A Summary of the
Black Hole
Perturbation Theory
Steven Hochman
Introduction
Many frameworks for doing perturbation theory
The two most popular ones
Direct examination of the Einstein equations ->
Zerilli-Regge-Wheeler equations for Schwarzschild.
Newman-Penrose formalism -> Bardeen-Press equation
for the Schwarzschild type, and the Teukolsky
equation for Kerr type black holes.
The Metric
In spherical polar coordinates the flat space
Minkowski metric can be written as
ds2 = −dt2 + dr2 + r2 dΩ2
where
r2 dΩ2 = r2 dθ2 + r2 sin2 θdφ2
Schwarzschild
The Schwarzschild metric is a vacuum solution
!
2M
2
ds = − 1 −
r
"
2
dr
2
2
$
dt2 + #
+
r
dΩ
1 − 2M
r
The coordinates above fail at R = 2M
Killing Vectors
Killing vectors tell us something about the
physical nature of the spacetime.
Invariance under time translations leads to
conservation of energy
Invariance under rotations leads to conservation of
the three components of angular momentum.
Angular momentum as a three-vector: one component
the magnitude and two components the direction.
Killing Vectors of
Schwarzchild
Two Killing vectors: conservation of the direction
of angular momentum -> we can choose pi = 2 for
plane
Energy conservation is shown in the timelike
Killing vector
µ
µ
K = (∂t ) = (1, 0, 0, 0)
Magnitude of the angular momentum conserved by the
final spacelike Killing vector
R = (∂φ ) = (0, 0, 0, 1)
µ
µ
Geodesics in
Schwarzschild
The geodesic equation can be written after some
simplification as
1
2
!
dr
dλ
"2
+ V (r) = ε,
The potential is
1
GM
L2
GM L3
V (r) = ! − !
+ 2−
2
r
2r
r3
The Event Horizon and
the Tortoise
Null cones close up
!
dt
2GM
=± 1−
dr
r
"−1
Replace t with coordinate that moves more slowly
t = ±r + constant
∗
where
!
r
r = r + 2GM ln
2GM
∗
"
−1
More Tortoise
ds =
2
!
2GM
1−
r
"
(−dt + dr ) + r dΩ
R = 2GM -> - infinity
Transmission Reflection
2
∗2
2
2
Kruskal Coordinates
32G M −r/2GM
2
2
2
2
ds = −
e
(−dT + dR ) + r dΩ
r
2
3
3
Null cones
T = ±R + constant
Unlike the tortoise the event horizon is not
infinitely far away, and is defined by
T = ±R
Vishveshwara
Kerr
!
2GM r
2
ds = − 1 −
ρ2
"
2
2GM
ar
sin
θ
ρ2 2
2
dt −
(dtdφ + dφdt) + dr
ρ2
∆
2
sin
θ 2
2
2 2
2
2
2
2
+ρ dθ +
[(r
+
a
)
−
a
∆
sin
θ]dφ
ρ2
where
∆(r) = r − 2GM r + a
2
2
and
ρ (r, θ) = r + a cos θ
2
Angular momentum
2
2
2
Einstein Field
Equation
Rµν
1
− Rgµν = 8πGTµν
2
Can also be written as
Rµν = 8πG(Tµν
1
− T gµν )
2
Perturbations
Rµν = 0
For a perturbation
!
gµν
= gµν + hµν
Inserting this in
Rµν + δRµν = 0
But
δRµν = 0
Schwarzschild
Perturbations
Regge and Wheeler - Spherical Harmonics
Stability? Gauge invariance? Physical Continuity?
“Ring down”
Zerilli - Falling particle
Tensor Harmonics
Separate the solution into a product of four
factors, each a function of a single coordinate.
This separation is best achieved by generalizing
the method of spherical harmonics already
established for vectors, scalars, and spinors.
Parity
Scalar functions have even parity.
Two kinds of vectors, each of different parity:
One the gradient of a the spherical harmonic and
has even parity. The pseudogradient of the
spherical harmonic, and has odd parity.
There are three kinds of tensors. One is given by
the double gradient of the spherical harmonic and
has even parity. Another is a constant times the
metric of the sphere, also with even parity. The
last is obtained by taking the double
pseudogradient; it has odd parity.

2
∂
∂
 its even(electric or polar) and odd(magnetic +G(t,
r)( dθ
−(cos
)]
into
or axial)
parity
parts θ)( sin1 θ )( ∂φ
2 )]


Sym
Sym
r2 [K(t, r) sin2 θ
Odd/Magnetic/Axial
parity = YLMSym
×



∂2

1
∂
∂
+G(t,
r)[(

0
0
−h0 (t, r)( sin θ )( ∂φ )
h0 (t, r)(sin θ)( ∂θ ) ∂φ2 )

∂
∂ θ)( )
+(sin θ)(
θ)(cos
 0
∂θ
0
−h (t, r)( 1 )( ∂ )
h (t, r)(sin
)
1
1

sin θ ∂φ
∂θ

2
2
1
∂
1
1
∂
∂
 Symof Sym
Due to the spherical symmetry
the background
equations
(**)sinand
do(cos θ)( ∂θ
h2 (t, r)[( sinmetric,
)(
) 2 h2 (t, r)[(
)( (**)) +
)

θ ∂θ∂φ
θ ∂φ∂φ

2
not mix terms belonging to different L and
parity.
L θon∂ the
−(cos
θ)( sin12M
−ofsin
)]
)( is∂ )]the projection
Black Hole Perturbation Theory
15
∂θ∂θ
θ ∂φ

