigate the excitation of scalar and vecto-rial modes in the metric perturbation, which could arise from modifications to the wave generation. We leave an investigation of these other effects to future work, but the results found here will still hold when these other modifi-cations are introduced.
The remainder of this paper is organized as follows: SectionIIintroduces the new bumpy framework to model modified gravity theories; Section III calculates the geo-desic equations that preserve the Kerr turning points; Section IV computes the orbital frequencies associated with the modified orbital motion; Section V calculates the implicit deformations introduced to the radiation-reaction force; SectionVIbuilds AK waveforms from the modified orbital frequencies and fluxes; and Section VII
concludes and points to future research.
Throughout this paper we use the conventions of Misner, Thorne and Wheeler [40]. We use Greek letters to denote spacetime indices, while Latin ones in the middle of the alphabet i, j;. . . stand for spatial indices only. We also use geometric units with G ¼ c ¼1. Background quantities are denoted with an overhead bar, while quantities associ-ated with geodesics in the Kerr metric are denoted with a subscript K.
II. BUMPY SPACETIMES FOR MODIFIED GRAVITY
In this section we discuss the parametrically deformed BH metric that we will later use to study geodesics. We begin by recapitulating the most important results of [34] for this paper. We then simplify this metric prescription by considering expansions in M=r 1, thus allowing us to compute explicit expressions for all components of the
metric deformation. Finally, we describe some properties of the new metric.
A. Spacetime construction
We decompose the metric tensor that is to describe the background spacetime of a supermassive BH as
g¼ gþ h; (1) where 1 is a ‘‘deformation’’ book-keeping parameter that reminds us that jhj=jgj 1. The background metric is assumed to be the Kerr metric g¼ gK, which in Boyer-Lindquist coordinates has components:
gKtt¼ 1 2Mr 2 ; gKt¼ 2M 2ar 2 sin 2; (2) gKrr¼ 2 ; gK¼ 2; gK¼ 2sin2; (3) for a BH with mass M and spin angular momentum directed along the symmetry axis of magnitude S ¼ M2a, where a is the dimensionless Kerr spin parameter. Equations (2) and (3) depend on the functions
2 r2þ a2M2cos2; (4) r2f þ M2a2; f 1 2M
r ; (5) ðr2þ M2a2Þ2 M2a2sin2: (6) We restrict attention to a certain class of metrics. We begin by requiring that the full metric be stationary and axisymmetric, although it need not solve the Einstein equations, i.e. it is a solution to a more general set of modi-fied gravity field equations that have a smooth GR limit: g! g as h! 0. In addition to the existence of a temporal and an azimuthal Killing vector, we also assume that a certain integrability condition holds (see Eq. (49) in [34]) such that the metric can be written in Lewis-Papapetrou form. The deformation of this metric g¼ gþ h, can be transformed to Boyer-Lindquist-like coordinates in which all components of the metric pertur-bation vanish except ðhtt; ht; hrr; hr; h; hÞ. These are the only components that are allowed to be nonzero.
With this at hand, we then force the full metric g to possess three constants of the motion: a conserved energy, azimuthal component of angular momentum and Carter constant. The first two are generated directly from the Killing vectors and are exact, while the last one is built from an approximate, second-order Killing tensor valid at least toOð2Þ. This Killing tensor is parametrized as
¼ lðkÞþ r2g: (7) The parametrization of the Killing tensor as in Eq. (7) identifies the coordinate r with a Boyer-Lindquist-like
radial coordinate. The vectors l and k are required to be null and is required to satisfy the Killing ten-sor equation, which determines the components of the null vectors up to some arbitrary functions of the radial
coordinate. This in turn determines the final form for the metric perturbation.
The nonvanishing components of the metric deforma-tion can be written as [34,41]
htt¼ a PDK2 PDK1 ht a 2 4 PDK1 @ht @r 2 M2a2rðr2þ M2a2Þ sin cos 2PDK1 hrþ ðr2þ M2a2Þ ^2 2PDK1 I þ 2M2a2r2sin2 PDK1 1þ ^ 2ðr2þ M2a2Þ 2PDK1 3 a sin 2 2 PDK3 PDK1 3þ 22 PDK4 PDK1 4 a2M 2 22sin2 PDK1 d1 dr a 2 2ð þ 2a2M3rsin2Þsin2 PDK1 d3 dr a2M 2 2ð2 4MrÞsin2 PDK1 d4 dr ; (8) hrr¼ 1I 1 3; (9) h¼ ðr2þ M2a2Þ2 M2a2 httþ M2a2I 2ðr2þ M2a2Þ a htþ M2a23 2 2sin2 Ma 3þ 2 2 M2a24; (10) @h @r ¼ 2 r 2hþ 2 M2a2sin cos 2 hrþ 2 @hr @ þ 2 r 2I 2r1þ 2 r 23; (11) @2ht @r2 ¼ 8 aM2sin cos 8 PDK5 PDK1 hr 4 aM2rðr2þ M2a2Þ sin cos 6 @hr @r þ 2 M2a2sin2 4 PDK6 PDK1 ht 2r 2 PDK7 PDK1 htþ 4 aM2rsin2 4 PDK15 PDK16 I 4aM2rsin2 4 PDK8 PDK1 1þ 4 aM2r 4 PDK9 PDK1 3þ 2sin 2 4 PDK10 PDK1 3 16aM2sin2 4 PDK11 PDK1 4 2 aM 4 PDK12 PDK1 d1 dr 2sin 2 4 PDK13 PDK1 d3 dr 2 aMsin2 4 PDK14 PDK1 d4 dr aMsin2 2 d21 dr2 sin2 4 ð þ 2a 2M3rsin2Þd23 dr2 aMð2 4MrÞsin2 4 d24 dr2 ; (12)
where ^2 r2 M2a2cos2 and PDKi are polynomials in r andcos, given explicitly in the Appendix of [34] (we have here adopted the deformed Kerr parametrization of [34]). The quantities i¼ iðrÞ are arbitrary functions of radius, while3¼ 3ðÞ is an arbitrary function of polar angle. The quantityI is defined as
I ¼Zdr2M 2a2sincos 2 hrþ2r1þ 2d1 dr : (13)
The metric perturbation component hr is free.
The parametrically deformed metric represents a family, some members of which are well-known BH solutions in modified gravity theories. For example, a certain choice of deformation parameters A leads to the slowly-rotating BH solution in dynamical CS gravity [20], while another choice leads to modified Schwarzschild BHs in quadratic gravity theories [35], as was shown in [34]. In both cases, these solutions derive from field equations that arise from a diffeomorphism invariant theory, with a well-defined
Lagrangian density. We refer the interesting reader to [34] for more details on the metric construction.
B. Simplification of the parameterization Let us simplify the metric perturbation of the previous section with the following criteria:
(1) Asymptotic flatness: Require that h! 0 at spatial infinity.
(2) Peeling: Require that jhj r2 or faster for r=M 1.
(3) Occam’s razor: Set the largest number of metric components to zero that are not needed to repro-duced known modified gravity predictions for the metric tensor in four dimensions [20,35].
