PHYSICAL REVIEW D 66, 024026 共2002兲
Reconstruction of black hole metric perturbations from the Weyl curvature
Carlos O. Lousto
Department of Physics and Astronomy, University of Texas at Brownsville, Brownsville, Texas 78520
and Instituto de Astronomı́a y Fı́sica del Espacio–CONICET, Buenos Aires, Argentina
Bernard F. Whiting
Department of Physics, P.O. Box 118440, University of Florida, Gainesville, Florida 32611-8440
共Received 18 March 2002; published 18 July 2002兲
Perturbation theory of rotating black holes is usually described in terms of Weyl scalars ␺ 4 and ␺ 0 , which
each satisfy Teukolsky’s complex master wave equation and respectively represent outgoing and ingoing
radiation. On the other hand metric perturbations of a Kerr hole can be described in terms of 共Hertz-like兲
potentials ⌿ in outgoing or ingoing radiation gauges. In this paper we relate these potentials to what one
actually computes in perturbation theory, i.e. ␺ 4 and ␺ 0 . We explicitly construct these relations in the nonrotating limit, preparatory to devising a corresponding approach for building up the perturbed spacetime of a
rotating black hole. We discuss the application of our procedure to second order perturbation theory and to the
study of radiation reaction effects for a particle orbiting a massive black hole.
DOI: 10.1103/PhysRevD.66.024026
PACS number共s兲: 04.25.Nx, 04.30.Db, 04.70.Bw
再冋
I. INTRODUCTION
The spherical symmetry of a Schwarzschild black hole
background allows for a multipole decomposition of metric
perturbations, even in the time domain. These were studied
originally by Regge and Wheeler 关1兴 for odd-parity perturbations and by Zerilli 关2兴 for the even-parity case. Moncrief
关3兴 has given a gauge-invariant formulation of the problem,
in terms of the three-metric perturbations. The two degrees
of freedom of the gravitational field are described in terms of
two waveforms ␾ 共even, odd兲 satisfying a simple wave equation
⫺
共even, odd兲
⳵ 2 ␾ (lm)
⳵t2
⫹
共even, odd兲
⳵ 2 ␾ (lm)
⳵ r* 2
0556-2821/2002/66共2兲/024026共7兲/$20.00
冋
册
4M ar
共 r 2 ⫹a 2 兲 2
⳵ tt ⫺
⳵ t␸
⌬
⌬
⫺2s 共 r⫹ia cos ␪ 兲 ⫺
⫹⌬ ⫺s ⳵ r 共 ⌬ s⫹1 ⳵ r 兲 ⫹
⫹
共even, odd兲
⫺V l 共even, odd兲共 r 兲 ␾ (lm)
⫽0.
冉
1
sin2 ␪
⫺
册
M 共 r 2 ⫺a 2 兲
⳵t
⌬
1
⳵ 共 sin ␪ ⳵ ␪ 兲
sin ␪ ␪
冊 冋
冎
册
a 共 r⫺M 兲 i cos ␪
a2
⳵ ⫹2s
⳵␸
⫹
⌬ ␸␸
⌬
sin2 ␪
⫺ 共 s 2 cot2 ␪ ⫺s 兲 ␺ ⫽4 ␲ ⌺T,
共1.1兲
Here r * ⬅r⫹2M ln(r/2M ⫺1), and V l 共even, odd兲(r) are the Zerilli and Regge-Wheeler potentials respectively.
Given the solution to the wave equation 共1.1兲 one can
reconstruct explicitly both the even and odd parity metric
perturbations of a Schwarzschild background in the ReggeWheeler gauge 关4,5兴. This permits a complete description of
the perturbative spacetime. But the Regge-Wheeler gauge is
unfortunately not asymptotically flat, and in order to extract
radiation information for a second order perturbative expansion one has to perform a new gauge transformation 关6兴.
Moreover, the desirable properties of the Regge-Wheeler
gauge being unique and invertible are lost in the case in
which the background is a rotating black hole, i.e. represented by the Kerr metric, where no effective multipole decomposition is yet known.
There is an independent formulation of the perturbation
problem derived from the Newman-Penrose formalism 关7兴
that is valid for perturbations of rotating black holes 关8兴. This
formulation fully exploits the null structure of black holes to
decouple the curvature perturbation equations into a single
wave equation that, in Boyer-Lindquist coordinates
(t,r, ␪ , ␸ ), can be written as
a 2 sin2 ␪ ⫺
共1.2兲
where M is the mass of the black hole, a its angular momentum per unit mass, ⌺⬅r 2 ⫹a 2 cos2␪, and ⌬⬅r 2 ⫺2M r⫹a 2 .
The source term T is built up from the energy-momentum
tensor 关8兴. Gravitational perturbations, corresponding to s⫽
⫾2, are compactly described in terms of contractions of the
Weyl tensor with a null tetrad. The components of the tetrad
共also given in Ref. 关8兴兲 are appropriately chosen along the
repeated principal null directions of the background spacetime 关see Eq. 共1.4兲 below兴. The resulting gauge and 共infinitesimally兲 tetrad invariant components of the Weyl curvature
are given by
␺⫽
再
␳ ⫺4 ␺ 4 ⬅⫺ ␳ ⫺4 C nm̄nm̄
␺ 0 ⬅⫺C lmlm
for s⫽⫺2,
for s⫽⫹2,
共1.3兲
where an overbar means complex conjugation and ␳ is given
in Eq. 共1.5兲 below. Asymptotically, the leading behavior of
the field ␺ represents either the outgoing radiative part of the
perturbed Weyl tensor (s⫽⫺2), or the ingoing radiative part
(s⫽⫹2).
