Author: Piotr T.
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.
Authors: Piotr T.
Chrusciel, Marc Herzlich
Comments: 27 pages, Tours preprint 13/2001, to appear in Pacific Jour. of Mathematics [http://arxiv.org/abs/math.DG/0110035]
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.
Authors: Piotr T. Chrusciel,
Comments: 48 pages, Tours preprint 14/2001, Advances in Theoretical and Mathematical Physics 19, Number 4, July 2001 (49 pages) [http://arxiv.org/abs/gr-qc/0110014]
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.
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.