arXiv:gr-qc/0010034 v2 26 Mar 2001
Perturbations of the Kerr spacetime in horizon penetrating coordinates
Manuela Campanelli
1
, Gaurav Khanna
2,3
, Pablo Laguna
2,4
, Jorge Pullin
2
, Michael P. Ryan
5
1. Albert-Einstein-Institut, Max-Planck-Institut f¨ur Gravitationsphysik,
Am M¨uhlenberg 1, D-14476 Golm, Germany
2. Center for Gravitational Physics and Geometry, Department of Physics,
The Pennsylvania State University, 104 Davey Lab, University Park, PA 16802.
3. Natural Science Division,
Southampton College of Long Island University,
Southampton NY 11968.
4. Department of Astronomy and Astrophysics, The Pennsylvania State University,
525 Davey Lab, University Park, PA 16802.
5. Instituto de Ci encias Nucleares, UNAM
Apartado Postal 70-543 M´exico 04510 D. F., M´exico.
We derive the Teukolsky equation for perturbations of a Kerr spacetime when the spacetime
metric is written in either ingoing or outgoing Kerr–Schild form. We also write explicit formulae for
setting up the initial data for the Teukolsky equation in the time domain in terms of a three metric
and an extrinsic curvature. The motivation of this work is to have in place a formalism to study
the evolution in the “close limit” of two recently proposed solutions to the initial value problem
in general relativity that are based on Kerr–Schild slicings. A perturbative formalism in horizon
penetrating coordinates is also very desirable in connection with numerical relativity simulations
using black hole “excision”.
I. INTRODUCTION:
There is considerable current interest in studying the collision of two black holes, since these events could be primary
sources of gravitational waves for interferometric gravitational wave detectors currently under construction. On the
theoretical side, it is expected that the problem of colliding two black holes will be tackled by some combination of
full numerical and semi-analytical methods. The first three dimensiona l collisions of black holes are starting to be
numerically simulated, albeit with very limited resolution and grid size. Long time stability of the codes is also an
issue. It is therefore of interest to have at hand approximate results which in certain regimes could be used to test
the codes. Among such approximation methods is the “close limit” a pproximation [1] in which the spacetime of a
black hole collision is repre sented as a single distorted black hole. This approximation has been used successfully to
test codes for the evolution of black hole space-times that are axisymmetric and tests are under way for “grazing”
inspiralling collisions [2]. Another realm of application of perturba tive calculations is to provide “outer bounda ries”
and to ex tend the reach of Cauchy codes into the radiation zone far away from the black holes, as was demonstrated
in [3]. Finally, perturbative codes can be used after a full non-linear binary black hole code has coalesced the holes
to continue the evolution in a simple and efficient fashion as was demonstrated in [4,5].
The perturbative approach requires specifying both a background metric and a coordinate system when performing
calculations. For evolutions in the time domain such as the ones we are considering, one also ha s to specify an initial
slice of the spacetime. In all perturbative evolutions performed up to now the background spacetime has either been
the Schwarzschild solution in ordinary coor dinates or the Kerr spacetime in Boyer–Lindquist coor dinates. These
backgrounds are adequate for instance , for the study of the evolution of “close limits” o f the Bowen–York [6] and
“puncture” [7] families of initial data, which reduce in the “ close limit” to those background space-times.
There have b een two recent proposals for alternative families of initial data that have some appealing features
[8,9]. Both these pr op osals are based on the use of the Kerr–Schild form of the Schwarzschild (or Kerr) solutions to
represent each of the black holes in the collision. Some of these solutions do not have an obvious close limit in which
they yield a s ingle black hole, although the close limit can be arranged with a simple modification of the original
proposal. In the cas e in which the close limit exists, the initial data appear as a perturbation of the Schwarzschild or
Kerr spacetime, but in Kerr–Schild coordinates. If one wishes to evolve perturbatively the s pacetime, this requires
having the per tur bative formalism set up on a background spacetime in Kerr-Schild coordinates. To our knowledge,
this has never been done in the past.
The use of Kerr– Schild coordinates appears quite desira ble in the context of numerical evolutions of black holes.
