# 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 [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.

### 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) [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.

### 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],
[http://arxiv.org/abs/gr-qc/0307109]

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.
*

*return to my home page*