2
Even and Odd
A Summary of the








∂
z − axis. To apply quantum
language
can say that∂ L,M
Sym
Sym to a classical
Sym problem,−hwe
2 (t, r)[(sin θ)( ∂θ∂φ ) − (cos θ)( ∂φ )
andharmonics
the parityfrom
are above
constants
of able
the motion.
Theperturbation
existence of hstill
another constant
Using the tensor
we are
to split
the
µν
M
Even/Electric/Polar Parity = YL ×
fromand
theodd(magnetic
circumstance
that
the background
into its even(electricfollows
or polar)
axial)
parity parts metric is independent of the cotime,
 or
∂
M
(1 −consider
2M/r)Ha0 (t,
r)
H
r)
h0 (t, r)(ω∂θ
h0 (t, r
T =
ct. On
can
perturbation
of1a(t,definite
frequency,
=) kc,
Odd/Magnetic/Axial
parity
= Ythis
L ×account we


r) h (t, r)( ∂ )
−1

Sym h (t, r)(sin
− 2M/r)
h1 (t, r
1
∂ the perturbation
∂have
so that
of
hµν(1will
aH
time
of the
2 (t, dependence
1

A Summary of the Black 0Hole
Perturbation
Theory
15
∂θ
0 every
−h0component
(t, r)(
)(
)
θ)(
)
0

sin
θ
∂φ
∂θ
(iωt)
(−ikT )
2

 the
=
e
.
proceed
to
determine
completely
form
of
the
r[K(t,
r)
r
G(t, r)[(
1We
∂thereforeSym
∂ Sym
 0form e


0
−h1 (t, r)( sin θ )(∂φ )
h1 (t, r)(sin θ)( ∂θ )


2
∂
Using the tensorharmonics
fromsolution
above we
are
able to
split L
theand
perturbation

µν frequency.
individual
of
parity,
M∂ 2values, hand
Ther)(
general
+G(t,
−(cos θ)(
1 specified
∂2
1
1
∂
2 )]
 Sym

dθ

Sym
h
(t,
r)[(
)(
)
h
(t,
r)[(
)(
)
+
(cos
θ)(
)
2
2

sin θ or∂θ∂φ
2parity parts
sin θ ∂φ∂φ
∂θ

into its even(electric 
or polar)
and will
odd(magnetic
axial) of
solution
be a superposition
these
individual
with
coefficients


2solutions
Sym
Sym
Sym adjusted r2 [K(t,

∂
1
∂


)]
−
sin
θ
)]
M −(cos θ)( sin2 θ )(
∂φ
∂θ∂θ
is no need to work
Odd/Magnetic/Axialparity

L ×
to =
fit Ythe
appropriate boundary
conditions and initial
values. There
+G(t, r
∂2
∂



Sym Sym
Sym
−h
(t,
r)[(sin
θ)(
)
−
(cos
θ)(
)
2
∂ as all will lead
∂ radial equation.
∂θ∂φ
∂φ In this case we will
an0 (t,
arbitrary
tor)(sin
the same
0
0with−h
r)( sin1 θ )(M∂φ
)
h0 (t,
θ)( ∂θ
)
+(sin θ)(c


M 1
∂ advantage that φ will completely
∂
 0
 from the calculations.
Even/Electric/Polar
Parity
=
× the
M 1=
with
disappear
L
0take
−h
(t,0Yr)(
)(the
) spherical symmetry
h1 (t, r)(sinofθ)(
)background

 metric, equations (**) and (**) do
sin
θ
∂φ
∂θ
Due
to
the



The odd Sym
waves
contain
three
unknown
functions:
2
2
1
∂ amount
1 of work 1remains.
∂
∂ odd ∂
∂ three unknown
A2 (t,
considerable
The
waves
contain

Sym
h
r)[(
)(
)
h
(t,
r)[(
)(
)
+
(cos
θ)(
)
(1
−
2M/r)H
(t,
r)
H
(t,
r)
h
(t,
r)(
)
h
(t,
r)(
)
2
0 not
0

0 M is
terms21belonging
L and ∂θparity.
of L on the
sin θmix
∂θ∂φ
sinto
θ different
∂φ∂φ
∂θ
∂φ the projection




2
∂
1
∂
∂
∂
functions
of
r:
(h
,
h
,
h
).
The
even
waves
contain
seven
unknown
functions:
−1


1
2


−(cos
)] ∂θ ) to a classical
)((1 0−)]2M/r)
Sym θ)(
H2 (t,−r)sinhθ1language
(t, r)(
h1 (t, r)(
z sin
− 2axis.
problem,
we

 can say that L,M
∂θ∂θ
∂φ
θ ∂φ To apply quantum



of gauge
2
(H
,
H
,
H
,
G,
K,
h
,
h
).
The
calculations
can
be
greatly
simplified
by
the
use
∂
∂
2
2
0
1
2
0
1
 another constant
Sym
Sym
r[K(t,
r)
r
G(t,
r)[(∂
/∂θ∂φ)
Sym
Sym
Sym
−h
(t,
r)[(sin
θ)(
)
−
(cos
θ)(
)
and
the
parity
are
constants
of
the
motion.
The
existence
of
still
2


∂θ∂φ
∂φ


transformations.
∂2
1
∂


+G(t,
r)(
−(cos
θ)(
)(
)]
follows
from
the
circumstance
that
the
background
metric
is
independent
of the cotime
M
2 )]
dθ
sin
θ
∂φ

Even/Electric/Polar 
Parity
=
Y
×
L


Sym T = ct. On thisSym
Sym
r2 [K(t,
r)ofsin
θ 
account we can
consider a perturbation
a2definite
 frequency, ω = kc

∂
∂

2
(1 −
2M/r)H
(t,
r)
H
(t,
r)
h
(t,
r)(
)
h
(t,
r)(
)
5.4.
Gauge
Transformations
0
1
0
0
∂
 dependence of the

∂θ
∂φ
so
that
every
component
of
the
perturbation
h
will
have
a
time
+G(t,
r)[(
)
2

