Rotating gravitational waves

Volume 151, number 9

PHYSICS LETTERS A

31 December 1990

Rotating gravitational waves Bahram Mashhoon Department o f Physics and Astronomy, University ~/ Mlssouri-( "olumbia, Columbia, MO 6521 I. ~ ?~\.f

and Hernando Quevedo lnstitutejor Theoretical Physics. University ~/ ( 'olo,~ne, I|'-5000 ( "ologne 4 l, Germany

Received 23 May 1990: accepted for publication 22 October 1990 Communicated by J.P. Vigier

We present a cylindrically symmetric solution of the gravitational field equations which describes the propagation of rotating gravitational waves in empty space. The radiation field can be represented as a superposition of incoming and outgoing waves. The singularities of the rotating waves are examined and their significance for the speed-of-light catastrophe is discussed.

Rotating electromagnetic waves are c o m m o n p l a c e in the laboratory; imagine, for instance, an electromagnetic radiation field set up in a cylindrical wave guide such that the waves travel a r o u n d the axis o f the coaxial cavity [ 1 ]. The confinement o f the electromagnetic field within the annular region is basically due to the possibility o f relative m o v e m e n t o f positive and negative charges within the cylindrical walls o f the cavity. In a similar configuration, gravitational waves would leak inward a r o u n d the axis o f s y m m e t r y as well as outward to infinity. This is a consequence o f the universality o f gravitational interaction. To construct a gravitational analog o f the rotating electromagnetic waves, it is therefore simpler to dispense with the cavity walls and let the inner cylinder shrink to the axis o f rotation and the outer cylinder recede to infinity. Vacuum solutions of the gravitational field equations corresponding to this intuitive picture will be considered in this work. We will show explicitly that solutions o f this type formally exist but are a c c o m p a n i e d by curvature singularities. The electromagnetic field equations are linear; therefore, a rotating electromagnetic wave in vacuum can be expressed as a linear superposition of familiar plane waves. A similar d e c o m p o s i t i o n is not possible for rotating gravitational waves due to the 464

nonlinearity of the field equations. This complicates the physical interpretation o f rotating gravitational waves, especially in connection with the formation o f singularities. In 1925, Beck discovered a class o f exact cylindrically symmetric solutions o f the gravitational field equations by a transformation of the Weyl fields using complex variables. He interpreted these solutions as representing the propagation o f cylindrical gravitational waves [2]. This class was rediscovered and studied by Einstein and Rosen [3] as an exa m p l e of nonlinear gravitational waves. F u r t h e r investigation o f this class o f solutions - by Bonnor [ 4 ], and by Weber and Wheeler [5], in particular - has led to the physical interpretation o f a s u b c l a s s o f the solutions in terms o f s i n g u l a r i t y - f r e e linearly polarized cylindrical gravitational waves that propagate inward in e m p t y space, i m p l o d e on the axis o f symmetry, and then propagate outward to spatial infinity. O f the two possible polarization states for gravitational radiation, all the waves in this class maintain a fixed state of linear polarization consistent with cylindrical symmetry. Jordan, Ehlers and K u n d t [6], and i n d e p e n d e n t l y K o m p a n e e t s [7], have generalized the line element for cylindrical waves to include the case o f two polarization states. Cylindrical gray-

0375-9601/~0/$ 03.50 (,3 1990 - Elsevier Science Publishers B.V. (North-Holland i

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itational waves of arbitrary polarization have been the subject of recent investigations (see, e.g., ref. [ 8 ] and the references cited therein). Further references to previous work on cylindrical waves are contained in ref. [9]. Consider a cylindrically symmetric spacetime with the line element

ds2=e2y-2~'(dt2_dp2 ) --/.t2e-2~'(fo d r + - e 2~'dz 2 ,

dq}) 2

( 1)

where 7, P, ~u, and 09 are functions of t and p only. Here t is a temporal coordinate, p is a radial coordinate that indicates the distance away from the axis of cylindrical symmetry, ~ denotes the azimuthal angle in a horizontal plane normal to this axis, z is a measure of distance along it, and c= 1. The spacetime represented by this metric contains two commuting spacelike Killing vector fields corresponding to invariance under rotation about, and translation along, the axis of cylindrical symmetry. The gravitational field equations for the metric ( 1 ) can be expressed as follows:

Oz~,,),,- (Fz~up),p = 0 ,

(2)

It.,, - lt pp = ½la~.p ,

(3)

O),p = l / / - 3e2>' ,

(4)

(u,,2 - u2,~ )y,, = u,, ~ ' - u,p q~ ,

(5)

(I'2,,- It p)7.p =/~,, ~ - / % g",

(6)

where partial differentiation is denoted by a comma,

~=/z(~,2 +~.p) 2 + ~t (/L,, +/t,pp) ,

(7)

qb= 2/t~u,,~,p + p,,p,

(8)

and 1 is a constant of integration. The integrability condition for eqs. (5) and (6) is satisfied via the remaining field equations ( 2 ) - ( 4 ) . For 1=0, lt=p is a solution ofeq. (3) and the line element with ~o=0 reduces to the standard metric for cylindrical waves with a fixed state of linear polarization. This standard form for the metric can also be achieved by means of an appropriate coordinate transformation when to depends only upon time. The function 09 is expected to be a measure of proper rotation of the waves. Therefore, it is necessary to impose the condition that ~Op~0 for rotating gravitational waves;

31 December 1990

otherwise, the metric simply represents cylindrical waves in a rotating coordinate system. The form of the gravitational field equations ( 2 ) (8) can be simplified by the introduction of radiation coordinates u = t - p and v= t+p representing retarded and advanced times, respectively. The resuiting differential equations with l~ 0 can be solved by demanding separation of radiation variables for the metric function #. We obtain

lt=a(u)fl(v) ,

(9)

~ = 3x/3/21n(c~/fl),

(lO)

y=½ l n ( ~2 (afl)3e%fl.,,),

(11)

and 8

~o.v-~o,. = 7 a ~,8. . . .

(12)

where a ( u ) and fl(v) represent arbitrary functions that are both either positive or negative. The metric functions ~, and y are entirely determined by a and ,8 according to eqs. ( 10) and ( 1 1 ), respectively, while ~o is given by a first-order differential equation ( 12) involving alp= o9~,-~o,. Thus o9 is determined up to an arbitrary function of time t= ( v + u ) / 2 . This arbitrariness corresponds to the possibility of transformation to a rotating system of coordinates. More general coordinate transformations can be used under certain conditions to eliminate the two arbitrary functions in the metric and thereby reduce it to a normal form [10]. This possibility will be investigated later in connection with the singularities of this gravitational field. In the normal form of the spacetime metric, a and,8 turn out to be linear functions of u and v, respectively. The scalar wave equation in the spacetime background characterized by the metric ( 1 ) is satisfied by the metric function ~ since (cf. eq. ( 2 ) ) (p~,..),,, + (p~u.v).. = 0 .

(13)

It follows that ~, may be interpreted as the potential of the gravitational wave. Introducing new functions U(u) and V(v) by a=exp(2x/~U),

,8=exp(2x/~V),

14)

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the potential can be written as

~u= U { u ) - l'(z~),

(15)

which indicates that the radiation may be represented as a linear superposition of incoming and outgoing waves of fixed linear polarization. The arbitrap. functions U and t" can always be so chosen as to correspond to a realistic pulse form. The R i e m a n n curvature tensor for the solution given by eqs. {9 ) ( 12 ) is algebraically special with vanishing scalar invariants and a unique null principal vector. The solution belongs to type IlI in the Petrov classification [ 1 1 ]. These results are consistent with the interpretation of the solution ( 9 ) - (12) in terms of propagating gravitational waves [ 12 ]. To demonstrate the rotation of the waves, imagine a test particle released from rest at an event in spacetime. In free fall, the particle is generally dragged in the azimuthal direction by the radiation field. Such a coordinate-dependent characterization of rotation without reference to a global set of inertial observers is necessary in this case because the spacetime is not asymptotically flat. The rotational dragging - determined by the proper azimuthal acceleration of the free test particle starting from rest - turns out to be proportional to l. This quantity may be assumed to be positive with no loss in generality. Thus, the constant length-scale l is associated with the rotation of waves. The rotating waves do not reduce to a proper limiting form in the absence of rotation; this is consistent with the fact that Beck's radiation field is of Petrov type I. In order to investigate the singularities of the solution ( 9 ) - ( 1 2 ) , we analyze the spacetime curvature measured by a free test observer on a "radial" path in a horizontal plane normal to the symmetry axis. The cylindrical symmetry of the gravitational field implies that for a free particle there are in general two constants of motion which may be interpreted as orbital angular m o m e n t u m about, and linear m o m e n t u m along, the axis of symmetry. We set both these constants equal to zero for the sake of simplicity, so that the geodesics u n d e r consideration are as radial as the rotation of the waves would allow. For instance, the gravitational radiation dragging of "'radial" null geodesics causes azimuthal motion given by dC~/dp= -T-¢o, where the upper (lower) sign refers to outgoing ( i n c o m i n g ) rays. The general