The components of the Boyer-Lindquist null tetrad for the
Kerr background are given by
66 024026-1
©2002 The American Physical Society
PHYSICAL REVIEW D 66, 024026 共2002兲
CARLOS O. LOUSTO AND BERNARD F. WHITING
共 l␣兲⫽
共 n␣兲⫽
共 m␣兲⫽
冉
冊
r 2 ⫹a 2
a
,1,0, ,
⌬
⌬
1
2 共 r ⫹a 2 cos2 ␪ 兲
2
共1.4a兲
共 r 2 ⫹a 2 ,⫺⌬,0,a 兲 ,
1
冑2 共 r⫹ia cos ␪ 兲
共1.4b兲
共 ia sin ␪ ,0,1,i/sin ␪ 兲 .
共1.4c兲
With this choice of the tetrad the nonvanishing spin coefficients are
␳ ⫽⫺
1
,
共 r⫺ia cos ␪ 兲
␲ ⫽ia ␳ 2
␮ ⫽ ␳ 2¯␳
sin ␪
冑2
⌬
,
2
¯
␥ ⫽ ␮ ⫹ ␳␳
,
␤ ⫽⫺¯␳
¯
␶ ⫽⫺ia ␳␳
cot ␪
2 冑2
sin ␪
冑2
,
,
␣ ⫽ ␲ ⫺ ¯␤ ,
共 r⫺M 兲
,
2
cially well suited for computing second order perturbations
of a Kerr hole and, once generalized in the presence of matter, can provide a first step toward computing radiation reaction 共self-force兲 corrections 关4兴 to the trajectory of a particle
orbiting a rotating black hole.
Vacuum second order perturbations have a direct application to the close limit expansion of the final merger stage of
binary black holes with comparable masses. Perturbation
theory in conjunction with limited full numerical simulations
has proved to be an extremely powerful tool to compute
waveforms from binary black holes from near the innermost
stable circular orbit 关17,18兴. The combination of radiation
reaction correction plus second order perturbations provides
a formidable tool for computing gravitational radiation from
binary black-hole–neutron-star systems, and is particularly
relevant to the computation of template banks for ground
based interferometers such as the Laser Interferometric
Gravitational Wave Observatory 共LIGO兲, VIRGO, or GEO
about to enter on-line, as well as space missions such as the
Laser Interferometer Space Antenna 共LISA兲, sensitive to supermassive black hole binary systems. Thus in order to incorporate these improvements to our theoretical predictions
it is imperative to know how to build up explicit metric
perturbations around a Kerr background.
II. FORMULATION OF THE PROBLEM
共1.5兲
We use the notation
and the only nonvanishing Weyl scalar in the background is
␺ 2 ⫽M ␳ .
3
g ␮ ␯ ⫽g ␮Kerr
␯ ⫹h ␮ ␯
共1.6兲
Analogously to the Zerilli-Regge-Wheeler waveforms, ␺ 4
can be directly used to compute energy and momentum radiated at infinity, but it remains to relate it to metric perturbations. Chandrasekhar 关9兴 studied a way to obtain metric
perturbations of the Kerr metric, but it was proved by Price
and Ipser 关10兴 that this choice is not a proper gauge, namely
it is an incomplete constraint on the coordinates.
Chrzanowski 关11兴 generalized the work of Cohen and
Kegeles 关12兴 on Hertz potentials to the gravitational perturbations of the Kerr metric. In Ref. 关11兴 explicit expressions
are given for homogeneous metric perturbations in the frequency domain. Wald 关13兴 subsequently showed that the expressions given in Ref. 关11兴 do not lead to real metric perturbations. Cohen and Kegeles 关14兴 then corrected their
expressions and gave explicit equations 共see Sec. II兲 relating
metric perturbations to a gravitational Hertz potential ⌿ that
satisfies Eq. 共1.2兲, but that is different from ␺ 4 or ␺ 0 . In
those works no explicit method was given for determining ⌿
in any specific astrophysical problem.
In Sec. III we provide the explicit formulas relating a
gravitational Hertz potential to ␺ 4 or ␺ 0 in the time domain
and, hence, to the given initial data defining the astrophysical
model one wants to evolve 共see Ref. 关15兴 for the 3⫹1 decomposition of the Weyl scalars兲. These allow one to define
the outgoing and ingoing radiation gauges that are asymptotically flat at future infinity and regular on the event horizon respectively. Such gauges have been found 关16兴 espe-
共2.1兲
to describe metric perturbations.
A. Ingoing and outgoing radiation gauges
Chrzanowski 关11兴, and Cohen and Kegeles 关14兴 found two
convenient gauges that allow one to invert the metric perturbations in terms of potentials ⌿ IRG or ⌿ ORG satisfying the
same wave equations as the Weyl scalars ␳ ⫺4 ␺ 4 or ␺ 0 respectively.
