[Paper Review] The mass of spacelike hypersurfaces in asymptotically anti-de Sitter space-times

Piotr T. Chruściel, Gabriel Nagy|ArXiv.org|Oct 2, 2001

Black Holes and Theoretical Physics18 references5 citations

TL;DR

This paper establishes a Hamiltonian definition of mass for spacelike hypersurfaces in asymptotically anti-de Sitter (AdS) spacetimes, proving that the resulting mass is a geometric invariant independent of the choice of background metric. It demonstrates that this Hamiltonian mass coincides with the Abbott-Deser mass under standard fall-off conditions, resolving long-standing ambiguities in mass definitions in AdS gravity and providing a unique, background-independent energy functional for such spacetimes.

ABSTRACT

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.

Motivation & Objective

• . To define a background-independent mass for spacelike hypersurfaces in asymptotically AdS spacetimes.
• . To resolve ambiguities in mass definitions arising from multiple possible background metrics of the form (1.1).
• . To establish the Hamiltonian mass as a geometric invariant, independent of the choice of asymptotic background.
• . To show equivalence between the Hamiltonian mass and the Abbott-Deser mass under standard fall-off conditions.
• . To provide a rigorous foundation for mass definitions in AdS gravity, including conformal and Komar-type approaches.

Proposed method

• . Uses a Hamiltonian framework to derive a mass expression via the integral of a superpotential Uαβ over the boundary at spatial infinity.
• . Defines the superpotential Uαβ using the space-time metric g, background metric b, and a Killing vector X of b.
• . Imposes fall-off conditions on g relative to b, ensuring convergence of the mass integral.
• . Analyzes the asymptotic isometry group of the background metric b to identify the relevant Killing vectors normal to the hypersurface S.
• . Compares the Hamiltonian mass to the Abbott-Deser mass using a tetrad formalism and asymptotic expansions in r.
• . Proves equivalence between the two masses by showing their superpotentials agree up to higher-order corrections under the condition α ≥ n/2.

Experimental results

Research questions

• RQ1 . Can a geometrically invariant mass be defined for spacelike hypersurfaces in asymptotically AdS spacetimes, independent of the choice of background metric?
• RQ2 . Does the Hamiltonian mass defined via the superpotential Uαβ yield a unique, physically meaningful energy functional in AdS spacetimes?
• RQ3 . Under what conditions does the Hamiltonian mass coincide with the Abbott-Deser mass?
• RQ4 . How do different mass definitions (e.g., conformal, Komar, coordinate-based) relate to the Hamiltonian mass in AdS?
• RQ5 . Is the Hamiltonian mass well-defined and convergent under standard fall-off conditions for the metric g relative to the background b?

Key findings

• . The Hamiltonian mass defined by the integral of the superpotential Uαβ is a geometric invariant for spacelike hypersurfaces in asymptotically AdS spacetimes.
• . The mass is independent of the choice of background metric b of the form (1.1), provided the asymptotic conditions are satisfied.
• . The Hamiltonian mass coincides exactly with the Abbott-Deser mass under the condition α ≥ n/2, where g falls off as o(1/rα).
• . The superpotential Uαβ asymptotically matches the Abbott-Deser superpotential V αβ up to o(rβ−2α) terms, with equality when β − 2α ≤ 0.
• . The asymptotic isometry group of the background metric b is one-dimensional when (M, h) has non-positive Ricci curvature and constant scalar curvature, ensuring a unique normal Killing vector field.
• . The results imply that conformal mass definitions in AdS are also invariant, as they are equivalent to the Hamiltonian mass under the same conditions.