Requirement (1) ensures that there is no constant piece to the metric deformation, such that the total metric g! g at 0. Since the Kerr metric is asymptotically flat, so would the total metric. Requirement (2) ensures that the BH mass is not renormalized by1=r corrections to the
ðt; tÞ or ðr; rÞ metric components. The norm in this require-ment is to be taken with the flat metric in spherical coor-dinates, such that the ð; Þ subsector is simply the metric on the 2-sphere. These two requirements ensure that the metric perturbation satisfies all Solar System constraints, as it introduces modifications at higher than leading, Newtonian-order in a weak-field expansion. Furthermore, any1=r correction to ðhtt; ht; hrrÞ would renormalize the BH mass or spin angular momentum, which would not be measurable. Requirement (3) automatically implies that (i) we can set hr¼ 0; and (ii) we can disallow pure angular deformations by setting3¼ 0.
Since the purpose of this paper is to construct ppE AK waveforms, which make extensive use of weak-field ex-pansions in M=r 1, we choose to parametrize the re-maining free functions as Taylor series:
A¼ X1 n¼0 A;n M r n ; 3¼ 1 r X1 n¼0 3;n M r n ; (14) where A ¼1 or 4 and i;n are dimensionless constants. We have pulled out a factor of1=r in the expansion of 3 because this quantity has dimensions of ½M1, as one can see from h in Eqs. (8)–(12). Notice that with these definitions the m;nconstants are dimensionless.
With this at hand, let us simplify the metric components, starting with hrr. The integralI can now be solved exactly: I ¼ 2
1. Via requirement (3) hrr¼ 1
2
: (15)
Requirements (1) and (2) force us to choose 1;0¼ 0 ¼ 1;1, since 2= ! 1 for r M.
The next simplest component to analyze is h, whose behavior is governed by Eqs. (8)–(12), which using the previous results simplifies to @h=@r ¼2r=ð2Þh. The solution to this equation is h¼ 4ðÞ2. Since g¼ 2, this correction would be leading order in the angular sector. By requirement (3), we disallow it and set 4 ¼ 0. This simplification, and the restrictions made when deriving the metric perturbation, fix the coordinate system in such a way that h¼ 0 and therefore g¼ 2þ Oð2Þ. The radial coordinate thus preserves, at lead-ing order, its physical interpretation in the Kerr metric—it may be interpreted as a circumferential radius in the equa-torial plane and becomes the oblate-spheroidal radial co-ordinate in flat-space at infinity. The fact that the radial coordinate has been fixed in this way will be important for interpretation of the waveforms we derive in this paper. This will be discussed further in Sec.VI.
To determine the remaining metric components, we must first solve the differential equation for ht, as the htt and h components depend explicitly on the former. This is an elliptic equation that could be solved numeri-cally. Since we seek analytic solutions only, however, we will solve it in the M=r 1 limit. To do so, we write
ht ¼ M XN n¼2 ht;nðÞ M n þ O 1 Nþ1 ; (16)
where ht;n are functions of that we will determine by solving Eqs. (8)–(12). We could have chosen to expand ht in a 1=r basis, but we empirically found that a 1= basis yields simpler results. Solving Eqs. (8)–(12) order by order in1=, we find that the first few nonzero terms are ht;2¼ sin2½3;3 að1;2þ 4;2Þ a23;1; (17) ht;3¼ sin2½ð23;3 3;4Þ þ að64;2 4;3þ 21;2 1;3Þ þ 2a23;1ð4cos2 3Þ (18)
ht;4 ¼ sin2½a43;1 ð4;2þ 1;2Þa3þ 2ð43;1cos2 3;3þ 43;1Þa2
þ ð1;4 84;2þ 64;3 4;4þ 21;3Þa þ 23;4 3;5; (19) ht;5¼ sin2f23;5þ að4;5 84;3þ 21;4þ 64;4 1;5Þ þ a2½23;4þ 3;3ð2 þ 5cos2Þ ð1=2Þ3;4cos2
þ a3½ð1=2Þcos2ð2 1;2 4;3 1;3 24;2Þ 4;3 1;3þ 44;2 þ a4ð43;1cos4 þ93;1cos2 43;1Þg; (20) ht;6¼ sin4½a2ð3;5 63;4þ 83;3Þ þ a3ð1;4 24;3þ 4;4 21;3Þ þ 23;3a4þ a5ð4;2þ 1;2Þ þ a63;1 þ sin2½2 3;6 3;7þ að64;5þ 21;5 84;4 1;6 4;6Þ þ a2ð43;4 33;5Þ þ a3ð2 1;4þ 21;3þ 64;3 24;4Þ 33;3a4 a5ð1;2þ 4;2Þ 3;1a6; (21) where we have here simplified the solution by setting
4;0¼ 0, 4;1¼ 0 and 3;2¼ 0. We will find that these conditions are necessary to ensure the metric is asymptoti-cally flat. We have also set 3;0¼ 0, which is required for the differential equation to be satisfied.
Let us now return to the httand hcomponents, which have been completely specified by the above solutions. At spatial infinity we find that
htt 24;0 2
4;1 8M4;0
h r 2
2aM23;2sin2 þOð1Þ; (23) when M=r 1. By requirements (1) and (2), this then implies that ð4;0; 4;1; 3;2Þ must all be set to zero.
In summary, the requirements of asymptotic flatness and the nonrenormalization of the mass, have forced us to the following conditions
3ðÞ ¼ 0; 1;0¼ 0; 1;1¼ 0; 3;0¼ 0; (24) 4;0¼ 0; 4;1¼ 0; 3;2¼ 0: (25)
With these choices, the metric perturbations ðh; hrÞ vanish, hrr is given by Eq. (15) and ht is given in the far field by Eq. (16), with the angular functions given in Eq. (21). The remaining components ðhtt; hÞ are given explicitly by Eqs. (8)–(12).
Let us now take the far-field expansion of all the metric perturbations: h¼X n h;n M r n : (26)
The first few nonzero terms are
htt;2¼ 1;2þ 24;2 2a3;1sin2; htt;3 ¼ 1;3 84;2 21;2þ 24;3þ 8a3;1sin2;
htt;4¼ 84;3 21;3þ 24;4þ 84;2þ 1;4 8a3;1sin2 þ a2ð1;2þ 24;2Þsin2 þ2a33;1cos2sin2; htt;5¼ 16a33;1sin4 þsin2½4a3;3þ a2ð1;3 21;2 124;2þ 24;3Þ 12a33;1
þ a2ð8
4;2þ 21;2Þ þ 1;5þ 24;5 21;4þ 84;3 84;4; (27) hrr;2¼ 1;2; hrr;3¼ 1;3 21;2; hrr;4¼ 1;4 21;3 41;2þ ð1=2Þ1;2a2ð1 cos2Þ;
hrr;5 ¼ a2sin2ð1;3þ 21;2Þ 1;5 21;4 41;3 81;2þ 2a21;2; (28) ht;2¼ Msin2½3;3þ að1;2þ 4;2Þ þ a23;1;
ht;3¼ 8Ma23;1sin4 þ Msin2½ð23;3 3;4Þ þ að64;2 4;3þ 21;2 1;3Þ þ 23;1a2; ht;4¼ Msin4½a2ð83;1 3;3Þ þ a3ð1;3 4;2Þ a43;1
þ sin2½ð2
3;4 3;5Þ þ að4;4 84;2þ 64;3 1;4þ 21;3Þ a23;3
ht;5¼ 16Ma43;1sin6 þ Msin4½a2ð23;3 3;4Þ þ a3ð4;3þ 104;2þ 21;2 1;3Þ þ 14a43;1 þ sin2½ð2
3;5 3;6Þ þ að1;5 84;3 4;5þ 21;4þ 64;4Þ 3;4a2þ a3ð21;2 64;2Þ 2a43;1; (29) h;2¼ 0; h;1 ¼ 0; h;0 ¼ 2M2a3;1sin4; h;1 ¼ 0;
h;2¼ M2sin4½2a3;3þ a21;2þ a33;1ð4 2cos2Þ;
h;3¼ 8M2a33;1sin6 þ M2sin4½að43;3þ 23;4Þ þ a2ð21;2 44;2þ 1;3Þ 4a33;1; (30)
Notice that although ðhtt; hrrÞ are indeed dimensionless, hthas units of length and hhas units of length squared, as expected since these are also the dimensions of the corresponding components of the Kerr metric.
