Positive energy theorem

General relativity
${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$
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The positive energy theorem (also known as the positive mass theorem) refers to a collection of foundational results in general relativity and differential geometry. Its standard form, broadly speaking, asserts that the gravitational energy of an isolated system is nonnegative, and can only be zero when the system has no gravitating objects. Although these statements are often thought of as being primarily physical in nature, they can be formalized as mathematical theorems which can be proven using techniques of differential geometry, partial differential equations, and geometric measure theory.

Richard Schoen and Shing-Tung Yau, in 1979 and 1981, were the first to give proofs of the positive mass theorem. Edward Witten, in 1982, gave the outlines of an alternative proof, which were later filled in rigorously by mathematicians. Witten and Yau were awarded the Fields medal in mathematics in part for their work on this topic.

An imprecise formulation of the Schoen-Yau / Witten positive energy theorem states the following:

Given an asymptotically flat initial data set, one can define the energy-momentum of each infinite region as an element of Minkowski space. Provided that the initial data set is geodesically complete and satisfies the dominant energy condition, each such element must be in the causal future of the origin. If any infinite region has null energy-momentum, then the initial data set is trivial in the sense that it can be geometrically embedded in Minkowski space.

The meaning of these terms is discussed below. There are alternative and non-equivalent formulations for different notions of energy-momentum and for different classes of initial data sets. Not all of these formulations have been rigorously proven, and it is currently an open problem whether the above formulation holds for initial data sets of arbitrary dimension.