In the ingoing radiation gauge 共IRG兲 we have
h ll ⫽h ␮ ␯ l ␮ l ␯ ⫽0;
h lm ⫽h ␮ ␯ l ␮ m ␯ ⫽0;
h ln ⫽h ␮ ␯ l ␮ n ␯ ⫽0,
h lm̄ ⫽h ␮ ␯ l ␮ m̄ ␯ ⫽0,
h mm̄ ⫽h ␮ ␯ m ␮ m̄ ␯ ⫽0,
共2.2兲
and the homogeneous 共for vacuum兲 metric components can
be written, in the time domain, in terms of solutions to the
wave equation for ␳ ⫺4 ␺ 4 only, as follows 关27兴:
024026-2
IRG
⫽⫺ 兵 共 ␦ ⫹ ¯␣ ⫹3 ␤ ⫺ ␶ 兲共 ␦ ⫹4 ␤ ⫹3 ␶ 兲 其 共 ⌿ IRG 兲 ⫹c.c.
h nn
共2.3a兲
IRG
h m̄m̄ ⫽⫺ 兵 共 D⫺ ␳ 兲共 D⫹3 ␳ 兲 其 共 ⌿ IRG 兲 ,
共2.3b兲
PHYSICAL REVIEW D 66, 024026 共2002兲
RECONSTRUCTION OF BLACK HOLE METRIC . . .
p⫹ia sin ␪⳵t and Z
p⫽⫺ 关 ⳵ ␪ ⫹s cot ␪⫺i csc ␪ ⳵ ␸ 兴
where L̄⫽Z
acting on spin-weight s, while for the outgoing radiation
gauge we have
1
IRG
¯ ⫺ ␶ 兲共 D⫹3 ␳ 兲
h (nm̄) ⫽⫺ 兵 共 ␦ ⫺ ¯␣ ⫹3 ␤ ⫺ ␲
2
⫹ 共 D⫹¯␳ ⫺ ␳ 兲共 ␦ ⫹4 ␤ ⫹3 ␶ 兲 其 共 ⌿ IRG 兲 ,
共2.3c兲
where c.c. stands for the complex conjugate part of the whole
object to ensure that the metric is real 关13,14兴. Note that in
this gauge the metric potential has the properties of being
transverse (h ␮ ␯ l ␮ ⫽0) and traceless (h ␮␮ ⫽0) at the future
horizon and past null infinity. It is thus a suitable gauge to
study gravitational radiation effects near the event horizon.
The complementary 共adjoint兲 gauge to the ingoing radiation gauge is the outgoing radiation gauge 共ORG兲, which
can be obtained from the IRG upon exchange of the tetrad
vectors l↔n, m̄↔m and the appropriate renormalization. It
satisfies
h nn ⫽0⫽h ln ⫽0⫽h nm ⫽0⫽h nm̄ ⫽0⫽h mm̄ .
共2.4兲
The metric potential has now the property of being transverse (h ␮ ␯ n ␮ ⫽0) and traceless (h ␮␮ ⫽0) at the past horizon
and future null infinity. It is thus an example of a suitable
asymptotically flat gauge in which to directly compute radiated energy and momenta at infinity.
In this gauge, the homogeneous metric components can
be written in terms of solutions to the wave equation for ␺ 0 ,
as
␳ ⫺4 ␺ 4 ⫽⌬ 2 ⌬ˆ ⌬ˆ ⌬ˆ ⌬ˆ 关 ⌬ 2 ⌿̄ORG 兴 ,
1
␺ 0 ⫽ 关 LLLL⌿̄ORG ⫹12␳ ⫺3 ␺ 2 ⳵ t ⌿ ORG 兴 .
4
III. EXPLICIT SOLUTION FOR NONROTATING BLACK
HOLES
A. Master equation
The key observation here is that for a⫽0 the differential
operator L acting on the potentials in Eqs. 共2.7兲 and Eqs.
共2.9兲 contains only angular derivatives. Since for the spherically symmetric background we can decompose the potentials into spherical harmonics of spin weight s and, from
关19兴, we have
s Ȳ lm ⫽ 共 ⫺ 兲
ZZZZ关 2 Ȳ lm 兴 ⫽ 共 ⫺ 兲 m⫹2
共3.1a兲
共 l⫹2 兲 !
Y
,
共 l⫺2 兲 ! ⫺2 l,⫺m
共3.1b兲
共 l⫹2 兲 !
Y
,
共 l⫺2 兲 ! 2 l,⫺m
共3.1c兲
we obtain a first order relationship among ␺ 4 or ␺ 0 and the
IRG or ORG potentials decomposed into multipoles
ˆ ⫹5 ␮ ⫺ ␮
¯ ⫺3 ␥ ⫺ ¯␥ 兲共 ¯␦ ⫺4 ␣ ⫹ ␲ 兲 其 共 ⌿ ORG 兲 ,
⫹共 ⌬
关 ␳ ⫺4 ␺ 4 兴 ⫾ ⫽⫾
共2.5c兲
ˆ ⫽n ␮ ⳵ ␮ , ␦
where the directional derivatives are D⫽l ␮ ⳵ ␮ ,⌬
␮
⫽m ⳵ ␮ , and the rest of the Greek letters represent the usual
notation for spin coefficients 关7兴.
