Local anisotropy in self-gravitating systems

The present status of the quasi-local mass-energy-momentum and angular momentum constructions in general relativity is reviewed. First the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasi-local quantities are recalled. Then the various specific constructions and their properties (both successes and defects) are discussed. Finally, some of the (actual and potential) applications of the quasi-local concepts and specific constructions are briefly mentioned. This review is based on the talks given at the Erwin Schrodinger Institute, Vienna, in July 1997, at the Universitat Tubingen, in May 1998, and at the National Center for Theoretical Sciences in Hsinchu and at the National Central University, Chungli, Taiwan, in July 2000.

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Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article

The present status of the quasi-local mass, energy-momentum and angular-momentum constructions in general relativity is reviewed. First, the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasi-local quantities, are recalled. Then, the various specific constructions and their properties (both successes and deficiencies are discussed. Finally, some of the (actual and potential) applications of the quasi-local concepts and specific constructions are briefly mentioned.

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Quasi-Local Energy-Momentum and Angular Momentum in General Relativity.

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.

Nonadiabatic charged spherical gravitational collapse

Attempts to estimate the influence of global cosmological expansion on local systems are reviewed. Here ``local'' is taken to mean that the sizes of the considered systems are much smaller than cosmologically relevant scales. For example, such influences can affect orbital motions as well as configurations of compact objects, like black holes. Also discussed are how measurements based on the exchange of electromagnetic signals of distances, velocities, etc. of moving objects are influenced. As an application, orders of magnitude of such effects are compared with the scale set by the apparently anomalous acceleration of the Pioneer 10 and 11 spacecrafts, which is ${10}^{\ensuremath{-}9}\phantom{\rule{0.3em}{0ex}}\mathrm{m}∕{\mathrm{s}}^{2}$. There is no reason to believe that the latter is of cosmological origin. However, the general problem of gaining a qualitative and quantitative understanding of how the cosmological dynamics influences local systems remains challenging, with only partial clues being so far provided by exact solutions to the field equations of general relativity.

Influence of global cosmological expansion on local dynamics and kinematics

The mass‐energy of spherically symmetric distributions of material is studied according to general relativity. An arbitrary orthogonal coordinate system is used whenever invariant properties are discussed. The Bianchi identity is shown to imply that the Misner‐Sharp‐Hernandez mass function is an integral of two combinations of Einstein's equations for any energy‐momentum tensor and that mass‐energy flow is conservative. The two mass equations thus found and the mass function provide a technique for casting Einstein's field equations into alternative forms. This mass‐function technique is applied to the general problem of the motion of a perfect fluid and especially to the examination of negative‐mass shells and their relation to singular behavior. The technique is then specialized to the study of a known class of solutions of Einstein's equations for a perfect fluid and to a brief treatment of uniform model universes and the charged point‐mass solution.

Spherical Symmetry and Mass‐Energy in General Relativity. I. General Theory

A solution of Einstein's field equations for the motion of a spherically symmetric distribution of perfect fluid is investigated in an isotropic comoving coordinate system. It is shown that this solution includes all solutions discussed by McVittie in which the density depends on both the radial and time coordinates. Necessary and sufficient conditions for the solution to be singularity‐free and for the density and pressure to be nonnegative and monotonically decreasing from the center of the material outwards to an outer boundary are found. The material is surrounded by empty space. Examples of both oscillating and ``bouncing'' solutions are produced. It is shown that the outer boundary of the material never penetrates the Schwarzschild radius in all singularity‐free solutions.

G. C. McVittie

Papers

Spherical Symmetry and Mass‐Energy in General Relativity. I. General Theory

Spherical Symmetry and Mass‐Energy in General Relativity. II. Particular Cases