Plane Impulsive Gravitational Waves Research Articles

Probing the Khan-Penrose colliding plane impulsive gravitational waves solution

Khan and Penrose obtained an exact solution for colliding plane impulsive gravitational waves with the remarkable feature that the spacetime before the collision is flat and after the collision is not only curved, but develops a future curvature singularity. The spacetime has been extensively investigated analytically by various authors using the Newman-Penrose equations. We feel that a separate insight is provided by using the method of Weber and Wheeler to calculate the momentum imparted to test particles by the colliding waves. In this paper we probe the curvature by considering the momentum imparted to test particles by the colliding gravitational waves so as to try to visualise the way the curvature increases in the chosen coordinates as the singularity is approached.

Colliding impulsive gravitational waves and a cosmological constant

We present a space--time model of the collision of two homogeneous, plane impulsive gravitational waves (each having a delta function profile) propagating in a vacuum before collision and for which the post collision space--time has constant curvature. The profiles of the incoming waves are $k\,\delta(u)$ and $l\,\delta(v)$ where $k, l$ are real constants and $u=0, v=0$ are intersecting null hypersurfaces. The cosmological constant $\Lambda$ in the post collision region of the space--time is given by $\Lambda=-6\,k\,l$.

Colliding Plane Impulsive Gravitational Waves

When two non-interacting plane impulsive gravitational waves undergo a head-on collision, the vacuum interaction region between the waves after the collision contains backscattered gravitational radiation from both waves. The two systems of backscattered waves have each got a family of rays (null geodesics) associated with them. We demonstrate that if it is assumed that a parameter exists along each of these families of rays such that the modulus of the complex shear of each is equal then Einstein's vacuum field equations, with the appropriate boundary conditions, can be integrated systematically to reveal the well-known solutions in the interaction region. In so doing the mystery behind the origin of such solutions is removed. With the use of the field equations it is suggested that the assumption leading to their integration may be interpreted physically as implying that the energy densities of the two backscattered radiation fields are equal. With the use of different boundary conditions this approach can lead to new collision solutions.

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.

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)].

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.

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 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.

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.