Relarivistic gravitational fields of rotating bodies

Volume 86A, number 2

PHYSICS LETTERS

2 November 1981

RELATWISTIC GRAVITATIONAL FIELDS OF ROTATING BODIES G. NEUGEBAUER Sektion Physik der Friedrich-Schiller-Universitdt, DDR-6900 Jena, German Dent Rep. Received 22 June 1981

The Einstein equations for the exterior fields of stationarily rotating axisymmetric sources axe solved by the inverse (scattering) method, i.e. the metric can be calculated from the “scattering” coefficients via a Zakhaxov—Shabat equation or a Gelfand—Levitan—Marchenko equation.

The exterior gravitational field of any stationarily rotating axisymmetric isolated source can be de. scribed by the line element [1] ds2 = e—2U(e2k dz di + p2 d~2) e2U(dt + ~ d~)2 —

(1) where p = Re z, ~ = Imz, ~ are cylindrical coordinates and t is the coordinate time. (A bar denotes complex conjugation.) The metric functions U, k, w depend only on z and ~ and must be calculated from Einstein’s vacuum equations Rik

=

0.

(2)

Several authors [2—6]have suggested applying the formalism of soliton physics to solve this problem. In this way the famous Kerr solution describing rotating black holes was rediscovered as a two-soliton solution [3]. The “multisoliton” solution has also been known forsome time [7]. Following the idea of the inverse (scattering) method we consider the linear eigenvalue problem [5,6,8] ax/az = (x + AiJi)A (z, ~) ______

ax/ai = (x + A~~,t’)B(z,~), —

(3)

asp/az = (,Li + A~)B(z,z),

is a spectral parameter depending on the coordinateindependent complex parameter K. By cross differentiation we obtain a system of differential equations (nonlinear “evolution” equations) forA and B, and this system is equivalent to eqs. (1) and (2) [5,6]. The functions ,li, x depend on K, z and 1, butK may be replaced by A. Once ~,(A)m~I.’(X, z, 1) and x(A) x (A, z, 1) have been found, the full metric and the Ernst function f can be deduced in the following way: e2U = 1~~+ ~

2’.)

w

a/c

~

— —

8P

—

1

o 031-9163/81/0000—0000/S 02.75 © 1981 North-Holland

‘—

.~—

~O)~

~/(—i)

5 (6)

A4 ,1, ‘(X)x ‘(A)

X-~oo

~‘(A)x(A) ‘

(7)

iP’(A)x’(A)

.

1 2D (P +‘..

(4)

“

(8)

~i,

where ii)1/2(K + iz)1/2

~

X’.1j~

Therefore we may call x the (pseudo-)potentials of the metric (1). We divide the construction of the potentials of our problem into two steps. (i) It can be shown that to every solution ~O(A,z, ~), x°(X,z, ~)of eqs. (3) a new solution ~Li(X, z, ~), x(A, z, 1) can be assigned by =

—

—

ai8p~iJ ~i(X)x(A)

a~1,/ai=(~,÷A—’x)A(z ~)

A(K, z, 1) = (K

J

2p X’(l)

=

a/c az

~—

h

‘

X=

[(GO

—

x°)D+~A) + (~O+ x°)D+(—A)1

(9)

)

2D(1) [(x° iP°)D.(A)+ (x°+ —

91

Volume 86A, number 2

PHYSICS LETTERS

where

2 November 1981

are also solutions of eqs. (3). Y~and ~2 form a corn-

D~(A)= ~/~~det(6ik

Qk(A + Ak) ±q’(A~)(A Ak)~,+ Ak)) —

plete system of solutions, and therefore we can write

‘P3=a(K)Y1+b(K)~2 onC, (11) The abbreviations Qk

=

x°(Ak)+ ~P°(Ak)+ak[x°(_Ak)_~J/°(_Ak)] x°(Ak) 11°(Xk)+ ak [x°(—Ak)+ ~/°(Ak)} —

(12) 2N

q(A)=fl A—A1 ~=i X+A1

Q,k1,...,2N),

(13)

