On the positivity of the Bondi mass

ANNALS

OF PHYSICS 151, 498-499

Abstracts

(1983)

of Papers to Appear

Non-equilibrium Statistical Mechanics in the General Theory Approximation. HENRY E. KANDRUP. Department of Barbara, California 93 106.

in Future

Issues

of Relativity. II. Linear Fields in a Kinetic Physics, University of California, Santa

The first paper in this series developed a new covariant approach to non-equilibrium statistical mechanics in general relativity. The object of this second paper is to apply that formalism to the consideration of the evolution of a collection of particles interacting via linear fields in a fixed curved background. The fundamental objects of the theory are then a many-particle distribution function and a many-particle conservation equation which this distribution satisfies. If one views a composite N-particle system as interacting one- and (N - I)-particle subsystems, one can derive exact coupled equations for appropriately defined reduced one- and (N - I)-particle distribution functions. The implementation of plausible assumptions, which are straightforward generalizations of standard non-relativistic “kinetic approximations,” permits the formulation of an approximate kinetic equation for the one-particle distribution function. In the non-relativistic limit, this equation reduces to a well-known expression. The explicit form of this equation is derived for three concrete examples: interactions via a Maxwell field. via a massive scalar field, and via a symmetric second rank tensor field. For a large class of interactions, of which these three examples are representative, the kinetic equation will admit a relativistic Maxwellian distribution as an exact stationary solution; and, for these interactions, an H-theorem may be derived.

On the Positivity of the Bondi Mass. California, Berkeley, California.

SIAN M. STUMBLES.

Department

of Mathematics,

University

of

A new way of showing local positivity of the Bondi mass is presented. The method used makes use only of the constraint equations. By looking at the constraint equations of general relativity on an asymptotically constant mean extrinsic curvature hypersurface. a mass aspect is picked out, whose average over the 2.sphere coincides with the Bondi mass. This mass aspect differs from Bondi’s mass aspect by terms linear in the news function. It is shown that the mass aspect so produced may be considered as a functional of the dependent data in the problem. This functional is shown to have Minkowski space as a critical point. The second variational of the functional about flat space is then shown to be positive. From this, one concludes that the functional is positive in a neighborhood of flat space. Hence the Bondi mass, too, is positive in a neighborhood of flat space.

Hadamard Renormalization in Curved Space-Time. MARIO A. CASTAGNINO. lnstituto de Fisica de Rosario. Av. Pellegrini 250, 2000 Rosario-Argentina and Instituto de Astlonomia y Fisica del Espacio, Casilla de Correo 67.Sue. 28, 1428 Buenos Aires-Argentina: ANI) DIEGO D. HARARI. Instituto de Astronomia y Fisica del Espacio, Casilla de Correo 67.Sucursal 28. 1428 Buenos AiresArgentina. The family of all Hadamard symmetric kernels for a quantum massive scalar field propagating at a classical curved background, which can be constructed in terms of purely local geometric quantities is derived. No such kernel exists in the massless limit, unless “restricted geometric objects,” which only have a meaning at conformally flat metrics, are used in its construction. The relation of this result with

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