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solution Ibr a "'radial" timelike geodesic at constant : is then

du

dT-~(F")

(16)

and dv -=~(/:,) dz

~

(17i

with dga/dt= -{o. Here r is the proper time along tile path and F ( u , z,) is a function given by

½12F.,,l.[, = ( fl ) , "(o~fl)~oz.,,/:(~.

(18}

Along the path, the magnitude of F differs from the proper time by only a constant. Let the measurements of an observer on a "'radial" geodesic path be referred to a local inertial frame determined by an orthonormal tetrad field that is parallel transported along the path. The temporal axis of the frame, T ' . is determined by the velocity vector of the observer and the spatial axes (XQ t". Z ~) are ideal gyroscope directions. In terms of the standard coordinates (t, t7. O, 5). we have

/"-=(/;~-/.'~',,} '(/-,.

/',.--(oF,.0}.

~!9~

where F,, = t.[, + b[,, and 1.,, = b[, - F,,. The cylindrical s y m m e t u of the radiation field allows us to choose Z ~ to be parallel to the axis of symmetry.

Z ' = ( 0 , 0, (L c "').

(20}

while ,}('~and Y" can be determined, in principle, flom the requirements of parallel propagation and orthonormality. Based on this flame, a Fermi coordinate system can be established in the neighborhood of the path. In terms of measurements performed by a "'radial" observer with respect to this local Fermi system, consider, for instance, the electric component of the Riemann curvature tensor along the Zaxis given by

K = Ra,~l~T " Z " T " Z ~ .

(2 1)

Here K determines the tidal influence of the radiation field on the Z - c o m p o n e n t of the relative motion of free test particles starting from rest in the Fermi frame along a direction parallel to the axis of cylindrical symmet W. It can be shown that

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K=C_

a 2 ,u

0¢2F 2, u

PHYSICS LETTERS A

3 a.fl, v - +C+ 4 o~flF,F~ ,

.

f12 /t " / 2 F 2, ~ '

(22)

where C+ are constants given by C+ = (9 + 4 x f 6 ) / 8 . In principle, the local tidal contribution of the gravitational waves can thus be determined. An investigation of the components of the Riemann curvature tensor projected onto a paralleltransported frame carried by an observer along a "radial" geodesic path reveals that singularities occur in general whenever c~, ~, o~,u, or fl,,, vanishes. It follows from eqs. (14) and ( 15 ) that the solution ( 9 ) - ( 12 ) is singular for any realistic gravitational waveform. The singularity is in the form of a null hypersurface which is the time development of a caustic cylinder. These results suggest that incoming gravitational waves develop caustics as a result of rotation, implode on the symmetry axis, and return to infinity. The axis of symmetry, which becomes singular at the m o m e n t of implosion of a caustic, acts as a reflector of gravitational waves. It is possible to reduce the gravitational field associated with rotating waves with arbitrary ~x and fl to a normal form in a piecewise manner. Consider, for this purpose, the coordinate transformation (t, p, 0, z ) - , (t', p', 0', z' ) with au' + ao= o~(u), by' + bo=fl(v), O'=~+f(u, v), and z'=z, where a ~ 0 , b ~ 0, ao and b0 are constants a n d f ( u , v) satisfies the differential equation 2

f " - f " = affl ( a f l - b a ) ( afl,~,-ba ,) .

