Disforming the Kerr metric

Starting from a recently constructed stealth Kerr solution of higher order scalar tensor theory involving scalar hair, we analytically construct disformal versions of the Kerr spacetime with a constant degree of disformality and a regular scalar field. While the disformed metric has only a ring singularity and asymptotically is quite similar to Kerr, it is found to be neither Ricci flat nor circular. Non-circularity has far reaching consequences on the structure of the solution. As we approach the rotating compact object from asymptotic infinity we find a static limit ergosurface similar to the Kerr spacetime with an enclosed ergoregion. However, the stationary limit of infalling observers is found to be a timelike hypersurface. A candidate event horizon is found in the interior of this stationary limit surface. It is a null hypersurface generated by a null congruence of light rays which are no longer Killing vectors. Under a mild regularity assumption, we find that the candidate surface is indeed an event horizon and the disformed Kerr metric is therefore a black hole quite distinct from the Kerr solution.

A Central Mass in a Stationary Vacuum without Spherical or Axial Symmetry

A vacuum spacetime with a central mass is derived as a stationary solution to Einstein's equations. The vacuum metric has a geodesic, shear-free, expanding, and twisting null congruence k and thus is algebraically special. The properties of the metric are calculated. In particular, it is shown that the spacetime has an event horizon inside which there is a black hole. The metric is neither spherically nor axially symmetric. It is therefore in interesting contrast with the majority of metrics featuring a central mass which have one or more of these symmetry properties. The metric reduces to the Schwarzschild case when a certain parameter is set to zero.

A METHOD TO CONSTRUCT 4-DIMENSIONAL SPACETIMES WITH A SPACELIKE CIRCLE ACTION

A general procedure to construct a 4-dimensional spacetime from a 3-dimensional time-oriented Lorentzian manifold and each of its timelike vector fields is exposed. It is based on the construction of the null congruence Lorentzian manifold. As an application, examples of stably causal spacetimes which obey the timelike convergence condition, are semi-symmetric, and admit an isometric spacelike circle action are obtained.

PARAMETRIC SPINOR APPROACH TO GRAVITY

It is shown that the gravitational field in general relativity has the properties of a parametric manifold, a mathematical structure generalizing the concept of gauge fields. A theory of parametric spinors is developed which contains, as a limiting case, spinor fields of unitary groups in Riemannian manifolds. Transition from the familiar [Formula: see text] spinor formalism to the parametric picture makes it possible to obtain a reparametrization-invariant decomposition of the gravitational equations with respect to a non-null congruence. This new formulation of the relativistic field equations appears to be amenable to canonical quantization, and provides more versatility in the treatment of phase-space variables.

A generalization of the Mariot theorem on congruences of null geodesics

It is shown that the property of a congruence of curves to consist of null geodesics can be defined in terms of a distribution of a co-dimension one, without reference to the conformal structure of the underlying differen­tiable manifold: if k is the vector field tangent to the congruence and k is a 1-form characterizing the distribution, then the congruence is said to be null if k ˩ k = 0 and geodesic if, and only if, k ∧£ k = 0. The geodesic property of the congruence, on an n -dimensional manifold, means that if F is an ( n —2)-form such that k ˩ F = 0 and k ∧ F = 0, then k ∧ d F = 0. A twisting geodesic null congruence on S 1 ᵡ S 2 l + 1 , associated with the Hopf fibration S 2 l + 1 → CP l , is constructed as an illustration.

A generalized Kerr-Schild solution of Einstein's equations

A very general exact solution of Einstein's equation Rik = σξiξk, is given in terms of the Kerr-Schild metric, gik = ηij + Hξiξk, where ξi is a shearfree geodetic null congruence. The Kerr-Schild solution for Rik = 0 is derived as a particular case.

Exact solutions of the field equations: twist-free pure radiation fields

This note is intended to give a rough survey of the results obtained in the study of twist-free pure radiation fields in general relativity theory. Here we are using the following Definition. A space-time ( V 4 of signature +2) is called a pure radiation field if it contains a distortion-free geodetic null congruence (a so-called ray congruence ), and if it satisfies certain field equations which we will specify below (e.g. Einstein’s vacuum-field equations). A (null) congruence is called twist-free if it is hypersurface-orthogonal (or ‘normal’). The results listed below were obtained by introducing special (‘canonical’) co-ordinates adapted to the ray congruence. Detailed proofs were given by Robinson & Trautman (1962) and by Jordan, Kundt & Ehlers (1961) (see also Kundt 1961). For the sake of completeness we include in our survey the subclass of expanding fields, and make use of some formulae first obtained by Robinson & Trautman.