Impulsive Gravitational Waves Research Articles

Interaction of null dust clouds fronted by plane impulsive gravitational waves. II

This paper extends the earlier discussion of the types of energy-momentum tensors Tμν that can exist in the region of interaction of two colliding plane impulsive gravitational waves, each followed by a null dust cloud. Two additional types of tensors Tμν are discussed. For each of the five types of interaction regions solutions to the Einstein field equations are obtained assuming solution of the vacuum equations are known.

Colliding impulsive waves in succession.

We formulate the initial-value problem for two colliding trains of impulsive gravitational waves. In the absence of a global, exact solution we show that in a region lying between the singularity ${u}^{2}+{v}^{2}=1$ and the wave trains, the solution is still the well-known Khan-Penrose solution.

Interaction of null dust clouds fronted by impulsive plane gravitational waves

This paper discusses the energy-momentum tensor T μν in the region of interaction of a space-time in which two colliding plane impulsive gravitational waves, each followed by a null dust cloud, exist. It is shown that in the interaction region T μν is of three types: (i) that of two noninteracting null dusts; (ii) that of a scalar field (equivalently an irrotational perfect fluid with energy density equal to pressure), and (iii) that of the sum of two independent non-interacting scalar fields [equivalently a complex scalar field or an anisotropic perfect fluid with energy density w and pressures (w,π,π)].

Upper limit on the flux of gravitational wave bursts

The data obtained with three highly sensitive gravitational wave antennas have been used to establish an improved observational upper limit on the flux of impulsive gravitational waves. Preliminary results under the assumption of a source located in the center of the Galaxy are also reported.

Collisions of gravitational and electromagnetic waves that do not develop curvature singularities

A five-parameter family of solutions is investigated, describing the collision of plane-fronted impulsive gravitational and shock electromagnetic waves. In the interaction region, to the future of the collision, it is a locally known solution of the Einstein–Maxwell electrovacuum equations of Petrov type D. The collision results in the formation of a Cauchy horizon. Extensions of the space-time are constructed beyond the Cauchy horizon and beyond certain two-dimensional surfaces that are mere coordinate singularities. It is found that in the extended space-time the following may occur: (i) no curvature singularities, (ii) two-dimensional spacelike curvature singularities, and (iii) two-dimensional timelike curvature singularities, according to the ranges of the parameters of the solution.

Collision of impulsive gravitational waves followed by dust clouds

The evolution of a space-time containing two colliding plane impulsive gravitational waves each of whose leading edges is followed by a distribution of null dust is determined. The conditions on the Ricci tensor that ensure that the evolution of such a space-time is unambiguous are determined. These are the same as those that apply in the planar case. The equations of motion of the medium contained in the region of interaction of the dust clouds are determined. These equations determine the change in energy density of each dust cloud as the interaction proceeds and involve the functions whose specifications ensure that the evolution of the space-time is unambiguous.

Emission and absorption of plane impulsive gravitational waves

Emission and absorption of plane impulsive waves of radiation by a thin shell of dust are studied. At the moment of interaction local conservation laws apply. An impulsive gravitational wave does not participate in the energy balance. Rather, it is associated with a change in isotropy of the source (absorber). Changes in the kinetic energy of the source exactly match the changes in its rest mass.

On the collision of planar impulsive gravitational waves

Exact continuous solutions to the Einstein field equations are determined under the assumption that the subspaces of space-time spanned by the variables XA (A=1,2) admit the three-dimensional group of motions of a two-plane, and the Ricci tensor RAB (A,B=1,2) vanishes. The space-time is assumed to contain two colliding planar impulsive gravitational waves. Each wave may be followed by a distribution of null dust. It is shown that the Cauchy data on a spacelike three-surface does not lead to a unique solution of the Einstein field equations unless additional requirements are imposed on the stress-energy tensor in the region of interaction of the waves.

Colliding plane gravitational waves: A class of nondiagonal soliton solutions.

