A. Di Prisco

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.

NONADIABATIC CHARGED SPHERICAL GRAVITATIONAL COLLAPSE

We present a complete set of the equations and matching conditions required for the description of physically meaningful charged, dissipative, spherically symmetric gravitational collapse with shear. Dissipation is described with both free-streaming and diffusion approximations. The effects of viscosity are also taken into account. The roles of different terms in the dynamical equation are analyzed in detail. The dynamical equation is coupled to a causal transport equation in the context of Israel-Stewart theory. The decrease of the inertial mass density of the fluid, by a factor which depends on its internal thermodynamic state, is reobtained, with the viscosity terms included. In accordance with the equivalence principle, the same decrease factor is obtained for the gravitational force term. The effect of the electric charge on the relation between the Weyl tensor and the inhomogeneity of the energy density is discussed.

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.

Shearfree cylindrical gravitational collapse

We consider diagonal cylindrically symmetric metrics, with an interior representing a general nonrotating fluid with anisotropic pressures. An exterior vacuum Einstein-Rosen spacetime is matched to this using Darmois matching conditions. We show that the matching conditions can be explicitly solved for the boundary values of metric components and their derivatives, either for the interior or exterior. Specializing to shearfree interiors, a static exterior can only be matched to a static interior, and the evolution in the nonstatic case is found to be given in general by an elliptic function of time. For a collapsing shearfree isotropic fluid, only a Robertson-Walker dust interior is possible, and we show that all such cases were included in Cockes discussion. For these metrics, Nolan and Nolan have shown that the matching breaks down before collapse is complete, and Tod and Mena have shown that the spacetime is not asymptotically flat in the sense of Berger, Chrusciel, and Moncrief. The issues about energy that then arise are revisited, and it is shown that the exterior is not in an intrinsic gravitational or superenergy radiative state at the boundary.

Dynamics of viscous dissipative gravitational collapse: A full causal approach

The Misner and Sharp approach to the study of gravitational collapse is extended to the viscous dissipative case in both the streaming out and the diffusion approximations. The dynamical equation is then coupled to causal transport equations for the heat flux, the shear, and the bulk viscosity, in the context of Israel–Stewart theory, without excluding the thermodynamics viscous/heat coupling coefficients. The result is compared with previous works where these later coefficients were neglected and viscosity variables were not assumed to satisfy causal transport equations. Prospective applications of this result to some astrophysical scenarios are discussed.

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.

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.