µν ∂


∂φ
∂
−1

∂
Sym
(1
−
2M/r)
H
(t,
r)
h
(t,
r)(
)
h
(t,
r)(
(iωt)
(−ikT
)
2
1
1


∂θ
∂φ
form e gauge,
= e which
. We
therefore
proceed
determine
completely
+(sin
θ)(cos
θ)(
) Whatthe
The Regge-Wheeler
we are
about
to 2walk to
through,
is unique.
weform of the


2

 ∂θ
Sym
Sym
r[K(t,
r)
r
G(t,
r)[(∂
/∂θ∂φ)
 frequency.
solution
of
specified
parity,
L and
M(**)
values,
and
aresymmetry
trying toindividual
do the
is create
gauge invariant
quantities
and
work
in
a do
fixed
gauge.
Actually,The genera
Due to 
the spherical
of
background
metric,
equations
(**)
and


∂2
1
∂

+G(t, r)(of
)] −(cos
θ)( sin θ solutions
)( ∂φ )] 
dθ2 these
solution
will
beparity.
athe
superposition
individual
coefficients
 with
these
two
tasks
are
one
and
same.
As
long
as
one
works
in
athe
uniquely
fixed
gauge, adjusted
not mix
terms
belonging
to
different
L
and
M
is
the
projection
of
L
on


2
2
Sym quantities
Sym
rin[K(t,
r) sinthat
θ one
 There

to one
fit the
appropriate
boundary
conditions
and
initial
values.
is no need to work
is Sym
dealing
with are
gauge invariant,
the
sense
 can translate
 To applythe
2
z − axis.
quantum
language
to
a
classical
problem,
we
can
say
that
L,M
∂


+G(t,
r)[(
) equation.

with
an
arbitrary
Mwithout
as all will
the
same
radial
In this case we wil
∂φ2
into any
gauge
one wants
changing
the
physical
problem.
In the following
and theparity are them
constants
of
the
motion.
The
existence
oflead
stillto
another
constant
∂
+(sin
θ)(cos θ)( disappear
)
takethe
M background
= 0 with themetric
advantage
that φ will
from the calculations
follows from the circumstance that
is independent
ofcompletely
the cotime,∂θ
A considerable
work
remains.
The
Due toTthe
spherical
symmetry
of can
the consider
background
metric,amount
equations
(**) and
(**) do
ct.
On this
account we
a seven
perturbation
of a of
definite
frequency,
ω =odd
kc, waves contain three unknown
The =even
waves
contain
unknown
functions:
of r:hM(his
h1 , hprojection
The of
even
waves
not mix
belonging
to different
Lperturbation
and parity.
L on
the contain seven unknown functions
0 , the
2 ).
so terms
that every
component
of thefunctions
µν will have a time dependence of the
(H0to
, Hproceed
H2 , G,toK,determine
h0 , h1 ). The
calculations
be of
greatly
(−ikT )
z − axis.
apply
language
a1 , classical
problem,
we
can
say that
L,M
formTo
e(iωt)
= equantum
. We
therefore
completely
thecan
form
the simplified by the use of gauge
transformations.
and the
parity are
constants
of the motion.
of and
still frequency.
another constant
individual
solution
of specified
parity, LThe
andexistence
M values,
The general
followssolution
from thewill
circumstance
that the background
metric is
independent
of the cotime,
be a superposition
of these individual
solutions
with coefficients
adjusted
5.4.a Gauge
Transformations
T = ct. On this account we can consider
perturbation
of a definite frequency, ω = kc,
Gauge Transformations
A Summary of the Black Hole Perturbation Theory
16
we will use calculations in the Regge-Wheeler gauge. Any result can be expressed in a
gauge of
invariant
manner
substitutingTheory
the Regge-Wheeler gauge quantities in terms
A Summary
the Black
Holeby
Perturbation
16 of
The
Regge-Wheeler
a is unique fixed gauge
a general
gauge. (See Gleiser gauge
1996 for proof)
of in
thethe
Black
Hole the
Perturbation
Theory
16
we will A
useSummary
calculations
Regge-Wheeler
gauge.
Any
result
can beviewed
expressed
in a
Different
waves
can
represent
same
physical
phenomena
in different
systems
ofmanner
coordinates.
Consider
an
coordinate
gauge
invariant
by substituting
theinfinitesimal
Regge-Wheeler
gauge transformation:
quantities in terms of
The
quantities
are
gauge
invariant
wegauge.
will use
in the Regge-Wheeler gauge. Any result can be expressed in a
a general
(Seecalculations
Gleiser
!α
α 1996
α for proof)
α
α
x
=
x
+
ξ
(ξ
!
x
).
(76) in terms of
gauge waves
invariant
manner
bythe
substituting
the phenomena
Regge-Wheeler
gauge
quantities
Different
can
represent
same
physical
viewed
in
different
Any result can be expressed
in a gauge invariant
α
aofgeneral
gauge.displacements
(See Gleiser
for coordinate
proof)
infinitesimal
ξ 1996
transform
like a vector.
In the new frame we shall
systemsThe
coordinates.
Consider
an infinitesimal
transformation:
manner by substituting the Regge-Wheeler gauge
have: Different
can represent the αsame αphysical phenomena viewed in different
!α
α waves
α
x =x +
(ξ ! x ).gauge
(76)
quantities
inξ terms of a general
systems of coordinates.
infinitesimal
coordinate transformation:
!
gµν
+ h!µν = gµνConsider
+ ξµ ;ν +ξνan
;µ +h
(77)
µν .
α
The infinitesimal displacements ξ transform like a vector. In the new frame we shall
!α
α
α
α
α
Consider
x
=
x
+
ξ
(ξ
!
x
).the Schwarzschild
(76)
Now
h
is
defined
as
the
difference
between
the
perturbed
metric
and
µν
have:
metric written in spherical coordinates. According to this definition, the difference in
α
!
!
The
infinitesimal
displacements
ξ
like a vector. In the new
g
+
h
=
g
+
ξ
;
+ξ
;
+h
.
(77) frame we shall
µν theµvalue
ν
ν µ
µνtransform
µν
µν have
the new frame
will
have:
new difference
old
Now hµν is defined ashthe
between the perturbed metric and the Schwarzschild(78)
µν = hµν + ξµ ;ν +ξν .
!
metric written in spherical
this definition, the difference in
gµν
+coordinates.
h!µν = gµν +According
ξµ ;ν +ξν ;to
(77)
µ +hµν .
This
result
can
be
interpreted
by
saying
that
the
infinitesimal
changes
in
the
coordinates
the new frame will have the value
of
the hhµν ,isundergo
aasgauge
transformation
quitethe
similar
to the metric
well known
gauge
Now
defined
the
difference
between
perturbed
and
the
Schwarzschild
µν
new
old
hµν = hfor
+ ξµelectromagnetic
;ν +ξν .
(78)
transformation
field. We use this to simplify the description
µν the
metric written in spherical coordinates. According to this definition, the difference in
of the hµν , undergo a gauge transformation quite similar to the well known gauge
The gauge for
transformation
can be field.
performed
on this
any toindividual
partial
wave.
transformation
the electromagnetic
We use
simplify the
description
Obviously
no real simplification
result unless the resulting wave still belongs to
of the perturbation
and make it will
unique.
the original
eigenvalues.
This requirement
the possible
for ξ α .partial
This vector
The gauge
transformation
can be limits
performed
on anychoices
individual
wave.
turns
out to no
be areal
spherical
harmonic
of the
same
L and
as thewave
partial
underto
Obviously
simplification
will
result
unless
theparity
resulting
stillwave
belongs
consideration.
Such a gauge
transformation
allowsthe
uspossible
to impose
additional
the original eigenvalues.
This
requirement limits
choices
for ξ α . simplifying
This vector
conditions
thea perturbation
hµν . We
havesame
therefore
turns out on
to be
spherical harmonic
of the
L andchosen
parity to
as eliminate
the partialthose
waveterms
under
which
contain the
derivatives
the highest order
with
to the
angles. simplifying
The final
consideration.
Such
a gauge of
transformation
allows
us respect
to impose
additional
α
radial
equations
then
simplify. Moreover,
the desired
gauge
transformation
can then
conditions
on the
perturbation
hµν . We have
therefore
chosen
to eliminateξ those
terms
be
foundcontain
by the the
use derivatives
of finite operations
only, without
arbitrary
and boundary
which
of the highest
order with
respectconstants
to the angles.
The final
conditions.
radial equations then simplify. Moreover, the desired gauge transformation ξ α can then
α
The gauge
thatoperations
simplifies the
odd
wave must
have the
be found
by thevector
use ofξ finite
only,general
without
arbitrary
constants
andform
boundary
Regge-Wheeler Gauge
conditions.
The
gauge
vector
odd
ξ 0 = 0;
ξα1 = 0; that
ξ µ = Λ(T, simplifies
r); "µν (∂/∂xν )YLM (θ, φ),the
(µ, ν = general
2, 3) (79)
The gauge vector ξ that simplifies the general odd wave must have the form
wave
has the form
according to the foregoing arguments. Moreover, the radial function Λ can be adjusted
ξ 0 = 0; ξ 1 = 0; ξ µ = Λ(T, r); "µν (∂/∂xν )YLM (θ, φ), (µ, ν = 2, 3) (79)
to annul the radial factor h2 (T, r).
The
final
canonical
form
for an odd
wave of
angular
momentum
L and
according
to the
foregoing
arguments.
Moreover,
thetotal
radial
function
Λ can
be adjusted
The
final
canonical
form
for
an
odd
wave
L,
projection
M=
0 is then
to annul the
radial
factor h2 (T, r).
is
M = 0
 L and
The final canonical form for an odd wave of total
0 angular
0
0momentum
h0 (r)