Using the inverse scattering method we construct a general class of nondiagonal solutions of the vacuum Einstein field equations describing the collision of two plane-fronted impulsive gravitational waves accompanied with gravitational shock waves with the same front. It is a two-parameter class of solutions: the first represents the angle between the directions of polarization of the two waves; the second determines the wave profile.

On the dispersion of cylindrical impulsive gravitational waves

An exact solution, describing the dispersion of a wave packet of gravita­tional radiation, having initially (at time t = 0) an impulsive character, is analysed. The impulsive character of the wave-packet derives from the space-time being flat, except at a radial distance ϖ = ϖ 1 (say) at t = 0, and the time-derivative of the Weyl scalars exhibiting δ-function singu­larities at ϖ = ϖ 1 , when t → 0. The principal feature of the dispersion is the development of a singularity of the metric function, v , and of the Weyl scalar, ψ 2 , when the wave, after reflection at the centre, collides with the still incoming waves. The evolution of the metric functions and of the Weyl scalars, as the dispersion progresses, is illustrated graphically.

On colliding waves that develop time–like singularities: a new class of solutions of the Einstein–Maxwell equations

Solutions 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 exact solution for colliding gravitational plane waves

A new exact solution of the vacuum Einstein equations describing the spacetime following the collision of two plane impulsive gravitational waves, each supporting a plane gravitational wave, is obtained. The solution has been extended prior to the instant of collision and the main features of the resulting space-time have been analyzed, using the Newman-Penrose formalism. It is shown that the result of the collision is the development of a singularity of the spacetime due to the simultaneous focussing of the two plane waves.

A new type of singularity created by colliding gravitational waves

An 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

The 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 є = 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 є = p .

On the collision of impulsive gravitational waves when coupled with fluid motions

An 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.

Some exact solutions of gravitational waves coupled with fluid motions

Some 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.

Errata

Proc. R. Soc. Lond. A 396, 55-74 (1984) On the Nutku-Halil solution for colliding impulsive gravitational waves By Subrahmanyan Chandrasekhar, F. R. S., and Valeria Ferrari Page 62, equation (65), first line, last term: replace d χ 2 by d x 2 ; third line: replace e - β by —e β , as in the preceding line. Page 65, equation (82), third line: replace ͞m i by m i . Page 66, equation (86): in the expression for γ a factor ∆ in front of (cot θ — cot ψ ) has been omitted and the expression in parentheses in the last term should read (sin ψ cos θ + cos ψ sin θ ). Page 68, equation (111), last line: replace by (1 — | E |) 2 by (1 — | E | 2 ) 2 . Page 70, equation (117), first line: replace ψ 4 by ψ 4 . Page 73, equation (131): replace γ in the first term by v .

On colliding waves in the Einstein—Maxwell theory

An 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 Nutku—Halil solution for colliding impulsive gravitational waves

The equations appropriate for space-times with two space-like Killing-vectors are set up, ab initio , and explicit expressions for the components of the Riemann, the Ricci, and the Einstein tensors in a suitable tetrad-frame are written. The equations for the vacuum are reduced to a single equation of the Ernst type. It is then shown that the simplest linear solution of the Ernst equation leads directly to the Nutku-Halil solution for two colliding impulsive gravitational waves with uncorrelated polarizations. Thus, in some sense, the Nutku-Halil solution occupies the same place in space-times with two space-like Killing-vectors as the Kerr solution does in space—times with one time-like and one space-like Killing-vector. The Nutku-Halil solution is further described in a Newman-Penrose formalism; and the expressions for the Weyl scalars, in particular, make the development of curvature singularities manifest. Finally, a theorem analogous to Robinson’s theorem (but much less strong) is proved for space-times with two space-like Killing-vectors.

Ultrarelativistic black-hole encounters

A method is given for investigating the ultrarelativistic encounter of two black holes which complements the usual low-speed approximation. It depends on the fact that the ultrarelativistic limit of a moving Schwarzschild black hole is a certain plane-fronted impulsive gravitational wave. This enables linearized theory on a curved background to be used. The solution is obtained, with the help of generalized function techniques, to the first order in the background energy, and the radiation pattern at conformal null infinity is examined. The whole approach has a strong connection with the theory of twistors.