(23)

This would be an admissible change o f coordinates if the associated Jacobian is nonzero; therefore, we must impose the conditions that a , ¢ 0 and fl,~.# 0. Under such a transformation, the metric in terms of the new coordinates is identical to a specific solution ( 9 ) - ( 1 2 ) in which a and fl are linear, i.e., a ( u ) = ao+au, fl(v)=bo+bv, and oJ=Ogo+8abp/l, where lcoo=4(abo-aob). Hence spacetime regions between the extrema of a and fl can be described in terms o f this normal form of the metric in which singularities occur at ao+au' = 0 and bo+bv' =0. The curvature singularities of the solution ( 9 ) (12) could be due to the highly idealized symmetry assumed for the derivation of the metric for rotating gravitational waves. It is unrealistic to suppose that a system with complete cylindrical symmetry should

31 December 1990

exist in nature. On the other hand, it is possible that the breakdown of the theory encountered here is related to the speed-of-light catastrophe discussed by Ardavan [ 13 ]. This catastrophe occurs in general in a linear classical theory when the source of radiation is a massless field. The singularity is expected to disappear when the nonlinearities inherent in the physical phenomenon under consideration are taken into account. Ardavan's treatment of gravitational waves generated by rotating electromagnetic waves is within the framework o f linearized theory of gravitational radiation. However, gravitational radiation can act as its own source. The investigation of general-relativistic nonlinearities provided the original motivation for our analysis. It is possible that self-interaction would prevent the formation of a singularity; otherwise, some of the predictions of the theory of gravitational radiation could not correspond to reality. Is the singularity associated with rotating gravitational waves a nonlinear analog of the singularity that appears in Ardavan's work? A more complete investigation involving realistic sources o f gravitational waves is necessary before this question could be conclusively answered.

Thanks are due to Professor F.W. Hehl for his support and encouragement. We are grateful to Professor W. Kinnersley for pointing out the possibility of reducing the rotating wave solution to a normal form. One of us (Hernando Quevedo) would like to thank the Department of Physics and Astronomy of the University of Missouri-Columbia for excellent hospitality while this work was done. This work was partly supported by the Deutsche Forschungsgemeinschaft, Bonn.

References

[ 1] F.E. Borgnis and C.H. Papas, Electromagnetic waveguides and resonators, in: Encyclopedia of physics, Vol. XVI, ed. S. Fliigge (Springer, Berlin, 1958) p. 326. [2] G. Beck, Z. Phys. 33 (1925) 713. [ 3 ] A. Einstein and N. Rosen, J. Franklin Inst. 223 ( 1937 ) 43; N. Rosen, Bull. Res. Council Israel 3 (1954) 328. [4] W.B. Bonnor, J. Math. Mech. 6 (1957) 203. [ 5 ] J. Weber and J.A. Wheeler, Rev. Mod. Phys. 29 ( 1957) 509. 467

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[ 6 ] P. Jordan, J. Ehlers and W. Kundt, Abh. Akad. Wiss. Mainl Math. Naturwiss. KI. 2 (1960). [7] A.S. Kompaneets, Sov. Phys. J ETP 7 (1958) 659. [8] S. Chandrasekhar, Proc. R. Soc. A 408 (1986) 209. [9] D. Kramer, H. Stephani, E. Herlt and M.A.H. MacCallum, Exact solutions of Einstein's field equations (Deutscher Verlag der Wissenschaften, Berlin, 1980 ). [ I 0 ] W. Kinnersley, private communication ( 1990 ~.

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[I 1 ] A.Z. Petrov, Sci. Not. Kazan State /lniversity 114 11954! 55: Einstein spaces ( Pergamon. Oxford, 1969 ! [12] F.A.E. Pirani, Phys. Rcv. 105 (1957) 1089. [13] H. Ardavan, in: Classical general relativit), eds. W.B Bonnor, J.N. Islam and M.A.H. MacCallum t Cambridge Univ. Press, Cambridge, 1984 ) p. 5: Phys. Rev. D 29 ( 1984 207: Proc. R. Soc. A 424 (1989) 113.