Positive energy theorem

The positive energy theorem states that an asymptotically flat spacetime satisfying suitable energy conditions has nonnegative total energy, with zero energy only for flat spacetime.

ADM mass: The total mass-energy of an asymptotically flat spacetime, defined from its behavior at spatial infinity. The theorem’s nonnegative total energy is commonly expressed as a bound on this mass.
General relativity: Einstein’s theory of gravity, in which spacetime geometry is dynamical and shaped by matter and energy. The theorem constrains the total energy allowed by this theory’s field equations.
Richard Schoen: An American mathematician known for work in differential geometry and geometric analysis. His collaboration with Yau produced the first general proof of the positive mass theorem.
Positive mass theorem for initial data sets: A version of the positive mass theorem for spacelike initial data, including nonzero momentum. It extends the energy bound from time-symmetric slices to general initial data.
Positive mass theorem in dimensions above seven: The study of mass positivity in high dimensions, where classical minimal-hypersurface arguments encounter singularities. It highlights a dimension limit of the original geometric proof techniques.
Dominant energy condition: An energy condition requiring matter’s energy density to be nonnegative and its energy flux to be causal. It is a standard hypothesis ensuring physically reasonable matter in the theorem.
Einstein field equations: Equations relating spacetime curvature to the distribution of matter and energy. They connect the matter conditions assumed by the theorem to spacetime geometry.
Shing-Tung Yau: A Chinese-American mathematician whose work includes major results in differential geometry. His work with Schoen established the theorem through minimal-surface methods.
Mass–energy inequality for charged black holes: A lower bound relating total mass and electric charge in asymptotically flat Einstein–Maxwell data. Charged versions refine the theorem by bounding mass in terms of conserved charge.
Singular minimal hypersurfaces: Area-minimizing hypersurfaces that develop singular points in sufficiently high dimensions. Their singularities complicate extensions of Schoen and Yau’s proof to higher dimensions.

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