The c oordinates penetrate the horizon of the holes without steep gradients in the metric components. This makes
them amenable to the numerical technique of singularity excision, which is sometimes viewed as the key to long term
binary black hole evolutions such as the ones needed for gravitational wave data analysis purp oses [10].
1
Since the Kerr–Schild coordinates are horizon-penetrating, developing a perturbative formalism in these coordinates
could, in principle, allow the study of per tur bations arbitrarily close to the horizon a nd even inside the horizon. The
traditional pertur bative formalisms, based on Schwarzschild and Boyer–Lindquist (in the Kerr case) c oordinates
cannot achieve this. Having a perturbative formalism that works close to the horizon is desirable in the c ontext of
the recently introduced “iso lated horizons” [11]. One could exhibit perturbatively the validity of s e veral new results
that are emerging in such context.
In this pape r we will develop perturbative equations in Kerr– Schild coordinates, taking advantage of the fact that
the Teukolsky formalism is coordinate invariant. We will end by constructing a perturbative equation that is well
behaved inside, across and outside the horizon.
The organization of this paper is as follows. In section II we review the Teukolsky formalism (including the setup of
initial data for Cauchy evolution). In section III we derive the Teukolsky equation in hor iz on penetrating Eddington–
Finkelstein coo rdinates and display a numerical evolution of it. In an appendix we discuss the equation in outgoing
Eddington–Finkelstein coordinates.
II. THE TEUKOLSKY EQUATION
Fortunately, the perturbative formalism due to Te ukolsky [12] is amenable to a reasonably straig htforward (but
computationally non-trivial) change of background. The Teukolsky formalism is based on the observation tha t the
Einstein equations written in the Newman–Penrose formalism naturally decouple in such a way that one obtains an
equation for the perturba tive portion of the Weyl spinor. In the no tation and conventions of Teukolsky (which in
turn follows those of Newman and Penrose), the resulting e quation (in vacuum) is,
[(∆ + 3γ −γ
∗
+ 4µ + µ
∗
) (D + 4ǫ − ρ) − (δ
∗
− τ
∗
+ β
∗
+ 3α + 4π) (δ − τ + 4β) − 3ψ
2
] ψ
4
= 0, (1)
where the quantities in brackets are computed using the background geometry and ψ
4
is a first order quantity in
perturbation theory. A similar equation can be derived for the ψ
0
component of the Weyl tensor. We will not
concentrate on this equation, however, since it has not proven as useful for evolutions in the time domain [13].
If we now particularize to a background spacetime given by the Kerr metric in Boyer–Lindquist coordinates,
ds
2
= (1 − 2Mr/Σ ) dt
2
+
4Mar sin
2
θ/Σ
dtdϕ −
Σ
△
dr
2
− Σdθ
2
− sin
2
θ
r
2
+ a
2
+ 2Ma
2
r s in
2
θ/Σ
dϕ
2
, (2)
where Σ = r
2
+ a
2
cos
2
θ and △ = r
2
− 2Mr + a
2