projection M = 0 is then


0
0
0
h
(r)
1
(−ikT )


hodd
=
e
(sin
θ)(∂/∂θ)P
(cos
θ)
×
L
µν

00 00 h00 (r)

 00
Sym Sym 0
01 (r) 

 0
0
0
h
odd
(−ikT
)


A Summary ofhthe
Black
Hole
Perturbation
Theory
17
=
e
(sin
θ)(∂/∂θ)P
(cos
θ)
×
L
µνthe Black Hole Perturbation Theory

A
Summary
of
17 even
0 the
0
0general

 0
The gauge vector that simplifies
0
The gauge transformation that simplifies even waves is Sym Sym 0
The gauge
transformation
that simplifies even waves is
wave
has
the form
M
M
ξ0 =
(θ, φ);
ξ1 =
(θ, φ);
(80)
0 M0 (T, r)YL M
1 M1 (T, r)YL M
ξ = M0 (T, r)YL (θ, φ);
ξ = M1 (T, r)YL (θ, φ);
(80)
M
2
M
ξ2 =
(θ, φ); ξ 3 =
(θ, φ).(81)
2 M (T, r)(∂/∂θ)YL M
3 M (T, r)(1/ sin θ)(∂/∂φ)Y
2
L M
ξ = M (T, r)(∂/∂θ)YL (θ, φ); ξ = M (T, r)(1/ sin θ)(∂/∂φ)YL (θ, φ).(81)
We adjust the factors M0 , M1 , and M to annul the factors G, h0 , and h1 in the
We adjust the factors M0 , M1 , and M to annul the factors G, h0 , and h1 in the
even perturbations,
thereby obtaining the
even waves
in the
formwave L, M = 0
The
final canonical
form
for
ancanonical
even
even perturbations, thereby obtaining the even waves in the canonical form