␺⫾
0 ⫽⫾
共 l⫹2 兲 ! ⫾
⫾
,
⌿ ⫺3M ⳵ t ⌿ IRG
4 共 l⫺2 兲 ! IRG
共3.2兲
共 l⫹2 兲 ! ⫾
⫾
⌿
⫹3M ⳵ t ⌿ ORG
, 共3.3兲
4 共 l⫺2 兲 ! ORG
where we have used the notation
B. Fourth order equations for the potential
From the evolution of the Teukolsky equation we can obtain ␺ 4 and ␺ 0 , the two gauge and tetrad invariant objects
representing outgoing and ingoing radiation respectively. To
relate the unknown potential ⌿ to them we use their definitions 共1.3兲 to obtain
1
␳ ⫺4 ␺ 4 ⫽ 关 L̄L̄L̄L̄⌿̄IRG ⫺12␳ ⫺3 ␺ 2 ⳵ t ⌿ IRG 兴 ,
4
⫺s Y l,⫺m ,
共2.5a兲
ORG
ˆ ⫹5 ␮ ⫺3 ␥ ⫹ ¯␥ 兲共 ⌬
ˆ ⫹ ␮ ⫺4 ␥ 兲 其 共 ⌿ ORG 兲 ,
⫽⫺ ␳ ⫺4 兵 共 ⌬
h mm
共2.5b兲
␺ 0 ⫽DDDD 关 ⌿̄IRG 兴 ,
m⫹s
pZ
pZ
pZ
p关 ⫺2 Ȳ lm 兴 ⫽ 共 ⫺ 兲 m⫺2
Z
1
ORG
ˆ ⫹ ␮ ⫺4 ␥ 兲
h (lm)
⫽⫺ ␳ ⫺4 兵 共 ¯␦ ⫺3 ␣ ⫹ ¯␤ ⫹5 ␲ ⫹¯␶ 兲共 ⌬
2
共2.9兲
In order to obtain an expression for the potentials in terms of
the known quantities ␺ 4 or ␺ 0 , one has to invert a fourth
order differential equation where ␺ 4 or ␺ 0 acts as a source
term. This will be the central task of our paper.
⫽⫺ ␳ ⫺4 兵 共 ¯␦ ⫺3 ␣ ⫺ ¯␤ ⫹5 ␲ 兲共 ¯␦ ⫺4 ␣ ⫹ ␲ 兲 其
h ORG
ll
⫻ 共 ⌿ ORG 兲 ⫹c.c.
共2.8兲
共2.6兲
共2.7兲
1
␺ ⫾ ⫽ 关 ␺ l,m ⫾ 共 ⫺ 兲 m ␺ l,⫺m 兴
2
共3.4兲
for all fields decomposed into multipoles.
Since ⌿ IRG and ⌿ ORG satisfy the master equation 共1.2兲
for spin s⫽⫿2 respectively we can eliminate from this
equation all time derivatives by replacing Eq. 共3.2兲 or Eq.
共3.3兲 and its time derivatives into the Teukolsky equation.
This leads to the following equation for the IRG-ORG potentials, both represented here by ⌿:
024026-3
PHYSICAL REVIEW D 66, 024026 共2002兲
CARLOS O. LOUSTO AND BERNARD F. WHITING
⌬ ⫺s ⳵ r 关 ⌬ s⫹1 ⳵ r ⌿ ⫾ 兴 ⫺
⫾2sr 共 ⍀ AS 兲
冉
r4
共 ⍀ AS 兲 2 ⌿ ⫾
⌬
冊
Mr
⫺1 ⌿ ⫾
⌬
⫺ 共 l⫺s 兲共 l⫹s⫹1 兲 ⌿ ⫾ ⫽F ⫾
共3.5兲
where
⌿⫽
⍀ AS ⫽
再
⌿ IRG
for
s⫽⫺2,
⌿ ORG
for
s⫽⫹2,
where ␭⫽(l⫺1)(l⫹2)/2. Note that y 2 gives rise to an algebraically special solution for ␺ 4 , (s⫽⫺2), at frequency
⫺
⫹
␻ AS
⫽⫺i⍀ AS and for ⌬ 2 ␺ 0 , (s⫽⫹2), at frequency ␻ AS
⫽
⫹i⍀ AS , while y 1 gives rise to an algebraically special solu⫹
⫺
and for ⌬ 2 ␺ 0 at frequency ␻ AS
.
tion for ␺ 4 at frequency ␻ AS
These two solutions satisfy the ‘‘plus’’ and ‘‘minus’’ parts
⫾
⫿
(s⫽⫹2) and vice-versa for ⌿ ORG
of Eq. 共3.5兲 for ⌿ IRG
(s⫽⫺2). A second set of independent solutions can be
found:
共3.6兲
1 共 l⫹2 兲 !
,
12M 共 l⫺2 兲 !
r4
r4
共 ⳵ t␺ ⫾ 兲⫿
共 ⍀ AS 兲共 ␺ ⫾ 兲
3M ⌬
3M ⌬
⫹2s
r
共 M r/⌬⫺1 兲共 ␺ ⫾ 兲 .