The perturbation is parameterized by a number of con-stants, depending on how many terms in M=r are kept relative to the leading-order Kerr metric: up toOðM2=r2Þ, the metric deformation is given by the 4 constantsB2 ð1;2; 3;1; 3;3; 4;2Þ; up to OðM3=r3Þ it is given by the 7 constants B2[ B3, whereB3 ð1;3; 3;4; 4;3Þ; up to OðM4=r4Þ it is given by the 10 constants B
2[ B3[ B4, where B4 ð1;4; 3;5; 4;4Þ; up to OðM5=r5Þ it is given by the 13 constants B2[ B3[ B4[ B5, where B5 ð1;5; 4;5; 3;6Þ. Later on in the paper, we will take certain
BN limits, where we mean we will only let the BN coef-ficients be nonzero and set all others to zero.
Lastly, note that known BH solutions in alternative theory of gravity can be reproduced with this parametriza-tion, as shown in [34]. In particular, in this paper we will frequently compare our results to that of dynamical CS gravity, where the metric of a slowly-rotating BH is iden-tical to Kerr, except in its ðt; Þ component, which is given by [20] gCSt¼ 2M 2ar 2 sin 2 þ5 8 aM5 r4 1þ12M 7r þ27 M2 10r2 sin2; (31)
where is a dimensionless coupling constant of the theory. This metric can be completely reproduced by the above parametrization by choosing all i;j¼ 0, except for 3ðrÞ, whose first nonvanishing terms are
3;5¼ 5
8a; 3;6¼ 6528a: 3;7¼ 709112a: (32) C. Physical properties of the deformed metric At this junction, one might wonder how the metric of Eqs. (8)–(12) may change physical properties of the space-time. For a stationary and asymptotically flat spacetime, the event horizon coincides with the Killing horizon, and it is given by the hypersurface where
g2t g2
gtt
g¼ 0: (33)
Let us now use the fact that at r ¼ rþ M þMð1a2Þ1=2, ¼ 0 and then h¼ ðr 2þ M2a2Þ2 M2a2 htt 2 ðr2þ M2a2Þ a ht: (34) With this at hand, linearizing Eq. (33) in leads to
0 ¼ 2ht Ma þ htt 2 þ 2 M2a2 : (35) This quantity clearly vanishes at r ¼ rþ, which then im-plies that the event horizon of the full metric remains at its Kerr value. Moreover, one can also show that the horizon coincides with the location of a coordinate singularity, by evaluating the components of the inverse metric.
Other quantities, however, will be different in the de-formed spacetime relative to their Kerr values. For ex-ample, it is likely that the geometry of the ergosphere, defined by the hypersurface where gtt ¼ 0, will be modi-fied. These and other properties of the spacetime metric will not be needed in this paper, and so we leave them to be explored elsewhere.
III. GEODESIC EQUATIONS
In this section, we study the geodesic equations associ-ated with the new metric we computed in the previous section. We begin by summarizing the most important results of [34], regarding the geodesic equations. We then fix the location of the turning points of the orbit and compute the modifications to the conserved quantities relative to the Kerr geodesic with the same turning points.
A. First-order equations
Let us define the dimensionless constants of the motion associated with this new metric as follows:
E t u
; ML u M2C u u ; (36) where t ¼ ½1; 0; 0; 0 is a timelike Killing vector, ¼ ½0; 0; 0; 1 is an azimuthal Killing vector and u ¼ ½_t; _r; _; _ is the 4-velocity, with overhead dots standing for differentiation with respect to proper time. The Carter constant is also often defined as Q C ðL aEÞ2. Here the approximate, second-order Killing tensor ¼ þ . The background Killing tensor is simply the Kerr one
¼ kð l Þþ r2g ; (37) with k and l the principal null directions of Kerr:
k ¼ r2þ M2a2 ;1; 0; Ma ; (38) l ¼ r2þ M2a2 ; 1; 0; Ma : (39)
The Killing tensor deformation is
½ kð l Þþ lð k Þþ 2h ð k Þl þ 3r2h : (40) where the deformed null vectors are
k ¼ r2þ M2a2 1þ 4; 1;0; aM 1þ 3 ; (41) l ¼ ½4;0; 0; 3; (42) and we have used the requirements of Sec.II Bto simplify these expressions.
The dimensionless constants of the motion, when eval-uated on the perturbed metric, become
E ¼ E þ ðhtuþ gt uÞ; (43) ML ¼ ML þ ðhuþ g uÞ; (44) M2C ¼ M2 C þ ð uuþ 2 uð uÞÞ; (45) and the normalization condition for the four-velocity is
0 ¼ huuþ 2 gu u; (46) since by definition 1 ¼ guu. In all of these equa-tions, u¼ ½_t; _r; _; _ is the unperturbed, Kerr four-velocity, while u is a perturbation ofOðÞ.
The second-order geodesic equations can be rewritten as a first-order set through Eqs. (43)–(46). To achieve this, one must make a gauge choice for the constants of the motion associated with the full spacetime ðE; L; CÞ. In [34], the authors chose to keep 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. (43)–(45) vanish. This choice, however, forces the turning points of the orbit to be different from the GR orbit with constants ð E; L; QÞ.
