Method of singular sources in application to electrovacuum gravitational Einstein Fields Текст научной статьи по специальности «Математика»

UDC 530.12:531.51

Method of Singular Sources in Application to Electrovacuum Gravitational Einstein Fields

Ts. I. Gutsunaev, A. A. Shaideman, A. Ya. Terletsky

Department of Theoretical Physics Peoples' Friendship University of Russia 6, Miklukho-Maklaya str., Moscow, Russia, 117198

By means of the method of singular sources it is possible to construct a generalization of gravitational magnetic dipole.

Key words and phrases: vacuum Einstein-Maxwell equations, axisymmetric solution, asymptotically flat metrics.

1. Introduction

Principally the static axially symmetric problem in general relativity has been formulated and developed in a most elegant manner by Weyl [1]. Among static solutions of the Einstein-Maxwell equations the first and at the time very important result was obtained by Reissner [2] and Nordstrom [3]. The further works by Majumdar [4] and Papapetrou [5] followed for the problem of the electrostatic field. By means of the method of singular sources it is possible to construct asymptotically flat metrics which reduce to the generalizations of the Schwarzschild metric in the absence of magnetism.

2. Basic Equations

The metric of the static axisymmetric gravitational field can be written in the canonical Weyl coordinates in the form

d s2 = 1 [e2j(dp2 + d z2) + p2dp2] - fdt2.

The fact that the static Einstein-Maxwell equations allow the existence of either the electric potential, or magnetic one, results from the stationary Einstein-Maxwell equations.

In this case we set and magnetostatic Einstein equations have the form

4 / 2 \ 2

uAu = (Vu)2 + ^ (VAs), viP^ VAs J =0. (1)

TT ^A rP 1 d d2 p _ d _ d , ^ . .

Here u = v/, A = f-n + - • — + —^, V = p0— + z0— (p0 and z0 are unit vectors) >' p op az2 az az

and As(p, z) is the magnetic component of the electromagnetic 4-potential.

The second equation in (1) can be viewed as the condition for the existence of a

new potential A's connected with As by relations

OAS = f OAs OA's = f OAs

a p p a a p a p

In that case Eqs (1) can be rewritten as

fAf = (Vf)2 + 2 /(VA's)2, fAA's = (VA's) • Vf. (3)

Received 31st May, 2012.

One can easily see that the electrostatic Einstein-Maxwell equations have the same form as Eqs. (3). Therefore, we can put A4 = A'3, where A4 is the electric component of the electromagnetic 4-potential Ai = [0, 0, 0, — A4(p, z)].

While A3 = A4 = 0 the Eqs (1), (2) turn to the Weyl vacuum static equations:

/A / = (V f)2. (4)

With the substitution f = e(4) becomes linear:

A . = dV + 1 dt + d^t (5)

dp2 p dp dz2

3. Method of Singular Sources

The right-hand side of (5) contains zero though actually there should be a certain singular unction characterizing the distribution of sources.

Let a(p, z) denote the mass density of such sources, and let us rewrite (5) in the form

1 d ( d*\ d2* A . .

ydpydp) + d? =—4™(p, ^ (6)

This equation has the solution

* = ^ / dV'. (7)

4n J \r — r'| v '

V

In the coordinates p, <p, z we have

dV' = p dp'd^' d z',

\r — r'\ = p2 + p'2 — 2pp • cos(v — ) + (z — d)2.

Since the left-hand side of (6) does not depend on <p, we can set = 0 in the integral.

If we choose

I > A — p0) I \

T(p ,z) =---a(po, z),

where p0 = const, 5(p' — p0) is Dirac's ¿-function, we obtain

2-k

*(p, z)= f , a)(p0>ZdZ' . (8)

J J Vp2 +p2 — 2p0p cos tf + (z — z ')2

Example 1. Let p0 = 0, a0(z') = 2S(z'). Integration of (8) then leads to

V = —(9)

i.e. to the Chazy-Curzon solution.

Example 2. Let a0(z') = 50 = const. With this choice we come to the Zipoy solution: _

* = lJ z_-moWp2+¿E™£) . (10)

2 \z + m,0 + yp2 + (z — «0 )2y

144 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 3, 2012. Pp. 142-146

If we put ö0 = 1 in (10), we obtain the Schwarzschild solution

_ 2mo / x = m - 1 y = c°s# \

r ' \z = m0xy, p = m0\J(x2 _ 1)(1 _ y2)J

Example 3.

a)

d z + \J p2 + z2

VP^T^72 p

* = J r^—7 = In-—-. (11)

0

It is the soliton solution b)

, , A [2, ^ > o

ao(z) =7(z) ^ x po = 0.

