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arXiv:gr-qc/9910099v3 31 Jan 2000

A NEW FORM OF THE KERR SOLUTION

CHRIS DORAN

1

Astrophysics Group, Cavendish Laboratory, Madingley Road,

Cambridge CB3 0HE, UK.

Abstract

A new form of the Kerr solution is presented. The solution involves

a time coordinate which represents the loca l proper time for free-falling

observers on a set of simple trajectories. Many physical phenomena

are particularly clear when related to this time coo rdinate. The chosen

coordinates also e nsure that the solution is well behaved at the horizon.

The solution is well suited to the tetrad formalism and a convenient null

tetrad is presented. The Dirac Hamiltonian in a Kerr background is

also given and, for one choice of tetrad, it takes on a simple, Hermitian

form.

PACS numbers: 04.20.Jb, 04.70.Bw

1 Introduction

The Kerr solution has been of central importance in astrophysics ever since

it was realised that accretion processes would tend to spin up a black hole

to near its critical rotation rate [1]. A number of forms of the Kerr solu-

tion currently exist in the literature. Most of these are contained in Chan-

drasekhar’s work [2], and useful summaries are contained in th e books by

Kramer et al. [3] and d’Inverno [4]. The purpose of this paper is to present a

new form of the solution which has already proved to be useful in numerical

simulations of accretion processes. The form is a direct extension of the

Schwarzschild solution when written as

ds

2

= dt

2

−

dr +

2M

r

1/2

dt

2

−r

2

(dθ

2

+ sin

2

θ dφ

2

). (1)

(Natural units have been employed.) This is obtained from the Eddington-

Finkelstein form

ds

2

=

1 −

2M

r

d

¯

t

2

−

4M

r

d

¯

t dr −

1 +

2M

r

dr

2

−r

2

(dθ

2

+ sin

2

θ dφ

2

).

(2)

by the coordinate transformation

t =

¯

t + 2(2Mr)

1/2

− 4M ln

1 +

r

2M

1/2

. (3)

1

e-mail: C.Doran@mrao.cam.ac.uk, http://www.mrao.cam.ac.uk/∼cjld1/

1

In both metrics r lies in the range 0 < r < ∞, and θ and φ take their usual

meaning.

The metric (1) has a nu mber of nice features [5], many of which extend

to the Kerr case. The solution is well-behaved at the horizon, so can be

employed safely to analyse physical processes near the horizon, and indeed

inside it [5]. An other useful feature is that the time t coincides with the

proper time of observers free-falling along radial trajectories starting fr om

rest at inﬁnity. This is possible because th e velocity vector

˙x

i

= (1, −(2M/r)

1/2

, 0, 0) ˙x

i

= (1, 0, 0, 0) (4)

deﬁnes a radial geo desic with constant θ and φ. The proper time along these

paths coincides with t, and the geodesic equation is simply

¨r = −M/r

2

. (5)

Physics as seen by these observers is almost entirely Newtonian, making this

gauge a very useful one for introducing some of the more diﬃcult concepts

of black hole physics. The various gauge choices leading to this form of

the Schwarzschild solution also carry through in the presence of matter

and provide a simple system for the study of the formation of spherically

symmetric clusters [6] and black holes [5].

A further useful feature of the time coordinate in (1) is th at it enables

the Dirac equation in a Schwarzschild background to be cast in a simple

Hamiltonian form [5]. Indeed, the full Dirac equation is obtained by adding

a single term

ˆ

H

I

to the free-particle Hamiltonian in Minkowski spacetime.

This additional term is

ˆ

H

I

ψ = i(2M/r)

1/2

(∂

r

ψ + 3/(4r)ψ) = i(2M/r)

1/2

r

−3/4

∂

r

(r

3/4

ψ). (6)

A useful feature of th is gauge is that the measure on surfaces of con s tant

t is the same as that of Minkowski spacetime, so one can employ stand ard

techniques from quantum theory with little m odiﬁcation. One subtlety is

that th e Hamiltonian is not self-adjoint due to the presence of the singularity.