Using this condition and Eq. (46), the geodesic equa-tions can be rewritten in first-order form:
2_t ¼ TKðr; Þ þ Tðr; Þ; (47) 4_r2 ¼ RKðrÞ þ Rðr; Þ; (48) 2 _ ¼ Kðr; Þ þ ðr; Þ; (49) 4 _2 ¼ KðÞ þ ðr; Þ; (50) where the Kerr potentials ðTK; RK;K;KÞ are given by
TK¼ aMðaM Esin2 M LÞ þ ðr2þ M2a2ÞP ; (51) RK¼ P2 ½M2 Q þ M2ða E LÞ2þ r2; (52) K¼ M2 Q M2L2cot2 M2a2cos2ð1 E2Þ; (53) K¼ aM E M L sin2 þaMP ; (54) where P Eðr2þ M2a2Þ M2a L. The perturbation po-tentials ð T; R; ; Þ are given by
Tðr; Þ ¼ðr 2þ M2a2Þ2 M2a2sin2 ht u þ 2aM2r h u ; (55) Rðr; Þ ¼½Aðr; Þr2þ Bðr; Þ; (56) ðr; Þ ¼ Aðr; ÞM2a2cos2 Bðr; Þ; (57) ðr; Þ ¼2aM 2r ht u 2 2Mr sin2 h u ; (58) and the functions Aðr; Þ and Bðr; Þ are given by
Aðr; Þ ¼2½h t_t þ h _u h u u ; (59)
Bðr; Þ ¼2½ð tt_t þ t_Þ utþ ð
t_t þ _Þ u
þ u u ; (60)
in which the four-velocity associated with the background trajectory u is ut _t ¼ 2T Kðr; Þ ur _r ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffi RKðrÞ q ; (61) u _ ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi KðÞ q ; u _ ¼ 2 Kðr;Þ; (62) and the four-velocity associated with the perturbation is
ut _t ¼ 2 T; u _ ¼ 2 : (63) Let us expand the perturbation to the geodesic equations in the far-field limit, using the metric in Eqs. (27)–(30). Doing so, we find
T ¼ M2X 1 n¼0 Tn M r n ; ¼ MX 1 n¼2 n M r n ; (64) R ¼ M4 X 1 n¼2 Rn M r n ; ¼M2X 1 n¼0 n M r n ; (65) where we note that ð Tn; Rn; n; nÞ are all dimen-sionless, unlike ðTK; T;K; Þ which have units of M2, ðRK; RÞ which have units of M4and ðK; Þ which have units of M. With this, the first nonvanishing perturbations are
T0¼ ð24;2þ 1;2 2a3;1sin2Þ E; T1 ¼ ð1;3þ 24;3þ 21;2Þ E;
T2¼ E½ð1;4þ 21;3þ 24;4þ 41;2Þ þ a2ð1;2þ 24;2Þ þ L½3;3 að4;2þ 1;2Þ a23;1; T3¼ E½ð1;5þ 24;5þ 41;3þ 21;4þ 81;2Þ þ ð1;3þ 24;3Þa2 L½3;4þ ð1;3þ 21;2þ 4;3Þa; T4¼ E½ð41;2þ 24;4þ 1;4Þa2þ ð24;6þ 81;3þ 21;5þ 161;2þ 1;6þ 41;4Þ
þ L½a23;3 3;5 ð4;4þ 1;4þ 41;2þ 21;3Þa;
T5¼ E½ð41;3þ 24;5 161;2þ 1;5Þa2þ ð41;5þ 321;2þ 81;4þ 24;7þ 1;7þ 21;6þ 161;3Þ
þ L½3;4a2 3;6þ ð41;3 4;5 1;5 21;4 81;2Þa þ 2a31;2; (66)
R2¼ ð24;2þ21;2Þ E2 2 E L3;11;2;
R1¼ E2ð44;2þ24;3þ21;3Þþ4 E L3;1þ21;21;3;
R0¼ E2½ð24;4þ 21;4 44;3Þþð44;2þ31;2Þa2 L E½2ð4;2þ1;2Þaþ23;3þ4a23;1 þ L2ð2a3;11;2Þþ21;31;4½a2þ Q1;2;
R1¼ E2½ð24;5þ 21;5 44;4Þþð21;2þ44;344;2þ31;3Þa2þ E L½ð41;221;324;3þ44;2Þa þð43;323;4Þþ43;1a2þ L2½21;2 1;34a3;1þ Qð21;21;3Þþ21;41;5a21;3; R2¼ E2½ð44;5þ24;6þ21;6Þ þ ð44;444;3þ31;4þ21;3Þa2þa4ð1;2þ24;2Þ
þ E L½ð24;221;2Þa3þð43;423;5Þþ ð24;4þ 44;3 41;321;4Þa4a23;3 2a43;1
þ L2½2a3;3þð1;4þ21;3Þþa21;2þ2a33;1þ Q½ð1;4þ21;3Þa21;2þ 21;51;6a21;4; (67) 0¼ 23;1a E2sin2 þ2 E L 3;1; n>0 ¼ 0; (68)
2¼ Eð3;1a2þ a1;2þ a4;2þ 3;3Þ 2a L3;1; 3¼ E½3;4þ ð4;3þ 1;3þ 21;2Þa;
4¼ E½ð21;3þ 41;2þ 1;4þ 4;4Þa þ a23;3þ 3;5 Lða21;2þ 2a3;3Þ;
5¼ E½ð1;5þ 81;2þ 21;4þ 4;5þ 41;3Þa þ 3;4a2þ 3;6 2a31;2 L½ð1;3þ 21;2Þa2þ 2a3;4; 6¼ E½ð4;6þ 161;2þ 81;3þ 41;4þ 1;6þ 21;5Þa þ 3;5a2þ 3;7 ð21;3þ 81;2Þa3
þ L½ð1;4 21;3 41;2Þ 2a3;5þ a41;2: (69) In the CS limit, these separated equations become
TCS ¼ 5 8M2 M r 4 La; (70) RCS¼ 5 4M4 M r 2 E L a; (71) CS ¼ 5 8M M r 4 Ea; (72)
to leading order and CS¼ 0, all of which agrees with the results found in [21].
We finish this section with a comment on this decom-posed set of first-order equations. The existence of an approximate, second-order Killing tensor allowed us to construct a Carter constant, which in turn allowed us to rewrite the second-order geodesic equations in first-order form. This does not necessarily imply that the resulting first-order equations will be decoupled (i.e., that the _r source term depends on r only and the _ source term depends on only). Nonetheless, in the far-field limit, we have just found empirically in the above equations that the resulting first-order equations do separate.
B. Weak-field expansion
The equations presented above are not strictly weak-field expansions, as the GR constants of the motion ð E; L; QÞ also
depend on the radius of the orbit. We choose to parametrize the orbit in terms of a semilatus rectum p, an eccentricity, e and an inclination angle, tp. The weak-field limit corre-sponds to taking p 1. We define these constants of the motion from the radial and azimuthal turning points of the motion, which are given by _rðrÞ ¼ 0 and _ðtpÞ ¼ 0, with
r¼ Mp=ð1 eÞ: (73) For Kerr, setting RKðrÞ ¼ 0 ¼ KðtpÞ, we find that
EK 1 þ 1 2pðe2 1Þ þ 3 8p2ð1 e2Þ2þ Oðp3Þ; (74) LK ffiffiffiffippsintpþ 1 2p1=2ðe2þ 3Þ sintp a pðe 2þ 3Þsin2 tp a 2 2p3=2ð3 þ e2Þ sintpcos2tp þ sintp 38p3=2ðe2þ 3Þ2þ a 2 p3=2ð1 þ e 2Þ a
QK cos2tp p þ ðe2þ 3Þ 2a p1=2ðe 2þ 3Þ sin tp þ 1 p½ðe 2þ 3Þ2þ a2ð3 þ e2Þsin2 tp þ 4 a2 p3=2 sintpð2 þ e 2Þð3 þ e2Þþ Oðp2Þ: (76)
We distinguish here between ð E; L; QÞ and EK, LK, QK, since we are defining EK etc. to be the constants of the motion for the Kerr orbit that has the same turning points as the geodesic in the deformed spacetime.