3M
冋
册
册
W共 r⬘兲
⬁ y 2共 r ⬘ 兲 2
共3.11兲
dr ⬘ ,
共3.12兲
⫹
⌿ IRG
共 r 兲 ⫽⫺y 1 共 r 兲
⫺
⌿ IRG
共 r 兲 ⫽z 2 共 r 兲
冕
冕
r
z 1共 r ⬘ 兲 F ⫹共 r ⬘ 兲
2M
⌬共 r⬘兲2
冕
⬁y
1共 r ⬘ 兲 F
⫹
⌬共 r⬘兲
r
共r⬘兲
2
r
y 2共 r ⬘ 兲 F ⫺共 r ⬘ 兲
2M
⌬共 r⬘兲2
⫹y 2 共 r 兲
冕
⬁z
r
2共 r ⬘ 兲 F
⫺
⌬共 r⬘兲
dr ⬘
dr ⬘ ,
共3.14兲
dr ⬘
共r⬘兲
2
dr ⬘ ,
共3.15兲
valid for each hypersurface where ␺ 0 or ␺ 4 , entering in F
given by Eq. 共3.5兲, are evaluated.
The equations for the ORG are obtained by exchanging
‘‘plus’’ and ‘‘minus’’ parts above 关cf. Eqs. 共A3兲 and 共A4兲兴,
and by adopting the corresponding dependence of F on the
components of ␺ 关see Eq. 共1.3兲兴.
IV. APPLICATIONS
共3.9兲
共3.10兲
共3.13兲
Note: z 1 and z 2 are not algebraically special solutions,
although they are each a homogeneous solution to their respective Teukolsky equation at an algebraically special frequency.
Making use of these solutions, the explicit expression for
the potential can be written as follows:
⫺z 1 共 r 兲
␭r ␭ 2 r 2 ␭ 2 共 ␭⫹1 兲 r 3 ⫺⍀ r *
⫹
⫹
e AS ,
M 3M 2
9M 3
共 ␭⫹1 兲 r 2 ⫹⍀ r *
r e AS ,
3M
r
dr ⬘ ,
⫽⌬ 共 r 兲 ⫽r 共 r⫺2M 兲 .
The single frequency appearing on the right hand side of
Eq. 共1.2兲 is precisely an algebraic special frequency. Algebraically special perturbations of black holes excite gravitational waves which are either purely ingoing or outgoing.
Hence only one of ␺ 4 or ␺ 0 is non-zero while the other
vanishes. Chandrasekhar 关21兴 has obtained the explicit form
of the algebraic special perturbations of the Kerr black hole.
This is a remarkable fact, because we can then use these
analytic solutions to the Teukolsky equation for algebraic
special perturbations to construct explicit solutions to Eq.
共3.5兲.
For Schwarzschild, Chandrasekhar 关21兴 gives two algebraic special solutions
y 2 共 r 兲 ⫽ 1⫺
冕
2M y 1 共 r ⬘ 兲 2
W 共 r 兲 :⫽y 1,2共 r 兲 ⳵ r z 1,2共 r 兲 ⫺z 1,2共 r 兲 ⳵ r y 1,2共 r 兲
共3.8兲
B. Solution
冋
z 2 共 r 兲 ⫽y 2 共 r 兲
W共 r⬘兲
r
where the Wronskian of the solutions is
Equation 共3.5兲 is our fundamental equation to solve for
the potential in terms of the known fields ␺ 4 or ␺ 0 that
appear in the source terms. The key observations here are to
note that, separately for the ‘‘plus’’ and ‘‘minus’’ parts: 共i兲 the
left-hand side of this equation contains the Teukolsky operator in the frequency domain for the algebraically special frequencies, ␻ ⫽⫾i⍀ AS , and 共ii兲 the source terms in Eq. 共3.8兲
are the Cauchy data for the Teukolsky operator in terms of
␺ 4 or ␺ 0 , precisely as they would appear in a Laplace transform approach to Eq. 共1.2兲 关see Eqs. 共A2兲, 共A3兲 of Ref. 关20兴兴.
Notably, we have arrived to this equation working in the time
domain, without any frequency decomposition.
y 1 共 r 兲 ⫽ 1⫹
冕
共3.7兲
and the source term is
F ⫾ ª⫺
z 1 共 r 兲 ⫽y 1 共 r 兲
A. Metric perturbations „IRG…
Using the Regge-Wheeler notation 关1兴 for metric perturbations, conditions 共2.2兲 read
024026-4
PHYSICAL REVIEW D 66, 024026 共2002兲
RECONSTRUCTION OF BLACK HOLE METRIC . . .
h l0 共 r,t 兲 (even,odd) ⫽⫺ 共 1⫺2M /r 兲 h l1 共 r,t 兲 (even,odd) ,
G l 共 r,t 兲 ⫽
⫹
2
K l 共 r,t 兲 ,
l 共 l⫹1 兲
⫺144 共 l⫹2 兲共 l⫺1 兲 M 3
共4.1兲
H l0 共 r,t 兲 ⫽H l2 共 r,t 兲 ⫽⫺H l1 共 r,t 兲 .
