a summary of the black hole perturbation theory hole perturbation... · equation for kerr type...

A Summary of the Black Hole

Perturbation TheorySteven 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.

In spherical polar coordinates the flat space Minkowski metric can be written as

where

The Metric

ds2 = !dt2 + dr2 + r2d!2

r2d!2 = r2d!2 + r2 sin2 !d"2

Schwarzschild

The Schwarzschild metric is a vacuum solution

The coordinates above fail at R = 2M

ds2 = !!

1! 2M

r

"dt2 +

dr2

#1! 2M

r

$ + r2d!2

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

Magnitude of the angular momentum conserved by the final spacelike Killing vector

Kµ = (!t)µ = (1, 0, 0, 0)

Rµ = (!!)µ = (0, 0, 0, 1)

Geodesics in Schwarzschild

The geodesic equation can be written after some simplification as

The potential is

V (r) =12!! !

GM

r+

L2

2r2! GML3

r3

12

!dr

d!

"2

+ V (r) = ",

The Event Horizon and the Tortoise

Null cones close up

Replace t with coordinate that moves more slowly

where

dt

dr= ±

!1! 2GM

r

"!1

t = ±r! + constant

r! = r + 2GM ln! r

2GM! 1

"

More Tortoise

R = 2GM -> - infinity

Transmission Reflection

ds2 =!

1! 2GM

r

"(!dt2 + dr!2) + r2d!2

Kruskal Coordinates

Null cones

Unlike the tortoise the event horizon is not infinitely far away, and is defined by

Vishveshwara

ds2 = !32G3M3

re!r/2GM (!dT 2 + dR2) + r2d!2

T = ±R + constant

T = ±R

Kerr

where

and

Angular momentum

ds2 = !!

1! 2GMr

!2

"dt2 ! 2GMar sin2 "

!2(dtd# + d#dt) +

!2

!dr2

+!2d"2 +sin2 "

!2[(r2 + a2)2 ! a2! sin2 "]d#2

!(r) = r2 ! 2GMr + a2

!2(r, ") = r2 + a2 cos2 "

Einstein Field Equation

Can also be written as

Rµ! !12Rgµ! = 8!GTµ!

Rµ! = 8!G(Tµ! !12Tgµ!)

Perturbations

For a perturbation

Inserting this in

But

Rµ! = 0

g!µ! = gµ! + hµ!

Rµ! + !Rµ! = 0

!Rµ! = 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.

Even and Odd

The odd waves contain three unknown functions:

The even waves contain seven unknown functions:

A Summary of the Black Hole Perturbation Theory 15

Using the tensor harmonics from above we are able to split the perturbation hµ!

into its even(electric or polar) and odd(magnetic or axial) parity parts

Odd/Magnetic/Axial parity = Y ML !

!

""""""""#

0 0 "h0(t, r)(1

sin " )(##$) h0(t, r)(sin !)( #

#" )

0 0 "h1(t, r)(1

sin " )(##$) h1(t, r)(sin !)( #

#" )

Sym Sym h2(t, r)[(1

sin " )(#2

#"#$) 12h2(t, r)[(

1sin " )(

#2

#$#$) + (cos !)( ##" )

"(cos !)( 1sin2 "

)( ##$)] " sin ! #2

#"#" )]

Sym Sym Sym "h2(t, r)[(sin !)( #2

#"#$)" (cos !)( ##$)

$

%%%%%%%%&

Even/Electric/Polar Parity = Y ML !

!

"""""""""""""#

(1" 2M/r)H0(t, r) H1(t, r) h0(t, r)(##" ) h0(t, r)(

##$)

Sym (1" 2M/r)!1H2(t, r) h1(t, r)(##" ) h1(t, r)(

##$

Sym Sym r[K(t, r) r2G(t, r)[("2/"!"#)

+G(t, r)( #2

d"2 )] "(cos !)( 1sin " )(

##$)]

Sym Sym Sym r2[K(t, r) sin2 !

+G(t, r)[( #2

#$2 )

+(sin !)(cos !)( ##" )

$

%%%%%%%%%%%%%&

Due to the spherical symmetry of the background metric, equations (**) and (**) do

not mix terms belonging to di!erent L and parity. M is the projection of L on the

z " axis. To apply quantum language to a classical problem, we can say that L,M

and the parity are constants of the motion. The existence of still another constant

follows from the circumstance that the background metric is independent of the cotime,

T = ct. On this account we can consider a perturbation of a definite frequency, $ = kc,

so that every component of the perturbation hµ! will have a time dependence of the

form e(i%t) = e(!ikT ). We therefore proceed to determine completely the form of the

individual solution of specified parity, L and M values, and frequency. The general

solution will be a superposition of these individual solutions with coe"cients adjusted

to fit the appropriate boundary conditions and initial values. There is no need to work

with an arbitrary M as all will lead to the same radial equation. In this case we will

take M = 0 with the advantage that # will completely disappear from the calculations.