We can now insert these relations into our expressions for ð T; R; ; Þ, but the resulting equations are quite horrendous. We will thus present results only for special cases in which certain i;jare nonvanishing. If onlyB2 ¼ ð1;2; 3;1; 3;3; 4;2Þ is nonvanishing (which corresponds to keeping only theOð1=r2Þ terms in h), then
TB2 ð1;2þ 24;2ÞM2 2M2a3;1sin2; (77) RB2 2M2r2p1=23;1sintpþ M2r2ð1;2þ 24;2Þ; (78) B2 2M2p1=23;1sintp 2M2a3;1sin2; (79) B2 2M 3ap1=2 r2 3;1sintp þM3 r2 ½að1;2þ 4;2Þ þ 3;3þ 3;1a 2: (80) In the higher-order cases, we can obtain a more general formula: if BN>2¼ ð1;N; 3;Nþ1; 4;NÞ is nonvanishing (which corresponds to keeping only theOð1=rNÞ terms in h), then (for N >2) TBN r2 M r N ð1;Nþ 24;NÞ; (81) RBN rM3 M r N3 ð1;Nþ 24;NÞ; (82) BN 0; (83) BN M M r N ½að1;Nþ 4;NÞ þ 3;Nþ1: (84) In the CS limit and to leading order in the weak-field, we find TCS 5 8 M6p1=2 r4 asintp; (85) RCS 5 4 M6p1=2 r2 asintp; (86) CS 0; (87) CS 58M 5 r4 a: (88) C. Orbital parametrization
If we define the orbit through the semilatus rectum, eccentricity and inclination parameter introduced above, then in the parametrically deformed spacetimes studied in this paper, the turning points are not in the same spatial location as the GR geodesic with the same ð E; L; QÞ. We can use the same parametrization of the orbit if we force the turning points to be in these locations. The correspond-ing constants of the motion are shifted from those in the Kerr spacetime with the given turning points by an amount ofOðÞ:
E ¼ EKþ E; (89) L ¼ LKþ L; (90) Q ¼ QKþ Q: (91) This only changes the ðTK; RK;K;KÞ potentials in the geodesic equations [Eqs. (47)–(50)], as ð E; L; QÞ cor-rections to ð T; R; ; Þ will be of Oð2Þ.
The shifts in the energy, angular momentum and Carter constant can be obtained by requiring that the turning points of the perturbed spacetime be the same as those of the GR Kerr spacetime. Expanding to OðÞ, the radial turning points lead to the conditions
0 ¼@RK @EK r E þ@RK @LK r L þ@RK @QK r Q þ Rðr; EK; LK; QKÞ; (92) while the location of the polar turning points provides the condition 0 ¼@K @EK tp E þ@K @LK tp L þ@K @QK tp Q þ ðtp; EK; LK; QKÞ: (93) These equations can be solved for ð E; L; QÞ to obtain
E ¼ Rþ Q ðtpÞ sin2 tp RðrþÞ ðR L þ cot2tpR QÞ R Q ðtpÞ sin2 tp RðrÞ ðRþ L þ cot2tpR þ QÞS1; (94) L ¼ R Q ðtpÞ sin2 tp RðrÞ ðRþ E 2a2Ecos2tpRþQÞ Rþ Q ðtpÞ sin2 tp RðrþÞ ðR E 2a2Ecos2tpRQÞS1; (95) Q ¼cot2tp2L L 2a2Ecos2tp E ðtpÞ; (96) where RE ¼ 2 p2 ð1 eÞ2þ a2 p2 ð1 eÞ2þ 2a2 p 1 e E 4a p 1 eL (97) RL ¼ 4a p 1 eE 2L p 1 e p 1 e 2 ; RQ ¼ p2 ð1 eÞ2 2 p 1 eþ a2 ; (98)
S ðRþE 2a2Ecos2tpRQþÞðRL þ cot2tpRQÞ ðRE 2a2Ecos2tpRQÞðRþL þ cot2tpRþQÞ; (99)
and all quantities are to be evaluated at ðEK; LK; QKÞ. Taking the weak-field limit of the above equations, when onlyB2 is nonvanishing, we find
EB2 1 2p3ð1 e2Þ2ð1;2þ 2a3;1sin2tpÞ; (100) LB2 1ffiffiffiffi p p ðasin3 tp3;1þ sintpð4;2þ 1;2=2ÞÞ; (101) QB2 2pffiffiffiffipsintp3;1; (102) while if onlyBNis nonvanishing for N >2 we have
EBN 1
2pNfN2ðeÞð1 e2Þ2ð1;Nþ 24;NÞ; (103) LBN 1
2p3=2NfNðeÞ sintpð1;Nþ 24;NÞ; (104) QBN p2NfNðeÞcos2tpð1;Nþ 24;NÞ; (105) where we have define the eccentricity function
fNðeÞ ¼ð1 þ eÞ
N ð1 eÞN ð1 þ eÞ2 ð1 eÞ2
: (106) If we take the CS limit, the leading-order pieces are
ECS ¼ 5
4ap11=2sintpð1 e2Þ2ð1 þ e2Þ; (107) LCS¼ 5
8ap4sin2tpðe2þ 3Þð3e2þ 1Þ; (108)
QCS ¼ 5
4ap7=2sintpcos2tpðe2þ 3Þð3e2þ 1Þ; (109) and we see that2 Q LCS¼ L QCS.
IV. FUNDAMENTAL FREQUENCIES
In this section, we calculate the fundamental frequencies of orbital motion for the parametrically deformed space-time. We first compute generic expressions for these fre-quencies, and then present results valid in the weak-field limit.
A. General results 1. Radial frequencies
The condition that the turning points of the radial motion are at r allows us to write the change to the right- hand side of the radial geodesic equation as
@RK @EK E þ @RK @LK L þ @RK @QK Q þ M 4 X1 n¼2 Rn M r n ¼Mp 1 e r r Mp 1 þ e M2 X 1 n¼2 vn M r n ; (110) where we have here factored out the two turning points r. Equating coefficients of rnallows us to derive a recursion relation for the constants vn:
v2¼ 2EK E; (111) v1þ 2p
v0þ 2p ð1 e2Þv1 p 2 ð1 e2Þv2 ¼ 2a2E K E 2LK L Q þ R2; (113) v1þ 2p ð1 e2Þv0 p 2 ð1 e2Þv1 ¼ 4aðaEK LKÞ E þ 4ðLK aEKÞ L þ 2 Q þ R1; (114) v2þ 2p ð1 e2Þv1 p 2 ð1 e2Þv0 ¼ a2 Q þ R0; (115) vnþ2þð1 e2p2Þvnþ1 p2 ð1 e2Þvn¼ Rn 8 n 1: (116)
These equations can be solved to derive successive coefficients. In practice, we will be interested in perturba-tions for which the series terminates at a finite N. If Rn N ¼ 0, then vn N ¼ 0 and the above equations pro-vide N þ2 equations for N unknowns. The two extra equations are actually redundant, as they give the E and L needed to keep the turning points fixed at r. These ð E; LÞ were already computed in the previous section.
The solution to these equations can be best studied by separating out a few special cases. Let us consider the situation where the series contains only one term, RN. Then, the above equations indicate that vN p2 and vn N p2þðnNÞ. Hence, all terms proportional to vn=rn have the same leading-order behavior in p. Now, let us consider the following special cases:
(i) If N 2, then the five coefficients (v2;. . . ; v2) are nonzero, but vn>2¼ 0.
(ii) If N <1, the coefficients are given by
v2¼ 2EK E; (117) v1¼ 4pEK E 1 e2 ; (118) v0¼ 2 p2ð3 þ e2Þ ð1 e2Þ2 þ a2 EK E þ2L L þ Q R2; (119) v1¼ 8p 3ð1 þ e2Þ ð1 e2Þ3 þ 4 pa2 1 e2
EK E 4aðaEK LKÞ E þ 4aEK L þ2ð2LK L þ QÞ p 1 e2 1 2p 1 e2 R2þ R1 ; (120) v2 ¼ 2ð5 þ 10e 2þ e4Þp4 ð1 e2Þ4 þ ð3 þ e2Þp2a2 ð1 e2Þ2 EK E þ 2p
1 e2½4aEK L 4aðaEK LKÞ E þp½ð3 þ e2Þp 4ð1 e2Þ ð1 e2Þ2 ð2LK L þ QÞ þ a2 Q ð3 þ e 2Þp2 ð1 e2Þ2 R2þ 21 ep 2 R1þ R0 ; (121)
where we have assumed all but one of the Ri’s are zero. (iii) If N 1, the previous five equations still hold, with
the Ri’s set to zero.
