Isometric Embeddings into the Minkowski Space and New Quasi-Local Mass

2008/05/31 by Mu-Tao Wang, Mu‐Tao Wang, Shing-Tung Yau +1 · 155 citations
Mathematics · Medicine · Physics and Astronomy · #Black Holes and Theoretical Physics #Computer science #Cosmology and Gravitation Theories #Geometry #Internal medicine #Isometric exercise #Mathematical analysis #Mathematics #Medicine #Minkowski space #Noncommutative and Quantum Gravity Theories #Physics #Pure mathematics #Space (punctuation) #gr-qc #math-ph #math.DG #math.MP

paper · pdf · doi:10.1007/s00220-009-0745-0

published in Communications in Mathematical Physics 288(3), 919-942 (Springer Science+Business Media) · 33 pages, final version. to appear in Comm. Math. Phys

arxiv created 2008/11/11 · openalex publication_date 2009/02/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The definition of quasi-local mass for a bounded space-like region in space-time is essential in several major unsettled problems in general relativity. The quasi-local mass is expected to be a type of flux integral on the boundary two-surface and should be independent of whichever space-like region it bounds. An important idea which is related to the Hamiltonian formulation of general relativity is to consider a reference surface in a flat ambient space with the same first fundamental form and derive the quasi-local mass from the difference of the extrinsic geometries. This approach has been taken by Brown-York and Liu-Yau (see also related works) to define such notions using the isometric embedding theorem into the Euclidean three-space. However, there exist surfaces in the Minkowski space whose quasilocal mass is strictly positive. It appears that the momentum information needs to be accounted for to reconcile the difference. In order to fully capture this information, we use isometric embeddings into the Minkowski space as references. In this article, we first prove an existence and uniqueness theorem for such isometric embeddings. We then solve the boundary value problem for Jang's equation as a procedure to recognize such a surface in the Minkowski space. In doing so, we discover new expression of quasi-local mass. The new mass is positive when the ambient space-time satisfies the dominant energy condition and vanishes on surfaces in the Minkowski space. It also has the nice asymptotic behavior at spatial and null infinity. Some of these results were announced in [29].

Citations

• On the proof of the positive mass conjecture in general relativity
• A new proof of the positive energy theorem

Cited by

• Horizons in noncompact fill-ins of nonnegative scalar curvature
• Small sphere limit of the quasi-local energy with anti de-Sitter space reference
• A note on the positivity of a quasi-local mass in general dimensions
• On energy and its positivity in spacetimes with an expanding flat de Sitter background
• The dihedral rigidity conjecture for n-prisms
• Total mass and limits of quasi-local mass at future null infinity
• A quasi-local Penrose inequality for the quasi-local energy with static references
• On the energy of a non-singular black hole solution satisfying the weak energy condition
• Quasi-local energy from a Minkowski reference
• Variation and rigidity of quasi-local mass
• The Positive Mass Theorem for Creased Initial Data
• Birkhoff's invariant and Thorne's Hoop Conjecture
• Quasi-local mass at the null infinity of the Vaidya spacetime
• Evaluating small sphere limit of the Wang-Yau quasi-local energy
• Boundary Effect of Ricci Curvature
• Rigidity and minimizing properties of quasi-local mass
• Four lectures on quasi-local mass
• Quasi-local mass on unit spheres at spatial infinity
• Conserved Quantities of harmonic asymptotic initial data sets
• On Weyl's embedding problem in Riemannian manifolds
• A localized spacetime Penrose inequality and horizon detection with quasi-local mass
• Quasilocal Energy in Kerr Spacetime
• Mass of asymptotically flat 3-manifolds with boundary
• On a spacetime positive mass theorem with corners
• Quasi-local energy with respect to de Sitter/anti-de Sitter reference
• Quasi-local mass at null infinity in Bondi-Sachs coordinates
• Classics illustrated: limits of spacetimes
• Limit of Quasilocal Mass at Spatial Infinity
• Quasi-local energy and the choice of reference
• A Gibbons–Penrose Inequality for Surfaces in Schwarzschild Spacetime
• Embeddings, Immersions and the Bartnik Quasi-Local Mass Conjectures
• A rigidity theorem for surfaces in Schwarzschild manifold
• The Jang equation and the positive mass theorem in the asymptotically hyperbolic setting
• Modifications of Gravity Via Differential Transformations of Field Variables
• Spinorial quasilocal mass for spacetimes with negative cosmological constant
• Quasi-Local Energy-Momentum and Angular Momentum in General Relativity. [europepmc]

Related

• A Quasi-Local Mass
• Quasi-local mass and isometric embedding with reference to a static spacetime
• A generalization of Liu-Yau's quasi-local mass
• Rigidity and minimizing properties of quasi-local mass
• Quasi-local masses in General relativity and their positivity: Spinor approach