PHYSICAL REVIEW D VOLUME 47, NUMBER 4 15 FEBRUARY 1993

Intermediate singularities in the interaction of gravitational and electromagnetic waves with cosmic strings

Patricio S. Letelier Departamento de Matematica Aplicada-Instituto de Matematica Estati'stica e C ihc ia da Computaqiio,

Universidade Estadual de Campinas, 13081 Campinas, SZo Paulo, Brazil (Received 15 January 1992)

The presence of intermediate or nonscalar singularities (infinite tidal forces) in spacetimes associated with a cosmic string interacting with either a plane-fronted gravitational wave or a pencil of electromag- netic radiation is studied.

PACS number(s): 04.30. +x, 04.20.Jb, 11.17. +y, 98.80.Cq

In a recent Letter [I], we analyzed the singular struc- ture of the curvature tensor associated with an exact solution of the Einstein field equation that represents a cosmic string interacting with either a plane-fronted gravitational wave (pp wave) or a pencil of electromag- netic radiation. We find the appearance of new types of singularities due to the interaction. The analysis was rather naive in the sense that the study of the singular be- havior was carried out using only the so-called physical con~ponents of the Riemann-Christoffel tensor in a par- ticular, albeit natural, vierbein. These physical com- ponents are invariant under a change of coordinates, but transform as a rank-four tensor under local Lorentz transformations that can be ill behaved in the singulari- ties under study. In other words, the vierbein used to write the curvature tensor can be very natural for the problem under study, but it may introduce new singulari- ties that in general may not be associated with physical singularities. However, there exists a privileged vierbein wherein the components of the curvature tensor are directly associated with physical quantities-the free- falling frame. The singularities of the curvature tensor measured in a frame attached to a free-falling particle represent infinite tidal forces, i.e., physical singularities. These type of singularities are known in the literature as nonscalar or intermediate singularities [2]. In the case under study, the polynomial scalars formed with the cur- vature tensor are null outside the string, and so we do not have strong curvature singularities. The study of scalar- wave propagation suggests that the intermediate singular- ities are unstable and may give origin to strong curvature singularities [3].

In this Brief Report, we study the appearance of inter- mediate singularities in spacetime with the line element

where H, A, and B are functions of u , x, and y; V is a function of x and y. The metric (1) is a special case of the general metric that admits a null vector with zero covari- ant derivative [4], and it is slightly less general than the one studied in Ref. [I] . In particular, we shall be interest-

ed in the case where the functions A and B are restricted by

A,, -B,,=O , A , , +B,, =O , (2)

and V is given by

V=2hlnr ,

where h is the string tension and r = ( x ~ + ~ ~ ) ' / ~ . When H, near r =0, is a regular solution of

the metric (1) represents a plane-fronted wave interacting with an infinite cosmic string that crosses perpendicularly the plane (x ,y ) at the origin. And when H i s given by

where w is a positive function of u and r , = [ ( x - x l ) 2 + ( y -y l )2]1/2; x 1 and y , are constants, and the metric (1) describes a pencil of electromagnetic radiation of zero cross section located at x = x l and y =y , in interaction with an infinite cosmic string placed at x =y =O. w represents the energy per unit of length of the beam of radiation.

A simple vierbein associated with (1) is

We have that

The physical components of the Riemann-Christofell ten- sor for the metric ( I ) with the restrictions (2) reduce to

where

@ 1993 The American Physical Society

1710 BRIEF REPORTS 47

We also have that, for pp waves in interaction with cos- mic strings,

I 2 +I, = O , (1 l a )

and, for a pencil of electromagnetic radiation [I], -

I , +I2 = - 8 ~ w 6 ( x - X I )6 (y -Y 1 )/Z/g2 . (1 lb)

We shall assume that the beam of electromagnetic radia- tion does not coincide with the cosmic string. Thus, near the string, we will also have that Eqs. (4) and (1 la) hold. Therefore, for gravitational waves and pencils of light, the functions P and Q near r = O also satisfy

The geodesic equation for the metric (1) yields

where the overdots mean a derivative with respect to s, and Wand so are integration constants.

Let F = ( u , d , x , y ) be the tangent vector to a generic geodesic; then, the covariant derivative of the vierbien along the geodesic is

Equations (13) and (14) tell us that the geodesic equa- tions, as well as the propagation of the vierbein, depend on the already known functions V and u and the two functions P and Q that satisfy (12). We also note that the behavior of the invariants (9a) depends on these last two functions. Once given explicitly, P and Q one can deter- mine A and B in terms of H:

or H in terms of A and B,

H = ~ J ( A -P,, ) d x + ( B - Q , , ) d y + A ( u ) , (16)

yhere the arbitrary functions A and 2 satisfy Eqs. (2) and H is another arbitrary function of the indicated argu-

ment. The existence of the quadratures is guaranteed by Eqs. (20) and (12).

Now we shall study the singular behavior of the curva- ture invariants near r = O when the functions P and Q are given by

where a and /3 are real constants, K is a complex con- stant, and f = x t i y . Really, we are interested in the be- havior of the curvature near r = O only; then, we can also consider expressions (17) as being the limit of more gen- eral functions P and Q in that region. We have that (17) contains three important special cases.