(iv) If N ¼1, 2, we have the additional equations 2p ð1 e2Þv2 p 2 ð1 e2Þv1 ¼ R1; (122) p2 ð1 e2Þv2 ¼ R2; (123) but these will be automatically satisfied for the E, L and Q derived earlier.
(v) If N >2, we have vn¼ 0 for n > N, v2; ; v2 are given by Eqs. (117)–(121) and
vNk¼ ð1e2Þ RN
2ep2þk½ð1þeÞkþ1ð1eÞkþ1 for0 k<N 2:
We note that vk is proportional to pkð2þNÞ.
The set of equations for the vn’s, E and L are linear, so to find the general solution we can proceed as follows. We denote by v2n , E2 and L2 the solution to the above equations with only R2 0 and define
Q2¼ cot2tp2LK L22a2EKcos2tp E2
ðtpÞ: (124)
Then, for N > 2, we denote by vN
n, EN and LN the solution to the above equations with only RN 0 and with
QN ¼ cot2tp2LK LN 2a2EKcos2tp EN: (125) The general solution is then given by vn¼ Pkvkn, E ¼ P k Ek, L ¼ P k Lk and Q ¼ P k Qk.
The radial geodesic equation is most conveniently inte-grated by changing variables. First, we define a new di-mensionless time parameter, :
d d 2 M d d; (126)
where is proper time (differentiation with respect to which was denoted by an overhead dot previously). Next, we parametrize the orbit in terms of the (dimension-less) semilatus rectum p, the eccentricity e and a phase anglec via
r ¼ Mp
1 þ e cosc: (127)
Equation (110) then becomes a differential equation forc, namely dc=d ¼pffiffiffiffiffiffiffiVc, where Vc ¼ VKc þ Vc and VKcðc; p; e; Þ ¼ 1 E 2 K ð1 e2Þ3½pð1 e2Þ p3ð1 þ e coscÞ ½pð1 e2Þ p 4ð1 þ e coscÞ; (128) Vcðc; p; e; Þ ¼ 1 ð1 e2Þ X1 n¼2 vn pnð1 þ e coscÞnþ2: (129) The Kerr potential depends on Mp3¼ ð1 e2Þr3 and Mp4 ¼ ð1 e2Þr4, where r3;4 are the other two turning points of the radial motion. These are given in terms of EK, QK, p and e by the expressions
p3 ¼ Y þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiY2 X; p4¼ Y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiY2 X (130) where Y 1 e 2 1 E2 K p; X ð1 e 2Þ3 1 E2 K a2QK p2 : (131) These relations imply that p3 1 and p4 a2in the weak field, so VKc ð1 E2KÞp2=ð1 e2Þ when p 1.
We can integrate the evolution equation for c over a complete orbital cycle to determine r, i.e., the time elapsed per radial cycle. Writing this in terms of the difference to the Kerr value,r¼ Krþ r, we obtain the correction to the radial period
r¼Z2 0 dc ffiffiffiffiffiffiffi Vc p Z2 0 dc ffiffiffiffiffiffiffiffiffiffi VKc p ¼ ð1 e2Þ7=2 ð1 E2 KÞ3=2 X1 n¼2 vn pn Z 0 ð1 þ e coscÞ nþ2 ½pð1 e2Þ p 3ð1 þ e coscÞ3=2 ½pð1 e2Þ p 4ð1 þ e coscÞ3=2dc; (132) which is valid for arbitrary radius at leading order in .
2. Polar frequencies
If we write z ¼cos2, the condition that the turning points of the polar motion are at z ¼cos2tp allows us to write the change to the right-hand side of the polar geode-sic equation as M2 ð1 zÞ½ Q zð Q þ 2LK LÞ þ2a2EK Ezð1 zÞ þ 2M2 3;1½EKLK aE2Kð1 zÞ ¼ M2cos2tp z 1 z ðt0þ t1zÞ; (133) and by matching powers of z we find
t0 ¼ 1 cos2 tp½ Q þ 23;1ðEKLK aE 2 KÞ; (134) t1 ¼ 2aðaEK E þ 3;1E2KÞ: (135) The coefficient of z then gives the relationship between Q, E and L we found in Eqs. (94)–(96).
Let us now rewrite the polar geodesic equation in a simpler way, by parametrizing the polar motion using
cos2 ¼cos2
tpcos2: (136) The polar equation then becomes
d d 2 ¼ VKð; p; e; Þ þ ðt0þ t1cos2tpcos2Þ; (137) VKð; p; e; Þ ¼ a2ð1 E2KÞðzþ cos2tpcos2Þ; (138) where zþ¼ QK a2ð1 E2KÞcos2tp: (139) As in the radial case, we can compute the perturbation to the period in by writing¼ Kþ and find
¼ 1 a3ð1 E2KÞ3=2 Z 0 t0þ t1cos2tpcos2 ðzþ cos2tpcos2Þ3=2 d: (140)
3. Azimuthal frequencies
We can write the azimuthal geodesic equation as d d ¼ 2 M d d ¼ rðrÞ þ ðÞ; (141) which allows us to define a radial and a polar contribution to the average advance of over a radial/polar period
;r¼ Z2 0 r½p=ð1 þ e coscÞ dc=d dc; (142) ;¼ Z2 0
½cos1ðcostpcosÞ
d=d d: (143) The average rate of advance of , !, can then be computed as ! ¼ d d ¼ ;r r þ ; : (144) We can write the individual contributions in the form ;r¼ K;rþ ;r and ;¼ K;þ ;in which we take the Kerr pieces to be
K;r¼ aM P EK ¼ aM2rEK aLK ; (145) K;¼MLsin2K: (146) Each of the corrections has two contributions—one from the change in the numerator, r etc., and one from the change in the denominator, dc=d. We obtain
;r¼ að1 e 2Þ3=2 ð1 E2 KÞ1=2 Z2 0
2pð1 þ e coscÞ E að1 þ e coscÞ2 L DrðcÞ½p2 2pð1 þ e coscÞ þ a2ð1 þ e coscÞ2 dc þ ð1 e2Þ3=2 ð1 E2 KÞ1=2 X1 n¼2 n pn Z2 0 ð1 þ e coscÞn DrðcÞ dc að1 e 2Þ7=2 ð1 E2 KÞ3=2 X 1 n¼2 vn pn Z 0
ð1 þ e coscÞnþ2f2pð1 þ e coscÞE
K að1 þ e coscÞ2LKg DrðcÞ3½p2 2pð1 þ e coscÞ þ a2ð1 þ e coscÞ2 dc; (147) DrðcÞ ¼ f½pð1 e2Þ p3ð1 þ e coscÞ½pð1 e2Þ p4ð1 þ e coscÞg1=2; (148) and ; ¼ 1 a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 E2 K q Z2 0 L ð1 cos2 tpcos2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zþ cos2tpcos2 q d 1 a3ð1 E2KÞ3=2 Z 0 ðt0þ t1cos2 tpcos2ÞLK ð1 cos2
tpcos2Þðzþ cos2tpcos2Þ3=2
d: (149)
In both the radial and polar corrections, the first term is the contribution from the change in the orbital constants E and L in the Kerr part of the numerator. The second term is the contribution from the change in the numerator due to the addition of the perturbation to the numerator. The third term is the contribution from the change in the denomina-tor. There is no second term in the contribution since the perturbation to _ is purely radial.