A considerable amount of work remains. The odd waves contain three unknown

functions of r: (h0, h1, h2). The even waves contain seven unknown functions:

(H0, H1, H2, G,K, h0, h1). The calculations can be greatly simplified by the use of gauge

transformations.

5.4. Gauge Transformations

The Regge-Wheeler gauge, which we are about to walk through, is unique. What we

are trying to do is create gauge invariant quantities and work in a fixed gauge. Actually,

these two tasks are one and the same. As long as one works in a uniquely fixed gauge,

the quantities one is dealing with are gauge invariant, in the sense that one can translate

them into any gauge one wants without changing the physical problem. In the following





!

""""""""#

0 0 "h0(t, r)(1

sin " )(##$) h0(t, r)(sin !)( #

#" )

0 0 "h1(t, r)(1

sin " )(##$) h1(t, r)(sin !)( #

#" )

Sym Sym h2(t, r)[(1

sin " )(#2

#"#$) 12h2(t, r)[(

1sin " )(

#2

#$#$) + (cos !)( ##" )

"(cos !)( 1sin2 "

)( ##$)] " sin ! #2

#"#" )]


#"#$)" (cos !)( ##$)

$

%%%%%%%%&


!

"""""""""""""#

(1" 2M/r)H0(t, r) H1(t, r) h0(t, r)(##" ) h0(t, r)(

##$)

Sym (1" 2M/r)!1H2(t, r) h1(t, r)(##" ) h1(t, r)(

##$

Sym Sym r[K(t, r) r2G(t, r)[("2/"!"#)

+G(t, r)( #2

d"2 )] "(cos !)( 1sin " )(

##$)]


+G(t, r)[( #2

#$2 )

+(sin !)(cos !)( ##" )

$

%%%%%%%%%%%%%&

















transformations.











!

""""""""#

0 0 "h0(t, r)(1

sin " )(##$) h0(t, r)(sin !)( #

#" )

0 0 "h1(t, r)(1

sin " )(##$) h1(t, r)(sin !)( #

#" )

Sym Sym h2(t, r)[(1

sin " )(#2

#"#$) 12h2(t, r)[(

1sin " )(

#2

#$#$) + (cos !)( ##" )

"(cos !)( 1sin2 "

)( ##$)] " sin ! #2

#"#" )]


#"#$)" (cos !)( ##$)

$

%%%%%%%%&


!

"""""""""""""#

(1" 2M/r)H0(t, r) H1(t, r) h0(t, r)(##" ) h0(t, r)(

##$)

Sym (1" 2M/r)!1H2(t, r) h1(t, r)(##" ) h1(t, r)(

##$

Sym Sym r[K(t, r) r2G(t, r)[("2/"!"#)

+G(t, r)( #2

d"2 )] "(cos !)( 1sin " )(

##$)]


+G(t, r)[( #2

#$2 )

+(sin !)(cos !)( ##" )

$

%%%%%%%%%%%%%&

















transformations.











!

""""""""#

0 0 "h0(t, r)(1

sin " )(##$) h0(t, r)(sin !)( #

#" )

0 0 "h1(t, r)(1

sin " )(##$) h1(t, r)(sin !)( #

#" )

Sym Sym h2(t, r)[(1

sin " )(#2

#"#$) 12h2(t, r)[(

1sin " )(

#2

#$#$) + (cos !)( ##" )

"(cos !)( 1sin2 "

)( ##$)] " sin ! #2

#"#" )]


#"#$)" (cos !)( ##$)

$

%%%%%%%%&


!

"""""""""""""#

(1" 2M/r)H0(t, r) H1(t, r) h0(t, r)(##" ) h0(t, r)(

##$)

Sym (1" 2M/r)!1H2(t, r) h1(t, r)(##" ) h1(t, r)(

##$

Sym Sym r[K(t, r) r2G(t, r)[("2/"!"#)

+G(t, r)( #2

d"2 )] "(cos !)( 1sin " )(

##$)]


+G(t, r)[( #2

#$2 )

+(sin !)(cos !)( ##" )

$

%%%%%%%%%%%%%&

















transformations.







Gauge Transformations

The Regge-Wheeler gauge a is unique fixed gauge

The quantities are gauge invariant

Any result can be expressed in a gauge invariant manner by substituting the Regge-Wheeler gauge quantities in terms of a general gauge

Consider


we will use calculations in the Regge-Wheeler gauge. Any result can be expressed in a

gauge invariant manner by substituting the Regge-Wheeler gauge quantities in terms of

a general gauge. (See Gleiser 1996 for proof)

Di!erent waves can represent the same physical phenomena viewed in di!erent

systems of coordinates. Consider an infinitesimal coordinate transformation:

x!! = x! + !! (!! ! x!). (76)

The infinitesimal displacements !! transform like a vector. In the new frame we shall

have:

g!µ" + h!µ" = gµ" + !µ;" +!" ;µ +hµ" . (77)

Now hµ" is defined as the di!erence between the perturbed metric and the Schwarzschild

metric written in spherical coordinates. According to this definition, the di!erence in

the new frame will have the value

hnewµ" = hold

µ" + !µ;" +!" . (78)

This result can be interpreted by saying that the infinitesimal changes in the coordinates

of the hµ" , undergo a gauge transformation quite similar to the well known gauge

transformation for the electromagnetic field. We use this to simplify the description

of the perturbation and make it unique.

