Bumpy black holes in alternative theories of gravity
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Vigeland, Sarah, Nicolás Yunes, and Leo Stein. “Bumpy black
holes in alternative theories of gravity.” Physical Review D 83
(2011). © 2011 American Physical Society.
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http://dx.doi.org/10.1103/PhysRevD.83.104027
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PHYSICAL REVIEW D 83, 104027 (2011)
Bumpy black holes in alternative theories of gravity
Sarah Vigeland, Nicolás 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
SARAH VIGELAND, NICOLÁS YUNES, AND LEO C. STEIN
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
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 ;
g rr ¼
2
;
¼
g
2
;
g
¼
2
sin ;
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 :
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
(1)
2
(3)
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 Þ, where 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 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 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
Þ;
TMat
ðx Þ ¼ pffiffiffiffiffiffiffi u u ð4Þ ½x zð
(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
via d ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d g ðzÞz_ z_ , u ¼ dz =d is the particle’s 4velocity, normalized via g u u ¼ 1 and ð4Þ is the
C u u :
E t u ;
(12)
L
(13)
u ;
(14)
It is conventional to define another version of the Carter
constant, namely,
104027-3
SARAH VIGELAND, NICOLÁS YUNES, AND LEO C. STEIN
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
2 _
(16)
t ¼ aðaEsin2 LÞ þ ðr2 þ a2 Þ ;
2
_ ¼ aE L þ aP ;
sin2
(17)
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.
r ¼ ½ðr2 þ a2 ÞE aL2 ½Q þ ðaE LÞ2 þ r2 ;
(18)
4 _ 2
4 _ 2
¼ Q cot2 L2 a2 cos2 ð1 E2 Þ;
(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
(20)
2Mr
4aMr SBK
SBK
hSBK
¼
2
c
1 ;
1
þ
tt
1
2
4
¼ ð2 c SBK
SBK
Þ
hSBK
1
1
t
hSBK
¼ 2ðSBK
c SBK
Þ
rr
1
1
2aMrsin2
2
;
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
;
2SBK
1
2
ðr þ a2 Þ2Mr
4
hSBK
¼ SBK
1þ
1
r
2 2 2
2
SBK 8a M r sin
hSBK ¼ ðSBK
c
Þ
2 c SBK
1
1
1
2
2Mr
2
2a2 Mrsin2
;
2
þ
2
;
2 2 4Mr
2Mrðr2 þ a2 Þ
asin2 ;
2
4
2Mr
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.
(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
2
A. From standard to generalized bumpy black holes
þ
hSBK ¼ 2ðSBK
c SBK
Þ
1
1
2;
4aMr
2Mrðr2 þ a2 Þ sin2
SBK
;
þ
2
2
2
2Mr 1
(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
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
Þ, to arbitrary functions
ðhBK
tt ; htr ; ht ; hrr ; hr ; h ; h
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
SARAH VIGELAND, NICOLÁS YUNES, AND LEO C. STEIN
hBK
tt ¼ 2
2
a 4 @hBK
a PBK
t
2
BK a sin
h
t
2M PBK
M PBK
4M
@r
1
1
PHYSICAL REVIEW D 83, 104027 (2011)
ð 2 4MrÞ2 @1 @3
a 2 2 PBK
@2 @5
3
sin
þ
þ
4M
@r
@r
@r
@r
PBK
PBK
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
2
1
2Mr
aMrsin2
1
1
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
ð1 3 Þ 2
hBK ¼ 1 þ
@2 hBK
t
@r2
þ
a2
2 ;
2ðr2 þ a2 Þ BK 2
2 sin2
ht þ 2 1 þ 2 ð1 þ 3 Þ ð2 þ 5 Þ;
a
a
a
a
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 ðr2
2
sin2
a 2
2
2
ðPBK
3 2a Mrsin Þð2 5 Þ þ sin 4 ;
2
2
2
a2 Þ 2
hBK
tt @hBK
PBK
PBK
PBK
PBK
t
10;n
11;n
12;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
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 Þ are functions of
since ðhBK
tt ; htr ; ht ; hrr ; hr ; h ; h
only ðr; Þ.