(and should not be confused with the Newman–Penrose quantity
∆ = n
µ
∂
µ
) and considers the Kinnersley tetrad,
l
µ
=
r
2
+ a
2
/△, 1, 0, a/△
, (3)
n
µ
=
r
2
+ a
2
, −△, 0, a
/(2Σ), (4)
m
µ
= [ia sin θ, 0, 1 , i/ sin θ] /(
√
2(r + ia cos θ)), (5)
for which the background Newman–Penrose quantities are,
ρ = −1/(r − ia cos θ), (6)
β = −ρ
∗
cot θ/(2
√
2), (7)
π = iaρ
2
sin θ/
√
2, (8)
τ = −iaρρ
∗
sin θ/
√
2, (9)
µ = ρ
2
ρ
∗
△/2, (10)
γ = µ + ρρ
∗
(r − M)/2, (11)
α = π − β
∗
, (12)
ψ
2
= Mρ
3
. (13)
and the differential operators D = l
µ
∂
µ
, ∆ = n
µ
∂
µ
, δ = m
µ
∂
µ
, then the resulting equation is the well known Teukolsky
equation,
2
"
r
2
+ a
2
2
△
− a
2
sin
2
θ
#
∂
2
ψ
∂t
2
+
4Mar
△
∂
2
ψ
∂t∂ϕ
+
a
2
△
−
1
sin
2
θ
∂
2
ψ
∂ϕ
2
−△
2
∂
∂r
1
△
∂ψ
∂r
−
1
sin θ
∂
∂θ
sin θ
∂ψ
∂θ
+ 4
M(r
2
− a
2
)
△
− r − ia cos θ
∂ψ
∂t
+4
a(r − M )
△
+
i cos θ
sin θ
∂ψ
∂ϕ
+ (4 cot
2
θ + 2)ψ = 0, (14)
where ψ = (r − ia c os θ)
4
ψ
4
. As was discussed in references [14], one c an write the initial data for the above equation
in terms of the pertur bative three metric and extr ins ic curva tur e . The formulae for the Teukolsky function and its
time derivative are,
ψ
4
= −
R
ijkl
+ 2K
i[k
K
l]j
(1)
n
i
m
j
n
k
m
l
+ 8
h
K
j[k,l]
+ Γ
p
j[k
K
l]p
i
(1)
n
[0
m
j]
n
k
m
l
(15)
−4 [R
jl
− K
jp
K
p
l
+ KK
jl
]
(1)
n
[0
m
j]
n
[0
m
l]
,
∂
t
ψ
4
= N
φ
(0)
∂
φ
(ψ
4
) − n
i
m
j
n
k
m
l
[∂
0
R
ijkl
]
(1)
(16)
+8n
[0
m
j]
n
k
m
l
h
∂
0
K
j[k,l]
+ ∂
0
Γ
p
j[k
K
l]p
+ Γ
p
j[k
∂
0
K
l]p
i
(1)
−4n
[0
m
j]
n
[0
m
l]
h
∂
0
R
jl
− 2K
p
(l
∂
0
K
j)p
− 2N
(0)
K
jp
K
p
q
K
q
l
+K
jl
∂
0
K + K∂
0
K
jl
]
(1)
+2{ψ
4
(l
i
∆ − m
i
¯
δ)N
i (0)
+ ψ
3
(n
i
¯
δ − ¯m
i
∆)N
i (0)
},
where
ψ
3
= −
R
ijkl
+ 2K
i[k
K
l]j
(1)
l
i
n
j
¯m
k
n
l
+ 4
h
K
j[k,l]
+ Γ
p
j[k
K
l]p
i
(1)
(l
[0
n
j]
¯m
k
n
l
− n
[0
¯m
j]
l
k
n
l
) (17)
−2 [R
jl
− K
jp
K
p
l
+ KK
jl
]
(1)
(l
[0
n
j]
¯m
0
n
l
− l
[0
n
j]
n
0
¯m
l
),
N
(0)
= (g
tt
kerr
)
−1/2
is the zeroth order lapse, n
i
in these equations should be taken to be related to that of the original
tetrad as n
0
= N
(0)
n
0
orig
, n
i
= n
i
orig
+ N
i (0)
n
0
. Latin indices run from 1 to 3, and the brackets are computed to only
first order (zeroth or der excluded). The derivatives involved in the ab ove express ions can be computed in terms of
the initial data on the Cauchy hypersurfa ce as,
∂
0
K = N
(0)
K
pq
K
pq
− ∇
2
N
(0)
, (18)
∂
0
R = 2K
pq
∂
0
K
pq
+ 4N
(0)
K
pq
K
p
s
K
sq
− 2K∂
0
K, (19)
∂
0
R
ijkl
= −4 N
(0)
K
i[k
R
l]j
− K
j[k
R
l]i
−
1
2
R
K
i[k
g
l]j
− K
j[k
g
l]i
(20)
+2g
i[k
∂
0
R
l]j
− 2g
j[k
∂
0
R
l]i
− g
i[k
g
l]j
∂
0
R + 2K
i[k
∂
0
K
l]j
− 2K
j[k
∂
0
K
l]i
,
and,
∂
0
K
ij
= N
(0)
h
¯
R
ij
+ KK
ij
− 2K
ip
K
p
j
− N
−1
(0)
¯
∇
i
¯
∇
j
N
(0)
i
(1)
. (21)
Remarkably, the above formulae are coordinate independent! Therefore the only adjustment needed to specify
initial data for the evolution equations we will derive in the next two sections is to inse rt the appropriate background
quantities in the above formulae.