The gauge transformation can be performed on any individual partial wave.

Obviously no real simplification will result unless the resulting wave still belongs to

the original eigenvalues. This requirement limits the possible choices for !!. This vector

turns out to be a spherical harmonic of the same L and parity as the partial wave under

consideration. Such a gauge transformation allows us to impose additional simplifying

conditions on the perturbation hµ" . We have therefore chosen to eliminate those terms

which contain the derivatives of the highest order with respect to the angles. The final

radial equations then simplify. Moreover, the desired gauge transformation !! can then

be found by the use of finite operations only, without arbitrary constants and boundary

conditions.

The gauge vector !! that simplifies the general odd wave must have the form

!0 = 0; !1 = 0; !µ = "(T, r); "µ"(#/#x")Y ML ($, %), (µ, & = 2, 3) (79)

according to the foregoing arguments. Moreover, the radial function " can be adjusted

to annul the radial factor h2(T, r).

The final canonical form for an odd wave of total angular momentum L and

projection M = 0 is then

hoddµ" = e("ikT )(sin $)(#/#$)PL(cos $)"

!

""""#

0 0 0 h0(r)

0 0 0 h1(r)

0 0 0 0

Sym Sym 0 0

$

%%%%&







x!! = x! + !! (!! ! x!). (76)


have:

g!µ" + h!µ" = gµ" + !µ;" +!" ;µ +hµ" . (77)




hnewµ" = hold

µ" + !µ;" +!" . (78)














conditions.


!0 = 0; !1 = 0; !µ = "(T, r); "µ"(#/#x")Y ML ($, %), (µ, & = 2, 3) (79)






!

""""#

0 0 0 h0(r)

0 0 0 h1(r)

0 0 0 0

Sym Sym 0 0

$

%%%%&







x!! = x! + !! (!! ! x!). (76)


have:

g!µ" + h!µ" = gµ" + !µ;" +!" ;µ +hµ" . (77)




hnewµ" = hold

µ" + !µ;" +!" . (78)














conditions.


!0 = 0; !1 = 0; !µ = "(T, r); "µ"(#/#x")Y ML ($, %), (µ, & = 2, 3) (79)






!

""""#

0 0 0 h0(r)

0 0 0 h1(r)

0 0 0 0

Sym Sym 0 0

$

%%%%&

Regge-Wheeler Gauge

The gauge vector that simplifies the general odd wave has the form

The final canonical form for an odd wave L, M = 0 is

The gauge vector that simplifies the general even wave has the form

The final canonical form for an even wave L, M = 0 is







x!! = x! + !! (!! ! x!). (76)


have:

g!µ" + h!µ" = gµ" + !µ;" +!" ;µ +hµ" . (77)




hnewµ" = hold

µ" + !µ;" +!" . (78)














conditions.


!0 = 0; !1 = 0; !µ = "(T, r); "µ"(#/#x")Y ML ($, %), (µ, & = 2, 3) (79)






!

""""#

0 0 0 h0(r)

0 0 0 h1(r)

0 0 0 0

Sym Sym 0 0

$

%%%%&







x!! = x! + !! (!! ! x!). (76)


have:

g!µ" + h!µ" = gµ" + !µ;" +!" ;µ +hµ" . (77)




hnewµ" = hold

µ" + !µ;" +!" . (78)














conditions.


!0 = 0; !1 = 0; !µ = "(T, r); "µ"(#/#x")Y ML ($, %), (µ, & = 2, 3) (79)






!

""""#

0 0 0 h0(r)

0 0 0 h1(r)

0 0 0 0

Sym Sym 0 0

$

%%%%&


The gauge transformation that simplifies even waves is

!0 = M0(T, r)Y ML (", #); !1 = M1(T, r)Y M

L (", #); (80)

!2 = M(T, r)($/$")Y ML (", #); !3 = M(T, r)(1/ sin2 ")($/$#)Y M

L (", #).(81)

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 canonical form

hevenµ! = e(!ikT )PL(cos ")!

!

""""#

H0(1" 2M/r) H1 0 0

Sym H2(1" 2M/r)!1 0 0

0 0 r2K 0

0 0 0 r2K sin2 "

$

%%%%&

There are therefore two unknown functions of r in the odd case (h0, h1) and four unknown

functions in the even case (H0, H1, H2, K).