2 > Z < 0

In this case we have

2 J VPrTS_z7)2 P

— oo

Example 4. ct(z') = ^(z') ■ a0(z'), p0 = 0,

a( t\ f1, _m<z <m 2 / z! gg ^

/ , ^(¿) = - '^0 ^ TT" ■ ri—2 . 10, _m>z'>m k \2m 1 + a^J

Here K is elliptic integral of the first kind. In this case

m

* — 1/ (12>

— m

If we put «o — 0 in (12), then we obtain Schwarzschild solution (10). Example 5. Let a(p0, z') — 1 S(z') &(z'), p — 0. In this case

,, n 2m MV (p+TQ)2+Z2J

*(p, z) — —------r. (13)

^ V(P + Po)2 + ¿2

If we put p0 — 0, then we obtain the Chazy-Curzon solution. Example 6. Let a(p0,z') — ê(z'). In this case

* — 1 / , 1 (J, ^pp0f d^. (14)

^ w v(p+p0)2+(*o2 IV(p+P0)2+(*-^o2/ v ;

— m v ' '

If we put p0 — 0, we obtain Schwarzschild solution. Example 7.

2 /^V (P+Po' z) = ~ / =-. (15)

K 0 y/(p + p0)2 + *'2

z

If we put po = 0, then we obtain the soliton solution (11). Example 8.

é(p' " = --./ V(p + ^ + * d(16)

—œ

If we put po = 0, then we obtain the soliton solution (11).

4. The Weyl—Bonnor—Papapetrou—Majumdar Class of Solutions of Einstein—Maxwell Equations

1. Let us consider the subclass of the Weyl electrovacuum solutions of the equations (3)

(1 - ^ A = fr - e2*)

1 - a2e2^ ' A3 = 1 - a2e2^ ' Aé = 0

_ _ (17)

The solution (17) includes also the most famous Weyl-Reissner-Nordstrom spherical-symmetric solution.

If we put ^ from (14) in (17), we obtain the generalization of the Reissner-Nordstrom solution.

2. The subclass of Papapetrou-Majumdar solution

u = ^^ , A!z = T^r, A^ = 0.

1 + é

1 + é

a) For é

Poz

1+

(p2 + z2)3/2

PoZ (p2 + z2)3/2_

we have the gravitational field of magnetic dipole [6]:

—2

a = Poz 3 (p2 + z2)3/2

1+

Poz

— 1

b) For é = -Po • — n az

K

4PP0

(p+po)2+^

j(p+po)2 +z2

(p2 + 2)3/2 we have the generalization of (18).

(18)

)

References

1. Weyl H. Zur Gravitationstheorie // Ann. Physik. — 1917. — Vol. 359, No 18. — Pp. 117-145.

2. Reissner H. Uber die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie // Ann. Physik. — 1916. — Vol. 355, No 9. — Pp. 106-120.

3. Nordstrom G. On the Energy of the Gravitational Field in Einstein's Theory // Proc. Kon. Ned. Anad. Wet. — 1918. — Vol. 20. — Pp. 1238-1245.

4. Majumbar S. D. A Class of Exact Solutions of Einstein's Field Equations // Phys. Rev. — 1947. — Vol. 72. — Pp. 390-398.

5. Papapetrou A. // Proc. R. Irish Acad. — 1947. — Vol. A51. — Pp. 191-204.

6. Gutsunaev T. I., Chernyaev V. A. Axysymmetric Gravitational Fields. — Moscow: MCXA-Press, 2004. — 168 p.

146 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 3, 2012. Pp. 142-146

УДК 530.12:531.51

Метод сингулярных источников в задачах электровакуума

Эйнштейна

Ц. И. Гуцунаев, А. А. Шайдеман, А. Я. Терлецкий

Кафедра теоретической физики Российский университет дружбы народов ул. Миклухо-Маклая, 6, Москва, Россия, 117198

С помощью метода сингулярных источников возможно построение обобщений известных электровакуумных решений.

Ключевые слова: вакуумные уравнения Эйнштейна—Максвелла, аксиально-симметричные решения, асимптотически плоские метрики.