This manifests itself as a decay in the wavefunction as current den sity is

sucked onto the singularity [5].

The time coordinate t in the metric of equation (1) has many of the

properties of a global, Newtonian time. This suggests that an attempt to

ﬁnd an analogue for the Kerr solution might fail due to its angular momen-

tum. The key to understanding how to achieve a suitable generalisation

is the realisation that it is only the local p roperties of t that make it so

convenient for describing the physics of the solution. The natural extension

for the Kerr solution is therefore to look f or a convenient set of reference

observers which generalises the idea of a family of observers on radial tra-

jectories. In Sections 2 and 3 we present a new form of the Kerr solution

2

and show that it has many of the desired properties. In Section 4 we give

various tetrad forms of the solution, and present a Hermitian form of the

Dirac Hamiltonian in a Kerr background. Throughout we use Latin letters

for spacetime indices and Greek letters for tetrad indices, and use the sig-

nature η

αβ

= diag (+ − − − ). Natural units c = G = ¯h = 1 are employed

throughout.

2 The Kerr Solution

The new f orm of the Kerr solution can be written in Cartesian-type coor-

dinates (t, x, y, z) in a manner analogous to the Kerr-Schild form [2, 4]. In

this coordinate system our new form of the solution is

ds

2

= η

ij

dx

i

dx

j

−

2α

ρ

a

i

v

j

+ α

2

v

i

v

j

dx

i

dx

j

(7)

where η

ij

is the Minkowski metric,

α =

(2Mr)

1/2

ρ

(8)

ρ

2

= r

2

+

a

2

z

2

r

2

, (9)

and a and M constants. The function r is given implicitly by

r

4

− r

2

(x

2

+ y

2

+ z

2

− a

2

) − a

2

z

2

= 0, (10)

and we restrict r to 0 < r < ∞, with r = 0 describing the disk z = 0,

x

2

+ y

2

≤ a

2

. The maximally extended Kerr solution (where r is allowed to

take n egative values) will not be considered here.

The two vectors in the metric (7) are

v

i

=

1,

ay

a

2

+ r

2

,

−ax

a

2

+ r

2

, 0

(11)

and

a

i

= (r

2

+ a

2

)

1/2

0,

rx

a

2

+ r

2

,

ry

a

2

+ r

2

,

z

r

. (12)

These two vectors p lay an important role in studying physics in a Kerr

background. Th ey are related to the two principal null directions n

±

by

n

±

= (r

2

+ a

2

)

1/2

v

i

± (αρv

i

+ a

i

). (13)

For computations it is useful to note that the contravariant components of

the spacelike vector in brackets are the s ame as those of −a

i

,

αρv

i

+ a

i

= −(r

2

+ a

2

)

1/2

0,

rx

a

2

+ r

2

,

ry

a

2

+ r

2

,

z

r

. (14)

3

The vector v

i

also plays a crucial role in separating the Dirac equation in

a Kerr background, and is the timelike eigenvector of th e electromagnetic

stress-energy tensor for the Kerr -Newman analogue of our form.

3 Spheroidal Coordinates

The nature of the metric (7) is more clearly revealed if we introduce oblate

spheroidal coordinates (r, θ, φ), where

cosθ =

z

r

0 ≤ θ ≤ π (15)

tanφ =

y

x

0 ≤ φ < 2π, (16)

so that ρ recovers its standard deﬁnition

ρ

2

= r

2

+ a

2

cos

2

θ. (17)

The use of the symbols r and θ here are standard, th ou gh one must be aware

that when M = 0 (ﬂat space) these reduce to oblate spheroidal coordinates,

and not spherical polar coordinates. This is clear from the fact th at r does

not equal

√

(x

2

+ y

2

+ z

2

).