This gauge helps tremendously in simplifying the di!erential equations involved.

It has the feature, however, that the perturbation in the metric increases with distance

from the center of attraction in the even part of hµ! . In the odd part it keeps an

unchanging order of magnitude. For the calculation of the radiation one needs a gauge

in which the magnitude of the perturbation falls o! as 1/r. Fortunately, we can create

a gauge for those requirements as well.

5.5. Solutions

Due to the verbose nature and specificity of the solutions of Wheeler and Zerilli

(compared to the purpose of this paper - I am not saying these solutions fail to be

invariant in some way), I shall describe the solutions in a more qualitative manner. The

solutions can be divided in a number of ways: static (k = 0) and non-static, L = 0, 1, 2...,

and of course the even and odd solutions.

We will, for these cases, discuss the solutions to the homogeneous equations and

the solutions for the case where there is a source term which is produced by a point

mass m0 falling on a geodesic of the background.

The low number of independent tensor harmonics for L = 0 and L = 1 even

(electric) type waves and the L = 1 odd (magnetic) type waves make the equations

possible to solve explicitly. These terms describe the changes produced by the

perturbation in mass, velocity, and angular momentum of the center of attraction. There

is no L = 0 odd (magnetic) type harmonic. While the L = 0 and L = 1 harmonics are

non-radiative, the harmonics of L # 2 describe the radiation. Unfortunately, it is not

possible, using these methods, to find their exact solution.

5.6. Odd/Magnetic Solutions

For odd waves there are three non-trivial equations. One of the equations is a

consequence of the two others, provided that the source term satisfies the divergence


The gauge transformation that simplifies even waves is

!0 = M0(T, r)Y ML (", #); !1 = M1(T, r)Y M

L (", #); (80)

!2 = M(T, r)($/$")Y ML (", #); !3 = M(T, r)(1/ sin2 ")($/$#)Y M

L (", #).(81)

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 canonical form

hevenµ! = e(!ikT )PL(cos ")!

!

""""#

H0(1" 2M/r) H1 0 0

Sym H2(1" 2M/r)!1 0 0

0 0 r2K 0

0 0 0 r2K sin2 "

$

%%%%&

There are therefore two unknown functions of r in the odd case (h0, h1) and four unknown

functions in the even case (H0, H1, H2, K).

This gauge helps tremendously in simplifying the di!erential equations involved.

It has the feature, however, that the perturbation in the metric increases with distance

from the center of attraction in the even part of hµ! . In the odd part it keeps an

unchanging order of magnitude. For the calculation of the radiation one needs a gauge

in which the magnitude of the perturbation falls o! as 1/r. Fortunately, we can create

a gauge for those requirements as well.

5.5. Solutions

Due to the verbose nature and specificity of the solutions of Wheeler and Zerilli

(compared to the purpose of this paper - I am not saying these solutions fail to be

invariant in some way), I shall describe the solutions in a more qualitative manner. The

solutions can be divided in a number of ways: static (k = 0) and non-static, L = 0, 1, 2...,

and of course the even and odd solutions.

We will, for these cases, discuss the solutions to the homogeneous equations and

the solutions for the case where there is a source term which is produced by a point

mass m0 falling on a geodesic of the background.

The low number of independent tensor harmonics for L = 0 and L = 1 even

(electric) type waves and the L = 1 odd (magnetic) type waves make the equations

possible to solve explicitly. These terms describe the changes produced by the

perturbation in mass, velocity, and angular momentum of the center of attraction. There

is no L = 0 odd (magnetic) type harmonic. While the L = 0 and L = 1 harmonics are

non-radiative, the harmonics of L # 2 describe the radiation. Unfortunately, it is not

possible, using these methods, to find their exact solution.

5.6. Odd/Magnetic Solutions

For odd waves there are three non-trivial equations. One of the equations is a

consequence of the two others, provided that the source term satisfies the divergence

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

For odd waves there are three non-trivial equations

Can be expressed as a wave equation known as the Regge-Wheeler Equation

In time domain

with

L = 0 no perturbation

L = 1 addition of angular momentum


condition. Thus we have a system of two first-order linear equations. The two first-

order equations can be expressed, after a series of substitutions, as a simple second-order

Schrodinger equation known as the Regge-Wheeler equation for odd perturbations:

d2!odd

dr!2+ k2(r)!odd = 0. (82)

This can also be written in the time-domain as

d2!odd

dr!2! d2!odd

dt2+ V (r)!odd = 0, (83)

with

V (r) = [!L(L + 1)/r2 + 6M/r3](1! 2M/r).

For L = 0 there is no odd parity type harmonic.

The L = 1 perturbation represents the addition of a angular momentum m0a, where

from the Zerilli perspective this can be seen as the conserved angular momentum of the

falling particle.