H0 (1 − 2M/r)

is
H1
0
0
H
(1
−
2M/r)
H
0
0


0
1
−1


Sym
H
(1
−
2M/r)
0
0
2
even
(−ikT )
−1


Sym
H
(1
−
2M/r)
0
0
hµνeven= e (−ikTP)L (cos θ) × 
2

2

hµν = e
PL (cos θ) ×
0
0
r
K
0


2
0
0
r K 2
02


0
0
0 r K
sin
θ
0
0
0 r2 K sin2 θ
There are therefore two unknown functions of r in the odd case (h0 , h1 ) and four unknown
The Choice of Gauge
There are now only two unknown functions for the
odd case and four for the even case
This helps tremendously with the differential
equations
But even perturbations increase with distance and
remain in unchanging magnitude for odd
1/r? We can choose another gauge (Radiation)
Solutions
Even/Electric/Polar
Odd/Magnetic/Axial
L = 0,1,2....
Static k=0
Solutions for L
values
There is no L = 0 odd/magnetic perturbation
L = 0, L = 1 even and L = 1 odd: the changes from
perturbations in mass, velocity, and angular
momentum, have exact solutions.
L >=2 describe the radiation, no exact solutions.
Odd/Magnetic
Solutions
A Summary
of the
Black
Hole Perturbation
Theory equations
For odd waves
there
are
three
non-trivial
condition. Thus we have a system of two first-order linear equations. The tw
18
Can be expressed
as
a
wave
equation
known
as
the
order equations can be expressed, after a series of substitutions, as a simple secon
Regge-Wheeler
Equation
condition.
Thus
we have known
a systemasofthe
twoRegge-Wheeler
first-order linear equation
equations. for
Theodd
twoperturbatio
firstSchrödinger
equation
A Summary of the Black Hole Perturbation Theory
A Summary
of the
Black Hole
Theory
18
order
equations
canPerturbation
be expressed,
after a series of substitutions, as a simple
second-order
2
d Ψodd
equation
known
the Regge-Wheeler
equation
2 first-order
condition.Schrödinger
Thus we have
a system
of twoas
linear equations.
The for
twoodd
first-perturbations:
dr∗2
+ k (r)Ψodd = 0.
order equations can be expressed,
d2 Ψodd after2 a series of substitutions, as a simple second-order
+ Regge-Wheeler
k (r)Ψodd = 0.equation for odd perturbations:
Schrödinger
equation
as
∗2 the
This
can known
alsodrbe
written
in the time-domain as
In time domain
2
Ψoddalso 2be written
Thisd can
as
d2 Ψodd
d2 Ψtime-domain
+ k (r)Ψ
= 0.in the
odd
odd
dr∗2
−
+ V (r)Ψ
odd
∗2 d2 Ψodd dt2
d2 Ψodd
dr
−
+asV (r)Ψodd = 0,
This can also be written in the
time-domain
dr∗2
dt2
with
d2 Ψodd
with
with
dr∗2
d2 Ψodd
−
+ V (r)Ψodd = 0,
dt2
2
L =
(82)
(83)
(83)
2
3
V
(r)
=
[−L(L
+
2 1)/r +
3 6M/r ](1 − 2M/r).
V (r) = [−L(L + 1)/r + 6M/r ](1 − 2M/r).
with
L =
= 0,
(82)
3
(r)
=is[−L(L
+ 1)/r
+
6M/r
](1
− 2M/r).
ForLL==0 V0there
there
is odd
no
odd
parity
type
harmonic.
For
no
parity
type harmonic.
LL
==
1 parity
perturbation
represents
the addition
of a angular
m0 a, where m0 a
The
1 perturbation
represents
the addition
of amomentum
angular momentum
For L = 0 thereThe
is no
odd
type harmonic.
the
perspective
this
can
themomentum
conserved
the
The perturbation
Lfrom
= 1 perturbation
represents
the
addition
ofseen
a angular
mangular
0 a, wheremomentum
from
theZerilli
Zerilli
perspective
thisbecan
beasseen
as the conserved
angular of
momentum
0 no
from the Zerilli
this can be seen as the conserved angular momentum of the
fallingperspective
fallingparticle.
particle.
falling particle.Static solutions of the odd type exist for L "= 0. For the odd equations with k = 0,
Static
solutions
of the
odd
exist
L "= 0.
For
1 addition
of
angular
Statichsolutions
of
the
odd
type exist
formomentum
L "= 0.type
For the
oddfor
equations
with
k =the
0, odd equations with
1 must vanish.
h1 must vanish.
h must
vanish.
with
dr
dt
1 perturbation
represents
the addition of a angular momentum m0 a, wh
For L = 0 The
thereLis=no
odd parity type
harmonic.
with
2
3
from
perspective
this can
seen asofthe
conserved
angular momentum
= [−L(L
+ 1)/r
+ 6M/r
](1
2M/r).
The
L the
=V (r)
1 Zerilli
perturbation
represents
the−be
addition
a angular
momentum
m0 a, whereof
2
3
V (r) = [−L(L + 1)/r + 6M/r ](1 − 2M/r).
falling
particle.
from
the
Zerilli
perspective
this can be seen as the conserved angular momentum of the
For L = 0 there is no odd parity type harmonic.
For
there
is no solutions
odd
parity of
type
harmonic.
Static
odd type
exist for
L "= 0. mFor
the odd equations with k =
falling
The
LL==10particle.
perturbation
represents
thethe
addition
of a angular
momentum
0 a, where
The
=
1solutions
perturbation
the
addition
of aangular
momentum
m0the
a,
where
hL1perspective
must
vanish.
from the Zerilli
this of
canrepresents
be seen
the
conserved
of
Static
the
odd as
type
exist
for
Langular
"= 0. momentum
For the odd
equations
with k = 0,
the Zerilli perspective this can be seen as the conserved angular momentum of the
fallingfrom
particle.
h1 must vanish.
fallingsolutions
