J. Ospino

SPHERICALLY SYMMETRIC DISSIPATIVE ANISOTROPIC FLUIDS: A GENERAL STUDY

The full set of equations governing the evolution of self-gravitating spherically symmetric dissipative fluids with anisotropic stresses is deployed and used to carry out a general study on the behavior of such systems, in the context of general relativity. Emphasis is given to the link between the Weyl tensor, the shear tensor, the anisotropy of the pressure, and the density inhomogeneity. In particular we provide the general, necessary, and sufficient condition for the vanishing of the spatial gradients of energy density, which in turn suggests a possible definition of a gravitational arrow of time. Some solutions are also exhibited to illustrate the discussion.

All static spherically symmetric anisotropic solutions of Einstein's equations

An algorithm recently presented by Lake to obtain all static spherically symmetric perfect fluid solutions is extended to the case of locally anisotropic fluids (principal stresses unequal). As expected, the new formalism requires the knowledge of two functions (instead of one) to generate all possible solutions. To illustrate the method some known cases are recovered.

Structure and evolution of self-gravitating objects and the orthogonal splitting of the Riemann tensor

The full set of equations governing the structure and the evolution of self-gravitating spherically symmetric dissipative fluids with anisotropic stresses is written down in terms of five scalar quantities obtained from the orthogonal splitting of the Riemann tensor, in the context of general relativity. It is shown that these scalars are directly related to fundamental properties of the fluid distribution, such as energy density, energy density inhomogeneity, local anisotropy of pressure, dissipative flux, and the active gravitational mass. It is also shown that in the static case, all possible solutions to Einstein equations may be expressed explicitly through these scalars. Some solutions are exhibited to illustrate this point.

Some analytical models of radiating collapsing spheres

We present some analytical solutions to the Einstein equations, describing radiating collapsing spheres in the diffusion approximation. Solutions allow for modeling physical reasonable situations. The temperature is calculated for each solution, using a hyperbolic transport equation, which permits to exhibit the influence of relaxational effects on the dynamics of the system.

ON THE STABILITY OF THE SHEAR-FREE CONDITION

The evolution equation for the shear is reobtained for a spherically symmetric anisotropic, viscous dissipative fluid distribution, which allows us to investigate conditions for the stability of the shear–free condition. The specific case of geodesic fluids is considered in detail, showing that the shear–free condition, in this particular case, may be unstable, the departure from the shear–free condition being controlled by the expansion scalar and a single scalar function defined in terms of the anisotropy of the pressure, the shear viscosity and the Weyl tensor or, alternatively, in terms of the anisotropy of the pressure, the dissipative variables and the energy density inhomogeneity.

Conformally flat anisotropic spheres in general relativity

The condition for the vanishing of the Weyl tensor is integrated in the spherically symmetric case. Then, the resulting expression is used to find new, conformally flat, interior solutions to Einstein equations for locally anisotropic fluids. The slow evolution of these models is contrasted with the evolution of models with similar energy density or radial pressure distribution but nonvanishing Weyl tensor, thereby bringing out the different role played by the Weyl tensor, the local anisotropy of pressure, and the inhomogeneity of the energy density in the collapse of relativistic spheres.

Anisotropic geodesic fluid spheres in general relativity

It is shown that unlike the perfect fluid case, anisotropic fluids (principal stresses unequal) may be geodesic, without this implying the vanishing of (spatial) pressure gradients. Then the condition of vanishing four acceleration is integrated in noncomoving coordinates. The resulting models are necessarily dynamic, and the mass function is expressed in terms of the fluid velocity as measured by a locally Minkowskian observer. An explicit example is worked out.

Dissipative collapse of axially symmetric, general relativistic, sources: A general framework and some applications

We carry out a general study on the collapse of axially (and reflection) symmetric sources in the context of general relativity. All basic equations and concepts required to perform such a general study are deployed. These equations are written down for a general anisotropic dissipative fluid. The proposed approach allows for analytical studies as well as for numerical applications. A causal transport equation derived from the Israel-Stewart theory is applied, to discuss some thermodynamic aspects of the problem. A set of scalar functions (the structure scalars) derived from the orthogonal splitting of the Riemann tensor are calculated and their role in the dynamics of the source is clearly exhibited. The characterization of the gravitational radiation emitted by the source is discussed.

EXPANSION-FREE CAVITY EVOLUTION: SOME EXACT ANALYTICAL MODELS

We consider spherically symmetric distributions of anisotropic fluids with a central vacuum cavity, evolving under the condition of vanishing expansion scalar. Some analytical solutions are found satisfying Darmois junction conditions on both delimiting boundary surfaces, while some others require the presence of thin shells on either (or both) boundary surfaces. The solutions here obtained model the evolution of the vacuum cavity and the surrounding fluid distribution, emerging after a central explosion, thereby showing the potential of expansion–free condition for the study of that kind of problems. This study complements a previously published work where modeling of the evolution of such kind of systems was achieved through a different kinematical condition.

Lemaitre-Tolman-Bondi dust spacetimes: Symmetry properties and some extensions to the dissipative case

We consider extensions of Lemaitre-Tolman-Bondi (LTB) spacetimes to the dissipative case. We have previously carried out a systematic study on LTB. This study is based on two different aspects of LTB. On the one hand, a symmetry property of LTB will be presented. On the other hand, the description of LTB in terms of some fundamental scalar functions (structure scalars) appearing in the orthogonal splitting of Riemann tensor will be provided. We shall consider as natural generalizations of LTB (hereafter referred to as GLTB) either those metrics admitting some similar kind of symmetry as LTB, or those sharing structure scalars with similar dependence on the metric.