In terms of (t, r, θ, φ) coordinates our new form of the Kerr solution is

ds

2

=dt

2

−

ρ

(r

2

+ a

2

)

1/2

dr + α(dt − a sin

2

θ dφ)

2

− ρ

2

dθ

2

− (r

2

+ a

2

) sin

2

θ dφ

2

. (18)

This neatly generalises the Schwarzschild form of equation (1), replacing

√

(2M/r) with

√

(2Mr)/ρ, and introducing a r otational component. The

line element can be simpliﬁed further by introd ucing the hyperbolic coor-

dinate η via a sinhη = r, though this can make some equations harder to

interpret and will not be employed here. The metric (18) is obtain ed from

the advanced Eddington-Finkelstein form of the Kerr solution,

ds

2

=

1 −

2Mr

ρ

2

dv

2

− 2 dv dr +

2Mr

ρ

2

(2a sin

2

θ)dv d

¯

φ + 2a sin

2

θ dr d

¯

φ

− ρ

2

dθ

2

−

(r

2

+ a

2

) sin

2

θ +

2Mr

ρ

2

(a

2

sin

4

θ)

d

¯

φ

2

, (19)

via the coordinate transformation

dt = dv −

dr

1 + (2Mr/(r

2

+ a

2

))

1/2

(20)

dφ = d

¯

φ −

a dr

r

2

+ a

2

+ (2Mr(r

2

+ a

2

))

1/2

. (21)

4

This transformation is well-deﬁned for all r, though the integrals involved

do not appear to have a simple closed form.

The velocity vector

˙x

i

= (1, −α(r

2

+ a

2

)

1/2

/ρ, 0, 0) ˙x

i

= (1, 0, 0, 0) (22)

deﬁnes an infalling geod esic with constant θ and φ, and zero velocity at

inﬁnity. The existence of these geodesics is a key property of the solution.

The time coordinate t now has the simple interpretation of recording the

local proper time for observers in free-fall along trajectories of constant θ

and φ. As in the s pherical case, many physical phenomena are simplest

to interpret w hen expressed in terms of this time coord inate. An example

of this is provided in the following section, where we show that the time

co ordinate produ ces a Dirac Hamiltonian which is Hermitian in form. The

diﬀerence between this free-fall velocity and the velocity v

i

(deﬁned by the

gravitational ﬁelds) also provides a local deﬁnition of the angular velocity

contained in the gravitational ﬁeld.

4 Tetrads and the Dirac Equation

The metric (18) lends itself very naturally to the tetrad formalism. From

the principal null directions of equation (13) one can construct the following

null tetrad, expressed in (t, r, θ, φ) coordinates,

l

i

=

1

r

2

+ a

2

(r

2

+ a

2

, r

2

+ a

2

−

2Mr(r

2

+ a

2

)

1/2

, 0, a) (23)

n

i

=

1

2ρ

2

(r

2

+ a

2

, −(r

2

+ a

2

) −

2Mr(r

2

+ a

2

)

1/2

, 0, a) (24)

m

i

=

1

√

2(r + ia cosθ)

(ia sinθ, 0, 1, i cscθ). (25)

In this frame the Weyl scalars Ψ

0

, Ψ

1

, Ψ

3

and Ψ

4

all vanish, and

Ψ

2

= −

M

(r − ia cosθ)

3

. (26)

A second tetrad, better suited to computations of matter geodesics, is

given by

e

0

i

= (1, 0, 0, 0)

e

1

i

= (α, ρ/(r

2

+ a

2

)

1/2

, 0, −αa sin

2

θ)

e

2

i

= (0, 0, ρ, 0)

e

3

i

= (0, 0, 0, (r

2

+ a

2

)

1/2

sinθ). (27)

This deﬁnes a frame for all values of the coordinate r, so is valid inside

and outside the horizon. Combined with the techniques described in [5] this

5

tetrad provides a very powerful way of analysing and visualising motion in

a Kerr background.