III. THE TEUKOLSKY EQUATION IN KERR–SCHILD COORDINATES:
INGOING EDDINGTON FINKELSTEIN COORDINATES
The initial da ta proposed by [8,9] is constructed using ingoing Eddingto n–Finkelstein (IEF) coordinates (strictly
sp e aking, since the initial data might include net angular momentum, one is really ta lk ing about the generalization
3
of IEF coordinates to the rotating case, commonly referred to as Kerr co ordinates). This is in part due to the fact
that these families of initial data are currently being evolved using a numerical code where the black holes are treated
using the “excision” technique. This technique requires coordinates that penetrate the horizon, such as the IEF ones.
The I EF coordinates (
˜
V , r, θ, ϕ) for the Kerr metric are defined through a redefinition of the time coordinate of the
Boyer–Lindquist coordinates as,
˜
V = t + r
∗
(22)
˜ϕ = ϕ +
Z
a
△
dr (23)
where r
∗
is the natural generalization to the Kerr case of the usual Schwarzschild “tortoise” coordinate, and is defined
by,
r
∗
=
Z
r
2
+ a
2
r
2
− 2Mr + a
2
dr. (24)
The codes curr ently being used to evolve the initial data in IEF are written in ter ms of a coordinate
˜
t =
˜
V −r. We
will therefore derive the Teukolsky equation in the (
˜
t, r, θ, ˜ϕ) coordinates. The Kerr metric in these coordinates,
ds
2
= (1 − 2M r/Σ) d
˜
t
2
− (1 + 2Mr/Σ) dr
2
− Σdθ
2
− sin
2
θ
r
2
+ a
2
+ 2Ma
2
r sin
2
θ/Σ
d ˜ϕ
2
−(4Mr/Σ) d
˜
tdr +
4Mra sin
2
θ/Σ
d
˜
td ˜ϕ + 2a sin
2
θ (1 + 2Mr/Σ) d˜rd ˜ϕ, (25)
In addition to changing coordinates, it is immediate to see that one needs also to change tetrads. The usual Kinnersley
tetrad is singular at the horizon, and therefore leads to a Teukolsky equation that is singular. However, it is easy to
fix this problem by just rescaling l
µ
by a factor of △ and dividing n
µ
by ∆. This does not change the orthogonality
properties of the tetrad, but makes it well defined
1
. Therefore we re-derive the Teukolsky equa tion using as new
tetrad vectors,
l
µ
= [△ + 4Mr, △, 0, 2a] (26)
n
µ
= [
1
2Σ
, −
1
2Σ
, 0, 0] (27)
This redefinition of the tetrad vectors changes the values of the Newman–Penrose scalars. The new values ar e ,
ǫ = r − M (28)
γ = µ = −
1
2
r + ia cos θ
Σ
2
(29)
ρ = −(r + ia cos θ)
△
Σ
(30)
with α, β, π, τ, ψ
2
remain unchanged. The r e sulting Teukolsky equation reads,
1
2
△ + 4Mr
Σ
∂
2
ψ
4
∂
˜
t
2
−
1
2
△
Σ
∂
2
ψ
4
∂r
2
− 2
Mr
Σ
∂
2
ψ
4
∂r∂
˜
t
−
a
Σ
∂
2
ψ
4
∂r∂ ˜ϕ
−
1
2
1
Σ
∂
2
ψ
4
∂θ
2
−
1
2
1
sin
2
θΣ
∂
2
ψ
4
∂ ˜ϕ
2
−
3a
2
(r − M ) cos
2
θ + r(7r
2
+ 4a
2
− 11M r) + 4ia cos θ△
Σ
2
∂ψ
4
∂r
−
r
2
(2r + 11M) + a
2
(2r − 3M) cos
2
θ − 2ia(r
2
+ 7Mr + a
2
cos
2
θ) c os θ
Σ(r − ia cos θ)
2
∂ψ
4
∂t
−
(r
2
+ 8a
2
− 9a
2
cos
2
θ) c os θ + 2iar(4 − 5 cos
2
θ)
2Σ(r − ia cos θ)
2
sin θ
∂ψ
4
∂θ
−
4ar(sin
2
θ − cos
2
θ) − 2i(r
2
+ 2a
2
− 3a
2
cos
2
θ) c os θ
Σ(r − ia cos θ)
2
sin
2
θ
∂ψ
4
∂ ˜ϕ
+
24M r sin
2
θ − 19r
2
+ (21r
2
− 9a
2
+ 7a
2
cos
2
θ) c os
2
θ − 2ia[(7r −6M) cos
2
θ − (5r − 6M)] cos θ
Σ(r − ia cos θ)
2
sin
2
θ
ψ
4
= 0. (31)
1
We wish to thank Steve Fairhurst and Badri Krishnan for suggesting this rescaling.
