No abstract

We obtain an expression for the active gravitational mass of a collapsing fluid distribution, which brings out the role of density inhomogeneity and local anisotropy in the fate of spherical collapse.

We analyse the effects of thermal conduction in a relativistic fluid, just after its departure from hydrostatic equilibrium, on a time scale of the order of thermal relaxation time. It is obtained that the resulting evolution will critically depend on a parameter defined in terms of thermodynamic variables, which is constrained by causality requirements

A general procedure to find static and axially symmetric, interior solutions to the Einstein equations is presented. All the so obtained solutions, verify the energy conditions for a wide range of values of the parameters, and match smoothly to some exterior solution of the Weyl family, thereby representing globally regular models describing non spherical sources of gravitational field. In the spherically symmetric limit, all our models converge to the well known incompressible perfect fluid solution.The key stone of our approach is based on an ansatz allowing to define the interior metric in terms of the exterior metric functions evaluated at the boundary source. Some particular sources are obtained, and the physical variables of the energy-momentum tensor are calculated explicitly, as well as the geometry of the source in terms of the relativistic multipole moments. The total mass of different configurations is also calculated, it is shown to be equal to the monopole of the exterior solution.

The static solutions of the axially symmetric vacuum Einstein equations with a finite number of Relativistic Multipole Moments (RMM) are described by means of a function that can be written in the same analytic form as the Newtonian gravitational multipole potential. A family of so-called MSA (Multipole-Symmetry Adapted) coordinates are introduced to perform the transformation of the Weyl solutions; a procedure for their calculation at any multipole order is given, and the results for a low order are shown.In analogy with a previous result [10] obtained in Newtonian gravity, the existence of a symmetry of a certain system of differential equations leading to the determination of that kind of multipole solutions in General Relativity is explored. The relationship between the existence of this kind of coordinates and the symmetries mentioned is proved for some cases, and the characterization of the MSA system of coordinates by means of this relationship is discussed.

A, recently presented, general procedure to find static and axially symmetric, interior solutions to the Einstein equations, is extended to the stationary case, and applied to find an interior solution for the Kerr metric. The solution, which is generated by an anisotropic fluid, verifies the energy conditions for a wide range of values of the parameters, and matches smoothly to the Kerr solution, thereby representing a globally regular model describing a non spherical and rotating source of gravitational field. In the spherically symmetric limit, our model converges to the well known incompressible perfect fluid solution.The key stone of our approach is based on an ansatz allowing to define the interior metric in terms of the exterior metric functions evaluated at the boundary source. The physical variables of the energy-momentum tensor are calculated explicitly, as well as the geometry of the source in terms of the relativistic multipole moments.Comment: 10 pages, 7 figures and 1 table. Published in Physical Revew D. arXiv admin note: substantial text overlap with arXiv:1607.0231

A family of vector fields that are the infinitesimal generators of determined one-parameter groups of transformations are constructed. It is shown that these vector fields represent symmetries of the system of differential equations interrelated by the axially symmetric Laplace equation and a certain supplementary equation. Group-invariant solutions of this system of equations are obtained by means of two alternative methods, and it is proved that these solutions turn out to be the family of axisymmetric potentials related to specific gravitational multipoles. The existence of these symmetries provides us with a generalization of the fact that the Newtonian Monopole is defined by the solution of the Laplace equation with spherical symmetry, and it allows us to extract from all solutions of this equation those with the prescribed Newtonian Multipole Moments.

A new exact asymptotically flat solution of the Einstein equations able to describe the exterior gravitational field of a static mass possessing a quadrupole moment is presented in explicit form. In contrast with the known solutions of this type earlier obtained by Erez and Rosen(1959) and Gutsunaev and Manko(1985), this solution represents a small deformation of the Schwarzschild solution.