We can now explicitly compute the metric perturbations
共2.3兲 or 共2.5兲 in terms of the computed ␺ 0 or ␺ 4 关28兴:
关 h (nm̄) 兴 l ⫽ 兵 h l0 共 r,t 兲 (even) ⫺ih l0 共 r,t 兲 (odd) 其
⫽
再冉
⫺
⫹
⫹
冑l 共 l⫹1 兲
冑2r
⫺36 r 2 C l M 兲 ⫻
and
⫺1 Y lm
关 h m̄m̄ 兴 (odd) ⫽
冊
y 1⬘
Cl
2
1
⫹
⫹
⫺ 2 关 ⌿ IRG
兴
ry 1 12M 共 1⫺2M /r 兲 r
⌬
ry 1
冉
冕
ry 2
⫺
冕
1
⌬2
r
y ⬘2
⌬
ry 2
⬁y
r
关 F ⫹ 兴 dr ⬘ ⫺
2M ⌬
关 F ⫺ 兴 dr ⬘ ⫺
2
3M 共 r⫺2M 兲
冎
⫹
共4.2兲
关 h nn 兴 l ⫽ 共 1⫺2M /r 兲 H l0 共 r,t 兲 0 Y lm
再
冑
⫻
共 l⫹2 兲 !
Y ,
共 l⫺2 兲 ! 0 lm
共4.3兲
冎冑
共 l⫹2 兲 !
共 l⫺2 兲 !
关 h m̄m̄ 兴
(even)
⫽
再
⫺1/6
共4.4兲
⫹
关 ⌿ IRG
兴共 12 M 2 ⫹r 2 C l 兲 y 1⬘
1/6
共4.5兲
y 2 r 共 r⫺2 M 兲 M
共 ⌬ 兲共 12 M 2 ⫺r 2 C l 兲
y 2 r 共 r⫺2 M 兲 M
冕
y2
r
2M ⌬ 2
关 F ⫺ 兴 dr ⬘
r
2
关 ␳ ⫺4 ␺ ⫺
4 兴共 ⫺36 r⫹84 M ⫺r C l 兲
共 r⫺2 M 兲 2 M
1 3 2
共 r C l ⫹12r 共 6 l 2 ⫹6 l⫺7 C l ⫺12兲 M 2
72
冕
⬁y
r
1
⌬
2
关 F ⫹ 兴 dr ⬘
冉 冊
⳵ r 共关 ⌿ ⫺ 兴 兲 ⫽
冉 冊
冎
共4.6兲
⫺2 Y lm .
⳵ry 1
⌬
关 ⌿⫹兴⫺
y1
y1
⳵ry 2
⌬
关 ⌿⫺兴⫹
y2
y2
冕冉
冊
冕冉 冊
⬁
r
y1
⌬
2
r
y2
2M
⌬2
关 F ⫹ 兴 dr ⬘ ,
关 F ⫺ 兴 dr ⬘ ,
共4.7兲
共4.8兲
and directly from Eq. 共3.2兲
⳵
r 关 ␳ ⫺4 ␺ ⫹
⫹
2 ⫺4
4 兴
r 关 ␳ ⳵ t␺ 4 兴
⳵r
⫺2/3
⫺2/3
共 r⫺2 M 兲 M
共 r⫺2 M 兲 2 M
2
关 ␳ ⫺4 ␺ ⫹
4 兴共 ⫺36r⫹84 M ⫹r C l 兲
⫺1/18
共 r⫺2 M 兲 2 M
共 r⫺2 M 兲 2 M 2 r
⳵ r 共关 ⌿ ⫹ 兴 兲 ⫽
y 1 r 共 r⫺2 M 兲 M
共 ⌬ 兲 共 12 M 2 ⫹r 2 C l 兲
y 1 r 共 r⫺2 M 兲 M
⫺
关 ⌿ IRG
兴
In writing these we have explicitly lowered the order of the
derivatives of the potential by making use of the following
identities:
⫺2 Y lm
⫽ 关 h m̄m̄ 兴 (even) ⫹ 关 h m̄m̄ 兴 (odd)
where
⫺2 Y lm ,
⫺144 共 l⫹2 兲共 l⫺1 兲 M 3 ⫹36 r 2 C l M 兲
where C l ⫽(l⫹2)!/(l⫺2)!, and
1
h 2 共 r,t 兲 l
G 共 r,t 兲 l ⫹i
关 h m̄m̄ 兴 l ⫽
2
r2
冎
⫺
关 ⌿ IRG
兴共 12 M 2 ⫺r 2 C l 兲 y 2⬘
⫺1/18
⫻ 冑2 共 l⫹2 兲共 l⫺1 兲 ⫺1 Y lm ,
␳2
⫹
⫽ 关 ⌿ IRG
兴
2
共 r⫺2 M 兲 2 M 2 r
r 2 关 ␳ ⫺4 ⳵ t ␺ ⫺
4 兴
Cl
2
1
⫺
⫺ 2 关 ⌿ IRG
兴
12M 共 1⫺2M /r 兲 r
关 ␳ ⫺4 ␺ ⫺
4 兴
1/6
⫹
关 ⌿ IRG
兴
⳵ ⫺4 ⫺
关␳ ␺4 兴
⳵r
⫺2/3
⫺2/3
共 r⫺2 M 兲 M
共 r⫺2 M 兲 2 M
3M 共 r⫺2M 兲
冊
再
⫹1/6
关 ␳ ⫺4 ␺ ⫹
4 兴
y2
1 3 2
共 r C l ⫹12r 共 6 l 2 ⫹6 l⫹7 C l ⫺12兲 M 2
72
⫾
⳵ t ⌿ IRG
⫽⫾
1
共 l⫹2 兲 !