Static solutions of the odd type exist for L "= 0. For the odd equations with k = 0,

h1 must vanish.

5.7. Even/Electric Solutions

For even waves, the ten Einstein field equations give seven nontrivial conditions:

one algebraic relation of two of the unknown functions; three first-order di"erential

equations; and three second-order di"erential equations. This leaves six equations

for the three remaining unknowns. Three of the first-order equations are su#cient to

determine a solution provided the divergence conditions on the source term are satisfied.

After some substitutions and other simplifications we can arrive at the second-order

Schrodinger equation known as the Zerilli equation for even perturbations:

d2!even

dr!2! d2!even

dt2+ V (r)!even = 0, (84)

with

V (r) =!1! 2M

r

" #1!2

#72M3

r5 ! 12Mr3 (L! 1)(L + 2)

!1! 3M

r

"$+ (L"1)L(L+1)(L+2)

r2

$,

and

! = L(L + 1)! 2 + 6Mr .

The L = 0 perturbation, being spherically symmetric, represents the augmentation

of the Schwarzschild mass by m0"0, a result required by Birkho"’s theorem. This means

the solution is time independent. From the Zerilli perspective, this addition m0"0 is the

mass-energy of the falling particle.

For L = 1, the nonzero hµ" 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.





d2!odd

dr!2+ k2(r)!odd = 0. (82)


d2!odd

dr!2! d2!odd

dt2+ V (r)!odd = 0, (83)

with

V (r) = [!L(L + 1)/r2 + 6M/r3](1! 2M/r).




falling particle.


h1 must vanish.









d2!even

dr!2! d2!even

dt2+ V (r)!even = 0, (84)

with

V (r) =!1! 2M

r

" #1!2

#72M3

r5 ! 12Mr3 (L! 1)(L + 2)

!1! 3M

r

"$+ (L"1)L(L+1)(L+2)

r2

$,

and

! = L(L + 1)! 2 + 6Mr .











d2!odd

dr!2+ k2(r)!odd = 0. (82)


d2!odd

dr!2! d2!odd

dt2+ V (r)!odd = 0, (83)

with

V (r) = [!L(L + 1)/r2 + 6M/r3](1! 2M/r).




falling particle.


h1 must vanish.









d2!even

dr!2! d2!even

dt2+ V (r)!even = 0, (84)

with

V (r) =!1! 2M

r

" #1!2

#72M3

r5 ! 12Mr3 (L! 1)(L + 2)

!1! 3M

r

"$+ (L"1)L(L+1)(L+2)

r2

$,

and

! = L(L + 1)! 2 + 6Mr .







Even/Electric Solutions

For even waves there are seven non-trivial equations: One algebraic relation, three first-order equations, and three second-order equations.

Can be expressed as a wave equation known as the Zerilli Equation

In time domain

with

L = 0 addition of mass

L = 1 shift of the cm





d2!odd

dr!2+ k2(r)!odd = 0. (82)


d2!odd

dr!2! d2!odd

dt2+ V (r)!odd = 0, (83)

with

V (r) = [!L(L + 1)/r2 + 6M/r3](1! 2M/r).




falling particle.


h1 must vanish.









d2!even

dr!2+ k2(r)!even = 0. (84)


d2!even

dr!2! d2!even

dt2+ V (r)!even = 0, (85)

with

V (r) =!1! 2M

r

" #1!2

#72M3

r5 ! 12Mr3 (L! 1)(L + 2)

!1! 3M

r

"$+ (L"1)L(L+1)(L+2)

r2

$,

and

! = L(L + 1)! 2 + 6Mr .





d2!odd

dr!2+ k2(r)!odd = 0. (82)


d2!odd

dr!2! d2!odd

dt2+ V (r)!odd = 0, (83)

with

V (r) = [!L(L + 1)/r2 + 6M/r3](1! 2M/r).




falling particle.


h1 must vanish.









d2!even

dr!2+ k2(r)!even = 0. (84)


d2!even

dr!2! d2!even

dt2+ V (r)!even = 0, (85)

with

V (r) =!1! 2M

r

" #1!2

#72M3

r5 ! 12Mr3 (L! 1)(L + 2)

!1! 3M

r

"$+ (L"1)L(L+1)(L+2)

r2

$,

and

! = L(L + 1)! 2 + 6Mr .





d2!odd

dr!2+ k2(r)!odd = 0. (82)


d2!odd

dr!2! d2!odd

dt2+ V (r)!odd = 0, (83)

with

V (r) = [!L(L + 1)/r2 + 6M/r3](1! 2M/r).




falling particle.


h1 must vanish.









d2!even

dr!2+ k2(r)!even = 0. (84)


d2!even

dr!2! d2!even

dt2+ V (r)!even = 0, (85)

with

V (r) =!1! 2M

r

" #1!2

#72M3

r5 ! 12Mr3 (L! 1)(L + 2)

