The Hamiltonian mass and asymptotically anti-de Sitter space-times

Author: Piotr T. Chrusciel
Comments: 6 pages, Proceedings of the symposium "100 years Werner Heisenberg", Bamberg, Septembre 2001, Fortschritte der Physik 50, 624-629 (2002)

This is a review, for the Bamberg Proceedings, of the results proved in the papers with Nagy and Herzlich, to be found below.

Paper: dvi (29kb), PostScript (156kb), tex (24kb), or pdf (156kb)

The mass of asymptotically hyperbolic Riemannian manifolds

The mass of asymptotically hyperbolic Riemannian manifolds

Authors: Piotr T. Chrusciel, Marc Herzlich
Comments: 27 pages, Tours preprint 13/2001, to appear in Pacific Jour. of Mathematics []

We present a set of global invariants, called ``mass integrals", which can be defined for a large class of asymptotically hyperbolic Riemannian manifolds. When the ``boundary at infinity" has spherical topology one single invariant is obtained, called the mass; we show positivity thereof. We apply the definition to conformally compactifiable manifolds, and show that the mass is completion-independent. We also prove a result, closely related to the problem at hand, that conformal completions of conformally compactifiable manifolds are unique.

Paper: dvi (126kb), PostScript (376kb), or pdf (305kb)

The mass of spacelike hypersurfaces in asymptotically anti-de Sitter space-times

The mass of spacelike hypersurfaces in asymptotically anti-de Sitter space-times

Authors: Piotr T. Chrusciel, Gabriel Nagy
Comments: 48 pages, Tours preprint 14/2001, Advances in Theoretical and Mathematical Physics 19, Number 4, July 2001 (49 pages) []

We give a Hamiltonian definition of mass for spacelike hypersurfaces in space-times with metrics which are asymptotic to the anti-de Sitter one, or to a class of generalizations thereof. We show that our definition provides a geometric invariant for a spacelike hypersurface embedded in a space-time. Some further global invariants are also given.

Paper: dvi (226kb), PostScript (595kb), or pdf (499kb)

The links above point to the latest version of the paper. A letter, that appeared in Classical Quantum Gravity 18, L61-L68 (2001) describing the results proved is also available [html, dvi (31kb), PostScript (151kb), or pdf (220kb)]. An earlier, slightly different, version of the letter can be found at hep-th/0011270. The differences are mainly in the presentation of the results. In the current version of the letter here we put more emphasis on the fact that the integrals we consider are Hamiltonians, and therefore defined uniquely up to a constant for each choice of background.

The Trautman-Bondi mass of initial data sets

Authors: P.T. Chrusciel, J. Jezierski, S. Leski
Comments: latex2e, 51 pages in A4, submitted for publication to Adv. Theor. Math. Phys., [ESI preprint 1354], []
We give a definition of mass for conformally compactifiable initial data sets. The asymptotic conditions are compatible with existence of gravitational radiation, and the compactifications are allowed to be polyhomogeneous. We show that the resulting mass is a geometric invariant, and we prove positivity thereof in the case of a spherical conformal infinity. When R(g) - or, equivalently, the trace of the extrinsic curvature tensor - tends to a negative constant to order one at infinity, the definition is expressed purely in terms of three-dimensional or two-dimensional objects.

Full-text: dvi (240kB), pdf (419kB), or ps (748kB)

You can find here my results on the analogous problem in the asymptotically flat case.

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