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On colliding waves that develop time-like singularities: a new class of solutions of the Einstein-Maxwell equation
No full textSolutions of the Einstein-Maxwell equations are found that provide generalizations of a solution discovered by Bell and Szekeres, which represents the collision of impulsive gravitational waves coupled with electromagnetic shock-waves in a conformally flat space-time. Starting with the Bell-Szekeres solution in a form more general than their original one (though equivalent to it) and applying to it a so-called Ehlers transformation, we obtain a new family of Petrov type-D space-times in which horizons form and subsequently two-dimensional time-like singularities develop. A second solution provides a generalization of the Bell-Szekeres solution in the same way as the axisymmetric distorted static black-hole solutions provide a generalization of the Schwarzschild solution. This second solution also forms a horizon but the time-like singularity that develops is three-dimensional. The mathematical theory that is developed seems specially adapted to the solution of these and related problems
A new type of singularity created by colliding gravitational waves
No full textAn exact solution is obtained for colliding plane impulsive gravitational waves accompanied by shock waves, which, in contrast to other known solutions, results in the development of a null surface which acts like an event horizon. The analytic extension of the solution across the null surface reveals the existence of time-like curvature singularities along two hyperbolic arcs in the extended domain, reminiscent of the ring singularity of the Kerr metric. Besides, the space-time, in the region of the interaction of the colliding waves, is of Petrov-type D and locally isometric to the Kerr space-time in a region interior to the ergosphere. Various other aspects of the solution are also discussed
On the collision of impulsive gravitational waves when coupled with null dust
No full textThe problem of colliding impulsive gravitational waves is considered when the region of space-time, after the instant of collision, is filled with a mixture of null dusts moving in opposite directions. The extension of the resulting space-time, to regions before the instant of collision, shows that null dust follows the leading edges of the impulsive waves, and, further, that one can arrange that the space-time in these regions is identical with what prevails when a perfect fluid with ε=p fills the region after the instant of collision. This ambiguity in the space-time, after the instant of collision, must be traced to an inherent ambiguity in the nature of null dust and its relation with a perfect fluid with ε=p
On colliding waves in the Einstein-Maxwell theory
No full textAn exact solution of the Einstein-Maxwell equations is obtained that represents a space-time which describes consistently the collision between two plane impulsive gravitational waves, each supporting an electromagnetic shock-wave. In obtaining the solution, the relationship, which had been established earlier, between the solutions describing stationary black-holes and solutions describing colliding plane-waves, is extended to the Einstein-Maxwell equations (and exploited). The case when the colliding waves are parallelly polarized is analysed in detail to exhibit the singularities and the discontinuities that occur on the null boundaries characteristic of this problem. It is found that the passage of the waves, prior to collision, produces a spray of gravitational and electromagnetic radiation and the collision results in the scattering and the focusing of the waves and the development of a space-time singularity. The solution that is obtained avoids in a natural way certain conceptual difficulties (such as the occurrence of the 'square root' of a δ-function and current sheets) that had been anticipated
On the collision of impulsive gravitational waves when coupled with fluid motions
No full textAn exact solution of Einstein's equations, with a source derived from a perfect fluid in which the energy density, ε, is equal to the pressure, p, is obtained. The solution describes the space-time following the collision of plane impulsive gravitational waves and is the natural generalization of the Nutku-Halil solution of the vacuum equations, in the region of interaction under similar basic conditions. A consistent extension of the solution, prior to the instant of collision, requires that the fluid in the region of interaction is the direct result of a transformation of incident null-dust (i.e. of massless particles describing null trajectories). The ultimate result of the collision is the development of a space-time singularity, the nature of which is strongly dependent on the amplitude and the character of the sound waves that are present. The distribution of ε that follows the collision has many intriguing features. The solution obtained in this paper provides the first example of an induced transformation of a massless into a massive particle
A perturbation analysis of the Bell-Szekeres space-time
No full textPerturbation analysis of the Bell-Szekeres space-time is provided both in the region where the scattering of the colliding plane-fronted
waves occurs (Region I) and in the regions prior to thc instant of collision (Regions II and III). A complete set of normal modes, expressed in
terms of spin-weighted spherical harmonics and bounded in the entire Region I, is obtained. These modes exhibit a behaviour of ever-increasing
frequency as the horizon (that develops as a result of the collision) is approached. Region II, bounded by the null lines, u=0(0≤v≤1)
and v=l(u≤0) (where u and v define the directions of propagation of the colliding waves) does not allow u-independent perturbations to all
orders; while all the u-dependent perturbations exhibit strong divergences along the entire line, v=l, u<0. This same description applies to
Region III with the roles of u and v interchanged. These behaviours of the perturbations, in the regions of the space-time prior to the instant of
collision, raise some serious conceptual questions.
The effect of sources on horizons that may develop when plane gravitational waves collide
No full textColliding plane gravitational waves that lead to the development of a horizon and a subsequent time-like singularity are coupled with an electromagnetic field, a perfect fluid (whose energy density, ε, equals the pressure, p), and null dust (consisting of massless particles). The coupling of the gravitational waves with an electromagnetic field does not affect, in any essential way, the development of the horizon or the time-like singularity if the polarizations of the colliding gravitational waves are not parallel. If the polarizations are parallel, the space-like singularity which occurs in the vacuum is transformed into a horizon followed by a three-dimensional time-like singularity by the merest presence of the electromagnetic field. The coupling of the gravitational waves with an (ε=p)-fluid and null dust affect the development of horizons and singularities in radically different ways: the (ε=p)-fluid affects the development decisively in all cases but qualitatively in the same way, while null dust prevents the development of horizons and allows only the development of space-like singularities. The contrasting behaviours of an (ε=p)-fluid and of null dust in the framework of general relativity is compared with the behaviours one may expect, under similar circumstances, in the framework of special relativity
Some exact solutions of gravitational waves coupled with fluid motions
No full textSome exact solutions of Einstein's equations are found which represent the interaction of gravitational waves with a perfect fluid in which the velocity of sound equals the velocity of light. These solutions, unlike the solutions representing the collision of impulsive gravitational waves, are bounded by a space-time singularity and have some resemblance to cosmological solutions: every time-like trajectory, extended into the past, encounters the singularity. Moreover, in the generic case, matter may be considered as being created at the singularity
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