involve the discrete spectrum A1 = A(K1, z, ~)and the constant “scattering” coefficients ak (i, k = 1, 2N) of eqs. (3). Ak The= 2Neigenvalues Ak liehalf-plane either on and the N unit circle, ~ (N in the upper in the lower half-plane), or arise as pairs with A 1= ±~i~ (k~i). In the first caseak = —ak,in the other oneä~= —a1~’. By setting A = 1 in (9) and (10) we immediately 2U f+f. arrive at the Ernst functionf, whence 2e The proofof 2 u°(D(_1) (14) ~.i1) ~~(l) / w= + ~p e where ~ follows from D~by replacing Qk by Q~1 ...,

—

X (1 Ak)(l + Ak)’, requires more manipulations. The metric coefficient exp(2k) is factor proportional to 0)D~(0)Dj~0) [9]. [The is alenghty exp(2k 0, Ak, ~P°(Ak), algebraic expression consisting of U x°(A~),Q~,q’(A~).1 The expressions (9) and (10), originally derived in a different form by means of Bäcklund transforma—

tions [5,7], are the potentialsof the nonlinear superposition of the metric belonging to the potentials ~, x°and an arbitrary number of “solitons” (Kerr—NUT solutions) [7,10,11]. (ii) Particular potentials ,,t,°,x°can be found by solving a regular Riemann—Hilbert problem (RHP) [121in the complex A-plane. Let

(15)

where C denotes the unit circle A = e~P(0 ~ p < 2ir). Note that we will not identify the K-intervals K1 = p ctg p~,0 ~Pi
—

Restrictions for a and b in the points p = 0, 2~rimmediately_follow from the_normalization conditions ~ii(—I) = x1(—i) = Xi(—i) = 0(A), ~‘i(—l) = 1 and (15). It can be x°(A)], shown that = [i~i =

IA I ~ I IAI< 1

‘l’l

(17)

is ‘F a solution of eqs. (3), if ‘Via ‘V2’ ‘I’3 obey (15) and if 1, ‘F3 have the following properties: 1’P (a)fr1r1)~’F1 [p~,.u_] and(K3r3) 3= [v÷, v_] ~ (IAI are1)analytic and in the functions interiorofof AC in (IXI the exterior < 1), respecof C tively. (b) ,~(A), K 3(A) are solutions of the regular (scalar) RHP K

(18) ~3 =a(K)~1 on C, 3(A)and K1(A) being analytic functions of A, when IAI
‘F1(A)= [~,i(A,z,~),xi(A,z,fl] be a solution of eqs. (3). Then ‘2

=

[~,Li1(—A, z, 2), —x1(—A, Z~Z

=

[xi(X1, z, 2), ~/i1(

and

92

1,z,

b(~ ~ ctg ço’) ~ ~1(—A’)r1(—X’) (20) a(~ p ctg ~p’) i~1(A’)r1(A’) The operation “lim” means: A A’ E C, IAI ~ 1, i.e. A goes from the outside to the unit circle C. Because of the analytic properties (a) and condition (19) the

N(A’)

2)]

=

—

—

—~

Volume 86A, number 2

PHYSICS LETTERS

2 November 1981

calculation of v~(A) and p~(A) is a regular RHP. Unlike other RHPs [3,131 our RHP decouples and leads to one-component integral equations. Starting e.g.

Eq. (24) with the “kernel” (25) is an analogue of the Gelfand—Levitan—Marchenko equation. For r1 = I and b = 0 (N 0, p = 1) we obtain

with the

x°(A,z, 2) = K 1(A, z, 2) (IAI ~ 1). In this case eqs.(5), (22) and (16) yield the static gravitational potential U,

to ~

=

second (minus) component of(l9) we are led a(A, z, 2)/a(—i, z, 2),

2ir d~’a(—A’,z,2)A’A a(A,z,i)=1—~---f A’N(A’)

I~ U = 2~

0

IAI> 1

2ir

—

2 —l ,AIna d~A’

,

ãa

=

1.