(a) K =O(P = Q = O ) . With these restrictions the cur- vature is null for V =O. For V =2h Inr we have that only I , is different from zero. The metric (1) in this case represents a string in a nontrivial accelerated system of coordinates.

(b) /3=0(P,,=P,,=Q,,=Q,,=O). In this case, when V =0, we do not have curvature, and for V=2hlnr we have that all the curvature invariants (8) are different from zero. Thus, in this particular case, we have a gravi- tational wave or a pencil of light supported by the string. When the string disappears, the radiation vanishes [5].

(c) a = O and B=1. For this specialization and V=O, we have a spacetime with constant-curvature invariants.

We shall assume that (13c) and (13d) with P and Q given by (17) are solved by

where a and n are constants to be determined; a is as- sumed to be complex and n real. We find

When K =0, the pure string case,

For h=O, the pure wave case, n = 1. For the generic case,

and the complex constant a is the solution of

We note that the case fl= 1 needs to be treated in a different way; we shall come back to this point later. We are aware that the assumption (18) restricts our analysis to a particular class of geodesics, but it is general enough for our purposes.

To complete the integration of the geodesic equation, we need to integrate (13b) so that once u , x , and y are known it is a simple quadrature (the result will not be presented here because it will not be used).

First, we shall study the pure wave case n = 1 and k=0; the geodesic equations as well as the equations of propagation of the vierbein are satisfied as long as

47 BRIEF REPORTS 171 1

( S , - S ) ~ + ~ goes to zero. Near x = y = O , I l + i I , = ( s o - ~ ) ~ + o - l . When a +8 2 1, near r = 0 the wave has regular curvature. This case represents the generic case of a nonsingular wave near the axis r =O. Note that for 1 > a+/3>O the wave has a singularity along the axis r = O that is not due to the string. Hence, in order to ana- lyze the formation of singularities due to the presence of the string, we shall assume a + B L 1.

Let us call C the generic member of this particular family of geodesics such that when s =so it touches the line r=O wherein the string is placed. Along C we find that the curvature invariants I, = -I2 and I3 are given by

where the bar means complex conjugation. Also, rela- tions (14) yield

Hence the vierbien fails to be in free fall along C at most by a factor

where q is the smallest of two numbers:

We will restrict our analysis to the case in which both numbers are positive. For cosmic strings originating in phase transitions in the very early Universe, h is estimat- ed to be between loW5 and When q > 0 , we have that near r = O ( s = s o ) the quantity (25) can be taken as small as desired. In other words, when we fall toward the string along C, we approach a parallel-propagated frame. The curvature invariants (23) will blow up as ( s , - s ) ~ 2

on the string. This fact indicates that on top of the conic singularity represented by (9b) we will also have an inter- mediate singularity that will be felt, by the observer fal- ling along C, as infinite tidal forces. In order to be sure that the approximation of the frame used represents well a free-falling frame near the singularity, we can further impose that

In this way we will have a good approximation to a

parallel-propagated frame before the effect of the singu- larity is felt.

Of course, one can also look for an exactly parallel- propagated frame along C, but in general this is a difficult task, since the finding of the required local Lorentz trans- formation involves the solving of nontrivial differential equations. In principle, the best that we can do is to find numerical solutions that by construction are approximate solutions that will give an approximate parallel- propagated frame; i.e., we shall have a situation similar to the one already described in the present Brief Report worsen by the typical difficulties of error control encoun- tered in numerical methods.

The geodesic equation for the cases a = O and /3= 1 can also be easily solved by assuming h << 1. From (13a), (13b), and (171, we get

Thus the solution that passes through r = O is

where Co is a complex constant. Near the string we have

Therefore, near r =0, we have that the frame fails to be parallel propagated by a factor ( so -s) and that the cur- vature invariants I, = -I, and I3 go to zero as ( S , - S ) ~ ~ ; i.e., in this case, we only have the string singularity represented by 14 .

The generic case with 8 = 0 [case (b)] deserves particu- lar attention. For this specialization we can also satisfy condition (27). Therefore we will have an intermediate singularity produced by the interaction of the string with either gravitational or electromagnetic radiation since, as mentioned before, if the string is removed, the spacetime becomes flat [cf. Eq. (23)]. Moreover, every spacetime that near the string behaves as (17) with 8 = 0 and a satis- fying q > 2 will develop an intermediate singularity sup- ported by the string. Equations (16) and (17) tell us that there is a great variety of metric functions that satisfy these conditions.

In summary, analysis of the singularities shows that if we start with a pure gravitational wave with no inter- mediate singularity on the r = O axis, a+/3> 1, the pres- ence of the string introduces a nonscalar singularity on top of the string. If we compare the results of the present analysis with that of Ref. [I], we conclude that the most interesting case of singularities produced by the interac- tion of either plane gravitational waves or pencils of elec- tromagnetic radiation with strings is also found for a large class of metric functions.

[I] P. S. Letelier, Phys. Rev. Lett. 66, 268 (1991). [4] See, for instance, P. S. Letelier, Gen. Relativ. Gravit. 11, [2] See, for instance, G. F. R. Ellis and B. G. Schmidt, Gen. 367 (1979).

Relativ. Gravit. 8, 915 (1977). [5] See also P. S. Letelier, Class. Quantum Grav. 9, 1707 [3] A. R. King, Phys. Rev. D 11, 763 (1975). (1992).