Comment On Colliding Plane Gravitational Waves

Volume 84A, number 7 PHYSICS LETTERS 17 August 1981

COMMENT ON COLLIDING PLANE GRAVITATIONAL WAVES M. HALILSOY1

Department of Physics, Middle East Technical University, Ankara, Turkey Received 27 May 1981

It is shown that a theorem proved on colliding plane gravitational waves is not correct.

It is the purpose of the present note to show that waves, which is however manifest prior to the collision. the theorem stated some time ago in ref. [1] in con- This is due to the fact that the plane wave nature of(2) is nection withcolliding plane gravitational waves is in- no more valid after the collision. The simplest case W correct. If we quote the equations from this reference, =0 corresponds to constant linear polarization [2,3].

the theorem states the following: In order to obtain solutions (W * 0) it is necessary that To any colliding gravitational plane-wave metric we must have W *0 also in regions prior to collision 2 — 2 M’du d ~ V’ ~2 + —v’ d 2~ ~ [4,5]. Provided this requirement is satisfied then a

con-ds — e v—e ~e C ~‘ ~ ~ ~ sistent matching of solutions at the boundaries becomes

one associates a new solution with W * 0, possible. The vacuum Einstein equations must be satis-fied everywhere including the boundaries and the re-cis2 2e_M du dv — e_U(e~’cosh W dx2 2 suiting solution must be nondiagonalizable. We

present-+e~’cosh W dy2 — 2 sinh W dx dy), ~ ed exact solutions to (2) before [4,5] which satisfied

the properties that two single pulses may be diagonaliz.

where ed separately whereas the two pulses cannot be

simul-AW taneously diagonalized in the same coordinate patch.

f

J cosh W(A2 cosh2W— 1)1/2 to the above-state theorem: (3a) and (3b) are

integrat-ed to yield (the results of ref. [1] are incorrect) A cosh WdW — =~~1 (3b)

~ (A~cosh2W— 1)1/2 tanh V=cosatanh V’,

and M=MA, A=const. (The factor A in the integrand tanh

w

v,

of(3b) is missing in ref. [1].) Stated in other words,

in the newly generated solution W and V are assumed where for convenience we introduced a new parameter to be functionally related. We shall show that whenever by cos a=A—1. Note also that the choice forM as M

Wand V are functionally dependent, which is the basic —M’A is also not correct in the same reference, but assumption of the theorem stated above, it turns out should be M=M’

that the metric becomes diagonalizable, hence the In conclusion, given a solution of(l) it seems that

theorem fails. through (4)and(U’=U, M’ =M) a new solution with

Before we do this we would like to point out, for W * 0 is generated. However, all this procedure does a better understanding, that the metric function W re- not give a solution other than (I): To see this, make a

presents the poiarization content of the colliding coordinate rotation, (5)

1Previous surname of the author was Hall, which is changed x=cos ~ +sin ~ y =—sin ~ag +cos

Volume 84A, number 7 PHYSICS LETTERS 17 August 1981

and observe, after simple algebra, that (2) reduces to reduces (6) to the Khan—Penrose solution. Since a—

(1) in the rotated coordinates (u,v,~, y’). In particular, measuresthe incident polarization of the waves in

col-if the incoming waves are impulsive waves the solution lision we conclude that a—j3 * 0 is the crucial quantity

generated by the above theorem reads explicitly which generates a nontrivial solution to (2) with W * 0. Any solution of (2) which involves a single constant

2 M C 2

ds = 2e— du dv — _— 1k I2 [Il— k~ dx (6) parameter (as the above theorem does) can be ruledout by a coordinate transformation. This completes

+11 +k12 dy2 +2i dx dy (k—

ii)],

where k=Cia(pw+qr), with the usual notations References

p=uO(u), q=vO(v), [1] D. Ray, Phys. Lett. 78A (1980) 315.

2 2 2 [2JK. Khan and R. Penrose, Nature (London) 229 (1971)

r21—p , W ‘l—q 185.

[3] P. Szekeres, J. Math. Phys. 13 (1972) 286.

and UandM correspond to Khan—Penrose values. [4] Y. Nutku and M. Hall, Phys. Rev. Lett. 39 (1977) 1379. Solution (6)is readily identified as the solution of ref. [5] M. Hall, J. Math. Phys. 20 (1979) 120,

[4] with the restriction a=~3.The same rotation (5)