⌿⫾ ⫺
␺⫾ .
12M 共 l⫺2 兲 ! IRG 3M ␳ 4 4
共4.9兲
B. Metric perturbations „ORG…
Using the Regge-Wheeler notation 关1兴 for metric perturbations, conditions 共2.4兲 read
024026-5
PHYSICAL REVIEW D 66, 024026 共2002兲
CARLOS O. LOUSTO AND BERNARD F. WHITING
h l0 共 r,t 兲 (even,odd) ⫽ 共 1⫺2M /r 兲 h l1 共 r,t 兲 (even,odd) ,
G l 共 r,t 兲 ⫽
2
K l 共 r,t 兲 ,
l 共 l⫹1 兲
H l0 共 r,t 兲 ⫽H l2 共 r,t 兲 ⫽H l1 共 r,t 兲 .
共4.10兲
The explicit metric perturbations can be found directly
from the previous subsection, Eqs. 共4.2兲–共4.6兲 upon exchanges of the tetrad contractions l→n and m→m̄, and the
consequent change in normalizations, s, and potentials, as
described throughout the paper.
V. DISCUSSION
We have explicitly computed the metric perturbations of a
nonrotating black hole in terms of the Weyl scalars ␺ 4 and
␺ 0 which can be computed directly by solving the Teukolsky
equation for any appropriate astrophysical scenario, given
the corresponding initial data. In doing so we had to invert
Eq. 共2.7兲 or Eq. 共2.9兲. This was performed making explicit
use of the multipole decomposition of the potential, Weyl
scalars and metric. The extension of this procedure to the
rotating background is not straightforward, but we have
learned some key features: The algebraic special solutions of
the Teukolsky equation will play a crucial role in finding the
solution for the Hertz potential in terms of ␺ 4 or ␺ 0 . One can
see this as follows. In order to invert Eq. 共2.7兲 for the potential we first seek out solutions of the homogeneous equation.
Hence ␺ 4 should vanish, but for the solution to be nontrivial,
␺ 0 , given by Eq. 共2.6兲, must not vanish 关22兴. These two
conditions are precisely the conditions that define the algebraically special solutions for the potential satisfying the
vacuum Teukolsky equation 共1.2兲. These two solutions 共in
the time domain兲 should allow one to build up the kernel that
inverts the fourth order equation 共2.7兲. An identical argument
applies for the ORG potential when working with Eqs. 共2.8兲
and 共2.9兲.
One main application of this formalism is to go beyond
first order perturbation theory and compute second order perturbations of rotating black holes. In Ref. 关16兴 there was
developed a formalism for the second order Teukolsky equation that takes the form of the first order wave operator acting on the second order piece of the Weyl scalar ␺ 4 , and a
source term built up as a quadratic combination of first order
perturbations. It is precisely to compute this source term that
one needs the explicit form of the metric perturbations. In
Ref. 关16兴 it was found that to describe the emitted gravitational radiation the ORG gauge is specially well suited.
Hence one has to solve the first order Teukolsky equation for
(1)
␺ (1)
4 and ␺ 0 , then later to build up the source of the second
order piece of the Weyl scalar ␺ (2)
4 关see Eq. 共9兲 in Ref. 关16兴兴.
A second important application of the reconstruction of
metric perturbations around the Kerr background is to compute the self-force of a particle orbiting a massive black hole
关23,24兴 and to compute corrected trajectories 关4兴 depending
on the perturbed metric and connection coefficients along the
particle world line. This task is left for a future paper. While
we know the form of the Teukolsky equation in the presence
of perturbative matter around a Kerr hole 关see Eq. 共1.2兲兴, we
need to generalize the equation satisfied by the potential and
the relationship between this potential and the metric perturbations, i.e. the generalization of Eqs. 共2.3兲 and 共2.5兲. In
particular we know that not all of conditions 共2.2兲 or 共2.4兲
can hold in the presence of matter since they are then incompatible with the Einstein equations 关25兴.
ACKNOWLEDGMENTS
We wish to thank H. Beyer, M. Campanelli and S. Detweiler for helpful discussions. This research has been supported in part by NSF Grant No. PHY-9800977 共B.F.W.兲
with the University of Florida. C.O.L. developed part of this
work in the Albert Einstein Institut, Germany, which B.F.W.
also thanks for hospitality.
APPENDIX: RELATION BETWEEN IRG AND ORG
POTENTIALS
We give here a relation expressing the potential ⌿ ORG in
terms of the result for ⌿ IRG . This was not given in 关11–14兴.