4
This equation can be used to study a variety of different things, including boundary conditions at the excision
region (inside the horizon), perturbations inside and near the horizon, and most significantly, we could use this
implementation to compare and perhaps even continue evolutions from full numerical codes that use Kerr–Schild
coordinates.
The Penetrating Teukolsky Code (PTC) evolves the following equation, where ψ = (r−ia cos(θ))
4
ψ
4
is the Teukolsky
function:
(Σ + 2Mr)
∂
2
ψ
∂
˜
t
2
− △
∂
2
ψ
∂r
2
− 6(r − M)
∂ψ
∂r
(32)
−
1
sin θ
∂
∂θ
sin θ
∂ψ
∂θ
−
1
sin
2
θ
∂
2
ψ
∂ ˜ϕ
2
− 4Mr
∂
2
ψ
∂
˜
t∂r
− 2a
∂
2
ψ
∂r∂ ˜ϕ
+
4i cot θ
sin θ
∂ψ
∂ ˜ϕ
−(4r + 4ia cos θ + 6M)
∂ψ
∂
˜
t
+ 2(3 cot
2
θ − csc
2
θ)ψ = 0.
To implement this equation numerically, we break the above equation down into a 2+1 dimensional one, using a
decomposition of the Teukolsky function into angular modes, ψ = Σψ
m
e
im ˜ϕ
. The Teukolsky equation for each m
mode now looks like the following:
(Σ + 2Mr)
∂
2
ψ
m
∂
˜
t
2
− △
∂
2
ψ
m
∂r
2
− (2aim + 6r − 6M)
∂ψ
m
∂r
(33)
−
1
sin θ
∂
∂θ
sin θ
∂ψ
m
∂θ
− 4Mr
∂
2
ψ
m
∂
˜
t∂r
− (4r + 4ia cos θ + 6M)
∂ψ
m
∂
˜
t
+(4 cot
2
θ − 2 + m
2
csc
2
θ − 4m cot θ csc θ)ψ
m
= 0.
We use L ax-Wendroff technique to numerically implement these simplified set of e quations exactly as done in [13]. A
full discussion of the applications of this code will be presented elsewhere. Here we just highlight in the figures how
the code indeed evolves perturbations inside and outside the horizon as expected.
IV. CONCLUSIONS
We have set up the black hole perturbation framework in Ke rr–Schild type coordinates. This framework will be
useful fo r comparisons with fully nonlinear numerical codes currently being implemented that run naturally in Kerr–
Schild coordina tes . We have also dis c ussed the numerical implementation of the perturbative evolution equation
and how to set up initial data in terms of the initial value data that will be available for black hole c ollisions.
Implementation of this framework for numerical computations is essentially complete.
This formalism allows us to study in a natural way perturbations close to the horizon and may also be of interest
to study and test in a concrete fashion several attractive properties of “isolated horizons” [11]. This formalism allows
us to make several predictions abo ut q uantities defined with notions intrinsic to the black hole (like a concept of local
mass and angular momentum), and their evolution. Several attractive formulae, for instance relating the ADM mass
to the “local horizon mass” and the “radiation content” can be worked out. Having a perturbative formalism that
operates correctly near and on the horizon will allow us to test the validity of these formulae. This issue is currently
under study.
V. ACKNOWLEDGMENTS
We wish to thank Dar´ıo N´u˜nez for pointing out several typos in an earlier version of this paper. This work
was supported in part by grants, NSF-INT-9722514, NSF-PHY-9423950, NSF-PHY-9800973, NSF-PHY-9800970,
NSF-PHY-9800973, and by funds of the Pennsylvania State University. M.C. holds a Marie Curie Fellowship (HPMF-
CT-1999-00334).
5