!1! 3M

r

"$+ (L"1)L(L+1)(L+2)

r2

$,

and

! = L(L + 1)! 2 + 6Mr .





d2!odd

dr!2+ k2(r)!odd = 0. (82)


d2!odd

dr!2! d2!odd

dt2+ V (r)!odd = 0, (83)

with

V (r) = [!L(L + 1)/r2 + 6M/r3](1! 2M/r).




falling particle.


h1 must vanish.









d2!even

dr!2+ k2(r)!even = 0. (84)


d2!even

dr!2! d2!even

dt2+ V (r)!even = 0, (85)

with

V (r) =!1! 2M

r

" #1!2

#72M3

r5 ! 12Mr3 (L! 1)(L + 2)

!1! 3M

r

"$+ (L"1)L(L+1)(L+2)

r2

$,

and

! = L(L + 1)! 2 + 6Mr .

Solutions for L>=2 Radiation

Can not solve the equations explicitly

Asymptotically at large r the perturbation is the sum or two traces tensor harmonics.

Using a Green's function formed from high frequency-limit solutions, we obtain amplitudes for the ingoing r=2M and outgoing r=infinity radiation for a particle falling radially into the black hole.

The amplitude peaks at approximately 3/16piM Integrating this, the estimated total energy radiated is

To determine distribution in time use Fourier

No static perturbations for L>=2



of the Schwarzschild mass by m0!0, a result required by Birkho!’s theorem. This means

the solution is time independent. From the Zerilli perspective, this addition m0!0 is the


For L = 1, the nonzero hµ! can be removed by a gauge transformation which can


When the perturbation is at large r, the shift looks like a transformation to the center-

of-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 di!erence between the Schwarzschild metric mass

and the new mass with addition "m. For L = 1, we find a solution that corresponds to

a displacement of the center of attraction by the amount "z.

5.8. Solutions for L ! 2 and Gravitational Waves

For L ! 2, we can not solve the equations explicitly. Asymptotically for large r in the

radiation gauge, the perturbation hµ! in the metric is the sum of two transverse traceless

tensor harmonics listed above as d and f. Asymptotically for large r the radial factors

have the form

hM0L(#, r) " #rAM(m)

L (#)ei"r! (86)

and

KML (#, r) " AM(e)

L (#)ei"r! . (87)

The amount of the escaping radiation, which is of more physical interest, is determined

completely by the amplitude coe"cients AM(m)L and AM(e)

L .

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

outgoing (at r = $) radiation for a particle falling radially into the black hole. The

amplitude, as a function of frequency #, increases like a power law of #. Amplitude

peaks at approximately # = 3/16$M , and decreases exponentially for high frequencies.

Integrating this, the estimated total energy radiated is (1/625)(m2o/M) times a factor

of order 1.

To determine the distribution of the energy in time rather than frequency, we must

form the Fourier integrals for even and odd waves, and construct the stress-energy tensor

from these time-dependent fields.

As Vishveshwara indicates, there seem to be no static perturbations for L ! 2 on

the Schwarschild metric.

StabilityThe 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.

Newman-Penrose Formalism

The second popular method for solving perturbation equations is the Newman-Penrose (NP) formalism.

The NP formalism is a notation for writing various quantities and equations that appear in relativity. It starts by considering a complex null tetrad

such that

The projections of the Weyl tensor (used heavily in NP formalism in place of G.. and R..) then become


5.9. Stability

The Schwarzschild metric background gives an equilibrium state. If the metric is

perturbed, however, will it remain stable? A sphere of water held together by

gravitational forces is stable against small departures from sphericity. A sphere of water

surrounded by a spherical shell of liquid mercury is also an equilibrium configuration for

gravitational force, but a situation of unstable equilibrium. The collapsed Schwarzschild

metric must be proven to be stable against small perturbations. It can not be assumed

a priori that these entities are allowed by nature to exist. The problem is fully

stated as: If the Schwarzschild metric is perturbed, will the perturbations create

undamped oscillations about the equilibrium state represented by the Schwarzschild

background, or will they grow exponentially with time and cause instability. The

stability of the Schwarzschild metric was originally studied by Regge and Wheeler in

1957. Their work presented the standard method of decomposing any given perturbation

on a spherically symmetric background metric into its normal modes using the tensor

spherical harmonics discussed above. A problem with the coordinates chosen by

them, however, prevented them from judging whether any divergence shown by the

perturbations at the surface (r = 2M) was real or due to the coordinate singularity

at r = 2M . Using the new 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

assumption. Thus perturbations with imaginary frequencies are physically unacceptable

and the metric is indeed stable.

5.10. Quasi-Normal Modes

5.11. The Transmission-Reflection Perspective

|x| =

!x if x ! 0;

"x if x < 0.

