Bartnik mass

Bartnik mass
Name	Bartnik mass
Field	Mathematical relativity
Introduced	1989
Introduced by	Robert Bartnik
Related	ADM mass, Bondi mass, Hawking mass, Brown–York mass

Bartnik mass The Bartnik mass is a quasi-local mass proposal in mathematical relativity introduced to quantify the gravitational mass of a bounded region in an asymptotically flat spacetime slice; it was defined by Robert Bartnik to address shortcomings of global invariants like the ADM mass and radiative notions such as the Bondi mass. Developed in the context of results related to the Positive energy theorem, the Bartnik construction links ideas from the Riemannian Penrose inequality, boundary-value problems for the Einstein equations, and geometric analysis techniques used in work by researchers connected to Richard Schoen, Shing-Tung Yau, and Edward Witten.

Definition and motivation

Bartnik proposed a definition motivated by efforts to give a geometric, coordinate-independent measure of mass for a compact body in a time-symmetric slice of an asymptotically flat Cauchy surface of an isolated system, inspired by problems studied in the aftermath of the Schwarzschild solution and the exploration of mass in the context of the ADM formalism of Arnowitt–Deser–Misner. The idea was to take a bounded region with prescribed boundary data and consider extensions that are complete, asymptotically flat, and satisfy the nonnegative scalar curvature condition central to the Positive energy theorem proven by Schoen and Yau and by Witten. The Bartnik mass arises as an infimum of ADM masses over such admissible extensions, echoing variational themes found in works by Gerhard Huisken, Tom Ilmanen, and contributors to the Riemannian Penrose inequality.

Mathematical formulation

In the original formulation one fixes a compact Riemannian 3-manifold with boundary and prescribed metric and mean curvature data motivated by the Israel–Wilson–Perjés and boundary matching conditions used in junction problems related to the Israel junction conditions. One considers the set of complete, asymptotically flat Riemannian 3-manifolds without horizons that contain an isometric embedding of the given boundary data and satisfy scalar curvature R >= 0 (as in contexts treated by Schoen, Yau, and Bartnik himself). The Bartnik mass m_B is then defined as the infimum of the ADM mass over this admissible class, paralleling minimization principles in geometric analysis familiar from the work of Michael Eichmair, Hugh Bray, and Christodoulou.

Properties and conjectures

The Bartnik mass satisfies monotonicity expectations and comparison properties conjectured in analog with the Hawking mass and the Brown–York mass, and it is expected to coincide with the ADM mass for regions that are themselves asymptotically flat. Key conjectures connect minimizers of the Bartnik variational problem to static vacuum extensions and to uniqueness results akin to those for the static vacuum Einstein equations investigated by Lichnerowicz, Bunting, and Masood-ul-Alam. Another central conjecture asserts that minimizers, if they exist, are realized by static, regular, horizon-free extensions that generalize the Schwarzschild metric exterior; this is related to rigidity statements in the Riemannian Penrose inequality proven in special cases by Bray and by Huisken and Ilmanen. Existence, regularity, and uniqueness of minimizers remain open in general, with partial results by Corvino, Miao, Anderson, and Smith.

Computation and examples

Explicit computation of the Bartnik mass is notoriously difficult. In highly symmetric cases one expects agreement with the mass parameter of the Schwarzschild solution and with quasi-local masses computed by methods of Brown and York or by flux integrals used in the ADM formalism. For minimal surface boundary data corresponding to time-symmetric black hole horizons, the Bartnik mass ties into the Riemannian Penrose inequality where explicit bounds by horizon area are available from techniques employed by Bray and Huisken–Ilmanen. Numerical and constructive examples have been investigated using gluing techniques pioneered by Corvino and Isenberg and conformal methods developed by Cantor and Adam Maxwell; these studies connect to computational work in mathematical relativity by groups at institutions such as Princeton University, Stanford University, and the Perimeter Institute.

Relation to other mass concepts in general relativity

The Bartnik mass is one member of a family of quasi-local mass proposals including the Hawking mass, Brown–York mass, Kijowski mass, Liu–Yau mass, and the Penrose quasi-local mass program; each addresses different desiderata motivated by studies of the Schwarzschild, Kerr, and radiating spacetimes like the Vaidya metric. While the ADM mass captures total energy at spatial infinity and the Bondi mass captures radiated energy at null infinity (as studied in work by Sachs and Bondi), the Bartnik mass attempts a geometric local-to-global linkage that interacts with existence theorems such as the Positive mass theorem and rigidity phenomena explored in the literature by Jang, Wald, and Penrose.

Applications and significance

Bartnik mass influences theoretical investigations into the localization of gravitational energy, the study of quasi-local inequalities like the Penrose inequality, and the analysis of boundary value problems for the Einstein constraint equations treated by York and Lichnerowicz. It serves as a conceptual bridge between geometric analysis, mathematical relativity, and numerical relativity programs pursued at centers such as Caltech and Cambridge University; progress on Bartnik-type problems informs uniqueness theorems, gluing constructions, and the rigorous understanding of isolated gravitating systems central to the research portfolios of scholars associated with Courant Institute, IAS, and national research funding bodies like the NSF.

Category:Mathematical relativity