PHYSICAL REVIEW D VOLUME 34, NUMBER 2 15 JULY 1986

Cylindrical gravitational waves with two degrees of freedom: An exact solution

T. Piran Racah Institute of Physics, The Hebrew University, Jerusalem 91 904, Israel

and The Institute for Advanced Study, Princeton, New Jersey 08540

P. N. Safier and J. Katz Racah Institute of Physics, The Hebrew Univer.~ity, Jerusalem 91904, Israel

(Received 17 June 1985)

The exact two-parameter solution of Einstein's equations described below represents ingoing and outgoing cylindrical gravitational waves with two degrees of polarization. It has been obtained from the Kerr metric by applying a well-known trick but, unlike the Kerr metric, it has no singularities.

There exists a well-known trick1 to obtain cylindrical time-dependent solutions from axially symmetric station- ary solutions of Einstein's equations. The trick is thus applicable to err's' metric with mass m and angular momentum3 ma#O in G =c =1 units. It gives an in- teresting new solution which comes about as follows. Start from Kerr's metric in ~ o ~ e r - ~ i n d ~ u i s t ~ coordinates x0,r,0,4. Go to "isotropic coordinates" xO,E,O,d with

Next go to cylindrical coordinates x O , p , ~ , b with

p=g sin6 and T=Z cos6 . (2)

Now employ the well-known trick of setting

and then introduce cylindrical coordinates T,R,Z ,d de- fined with

and #=Z-'M#J. As a result of these transformations we obtain from the

Kerr solution a metric which has the Jordan-Ehlers- Kundt-Kompaneetz5 (JEKK) form:

where I-, Y, and R are functions of U = T - R and V = T + R that appear in the following combinations:

h , - -5-1 [ ( a 2 + u ~ ) " ~ - u],

~ , = Z - ' [ ( Z ~ + V ~ ) ~ / ~ + V ] . (6)

r, Y, and R are given by

- 5(h , h, )-'I2( A, + h, l2

~ [ a ~ ( l - h , h ~ ) ~ + ( h , + h , i ~ ] - ~ ] ,

in which

~ = l + h , h , + 2 [ ( 1 -a-2)h,h,]1'2 . (8)

We shall employ the chart { ( T,z ,R ,@)E R 4 : - w < T , Z < cc ,O 5 R < cc ,O 5 @ < 271-1, where spacetime points with @ = 277- are identified with those with @ = O in the obvious way. The parameter a = M / a varies from 1 to co and for a= 1, that is, m =0, the space is flat. a determines the total energy of the waves (see below). The other parameter Z ranges from 0 to w and plays the role of a length scale.

The metric (7) is the only known analytic cylindrical- wave solution of the form ( 5 ) with two degrees of freedom or, what is the same, two polarizations. Solutions with one degree of freedom were found a long time ago.6 A plane-wave solution with two degrees of freedom has also been found.' That solution can be obtained from the Kerr metric by a procedure8 which is quite different from ours.

Our new metric is regular everywhere. On the axis R = O there is no conical singularity. At past and future infinite I?( T = t cc 1, the spacetime is Minkowski flat. At spatial infinity I'(R = W ) it is conical:

[notice that z is the one defined in Eq. (311. At future null infinity 4+( CT = cc ), @=O, O = - W , and

eZr=(a2+hu2) / (1 + h u 2 ) . (10)

At past null infinity 4-( V = - cc the metric is similar with h, instead of h, (Ref. 9). Einstein's energy flux per unit height through a surface ( U, V) = const over the flat background is"

33 1 @ 1986 The American Physical Society

T. PIRAN, P. N. SAFIER, AND J. KATZ

FIG. 1. aR r in units of its maximum value in terms of R /dand T / Z in the ranges 0 2 R /Z 20 and - 10 _< T / Z < 10. (a) is for a= 1.01 and (b) for a= 10. a R r ( m a x ) scales like (a2- 1 )/Z for a= 1 and like l /Z for a > > 1. The tendency towards a singular behavior at R = O for a- m appears already for a= 10.

1 2r E = T ( e - 1 ) . (1 1) emerges at X - , reaches its highest concentration near the axis R =O at time T =0, and is reflected out to N + . The total energy that comes in and goes out is $(a2- 1 ).

The metric (7) thus describes a flow of ingoing-outgoing Figure 1 displays aR T, which is related to the energy packets of waves on a flat background. A wave packet flux; it is positive definite:

The behavior of aR r is representative of the behavior of other quantities in the metric.

This research was partially supported by National Science Foundation Grant No. PHY 84-0721 9 to the Institute of Advanced Study.

'W. Kinnersley, in General Relativity and Gravitation, edited by G. Shaviv and J. Rosen (Wiley, New York, 19751, p. 109.

2R. Kerr, Phys. Rev. Lett. 11, 237 (1963). 3iT=0, which corresponds to the Schwarzschild solution, 1s

uninterestingly pathological. 4R. H. Boyer and R. W. Lindquist, J. Math. Phys. 8, 265 (1967). 5P. Jordan, J. Ehlers, and W. Kundt, Akad. Wiss. Lit. Mainz,

Abh. Math. Natunviss. K1. Jahrg. No. 2 (1960); A. S. Kom- paneetz, Zh. Eksp. Theor. Fiz. 34, 953 11958) [Sov. Phys. JETP 7, 659 (1958)l.

6 ~ . Einstein and N. Rosen, J. Franklin Inst. 223, 43 (1937). 'Y. Nutku and M. Halil, Phys. Rev. Lett. 39, 1379 (1977). %. Chandrasekhar and V. Ferrari, Proc. Soc. R. London A396,

55 (1984). The Nutku-Halil solution is obtained by using what Chandrasekhar calls a, a wave function in which the role of t and r are interchanged. In our case a is simply r2, which is very different.

91n spite of the lack of asymptotic flatness in the z direction, we can define null infinity for radially going rays.

l0See N. Rosen, Phys. Rev. 110, 291 (1958); see also J. Board- man and P. G . Bergmann, Phys. Rev. 115, 1318 (1959). The fastest way to obtain the result is to use the formulas of J. Katz, Class. Quantum Gravit. 2, 423 (1985). Notice that R, which diverges at Jt and 9- does not enter in the energy formula.

FIG. I. a ~ r in units of its maximum value in terms of R/Zand T/Zin the ranges O<R/Z<20 and -105 T/Z< 10. (a) is for a=1.01 and ib) for a=10. a,r(max) scales like (a2-I) /Z for a = l and like I/< for a>>l . The tendency towards a singular behavior at R =O for a- m appears already for a= 10.