6. Newman-Penrose Formalism

The second popular method for solving perturbation equations is the Newman-Penrose

(NP) formalism. The NP formalism is a notation for writing various quantities and

equations that appear in relativity. It starts by considering a complex null tetrad

("#l ,"#n ,"#m,"#m) such that,

"#l ·"#n = 1 = ""#m ·"#m. (88)

A notation is introduced for the directional derivatives along tetrad vectors:

D = lµ!µ, ! = nµ!µ, " = mµ!µ, "! = mµ!µ. (89)

And #, $, %, &, ',(, µ, ), *, +,,, - are a notation for the spin coe"cients of the null tetrad.


5.9. Stability

The Schwarzschild metric background gives an equilibrium state. If the metric is

perturbed, however, will it remain stable? A sphere of water held together by

gravitational forces is stable against small departures from sphericity. A sphere of water

surrounded by a spherical shell of liquid mercury is also an equilibrium configuration for

gravitational force, but a situation of unstable equilibrium. The collapsed Schwarzschild

metric must be proven to be stable against small perturbations. It can not be assumed

a priori that these entities are allowed by nature to exist. The problem is fully

stated as: If the Schwarzschild metric is perturbed, will the perturbations create

undamped oscillations about the equilibrium state represented by the Schwarzschild

background, or will they grow exponentially with time and cause instability. The

stability of the Schwarzschild metric was originally studied by Regge and Wheeler in

1957. Their work presented the standard method of decomposing any given perturbation

on a spherically symmetric background metric into its normal modes using the tensor

spherical harmonics discussed above. A problem with the coordinates chosen by

them, however, prevented them from judging whether any divergence shown by the

perturbations at the surface (r = 2M) was real or due to the coordinate singularity

at r = 2M . Using the new 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

assumption. Thus perturbations with imaginary frequencies are physically unacceptable

and the metric is indeed stable.

5.10. Quasi-Normal Modes

5.11. The Transmission-Reflection Perspective

|x| =

!x if x ! 0;

"x if x < 0.

6. Newman-Penrose Formalism

The second popular method for solving perturbation equations is the Newman-Penrose

(NP) formalism. The NP formalism is a notation for writing various quantities and

equations that appear in relativity. It starts by considering a complex null tetrad

("#l ,"#n ,"#m,"#m) such that,

"#l ·"#n = 1 = ""#m ·"#m. (88)

A notation is introduced for the directional derivatives along tetrad vectors:

D = lµ!µ, ! = nµ!µ, " = mµ!µ, "! = mµ!µ. (89)

And #, $, %, &, ',(, µ, ), *, +,,, - are a notation for the spin coe"cients of the null tetrad.


The projections of the Weyl tensor (used heavily in NP formalism in place of Gµ!

and Rµ!) then become

!0 = !Cµ!"#lµm!l"m# !1 = !Cµ!"#lµn!l"m# !2 = !Cµ!"#lµm!m"n#

!3 = !Cµ!"#lµn!m"n# !4 = !Cµ!"#nµm!n"m#.

7. Kerr Perturbations

The Kerr spinning black hole metric is non-diagonal, so it has cross-terms between

spatial and time coordinates. Due to the complexity of the Kerr metric, it becomes

di"cult to use the Einstein equations directly to get a solvable perturbation equation. To

obtain the perturbation equation for rotating black holes, Teukolsky used the Newman-

Penrose formalism. Skipping over much laborious calculation we arrive at the Teukolsky

equation.!

(r2+a2)2

! ! a2 sin !"

$2"$t2 + 4Mar

!$2"$t$% +

!a2

! !1

sin2 &

"$2"$'2

!#!s $$r

##s+1 $"

$r

$! 1

sin &$$&

#sin ! $"

$&

$! 2s

!a(r!M)

! + i cos &sin2 &

"$"$'

!2s!

M(r2!a2)! ! r ! ia cos !

"$"$t + (s2 cot ! ! s)! = 0,

where once again

#(r) = r2 ! 2GMr + a2. (90)

If s = 2, ! =! 0 and if s = !2, ! = "!4!4 where " = !1(r!ia cos &) .

This equation is considerably more involved than the Zerilli equation. While it is

not possible to achieve angular separation in the time domain, in the frequency domain

it is separable. If you assume

! = e!i(teim'S(!, #)R(r,#), (91)

the S’s become spherical functions SML (!a2#, cos !).

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.

8. Conclusion

One More Day.

References

[1] B.F. Whiting, Class Notes from General Relativity I (Transcribed by various students, Gainesville,2008).[2] S.M. Carroll, Spacetime and Geometry (Addison-Wesley, San Fransisco, 2004).[3] J.B. Hartle, Gravity (Addison-Wesley, San Fransico, 2003).[4] B.F Schutz, A First Course in General Relativity (Cambridge University Press, Cambridge, 1985).[5] B.F Schutz, Geometrical Methods of Mathematical Physics (Cambridge University Press, Cambridge,

Kerr PerturbationsDue to the complexity of the Kerr metric, it becomes difficult to use the Einstein equations directly to get a solvable perturbation equation.

