Distorting metric with vanishing twist and divergence

Volume 46A, number 4

PHYSICS

DISTORTING

METRIC

31 December

LETTERS

WITH VANISHING

1973

TWIST AND DIVERGENCE

B.P. YADAV Physical Research Laboratory, Received

Ahmedabad-380009,

9 October

India

1973

In linear relativity, we obtain a gravitational field having irrotational and shearing geodesic rays with vanishing expansion for asymptotically flat vacuum space-time. Such a field is not allowed by field equations in nonlinear theory.

We deal here only with empty space-times, that is, with four dimensional Riemann spaces of signature-2 with vanishing Ricci tensor. We use the coordinate system and other results of spin-coefficient technique developed by Newman-Penrose [l] and Newman-Unti [2] . In Einstein’s nonlinear relativity, significant metrics are obtained for gravitational fields containing geodesic rays [3-81 . In nonlinear theory a gravitational field with shearing geodesic rays and vanishing expansion and rotation is not allowed by field equations [8]. We examine whether such a situation is possible for asymptotically flat vacuum gravitational fields in linear theory. Coordinate x0 = u is the retarded timelike coordinate and it is a parameter. The null hypersurfaces are designated by u = const. x1 = r is an affine parameter along each geodesic of the null congruence. Angular coordinates x2 = 0 and x3 = cplabel a particular null geodesic in each hypersurface. (Zp, nu, rnp’,ii+‘) is a tetrad of null vectors where I” and np are real and fi” is the complex conjugate of complex vector mp. This tetrad satisfies orthogonality conditions lpn@ = - m5i, = 1, and other scalar products are zero. Spin coefficients u and p can be defined as p=l mCLe* mPmv, /w I.r;v where u is the complex shear and p is given by a=1

(1)

p=f3+iw, where f3 and w are, respectively, 0 = lp./l >

w = (I,p;v, Py’*

ln Newman-Penrose

expansion

and twist of the field defined by

.

formalism

[ 1 ] Zp is a geodesic ray [9, lo] if [ 1 , 51

(2)

9,=0, where \k, is one of the five independent

components

of Weyl tensor defined as

9, = -C~vPoIPmvIpmO . We write the following equations Dp =p*+.,

,

from Newman-Penrose’s

work [ 1]

Do=2pa+\ko.

(3,4)

When the field possesses geodesic rays, eq. (4) becomes Du=2pu.

(5)

When we solve the linearised form of eqs. (3) and (5), we get the solutions (5= uO(,JQ)r-*

)

P’O,

(6 7) 253

Volume

46A, number

4

PHYSICS

31 December

LETTERS

1973

where the term PO~‘-~ is eliminated by the coordinate transformation r’ = r -~ p”(u, 0, ip). A comparison of relations (1) and (7) shows that expansion f? and rotation w are separately equal to zero. For this case, the metric takes the form gOLgum=O

g 01 = 1 %

0 dr{fi

’

rrdr --

,

where m, n = 2,3 and y. T are other spin coefficients y = $($.,

n’ln” - mp;J7PnV)

7=I

,

Our metric (8) contains hypersurface Sperical rays exist if p2foC.

mlJny

!J;v

orthogonal

distorting

geodesic rays with vanishing expansion

PfO

and cylindrical p2=.a

defined by

(9)

rays can exist if

#O.

(10)

Now it is obvious from (6) and (7) that neither spherical not cylindrical

field,

References [ 1 ] E. Newman

and R. Penrose,

J. Math. Phys. 3 (1962)

565.

[ 2) E. Newman and T. Unit, J. Math. Phys. 3 (1962) 891. [3] [4] 15) [6] [7] [S] [9] [lo]

254

W. Kundt, Z. Physik 163 (1961) 77. I. Robinson and A. Trautman, Proc. Roy. Sot. A265 (1962) 463. E. Newman and L. Tamburino, J. Math. Phys. 3 (1962) 902. R. Kerr, Phys. Rev. Lett. 11 (1963) 237. E. Newman, L. Tamburino and T. Unti, J. Math. Phys. 4 (1963) 915. T. Unit and R. Torrence, J. Math. Phys. 7 (1966) 535. R. Penrose, Ann. Phys. 10 (1960) 171. R. Sachs, Proc. Roy. Sot. A264 (1961) 309.

rays can exist in our present gravitational