particle.of the odd type exist for L "= 0. For the odd equations with k = 0,
Static
5.7.
Even/Electric
Static
solutions
of the oddSolutions
type exist for L "= 0. For the odd equations with k = 0,
h1 must vanish.
Even/Electric
Solutions
h5.7.
vanish.
1 must
Even/Electric
Solutions
For even
waves,are
the ten
Einstein
field equations give seven nontrivial conditio
For even waves
there
seven
non-trivial
5.7. Even/Electric
Solutions
one
algebraic
relation
of
two of
theequations
unknown
functions;
first-order
differen
For
even
waves,
the
ten Einstein
field
give
seven three
nontrivial
conditions:
equations:
One
algebraic
relation,
three
first5.7. Even/Electric
Solutions
equations;
and three
three
second-order
equations.
This leaves
six equati
even
the ten
Einstein
equations
givedifferential
seven functions;
nontrivial
conditions:
onewaves,
algebraic
relation
offield
two
ofsecond-order
the unknown
three first-order
differential
orderForequations,
and
equations.
For even waves, the ten Einstein field equations give seven nontrivial conditions:
one algebraic
relation
of two
of the
unknown
functions;
three equations.
first-order
differential
for
theand
three
remaining
unknowns.
Three
of
the first-order
equations
sufficient
equations;
three
second-order
differential
This
leaves six are
equations
one
algebraic
relation
of
two
of
the
unknown
functions;
three
first-order
differential
equations; and
three second-order
differential
equations.
This leaves
six equations
determine
a
solution
provided
the
divergence
conditions
on the
source
are satisfi
for
the
three
remaining
unknowns.
Three
of
the
first-order
equations
are term
sufficient
to
equations;
and three
second-order
equations.
Thisare
leaves
six the
equations
for the
three remaining
unknowns.
Three differential
of equation
the first-order
equations
sufficient
to
Can be
expressed
as
a
wave
known
as
Afteraremaining
some
substitutions
and
simplifications
we
can
arrive
thesatisfied.
second-or
determine
solution
provided
the
divergence
conditions
on
the
source
term
for
the
three
unknowns.
Three
ofother
the on
first-order
equations
are
sufficient
to atare
determine
a
solution
provided
the
divergence
conditions
the
source
term
are
satisfied.
Zerilli determine
Equation
Schrödinger
equation
known
the
Zerilli
for arrive
even
perturbations:
a solution
provided
divergence
on equation
theatsource
term
are satisfied.
Aftersubstitutions
some
substitutions
and
otherasconditions
simplifications
we
at the second-order
After some
and
otherthe
simplifications
we can
arrive
the can
second-order
After
some
substitutions
and
otheras
simplifications
weperturbations:
can arrive
at theperturbations:
second-order
2known
Schrödinger
equation
known
asdthe
Zerilli
equation
for even
Schrödinger
equation
the
Zerilli
equation
for
even
Ψ
even
2 equation for even perturbations:
Schrödinger
equation known as
the+Zerilli
k
(r)Ψeven = 0.
(
2
∗2
d Ψeven d22Ψ
dr
even
2 + k (r)Ψ
(84)
even = 20.
d
Ψ
∗2
even
(84)
2 + k (r)Ψeven = 0.
dr
+
k (r)Ψ
0. the time-domain as
(84)
∗2be
even = in
dr
This can
also
written
∗2
dr
This can
also be written in the time-domain as
In time
domain
2
Thiscancan
also
be2 written
in
thedtime-domain
as
This
also
be written
in2 Ψ
the
time-domain
d
Ψevenas
even
2
d Ψeven d Ψeven
− = 0,
+ V (r)Ψeven = 0,
(
2− 2
2+ V
(85)
∗2(r)Ψ
2
even dt2
d
Ψ
d
Ψ
dr
∗2
2
even
even
dΨ
d+
Ψeven
dr
dt−
even
V (r)Ψ
= 0,
(85)
−
+even
V (r)Ψ
(85)
∗2
2
even = 0,
dr
dt
∗2
2
with
dr
dt
with
with !
" # # !3
!
"$
"# #
! $
"$
$
(L−1)L(L+1)(L+2) 3M
3
2M
1 72M
12M
3M
with
(L−1)L(L+1)(L+2)
2M
1
72M
12M
"
#
#
!
"$
$
V (r) = 1 − !rV (r)
−
(L
−
1)(L
+
2)
1
−
+
,
3
− r3 (L
1)(L +(L−1)L(L+1)(L+2)
+
,
λ2 = r 511 −
r −3M
r 22) 1 −
2
5
2
72Mrr3
12M
with
λ
r
r
r
V (r) = 1 −
−
(L
−
1)(L
+
2)
1
−
+
,
! 2M
"
#
#
!
"$
$
2
r
λ2
r5
r3 3
r
2M
1 72M
12M
3Mr
and
V (r) = 1 − r
− r3 (L − 1)(L + 2) 1 − r
+ (L−1)L(L+1)(L+2)
,
2
5
λ
r
r2
and
and
6M
.
λ
=
L(L
+
1)
−
2
+
r
and
6M
. + 1) − 2 + 6M .
λ = L(L + 1) −
λ 2=+L(L
r
r
L = 0 addition of mass
L = 1 shift of the cm
λ = L(L + 1) − 2 +
6M
.
r
the solution is time independent. From the Zerilli perspective, this addition m0 γ0 is the
mass-energy of the falling particle.
For L = 1, the nonzero hµν can be removed by a gauge transformation which can
be interpreted by a distant observer as a shift of the origin of the coordinate system.
When the perturbation is at large r, the shift looks like a transformation to the centerof-momentum system where the particles orbit each other with distances from the center
of momentum which are in inverse proportion to their relativistic masses.
In the static case where k = 0, one of the unknown functions vanishes, namely H1 .
For L = 0 the solution is trivial; the difference between the Schwarzschild metric mass