A f urther tetrad is provided by reverting to the original Cartesian -type

co ordinates of equation (7) and writing

e

µ

i

= δ

µ

i

−

α

ρ

v

i

a

j

η

jµ

, (28)

where v

i

and a

i

are as deﬁned at equations (11) and (12). The inverse is

found to be

e

µ

i

= δ

i

µ

+

α

ρ

η

ij

a

j

δ

k

µ

v

k

. (29)

This ﬁnal form of tetrad is the s implest to use when constru cting the Dirac

equation in a Kerr background. We will not go through the details here

but will just present the ﬁnal form of the equ ation in a Hamiltonian form.

Following the conventions of Itzykson and Zuber [7] we denote the Dirac-

Pauli matrix representation of the Dirac algebra by {γ

µ

} and write α

i

=

γ

0

γ

i

, i = 1 . . . 3. Since e

µ

0

= δ

0

µ

, premultiplying the Dirac equation by γ

0

is

all that is requ ired to bring it into Hamiltonian f orm. When this is done,

the Dirac equation in a Kerr background becomes

i∂

t

ψ = −iα

i

∂

i

ψ + mγ

0

ψ +

ˆ

H

K

ψ (30)

where

ˆ

H

K

ψ =

√

2M

ρ

2

(r

3

+ a

2

r)

1/4

i∂

r

(r

3

+ a

2

r)

1/4

ψ

− a cosθ r

1/4

α

φ

i∂

r

r

1/4

ψ

−

a cosθ

2

(r

2

+ a

2

)

1/2

γ

5

ψ

(31)

and

α

φ

= −sinφ α

1

+ cosφ α

2

. (32)

The measure on hypersurfaces of constant t is again the same as that of

Minkowski spacetime, since the covariant volume element is s imply

dx dy dz = ρ

2

sinθ dr dθ dφ. (33)

As with the Schwarzschild case the interaction Hamiltonian

ˆ

H

K

is not self-

adjoint when integrated over these hyp ersurfaces. This is because the sin-

gularity causes a boundary term to be present when the Hamiltonian is

integrated.

6

5 Conclusions

The Kerr solution is of central importance in astrophysics as ever more com-

pelling evidence points to the existence of black holes rotating at near their

critical rate [8]. Any form of the solution wh ich aids physical understanding

of rotating black holes is clearly ben eﬁ cial. The form of the solution pre-

sented here has a number of features which achieve this aim. The solution is

well suited for s tu dying processes near the horizon, and the compact form of

the spin connection for the tetrad of equ ation (28) makes it particularly good

for numerical computation. It shou ld also be noted that this gauge admits

a simple generalisation to a time-dependent form which looks well-suited to

the study of accretion and the formation of rotating black holes.

A m ore complete exposition of the features of this gauge, includin g the

derivation of the Dirac Hamiltonian will be presented elsewhere. One reason

for not highlighting more of the advantages here is th at many of the the-

oretical manipulations which exploit these properties have been performed

utilising Hestenes’ spacetime algebra [5, 9 ]. This language fully exposes

much of the intricate algebraic structure of the K err solution and brings

with it a number of insights. These are hard to describ e without employ-

ing spacetime algebra and so will be presented unadulterated in a separate

paper.

The fact that the time coordinate measured by a family of free-falling

observers brings the Dirac equation into Hamiltonian form is suggestive of a

deeper principle. This form of the equations also permits many techniques

from quantum ﬁeld theory to be carried over to a gravitational background

with little modiﬁ cation. The lack of self-adjointn ess due to the source itself

is also natural in this framework, as the singularity is a natural s ink for

the current. In the non-rotating case the physical processes resulting from

the presence of this sink are quite simple to analyse [5]. The Kerr case is

considerably more complicated, due both to the nature of the ﬁelds inside

the inner horizon, and to the structure of the singularity. One interesting

point to note is that the sink region is described by r = 0, and so represents

a disk, rather than just a ring of matter. T his in part supports the results

of earlier calculations described in [8], though much work remains on th is

issue.

Acknowledgements

CD is supported by an EPSRC Fellowship. The author is grateful to An-

thony Lasenby and Anthony Challinor for helpful discussions.

References

7

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8