The constants of the motion then become
E ¼ E þ ðh t u þ g t u Þ;
L ¼ L þ ðh
u
þ g u Þ;
C ¼ C þ ð u u þ 2 u ð uÞ Þ;
(36)
(37)
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:
Aðr; Þ ¼ 2½ht t_ þ h _ u h u u ;
Bðr; Þ ¼ 2½ðtt t_ þ t _ Þt_ þ ðt t_ þ 2 t_
M
jj ¼ jj
(39)
þ
¼ TK ðr; Þ þ Tðr; Þ;
¼ RK ðrÞ þ Rðr; Þ;
_ ¼ K ðr; Þ þ ðr; Þ;
4 r_ 2
2
4 _2
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
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)]:
(38)
since by definition 1 ¼ g u u . In these equations,
u ¼ ½t_ ; r_ ; _ ; _ is the unperturbed, Kerr 4-velocity, while
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
_ Þ _ þ u u :
while the normalization condition becomes
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
2
2aMr
2Mr
ht u ðr; Þ ¼
h
sin2
;
u
(42)
Let us first consider the most general spacetime metric
that one can construct for a stationary and axisymmetric
104027-7
SARAH VIGELAND, NICOLÁS YUNES, AND LEO C. STEIN
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
a
½b c
t Ra t
d
¼0¼
a
½b c
Ra t
d
2
2
d
:
þ 2 ðd
2
þ dz2 Þ;
(50)
2
ðrþ þ r Þ2 4M2 þ M2aa2 ðrþ r Þ2
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
þ ðz M2 a2 Þ2 . One can map from
where r ¼
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 , ,
and q Vw and perturb the metric functions via
V ¼ V þ V;
¼ þ ;
q ¼ q þ q;
¼ þ :
g ¼ g þ hDK
;
(53)
where the perturbation is given by
(49)
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]
V ¼
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:
ds2 ¼ Vðdt wd Þ2 þ V 1
PHYSICAL REVIEW D 83, 104027 (2011)
(52)
hDK
¼ q;
t
hDK
r
ðr MÞ2 sin2
;
¼ ðr MÞ cos sin ð Þ;
hDK
¼
2 þ hDK
rr ¼ cos
hDK ¼
sin2
ðr MÞ2
þ
cos2
(54)
;
2
2
sin
f4aMrq
2Mr
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; ; 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 ; hDK Þ to be arbitrary functions of
ðhDK
tt ; hrr ; hr ; h
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
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 PHYSICAL REVIEW D 83, 104027 (2011)
where i i ðrÞ are arbitrary functions of radius,
while lrDK and hDK
are arbitrary functions of both ðr; Þ.
r
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
2 PDK
2 PDK
M PDK
2M PDK
@r
PDK
1
1
1
1
1
2 ðr2 þ a2 Þ
a sin2 PDK
2 PDK
a2
3
4
þ
3
3
4
2 DK
2
2 PDK
M
2M
PDK
P1
1
1
þ
2 2 sin2
PDK
1
d1
dr
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
h
þ
I
þ
þ
4 ;
hDK ¼ 3
3
tt
t
a
a
a2
a2
a2
a2
@hDK 2r DK 2a2 sin cos DK
@hDK
2r
2r
r
¼ 2h þ
þ 2 I 2r1 þ 2 3 ;
h
þ
2
r
2
@r
@
@2 hDK
t
@r2
¼
2
2
DK
2
2
DK
8aM sin cos PDK
PDK
5
6