and the new mass with addition δm. For L = 1, we find a solution that corresponds to
a displacement of the center of attraction by the amount δz.
Solutions for L>=2
Radiation
not solve
5.8. Solutions for L ≥Can
2 and Gravitational
Waves
the equations explicitly
For L ≥ 2, we can not solve the equations explicitly. Asymptotically for large r in the
radiation gauge, the perturbation
hµν in the metric is at
the sum
of two transverse
traceless
Asymptotically
large
r the
perturbation
tensor harmonics listed above as d and f. Asymptotically for large r the radial factors
sum or two traces tensor harmonics.
have the form
M (m)
hM
0L (ω, r) ≈ −rAL
(ω)eiωr
is the
∗
(86)
Using a Green's function formed
from high
and
frequency-limit solutions, we obtain amplitudes for
M (e)
KLM (ω, r)
≈ AL ingoing
(ω)eiωr .
the
r=2M and outgoing (87)
r=infinity radiation
for radiation,
a particle
falling
radially
The amount of the escaping
which is of more
physical interest,
is determined into the black
M (m)
M (e)
completely by the amplitude
coefficients AL
and AL .
hole.
∗
Using a Green’s function formed from the high frequency-limit solutions of the
homogeneous equations, we obtain amplitudes for the ingoing (at r = 2M ) and
amplitude
peaks
attheapproximately
3/16piM
outgoing (at r = ∞) The
radiation
for a particle falling
radially into
black hole. The
amplitude, as a function
of frequency ω, increases
like a power
of ω. Amplitude total energy
Integrating
this,
the lawestimated
peaks at approximately ω = 3/16πM , and decreases exponentially for high frequencies.
radiated is
Integrating this, the estimated total energy radiated is (1/625)(m2o /M ) times a factor
of order 1.
To determine theTo
distribution
of the energy in
time rather than frequency,
must
determine
distribution
inwe time
use Fourier
form the Fourier integrals for even and odd waves, and construct the stress-energy tensor
from these time-dependent fields.
static
for
As VishveshwaraNo
indicates,
there seemperturbations
to be no static perturbations
for L L>=2
≥ 2 on
Stability
The Schwarzschild metric background gives an
equilibrium state.
If the metric is perturbed, however, will it remain
stable?
The collapsed Schwarzschild metric must be proven
to be stable against small perturbations.
A problem with coordinates chosen by Regge-Wheeler
prevented from judging whether any divergence shown
by the perturbations at the surface was real or due
to the coordinate singularity at r=2M. Using new
Kruskal coordinates, Vishveshwara was able to
determine background metric finite at the surface
and the divergence of the perturbations with
imaginary frequency time dependence violate the
small perturbation assumption. Thus perturbations
with imaginary frequencies are physically
unacceptable and the metric is indeed stable.
on a spherically symmetric background metric into its normal modes using the tensor
that the background metric is indeed finite at the surface and
spherical harmonics discussed above. A problem with the coordinates chosen by
perturbations with imaginary frequency time dependence violate
them, however, prevented them from judging whether any divergence shown by the
perturbations
withtoimaginary
frequencies
are p
perturbations atassumption.
the surface (rThus
= 2M
) was real or due
the coordinate
singularity
andthe
thenew
metric
is indeed
stable.
at r = 2M . Using
Kruskal
coordinates,
Vishveshwara was able to determine
that the background metric is indeed finite at the surface and the divergence of the
perturbations with
imaginary
frequency time
dependence violate the small perturbation
5.10.
Quasi-Normal
Modes
assumption. Thus perturbations with imaginary frequencies are physically unacceptable
The
Transmission-Reflection Perspective
and the metric is5.11.
indeed
stable.
!
Newman-Penrose
Formalism
x
if x ≥ 0;
The second 5.10.
popular
method
solving perturbation
Quasi-Normal
Modes for
|x| =
−x if x < 0.
equations is
the
Newman-Penrose
(NP)
5.11. The Transmission-Reflection Perspective formalism.
!
x
if x ≥ 0;
6. Newman-Penrose
Formalism
|x| =a
is
The NP formalism
notation
−x if x < 0. for writing various
quantities and equations
that
appear
insolving
relativity.
The second
popular
method for
perturbation equations is
6. Newman-Penrose
Formalism
It starts by
considering
a complex
tetrad
(NP) formalism.
The NPnull
formalism
is a notation for writing
such that
equations
that
appear
in relativity.
It isstarts
by considering
The second popular
method
for
solving
perturbation
equations
the
Newman-Penrose
−
→ −
−
→
→
−
→
(
l
,
n
,
m,
m) suchisthat,
(NP) formalism. The NP formalism
a notation for writing various quantities and
equations that appear in relativity.
It starts by−
considering