(26)

0

,

(21)

where a(—A’) = lim a(—A). In the limit A—’~—A” E C, Al> 1, eq. (21) becomes a Zakharov—Shabat integral equation for a(—A), A E C. From any solution of this equation we obtain a(A, z, 2) via (21) and from a the desired pseudopotential

Therefore we may interpret In a as a line mass density distributed along the symmetry axis and b as a distribution of angular momentum. Via the pseudopotentials x the “scattering” data Ak, ak; a(K), b(K) enter the metric (1). They represent the integration “constants” (functions) of the axisymmetric stationary Einstein fields. The connection between these data and boundary conditions for ~‘,

x°(A,z,Z)=K

1T1U(A,z,Z)Eo(—1,z,2)]’ , AI>1. This expression as well as N(A’) in (21) involves the functions K1(A), r1 (A) defined by (b) and (c). For a wide class of coefficients a(~ p ctg p’) the solution of the RFIP (18) can be shown to be given by —

2ir

K1(A, z, 1) = exp

(~!._

j’

‘A+l)lna d~’(A’ + 1)(A’ A))’

(K3

(22)

is the same expression with Al < 1). The solution

r1 (A, z, 2) of the eigenvalue problem formulated in (c) is known in principle. A completely analogous procedure (N —N) yields z, 2) (A> 1). An alternative way to solve the RHP (19) leads to an algebraic system of equations. The analytic function a(A, z, 2) (IA I > 1) admits the representation -*

a= 1

+ ~

m1

(23)

Cm(Z, 2)A~m .

Fr

where Fr

=

—

m1 E CmFrm(_l)

=

0,

V.A. Belinski and V.E. Zakharov,Zh. Eksp. Teor. Fiz. 75 (1978) 1953; 77 (1979)3. B.K. Harrison, Phys. Rev. Lett. 41(1978)1197. G. Neugebauer, J. Phys. A12 (1979) L67;Ai3 (1980)

1737. Y. Nakamura, Math. Japon. 24 (1979) 469. [7] G. Neugebauer, J. Phys. A13 (1980) L19. [8] For a group-theoretical approach developed by Geroch, Kinnersley et a!. see: W.M. Kinnersley, Lecture notes in physics, Vol. 135 (Springer, Berlin, 1980) p. 432; C. Hoenselaers, W. Kinnersley and B. Xanthopoulos, J. Math. Phys. 20 (1979) 2530;

[6]

(24)

(25)

[12] V.E. Zakharov, S.V. Manakov, S.P. Novikov and L.P. Pitayevski, Teoria solitonov; metod obratnoi sadachi [13] 1. (Moscow, Hauser and 1980). F.J. Ernst, J. Math. Phys. 21(1980)1126.

general relativity, Gen. Rd. Gray. (1981), to be published.

2ir

~f

[3] [4] [5]

C.M. Cosgrove, J. Math. Phys. 21(1980) 2417.

m

—

[1] A. Papapetrou, Ann. der Phys. 12 (1953) 309. [2] D. Maison, Phys. Rev. Lett. 41(1978)521.

[9] D. Kramer, in: 9th Intern. Conf. on General relativity and gravitation (July 14—19, 1980), Abstracts of contributed papers for the discussion groups, Vol. 1. [10]D. Kramer and G. Neugebauer, Phys. Lett. 75A (1980) [ii] 259. G. Neugebauer and D. Kramer, Soliton concept in

Again from (19), we have the following linear algebraic equations for Cm (z, 5): Cr

The author wishes to acknowledge his gratitude to Dietrich Kramer and Reinhard Meinel for many stimulating discussions. References

—

Al > 1

themetric(1)isstffltobe established.

d~’N(A’)e’~’,

=

,2

93