To obtain the perturbation equation for rotating black holes, Teukolsky used the Newman-Penrose formalism. Skipping over much laborious calculation we arrive at the Teukolsky equation.

where

While not possible to achieve angular separation in the time domain, in the frequency domain it is separable.





!3 = !Cµ!"#lµn!m"n# !4 = !Cµ!"#nµm!n"m#.







equation.!

(r2+a2)2

! ! a2 sin !"

$2"$t2 + 4Mar

!$2"$t$% +

!a2

! !1

sin2 &

"$2"$'2

!#!s $$r

##s+1 $"

$r

$! 1

sin &$$&

#sin ! $"

$&

$! 2s

!a(r!M)

! + i cos &sin2 &

"$"$'

!2s!

M(r2!a2)! ! r ! ia cos !

"$"$t + (s2 cot ! ! s)! = 0,

where once again

#(r) = r2 ! 2GMr + a2. (90)





! = e!i(teim'S(!, #)R(r,#), (91)





8. Conclusion

One More Day.

References






!3 = !Cµ!"#lµn!m"n# !4 = !Cµ!"#nµm!n"m#.







equation.!

(r2+a2)2

! ! a2 sin !"

$2"$t2 + 4Mar

!$2"$t$% +

!a2

! !1

sin2 &

"$2"$'2

!#!s $$r

##s+1 $"

$r

$! 1

sin &$$&

#sin ! $"

$&

$! 2s

!a(r!M)

! + i cos &sin2 &

"$"$'

!2s!

M(r2!a2)! ! r ! ia cos !

"$"$t + (s2 cot ! ! s)! = 0,

where once again

#(r) = r2 ! 2GMr + a2. (90)





! = e!i(teim'S(!, #)R(r,#), (91)





8. Conclusion

One More Day.

References


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

[1] B.F. Whiting, Class Notes from General Relativity I (Transcribed by various students - usually Shawn Mitryk, Gainesville, 2008). [2] S.M. Carroll, Spacetime and Geometry (Addison-Wesley, San Fransisco, 2004). [3] J.B. Hartle, Gravity (Addison-Wesley, San Fransisco, 2003). [4] B.F Schutz, A First Course in General Relativity (Cambridge University Press, Cambridge, 1985). [5] B.F Schutz, Geometrical Methods of Mathematical Physics (Cambridge University Press, Cambridge, 1980). [6] S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1992). [7] S. Chandrasekhar, Selected Papers, Volume 6: The Mathematical Theory of Black Holes and of Colliding Plane Waves (University of Chicago Press, Chicago, 1991). [8] D.J. Griffiths, Introduction to Electrodynamics, Third Edition (Prentice-Hall, Upper Saddle River, 1999). [9] D.J. Griffiths, Introduction to Quantum Mechanics, Second Edition (Prentice-Hall, Upper Saddle River, 2005). [10] T. Regge and J.A. Wheeler, Stability of a Schwarzschild Singularity Phys. Rev. 108, 1063 (1957). [11] F.J. Zerilli, Gravitational Field of a Particle Fal ling in a Schwarzschild Geometry Analyzed in Tensor Harmonics Phys. Rev. D 2, 2141 (1970). [12] F.J. Zerilli, Effective Potential for Even-Parity Regge-Wheeler Gravitational Perturbation Equations Phys. Rev. Lett 24, 737 (1970). [13] C.V. Vishveshwara, Stability of the Schwarzschild Metric Phys. Rev. D 1, 2870 (1970). [14] S. Chandrasekhar, On the Equations Governing the Perturbations of the Schwarzschild Black Hole Proc. R. Soc. 343, 289 (1975). [15] S. Chandrasekhar, and S. Detweiler, The Quasi-Normal Modes of the Schwarzschild Black Hole Proc R. Soc. 344, 441 (1975). [16] S. Chandrasekhar, On One-Dimensional Potential Barriers Having Equal Reflection and Transmission Coefficients Proc. R. Soc. 369, 425 (1980). [17] S. Chandrasekhar, and S. Detweiler, On the Equations Governing the Axisymmetric Perturbation of the Kerr Black Hole Proc R. Soc. 345, 145 (1975). [18] J.M. Bardeen, and W.H. Press, Radiation Fields in the Schwarzschild Background J. Math. Phys. 14, 7 (1972). [19] J.M. Stewart M. Walker, Perturbations of Space-Times in General Relativity Proc. R. Soc. 341, 49 (1974). [20] S.A. Teukolsky, Perturbations of a Rotating Black Hole Astrophys J. 185, 635 (1973).

a summary of the black hole perturbation theory hole perturbation... · equation for kerr type...

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