a complex null tetrad
−
→ −
−
→
→
→
−
→ −
l · n = 1 = − m · m.
−
→
→
( l ,→
n,−
m, m) such that,
−
→A−
−
→
→
is
for the directional derivatives along
l ·→
nnotation
= 1 = −−
m introduced
· m.
(88) tet
A notation is introduced for the D
directional
vectors:
= lµ ∂µ , derivatives
∆ =along
nµ ∂µtetrad
,
δ = mµ ∂µ ,
A Summary of the Black Hole Perturbation Theory
21
µ
= l ∂α,
∆κ,
=λ,
n µ,
∂µ ,ν, π, (used
δσ,=τmare
∂µheavily
, notation
δ ∗ = for
min
∂the
The projections ofD And
the
tensor
µ . spin (89)
µ ,Weyl
β,
γ,
&,
ρ,
a
coeffic
The projections of the Weyl tensor (used heavily in NP formalism in place of Gµν
NP formalism in place of G.. and R..) then become
and Rµν ) then become And α, β, γ, &, κ, λ, µ, ν, π, ρ, σ, τ are a notation for the spin coefficients of the null tetrad.
µ
µ
µ
Ψ0 = −Cµνρσ lµ mν lρ mσ Ψ1 = −Cµνρσ lµ nν lρ mσ Ψ2 = −Cµνρσ lµ mν mρ nσ
Ψ3 = −Cµνρσ lµ nν mρ nσ
Ψ4 = −Cµνρσ nµ mν nρ mσ .
7. Kerr Perturbations
and Rµν ) then become
σ
µ ν ρ σ
Ψ0 = −Cµνρσ
lµ mν lρofmthe
Ψ1 Hole
= −C
l m
Ψ2 = −Cµνρσ lµ mν mρ n21σ
A Summary
Black
Perturbation
µνρσ l n Theory
ν ρ σ
µ ν ρ σ
Ψ3 = −CµνρσThe
lµ nprojections
m n of the Weyl tensor (used heavily Ψ
=
−C
n
m nof m
4
µνρσ
in NP formalism in place
Gµν .
Kerr Perturbations
and Rµν ) then become
7. Kerr Perturbations
Ψ0 = −Cµνρσ lµ mν lρ mσ Ψ1 = −Cµνρσ lµ nν lρ mσ Ψ2 = −Cµνρσ lµ mν mρ nσ
Due to the complexity
of
Ψ3 = −Cµνρσ
lµ nνthe
mρ nσ Kerr metric, it
Ψ4 = −Cµνρσ nµ mν nρ mσ .
becomes
use the Einstein
The Kerr spinning
blackdifficult
hole metricto
is non-diagonal,
so it hasequations
cross-terms between
Kerrget
Perturbations
directly7. to
a solvable perturbation equation.
spatial and time coordinates. Due to the complexity of the Kerr metric, it becomes
Theequations
Kerr spinning
black hole
metric
is non-diagonal,
so it has cross-terms
between
difficult to useTo
the obtain
Einstein
directly
to
get
a
solvable
perturbation
equation.
To
the
perturbation
equation
for
rotating
spatial and time coordinates. Due to the complexity of the Kerr metric, it becomes
obtain the perturbation
equation
for
black directly
holes,Newman-Penrose
the Newmanblack holes,
used
the
difficult toTeukolsky
use
therotating
Einstein equations
toTeukolsky
get a solvable used
perturbation
equation. To
formalism.
over
much
laborious
calculation
obtainSkipping
the perturbation
equation
for
rotating
blackwe
holes,
Teukolsky
usedTeukolsky
the NewmanPenrose formalism.
Skipping
over
much laborious
calculation
arrive
at the
Penrose
Skipping over much
laborious calculation we arrive at the Teukolsky
we arrive
at formalism.
the Teukolsky
equation.
equation.
!
equation.
"
!
"
! 2∂ 2 Ψ
" 2
2
(r2 +a2 )2
∂ !2 Ψ
∂ 2 Ψ" ∂ 2 Ψ a4M
2
(r2 +a2 )24M ar
ar ∂ 2 Ψ1
a
1
∂ Ψ
2
−
a
sin
θ
+
+
−
−
a
sin
θ
+
−
2
2
2 +
2
2
∆
∂t
∆
∂t∂ψ
∆
∆
θ ∂φ!sin θ ∂φ"2
$ $ ∆ # !sin $
"
#
$ ∂t −s ∂ #∆#s+1∂t∂ψ
a(r−M )
∂Ψ
1 ∂
∂Ψ
i
cos
θ
∂Ψ
cos θ+ ∂Ψ
−∆1 ∂r∂ ∆ ∂r ∂Ψ
− sin θ ∂θ sin θa(r−M
−)2s i ∆
−s ∂
s+1 ∂Ψ
2θ
∂θ
∂φ
sin
−∆ ∂r ∆ ∂r − sin!θM∂θ
sin
θ
−
2s
+
"
2θ
2
2
∂θ
∆
∂φ
sin
(r "
−a )
∂Ψ
2
!
−2s
−
r
−
ia
cos
θ
+
(s
cot θ − s)Ψ = 0,
2
2
∆
∂t
M (r −a )
∂Ψ
2
−2s
− r − ia cos θ ∂t + (s cot θ − s)Ψ = 0,
∆
where
where once again
where once again
∆(r) = r2 − 2GM r + a2 .
(90)
−1
2 If s =
2 to
2, Ψ =Ψ 0 and
if s =achieve
−2, Ψ = ρ−4 Ψangular
.
While
not
possible
separation
in (90)
4 where ρ = (r−ia
cos θ)
∆(r) = r − 2GM r + a .
This equation
more involveddomain
than the Zerilli
While it is
the time domain,
inis considerably
the frequency
it equation.
is
not possible to achieve
−1 time domain, in the frequency domain
−4 angular separation in the
separable.
If s = 2, Ψ =Ψ 0 and if sit=is −2,
Ψ = ρ Ψ4 where ρ = (r−ia cos θ) .
separable. If you assume
This equation is considerably more−iωtinvolved
than the Zerilli equation. While it is
imφ
Ψ=e
e S(θ, ω)R(r, ω),
(91)
not possible to achieve angular separation in the time domain, in the frequency domain
M
2
Connections
When a=0 in the Teukolsky equation you are then
left with the Bardeen-Press equation for
Schwarzchild black holes. The Bardeen-Press
equation contains in its real and imaginary parts
the Zerilli and the Regge-Wheeler equations
respectively.
References
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Gainesville, 2008).
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[4] B.F Schutz, A First Course in General Relativity (Cambridge University Press, Cambridge, 1985).
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(University of Chicago Press, Chicago, 1991).
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[17] S. Chandrasekhar, and S. Detweiler, On the Equations Governing the Axisymmetric Perturbation of the Kerr Black
Hole Proc R. Soc. 345, 145 (1975).
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