DK 4aMrðr þ a Þ sin cos @hr þ 2a sin
DK 2r P7 hDK
h
h
r
t
8
4
2 PDK t
6
@r
PDK
PDK
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
4
4
4
4
4
PDK
PDK
PDK
PDK
PDK
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 4 PDK dr
4
4
2
2
4
P1
P1
dr
dr
1
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
dr
2
2a sin cos
2
hDK
r
þ 2r1 þ
2
d1
: (57)
dr
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
SARAH VIGELAND, NICOLÁS YUNES, AND LEO C. STEIN
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
; ht Þ to
r ;h
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
2
d ;
(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
¼ 0, but also enr
sures 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
2
2
2
0
DK þ aM sin 2aMrðr þ a Þcos sin ;
¼
h
hDK
1
2
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
@
0
hDK ¼ hDK þ r1 a2 sin cos 2 þ 2 ;
@
2
sin
0
hDK ¼ hDK þ 2 fr5 a2 Mr2 þ a2 cos2 ½2rðr2 þ M2 Þ
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
2
0
DK þ aM sin ;
hDK
¼
h
1
t
t
2
a2 r
@1
0
DK
hDK
;
rr ¼ hrr þ 2 1 þ
@r
0
hDK ¼ hDK 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
¼ 0. This can be achieved
r
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 :
hDK ¼ hDK þ
We have thus found a generic diffeomorphism that maps
a DK metric to one that has the same nonzero components
104027-10
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
such that hDK
¼ hBK
tr ¼ htr and hr
r . This implies
1
2Mr d0 aMrsin2 d3
1
; (66)
0 ¼ hBK
tr þ
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
quantity gCS 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
½htr cos ð2a2 Mr a2 r2 a4 Þ
2
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
2MrahBK
r ;
4
2aMrsin2 hBK
tr :
hBK
r ¼
F2 ðrÞð
2
2
2Mrasin2 F2 ðrÞ
2
F3 ðrÞ
2
½
4
þ a4 ð
;
2
þ 2MrÞsin2 :
r2 þ a2
a
CS
; 1; 0; þ g ;
¼
2
2
r þa
a
; 1; 0; þ gCS ;
l ¼
0 ¼ 2
Z
F2 ðrÞdr;
3 ¼ 2
Z
F3 ðrÞdr:
(71)
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]
B. To alternative theories
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 1. Dynamical Chern-Simons modified gravity
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.
(70)
BK
Provided hBK
components can be written in the
tr and hr
above form, the generating function becomes
(73)
k
(69)
2MrÞ þ 2Mrasin2 F3 ðrÞ
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
gCS ¼ CS aM4
hQG
rr
104027-11
SARAH VIGELAND, NICOLÁS YUNES, AND LEO C. STEIN
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
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
2
2Mr
þ 4Mra2 þ 4Mr3 Þsin2
2
;
4
1 asin2 ð 2 þ 4MrÞ
;