4. Temporal frequencies
The motion in t can be treated in the same way as the motion in . With the analogous definitions, the average rate of advance of time with respect to , !T, is
!T¼ dt d ¼ T;r r þ T; ; (150) where T;r¼ TK;rþ T;r and T;¼ TK;þ T;. The Kerr pieces of this rate of change are TK;r¼ ðr2þ M2a2Þ P ; TK;¼ aMðaMEKsin2 MLKÞ; (151)
while theOðÞ corrections are
T;r M ¼ ð1e2Þ3=2 ð1E2 KÞ1=2 Z2 0
½p2þa2ð1þecoscÞ2f½p2þa2ð1þecoscÞ2 Eað1þecoscÞ2 Lg ð1þecoscÞ2D
rðcÞ½p22pð1þecoscÞþa2ð1þecoscÞ2
dc þð1e2Þ3=2 ð1E2 KÞ1=2 X1 n¼0 Tr n pn Z2 0 ð1þecoscÞn DrðcÞ dc ð1e2Þ7=2 ð1E2 KÞ3=2 X 1 n¼2 vn pn Z 0
ð1þecoscÞn½p2þa2ð1þecoscÞ2
DrðcÞ3½p22pð1þecoscÞþa2ð1þecoscÞ2fEK½p
2þa2ð1þecoscÞ2aL
Kð1þecoscÞ2gdc; (152) and T; M ¼ 1 a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 E2 K q Z2 0
a L a2ð1 cos2tpcos2Þ E ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zþ cos2tpcos2 q d 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3;1EK 1 E2 K q Z2 0 ð1 cos2 tpcos2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zþ cos2tpcos2 q d 1 a3ð1 E2KÞ3=2 Z 0 ðt0þ t1cos2
tpcos2Þ½aLK a2EKð1 cos2tpcos2Þ ðzþ cos2tpcos2Þ3=2
d; (153)
in which DrðcÞ was defined in Eq. (149). The origin of each term is as described in the azimuthal case. We note that Tr
n denotes the radial part of the perturbation to the temporal potential. This is equal to Tn, except T0r¼ ð1;2þ 24;2ÞM2.
5. Combining results
The orbital frequencies quoted so far were written in terms of the parameter. We compute the perturbations to the physical frequencies, i.e., those expressed in Boyer-Lindquist time, by first defining
T ¼ T;rþ rT;; (154) ¼ ;rþ r;; (155) and then writing
¼ hd di hdt di ¼ T; r¼ 2 T ; ¼ 2 r T : (156) The factor of2 is included in the definitions of rand so that all three frequencies are angular frequencies, ex-pressed as radians per second. From the preceding equa-tions, we may find the perturbations to the physical frequencies to be ¼ T T ; r r ¼ T T ; (157) ¼ r r T T : (158) The frequencies that we need for the construction of AK waveforms, using the notation of Barack and Cutler [12], are the orbital frequency, 2, the perihelion precession frequency, _~, and the orbital plane precession frequency, _ (see Sec.VI). These are given by
2 ¼ r; _~ ¼ r; _ ¼ ; (159) and therefore the changes in the frequencies can be com-puted from ¼ T T ; (160) _~ _~ ¼ ðr Þ r T T ; (161) _ _ ¼ ð 2rÞ 2r T T : (162) B. Weak-field expansion 1. Radial frequencies
Denoting by vBN;i the leading order in p piece of the parameter viunder assumptionsBN, we find forB2
vB2;2 ¼ ð1 e 2Þ2 p3 ½1;2þ 2asin 2 tp3;1; (163) vB2;1 ¼ 2ð1 e 2Þ p2 ½1;2þ 2asin 2 tp3;1; (164) vB2;0 ¼ð1 e 2Þ p ½1;2 2asin 2 tp3;1; (165) and the other vB2;i’s are subdominant in the sense that vB2;i=pi is higher order in 1=p. We see that for these dominant terms vB2;i=pi Oð1=pÞ, which motivates us to define p-independent terms ~vB2;i ¼ vB2;i=pi1. For BN with N >2, we find vBN;i=pi Oð1=pN2Þ so we can define the p-independent quantities ~vBN;i ¼ vBN;i=piþ2N. We find vBN;i 0 for i > N 4,
~vBN;2 ¼ ð1 e2Þ2fN2ðeÞð1;Nþ 24;NÞ; (166) and for 1 i N 4
~vBN;i ¼ 2ð1 e2ÞfNi3ðeÞð1;Nþ 24;NÞ; (167) in which fNðeÞ is as defined in Eq. (106).
The corresponding expressions in the CS limit are ~vCS;2¼ 52ð1 e2Þ2ð1 þ e2Þa sintp; (168)
~vCS;1¼ 5ð1 e2Þð1 þ e2Þa sintp; (169) ~vCS;0¼ 54ð1 e2Þð3 þ e2Þa sintp; (170) ~vCS;1¼ 52ð1 e2Þa sintp; (171) ~vCS;2¼ 54ð1 e2Þa sintp; (172) where in this case ~vCSN;i ¼ vCSN;i=pi7=2.
With this identification of the leading-order parts of the vi’s, the integral for the change to the radial frequency reduces to r¼ pKð1 e2Þ1 X 1 n¼2 ~vnInþ2; (173) in which In¼ Z 0 ð1 þ e coscÞ ndc; (174)
the ~vn’s are as defined above and K represents the scaling of the frequency correction with p, which is K ¼5=2 for B2, K ¼ N 1=2 for BNwith N 3 and K ¼ 5 in the CS limit. We note that although we have written 1 as the upper limit of the sum for convenience, in general we will be able to terminate the summation as described above, e.g., at n ¼0 for B2, at n ¼ N 4 for BN with N 3 and at n ¼2 for CS. The integral Inis evaluated in AppendixA and shown to be equal to
In¼ X bðn=2Þc k¼0 e 2 2k n! ðn 2kÞ!ðk!Þ2; (175) in which bxc denotes the largest integer smaller than x. For the various special cases we have been considering else-where, the leading-order shifts in the radial frequency can be simplified to B2r ¼ 2p5=2½ð4 3e2Þ1;2 2að8 e2Þsin2tp3;1; (176) B3r ¼ ð3 e2Þ 2p5=2 ð1;3þ 24;3Þ; (177) B4r ¼ ð8 e2Þ 2p7=2 ð1;4þ 24;4Þ; (178) B5r ¼ ð15 þ 5e2 e4Þ 2p9=2 ð1;5þ 24;5Þ; (179) CSr ¼ 5 32p5asintpð12 e2Þð8 þ 9e2Þ: (180) 2. Polar frequencies
For p 1, we find zþ ðQ þ L2Þ=a2ð1 E2Þ and Q þ L2 p, so VKð; p; e; Þ p. The weak-field limit is therefore p3=2 t0þt1cos 2 tp 2 ; (181) which for assumptionsB2 gives
B2
p3=2f1;2þ 24;2þ ½3acos 2
tp 4a3;1g; (182) for assumptionsBN with N 3 gives
BN
pN1=2fNðeÞð1;Nþ 24;NÞ; (183) and in the CS limit gives
CS 5
