Jose M. M. Senovilla

Geometry of general hypersurfaces in spacetime: junction conditions

The authors study embedded general hypersurfaces in spacetime, i.e. hypersurfaces whose timelike, spacelike or null character can change from point to point. Inherited geometrical structures on these hypersurfaces are defined by two distinct methods: the first one, in which a rigging vector (a vector not tangent to the hypersurface anywhere) induces the standard rigged connection; and the other one, more adapted to physical aspects, where each observer in spacetime induces a completely new type of connection that is called the rigged metric connection which is volume preserving. The generalizations of the Gauss and Codazzi equations are also given. With the above machinery, they attack the problem of matching two spacetimes across a general hypersurface. It is seen that the preliminary junction conditions allowing for the correct definition of Einsteins equations in the distributional sense reduce to the requirement that the first fundamental form of the hypersurface be continuous, because then there exists a maximal C1 atlas in which the metric is continuous. The Bianchi identities are then proven to hold in the distributional sense. Next, they find the proper junction conditions which forbid the appearance of singular parts in the curvature. These are shown to be equivalent to the existence of coordinate systems where the metric is C1. Finally, they derive the physical implications of the junction conditions: only six independent discontinuities of the Riemann tensor are allowed. These are six matter discontinuities at non-null points of the hypersurface. For null points, the existence of two arbitrary discontinuities of the Weyl tensor (together with four in the matter tensor) are also allowed. The classical results for timelike, spacelike or null hypersurfaces are trivially recovered.

Axial symmetry and conformal Killing vectors

Axisymmetric spacetimes with a conformal symmetry are studied and it is shown that, if there is no further conformal symmetry, the axial Killing vector and the conformal Killing vector must commute. As a direct consequence, in conformally stationary and axisymmetric spacetimes, no restriction is made by assuming that the axial symmetry and the conformal timelike symmetry commute. Furthermore, the authors prove that in axisymmetric spacetimes with another symmetry (such as stationary and axisymmetric or cylindrically symmetric spacetimes) and a conformal symmetry, the commutator of the axial Killing vector with the two others must vanish or else the symmetry is larger than that originally considered. The results are completely general, and do not depend on Einsteins equations or any particular matter content.

Theorems on Shear-Free Perfect Fluids with Their Newtonian Analogues

In this paper we provide fully covariant proofs of some theorems on shear-free perfect fluids. In particular, we explicitly show that any shear-free perfect fluid with the acceleration proportional to the vorticity vector (including the simpler case of vanishing acceleration) must be either non-expanding or non-rotating. We also show that these results are not necessarily true in the Newtonian case, and present an explicit comparison of shear-free dust in Newtonian and relativistic theories in order to see where and why the differences appear.

On the definition of cylindrical symmetry

The standard definition of cylindrical symmetry in general relativity is reviewed. Taking the view that axial symmetry is an essential prerequisite for cylindrical symmetry, it is argued that the requirement of orthogonal transitivity of the isometry group should be dropped, this leading to a new, more general definition of cylindrical symmetry. Stationarity and staticity in cylindrically symmetric spacetimes are then defined, and these issues are analysed in connection with orthogonal transitivity, thus proving some new results on the structure of the isometry group for this class of spacetimes.

Singularity-free space-time

We show that the solution published in the paper by Senovilla [Phys. Rev. Lett. 64, 2219 (1990)] is geodesically complete and singularity-free. We also prove that the solution satisfies the stronger energy and causality conditions, such as global hyperbolicity, the strong energy condition, causal symmetry, and causal stability. A detailed discussion about which assumptions in the singularity theorems are not satisfied is performed, and we show explicitly that the solution is in accordance with those theorems. A brief discussion of the results is given.

On the extension of Vaidya and Vaidya-Reissner-Nordström spacetimes

A revision of Israels extension of Vaidyas spacetime is proposed in order to apply the procedure to more general spacetimes. Israels procedure is reformulated to obtain the extension even if the only available function is the mass in the non-extended region. The additional demands on analiticity and energy conditions are discussed. Eventually, the method is applied to the extension of Vaidya-Reissner-Nordstroms spacetime. The different cases arising are described and their conformal diagrams presented.

(Super) n -Energy for Arbitrary Fields and its Interchange: Conserved Quantities ∗

Inspired by classical work of Bel and Robinson, a natural purely algebraic construction of super-energy (s-e) tensors for arbitrary fields is presented, having good mathematical and physical properties. Remarkably, there appear quantities with mathematical characteristics of energy densities satisfying the dominant property, which provides s-e estimates useful for global results and helpful in other matters. For physical fields, higher order (super)n-energy tensors involving the field and its derivatives arise. In special relativity, they provide infinitely many conserved quantities. The interchange of s-e between different fields is shown. The discontinuity propagation law in Einstein–Maxwell fields is related to s-e tensors, providing quantities conserved along null hypersurfaces. Finally, conserved s-e currents are found for any minimally coupled scalar field whenever there is a Killing vector.

Matching of the Vaidya and Robertson-Walker metric

The authors give the necessary conditions for the matching of a general Robertson-Walker geometry to general spherically symmetric radiating metric. They also found the conditions for the matching of a Vaidya metric (1951) to a general Robertson-Walker metric. The possible applications of the results to the stellar collapse and to the study of local inhomogeneities in a cosmological context are considered. An alternative interpretation of the energy-momentum tensor of the Robertson-Walker part of spacetime is given in such a way that the physical processes can be better understood.

New family of stationary and axisymmetric perfect-fluid solutions

The author presents a new family of stationary and axisymmetric perfect-fluid solutions. The metrics are defined by the solutions to a system of two coupled ordinary differential equations of second order on three unknown functions. Therefore, they depend on an arbitrary function of one coordinate. The fluid has, in general, differential rotation. The equation of state is rho = rho +const. Some properties of the solutions are described and an explicit example given in terms of elementary functions is presented.

New and inhomogeneous cosmological models from the generalized Kerr--Schild transformation

New perfect-fluid solutions of Einsteins field equations have been found by means of the generalized Kerr--Schild transformation for perfect-fluid metrics. All these solutions are Petrov type D with the velocity vector of the fluid out of the two-space spanned by the two null principal directions of the Weyl tensor. In general, the solutions have just one Killing vector, but some special cases admit two commuting Killing vectors. All the solutions can be interpreted as inhomogeneous cosmological models, and an interesting property is that they contain some Friedman--Robertson--Walker metrics as special cases. Thus, they can be seen as inhomogeneous generalizations of the standard FRW cosmologies.