We begin by defining a field ␹ , with spin weight ⫺2,
through
1
⌿ IRG ⫽ 关 L̄L̄L̄L̄¯␹ ⫹12␳ ⫺3 ␺ 2 ⳵ t ␹ 兴 ,
4
共A1兲
analogous to Eq. 共2.9兲. The solution for ⌿ ORG is
⌿ ORG ⫽DDDD 关 ¯␹ 兴 ,
共A2兲
also the relation 共2.6兲 between ␺ 0 and ⌿ IRG . A potential
degeneracy for algebraically special modes was discussed
briefly in 关26兴. The explicit solution for the modes ␹ lm can be
written, in terms of ␹ ⫾ given by Eq. 共3.4兲, as
␹ ⫹ 共 r 兲 ⫽z 2 共 r 兲
冕
r
y 2共 r ⬘ 兲 V ⫹共 r ⬘ 兲
2M
⌬共 r⬘兲2
⫹y 2 共 r 兲
冕
␹ ⫺ 共 r 兲 ⫽⫺y 1 共 r 兲
冕
⫺z 1 共 r 兲
冕
⬁z
2共 r ⬘ 兲 V
⫹
⌬共 r⬘兲
r
dr ⬘
共r⬘兲
2
dr ⬘ ,
r
z 1共 r ⬘ 兲 V ⫺共 r ⬘ 兲
2M
⌬共 r⬘兲2
⬁y
r
1共 r ⬘ 兲 V
⫺
共r⬘兲
⌬共 r⬘兲2
共A3兲
dr ⬘
dr ⬘ ,
共A4兲
cf. Eqs. 共3.14兲 and 共3.15兲 above. Here V ⫾ are given by
V ⫾ª
r4
r4
⫾
⫾
兲⫿
兲
共 ⳵ t ⌿ IRG
共 ⍀ AS 兲共 ⌿ IRG
3M ⌬
3M ⌬
⫹4
r
⫾
兲,
共 M r/⌬⫺1 兲共 ⌿ IRG
3M
similar to Eq. 共3.8兲, but note the sign differences.
024026-6
共A5兲
PHYSICAL REVIEW D 66, 024026 共2002兲
RECONSTRUCTION OF BLACK HOLE METRIC . . .
关1兴
关2兴
关3兴
关4兴
关5兴
关6兴
关7兴
关8兴
关9兴
关10兴
关11兴
关12兴
关13兴
关14兴
关15兴
关16兴
T. Regge and J. Wheeler, Phys. Rev. 108, 1063 共1957兲.
F.J. Zerilli, Phys. Rev. D 2, 2141 共1970兲.
V. Moncrief, Ann. Phys. 共N.Y.兲 88, 323 共1974兲.
C.O. Lousto, Phys. Rev. Lett. 84, 5251 共2000兲.
C.O. Nicasio, R. Gleiser, and J. Pullin, Gen. Relativ. Gravit.
32, 2021 共2000兲.
R.J. Gleiser, C.O. Nicasio, R.H. Price, and J. Pullin, Phys. Rep.
325, 41 共2000兲.
E. Newman and R. Penrose, J. Math. Phys. 3, 566 共1962兲.
S.A. Teukolsky, Astrophys. J. 185, 635 共1973兲.
S. Chandrasekhar, The Mathematical Theory of Black Holes
共Oxford University Press, Oxford, England, 1983兲.
R.H. Price and J.R. Ipser, Phys. Rev. D 44, 307 共1991兲.
P.L. Chrzanowski, Phys. Rev. D 11, 2042 共1975兲.
J.M. Cohen and L.S. Kegeles, Phys. Rev. D 10, 1070 共1974兲.
R.M. Wald, Phys. Rev. Lett. 41, 203 共1978兲.
L.S. Kegeles and J.M. Cohen, Phys. Rev. D 19, 1641 共1979兲.
M. Campanelli, C.O. Lousto, J. Baker, G. Khanna, and J. Pullin, Phys. Rev. D 58, 084019 共1998兲; 62, 069901共E兲 共2000兲.
M. Campanelli and C.O. Lousto, Phys. Rev. D 59, 124022
共1999兲.
关17兴 J. Baker, B. Brugmann, M. Campanelli, C.O. Lousto, and R.
Takahashi, Phys. Rev. Lett. 87, 121103 共2001兲.
关18兴 J. Baker, M. Campanelli, C.O. Lousto, and R. Takahashi, Phys.
Rev. D 65, 124012 共2002兲.
关19兴 J.N. Goldberg, A.J. MacFarlane, E.T. Newman, F. Rohrlich,
and E.C.G. Sudarshan, J. Math. Phys. 8, 2155 共1967兲.
关20兴 M. Campanelli and C.O. Lousto, Phys. Rev. D 58, 024015
共1998兲.
关21兴 S. Chandrasekhar, Proc. R. Soc. London A392, 1 共1984兲.
关22兴 R.M. Wald, J. Math. Phys. 14, 1453 共1973兲.
关23兴 Y. Mino, M. Sasaki, and T. Tanaka, Phys. Rev. D 55, 3457
共1997兲.
关24兴 T.C. Quinn and R.M. Wald, Phys. Rev. D 56, 3381 共1997兲.
关25兴 L. Barack and A. Ori, Phys. Rev. D 64, 124003 共2001兲.
关26兴 B.F. Whiting, J. Math. Phys. 30, 1301 共1989兲.
关27兴 We have made the choice that ⑀ ⫽0.
关28兴 For Newman-Penrose scalars, as in Eqs. 共4.2兲–共4.6兲, even and
odd really refer to contributions to the real and imaginary parts
of the whole scalar, not specifically to the individual coefficients in a mode sum decomposition.
024026-7