4p5asintpð3 þ e2Þð1 þ 3e2Þ: (184) 3. Azimuthal frequencies
In the weak-field, the leading-order part of ;r comes from the terms
;r¼ 4a E p3=2 þ 2p Xffiffiffiffip 1 l¼N l plþ1 Xbl 2c k¼0 e 2 2k l! ðl 2kÞ!ðk!Þ2 2a ð1 e2ÞpKþ1 X1 n¼2~vn Inþ3; (185) in which we are using Nto denote the first nonzero l, Inis as defined above and K has the same meaning as in the radial case: K ¼5=2 for B2, K ¼ N 1=2 for BN with N 3 and K ¼ 5 in the CS limit. Under assumptions B2, the second term can be seen to dominate, N ¼ 2 and 2 2a ffiffiffiffipp sintp3;1. We therefore find
B2 ;r¼ 2 p2a3;1ðe 2þ 2Þ sin tp: (186) Under assumptionsBN with N 3, the second and third terms both contribute and we find
BN>2;r ¼ 1 pNþ1=2 2½3;Nþ1þ að1;Nþ 4;NÞIN þ 2að1 e2ÞfN2ðeÞ þ 2X N1 k¼2 fNkðeÞIk ð1;Nþ 24;NÞ ; (187)
which for the first few N’s gives B3;r¼ 1 p7=2ðð2 þ 3e 2Þ 3;4þ að5 þ 3e2Þ1;3 þ að8 þ 3e2Þ 4;3Þ; (188) B4;r¼ 1 4p9=2ðð8 þ 24e2þ 3e4Þ3;5 þ að40 þ 36e2þ 3e4Þ 1;4 þ 3að24 þ 16e2þ e4Þ 4;4Þ; (189) B5;r ¼ 1 4p11=2ðð8 þ 40e2þ 15e4Þ3;6 þ 2að34 þ 50e2þ 9e4Þ 1;5 þ að128 þ 160e2þ 21e4Þ
4;5Þ: (190) In the CS limit we find that the second term again domi-nates, which gives
CS;r¼ 5 4p9=2a 1 þ 3e2þ 3 8e4 : (191) The dominant contribution to ; is given by
; ¼ p3=2 2QK L cos2tpt0LK cos2 tpsintp þ2a2ð1 e2Þ p L þ t1LK 1 1 sintp ; (192) With theB2 assumptions, we find
B2;¼ 2a
p sintp3;1: (193) For assumptionsBN with N 3 and in the CS limit, we have zt0¼ Q and using Eqs. (94)–(96), we see that
2QK L LK Q cos2 tp ¼ 2 QK cos2 tp L2K sin2 tp L þ2a2EKLK E; ¼ 2a2 p ð1 e 2Þ L þ 2a2E KLK E: (194) Both terms therefore contribute at the same order giving
BN>2; ¼a 2ð1 e2Þ pNþ1 ½fNðeÞ þ ð1 e 2Þf N2ðeÞ ð1;Nþ 24;NÞ sintp; (195) which for the first few N’s gives
B3;¼ 2a 2ð1 e2Þ p4 ð1;3þ 24;3Þ sintp; (196) B4;¼ 3a 2ð1 e2Þ p5 1 þe2 3 ð1;4þ 24;4Þ sintp; (197) B5;¼ 4a2ð1 e2Þ p6 ð1 þ e 2Þð 1;5þ 24;5Þ sintp: (198) In the CS case, we find, at leading order in p,
CS;¼ 25 4 a3 p13=2ð1 e 2Þ1 þ 2e2þe4 5 sin2 tp: (199) If we linearize in a, for consistency with the order to which the CS limit was derived, this term can be seen to vanish CS
;¼ 0.
4. Temporal frequencies
In the weak-field, the leading-order part of T;r comes from the terms
T;r M ¼ 2 p3=2 E ð1 e2Þ3=2þ 2 ffiffiffiffi p p X1 l¼NT Tl plþ1 X bl=2c k¼0 e 2 2k l! ðl 2kÞ!ðk!Þ2 1 ð1 e2ÞpK2 X N4 n¼2~vn In; (200) in which NT denotes the first nonzero Tnand In, K have the same meaning as in the radial case and as in the preceding section.
We can now evaluate the dominant contribution to T;r in the weak-field for the various cases we have been considering. For assumptions B2, the second and third terms dominate and we find
T;rB2 ¼M p1=2 1;2þ 44;2 2asin2tp3;1 þ 3 ð1 e2Þ1=2ð1;2þ 2asin2tp3;1Þ : (201) Under assumptions BN for N 3, the third term domi-nates. For the first few N’s we obtain
T;rB3 ¼ 3M 2p1=2ð1 e2Þ1=2ð24;3þ 1;3Þ; (202) TB4 ;r¼ M p3=2 3 ð1 e2Þ1=2þ 1 ð24;4þ 1;4Þ; (203) T;rB5 ¼ 3M 2p5=2 3 þ e2 ð1 e2Þ1=2þ 2 ð24;5þ 1;5Þ: (204) In the CS limit, at leading order in p, the third term again dominates and we obtain
TCS;r¼ 15M 8 a p3 sintp 4ð1 þ e2Þ ð1 e2Þ1=2þ ð4 þ e2Þ : (205) The dominant contribution to T; comes from the terms T; M ¼ 2 a L ffiffiffiffi p p 2a3;1EKð2 cosffiffiffiffi 2tpÞ p p aLK p3=2 t0þcos 2 tpt1 2 þa2EK p3=2 1 cos2tp 2 t0þ 1 3cos2tp 4 cos2 tpt1 : (206) Under assumptions B2, the second term dominates and we find TB2 ; M ¼ 2affiffiffiffi p p 3;1ð2 cos2 tpÞ: (207) Under assumptionsBNfor N 3, the final term dominates giving T;BN>2 M ¼ a2 2pN1=2fNðeÞð2 cos2tpÞð24;Nþ 1;NÞ: (208) At leading order in p the same term dominates in the CS limit to give TCS; M ¼ 5 8 a3 p5 ð2 cos 2
tpÞ sintpðe2þ 3Þð3e2þ 1Þ: (209) If we linearize in a for consistency, then we find T;CS ¼ 0 in the CS limit.
5. Physical frequencies
In the weak-field Kerr metric, the leading-order parts of the frequencies are given by
rK¼ 2 ffiffiffiffi p p K¼ 2 ffiffiffiffi p p ; (210) ;rK¼ 2 a ffiffiffiffi p p ;K¼ 2; (211) T;rK M ¼ 2 p 1 e2 3=2 ; (212) T;K M ¼ 2a sintp; (213) MK¼ MrK ¼ MK¼1 e 2 p 3=2 ; (214) TK ¼ 42Mpð1 e2Þ3=2; (215) rK K ¼ 6p3=2; (216) K 2rK¼ 82ap2; (217) from which we obtain
M _~K¼ 3ð1 e 2Þ3=2 p5=2 ; M _ K¼ 2a ð1 e2Þ3=2 p3 : (218) We can now put together the pieces from the preceding sections to derive the leading-order corrections to the three frequencies, which we require in order to construct the modified AK waveforms.
Orbital frequency—
The perturbation to the orbital frequency,2, is M ð2Þ M r¼ Mr T T ¼ ð1 e2Þ3 2p5=2 T ;r Mpffiffiffiffip þ T; Mpffiffiffiffip þ a sintp½ r : (219)
Under assumptions B2, the first two terms dominate and we find M B2r ¼ ð1 e 2Þ3 2p7=2 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 1 e2 p 1;2þ 44;2 2a1 þ4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 1 e2 p sin2 tp 3;1 : (220) Under assumptions BN with N 3, the first term alone dominates and we find
M BN>2r ¼ ð1 e 2Þ2 2pNþ1=2 X N4 n¼2 ~vnIn: (221) For the first few N’s we have
