Jerry B. Griffiths

Physical interpretation of vacuum solutions of Einstein's equations. Part II. Time-dependent solutions

The study of interpretations of the vacuum solutions of Einsteins field equations is continued by considering some well known time-dependent solutions. Among these are metrics representing accelerating particles, cylindrical and plane waves and cosmological solutions.

Impulsive waves in de Sitter and anti-de Sitter spacetimes generated by null particles with an arbitrary multipole structure

We describe a class of impulsive gravitational waves which propagate either in a de Sitter or an anti-de Sitter background. They are conformal to impulsive waves of Kundts class. In a background with positive cosmological constant they are spherical (but non-expanding) waves generated by pairs of particles with arbitrary multipole structure propagating in opposite directions. When the cosmological constant is negative, they are hyperboloidal waves generated by a null particle of the same type. In this case, they are included in the impulsive limit of a class of solutions described by Siklos that are conformal to pp-waves.

Interpreting the C-metric

The basic properties of the C-metric are well known. It describes a pair of causally separated black holes which accelerate in opposite directions under the action of forces represented by conical singularities. However, these properties can be demonstrated much more transparently by making use of recently developed coordinate systems for which the metric functions have a simple factor structure. These enable us to obtain explicit Kruskal–Szekeres-type extensions through the horizons and construct two-dimensional conformal Penrose diagrams. We then combine these into a three-dimensional picture which illustrates the global causal structure of the spacetime outside the black hole horizons. Using both the weak field limit and some invariant quantities, we give a direct physical interpretation of the parameters which appear in the new form of the metric. For completeness, relations to other familiar coordinate systems are also discussed.

Uniformly accelerating black holes in a de Sitter universe

A class of exact solutions of Einstein’s equations is analysed which describes uniformly accelerating charged black holes in an asymptotically de Sitter universe. This is a generalisation of the C-metric which includes a cosmological constant. The physical interpretation of the solutions is facilitated by the introduction of a new coordinate system for de Sitter space which is adapted to accelerating observers in this background. The solutions considered reduce to this form of the de Sitter metric when the mass and charge of the black holes vanish.

Accelerating and rotating black holes

An exact solution of Einsteins equations which represents a pair of accelerating and rotating black holes (a generalized form of the spinning C-metric) is presented. The starting point is a form of the Plebanski–Demianski metric which, in addition to the usual parameters, explicitly includes parameters which describe the acceleration and angular velocity of the sources. This is transformed to a form which explicitly contains the known special cases for either rotating or accelerating black holes. Electromagnetic charges and a NUT parameter are included, the relation between the NUT parameter l and the Plebanski–Demianski parameter n is given, and the physical meaning of all parameters is clarified. The possibility of finding an accelerating NUT solution is also discussed.

Accelerating Kerr–Newman black holes in (anti-)de Sitter space-time

A class of exact solutions of the Einstein-Maxwell equations is presented which describes an accelerating and rotating charged black hole in an asymptotically de Sitter or anti-de Sitter universe. The metric is presented in a new and convenient form in which the meaning of the parameters is clearly identified, and from which the physical properties of the solution can readily be interpreted.

Generalized Kundt waves and their physical interpretation

We present the complete family of spacetimes with a non-expanding, shear-free, twist-free, geodesic principal null congruence (Kundt waves) that are of algebraic type III and for which the cosmological constant (Λc) is non-zero. The possible presence of an aligned pure radiation field is also assumed. These spacetimes generalize the known vacuum solutions of type N with arbitrary Λc and type III with Λc = 0. It is shown that there are two, one and three distinct classes of solutions when Λc is respectively zero, positive and negative. The wave surfaces are plane, spherical or hyperboloidal in Minkowski, de Sitter or anti-de Sitter backgrounds respectively, and the structure of the family of wave surfaces in the background spacetime is described. The weak singularities which occur in these spacetimes are interpreted in terms of envelopes of the wave surfaces.

A note on the parameters of the Kerr-NUT-(anti-) de Sitter spacetime

Different forms of the metric for the Kerr–NUT–(anti-)de Sitter spacetime are being widely used in its extension to higher dimensions. The purpose of this note is to relate the parameters that are being used to the physical parameters (mass, rotation, NUT and cosmological constant) in the basic four-dimensional situation.

Boosted static multipole particles as sources of impulsive gravitational waves

It is shown that the known solutions for nonexpanding impulsive gravitational waves generated by null particles of arbitrary multipole structure can be obtained by boosting the Weyl solutions describing static sources with arbitrary multipole moments, at least in a Minkowski background. We also discuss the possibility of boosting static sources in (anti\char21{})de Sitter backgrounds, for which exact solutions are not known, to obtain the known solutions for null multipole particles in these backgrounds.

Global aspects of accelerating and rotating black hole spacetimes

The complete family of exact solutions representing accelerating and rotating black holes with possible electromagnetic charges and a NUT parameter is known in terms of a modified Plebanski–Demianski metric. This demonstrates the singularity and horizon structure of the sources but not that the complete spacetime describes two causally separated black holes. To demonstrate this property, the metric is first cast in the Weyl–Lewis–Papapetrou form. After extending this up to the acceleration horizon, it is then transformed to the boost-rotation-symmetric form in which the global properties of the solution are manifest. The physical interpretation of these solutions is thus clarified.