4
2
11
7
PBK
8;n ¼ ð8r 4r
12a4 r5
2
4
þ 16r9 M2 þ 2r9
16r3 a6
2
6a8 r
2
2
44a6 M
36a2 Mr6
2
8a10 M 68a4 Mr4
2
þ 40a2 Mr4
4
16r8 Ma2 16r7 a2 M2 48r3 a6 M2 þ 16r4 Ma6 56a2 M2 r3
þ 32r5 a6 þ 8r3 a8 þ 60a4 M
4r8 M
2
12r3 a4
þ 2r3 a2
6
þ 5a4 r
6
4
18a2 M
4
12a2 r5
þ 12a4 M
4
4Mr6
6 Þsin2
6
4
2
þ 128a4 M2 r3
þ 32r9 a2 3r5
56a6 rM2
2
8r10 M 20a2 rM2
6
þ 72a4 rM2
4
þ 48r7 a4
80a4 M2 r5 þ 8a8 M þ 16a8 M2 r þ 36a8 M
2
32r7 M2
4a6 r
þ 24M2 r5
4
;
104027-13
4
þ 64a2 M2 r5
6
2
44a6 M
4
þ 12Mr4
2
6
8M2 r3
6
4
2
SARAH VIGELAND, NICOLÁS YUNES, AND LEO C. STEIN
6
PBK
8;d ¼ ð4r þ 2
a 4a2
4 2
þ 8Mra4 4a6 Þ
2
8a2
2
Mr þ 2a4
PHYSICAL REVIEW D 83, 104027 (2011)
þ 8r4 a2 2
4
2
4
Mr 2r4
4;
2
2 5
PBK
9;n ¼ asin ð16M r
þ 8M2 r3
2
4
4Mr4
4
20Mr6
2
32a2 M2 r5 þ 24r8 M 24a4 r5 8a8 M 8r9
þ 56a2 Mr6 þ 40a4 Mr4 24r7 a2 8r3 a6 þ 8a6 M þ 16a6 M2 r 12a4
PBK
9;d
¼
þ 12a2
4
þ 6a6 r
2
4 ð4r6
M 48a4 M2 r3 þ 8ra2 M2
þ r3
þ2
4 a2
þ 4a4 2
6
þ 48a2 M2 r3
4a2 2 2
4
3a2 r
40a4 M2 r
8a2 2 Mr
þ
2
þ 16r7 M2 þ 2a2 M
6
48a2 Mr4
þ
2a4 2
8r4 a2
2
6
4
M þ 28a6 M
þ 18a2 r5
2
2
36a4 M
þ 6r7
2
2
þ 18r3 a4
2
Þ;
4
2
4 Mr
2r4
2
þ 4a4 þ 8Mra4 4a6 Þ;
2
4
PBK
10;n ¼ sin ð12rMa
4
16M2 r8 þ 6a2
PBK
10;d
48a2 r8 þ 6r4 a2
6
Mr þ 24r4 a4
32r5 Ma2
2
þ 8r8
þ 16r3 Ma4
2
þ 24a2 r6
2
2
4
48a4 r6 þ 16a6 M2 þ 32a6 M3 r þ 8a6
þ 32r9 M 32a2
16a8 M2 þ 96a2 M2 r4
2
6Mr3
þ 3r4
6
6
2
4
Mr3 8a8
2
þ 8a6 M2
2
8a2 M2
þ 40M2 r6
2
2
6
16r10
þ 16rMa6
2
16r4 a6 þ 32r5 Ma4 80r6 a2 M2 48r4 a4 M2
þ 2r6
4
þ 5a4
6
þ 64r7 Ma2 þ 24Mr5
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
4
24M2 r4
4
þ 4a4 þ 8Mra4 4a6 Þ;
2
3
PBK
11;n ¼ 8aMsin ðr
3a4 M
PBK
11;d ¼
4 ð4r6
2
4
5
r þ 4r7 3ra2
2 5
8a2 Mr4 þ 3a4 r
þ2
4 a2
4a2
2
2
4
þ 6a2 r3
2
þ a2
2Mr6 4a4 r3 þ 3Mr4
8a2
þ 2a4
2 Mr
2
4
2
M þ a2
2
M þ 2a2
2
M2 r
þ 2a4 M þ 4a4 M2 r 2a6 MÞ;
þ 8r4 a2 4
2
2r4
2
þ 4a4 a 18r4
2
þ 16r6
2r4
2
þ 4a4 þ 8Mra4 4a6 Þ;
4 Mr
þ 8Mra4 4a6 Þ;
2
PBK
12;n ¼ 2rð28a
2
56a2
2
Mr þ 18a4
2
þ
6
þ2
4
þ4
4
Mr þ 4
4 2
þ 32r4 a2 þ 16a4 þ 32Mra4 16a6 Þ;
PBK
12;d ¼
2 ð4r6
þ2
4 a2
2
2
2
PBK
13;n ¼ 2a sin ð8a
PBK
13;d ¼
PBK
14 ¼
4
ð4r6 þ 2
4 ð4r6
þ2
4a2
2
þ 16a2
a 4a2
4 2
4 a2
2
2
8a2
2 Mr
8a4
8a2
4a2 2 2 Mr
2
þ 2a4
2
6
Mr þ 2a4
2
þ 8r4 a2 6
2
4
12
þ 8r4 a2 4
2
4 Mr
4
4 Mr
þ8
2
4 a2
4
þ 24r4
Mr 2r4
2
2
16r6 16r4 a2 Þ;
þ 4a4 þ 8Mra4 4a6 Þ;
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
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]:
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SARAH VIGELAND, NICOLÁS YUNES, AND LEO C. STEIN
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.
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