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.
The original proof of the theorem for ADM mass was provided by Richard Schoen and Shing-Tung Yau in 1979 using variational methods and minimal surfaces.
Edward Witten gave another proof in 1981 based on the use of spinors, inspired by positive energy theorems in the context of supergravity.
An extension of the theorem for the Bondi mass was given by Ludvigsen and James Vickers, Gary Horowitz and Malcolm Perry, and Schoen and Yau.
Gary Gibbons, Stephen Hawking, Horowitz and Perry proved extensions of the theorem to asymptotically anti-de Sitter spacetimes and to Einstein–Maxwell theory.
, the mass of the spacetime satisfies (in Gaussian units) with equality for the Majumdar–Papapetrou extremal black hole solutions.
One says that a Lorentzian manifold (M, g) is a development of an initial data set (M, g, k) if there is a (necessarily spacelike) hypersurface embedding of M into M, together with a continuous unit normal vector field, such that the induced metric is g and the second fundamental form with respect to the given unit normal is k. This definition is motivated from Lorentzian geometry.
Given a Lorentzian manifold (M, g) of dimension n + 1 and a spacelike immersion f from a connected n-dimensional manifold M into M which has a trivial normal bundle, one may consider the induced Riemannian metric g = f *g as well as the second fundamental form k of f with respect to either of the two choices of continuous unit normal vector field along f. The triple (M, g, k) is an initial data set.
According to the Gauss-Codazzi equations, one has where G denotes the Einstein tensor Ricg - 1/2Rgg of g and ν denotes the continuous unit normal vector field along f used to define k. So the dominant energy condition as given above is, in this Lorentzian context, identical to the assertion that G(ν, ⋅), when viewed as a vector field along f, is timelike or null and is oriented in the same direction as ν.
One considers an initial data set (M, g, k) which may or may not have a boundary; let n denote its dimension.
Such connected components are called the ends of M. Let (M, g, 0) be a time-symmetric initial data set satisfying the dominant energy condition.
Suppose that (M, g) is an oriented three-dimensional smooth Riemannian manifold-with-boundary, and that each boundary component has positive mean curvature.
then m must be positive unless the boundary of M is empty and (M, g) is isometric to ℝ3 with its standard Riemannian metric.
This can be interpreted in a purely mathematical sense as a strong form of "asymptotically flat", where the coefficient of the |x|−1 part of the expansion of the metric is declared to be a constant multiple of the Euclidean metric, as opposed to a general symmetric 2-tensor.
Note also that Schoen and Yau's theorem, as stated above, is actually (despite appearances) a strong form of the "multiple ends" case.
Let (M, g, k) be an initial data set satisfying the dominant energy condition.
implies that n = 1, that M is diffeomorphic to ℝ3, and that Minkowski space ℝ3,1 is a development of the initial data set (M, g, k).
be an oriented three-dimensional smooth complete Riemannian manifold (without boundary).
and the maximal globally hyperbolic development of the initial data set
However, a third paper by Schoen and Yau[4] shows that their 1981 result implies Witten's, retaining only the extra assumption that
[5] Schoen and Yau's 1979 result and proof can be extended to the case of any dimension less than eight.
[6] More recently, Witten's result, using Schoen and Yau (1981)'s methods, has been extended to the same context.
[7] In summary: following Schoen and Yau's methods, the positive energy theorem has been proven in dimension less than eight, while following Witten, it has been proven in any dimension but with a restriction to the setting of spin manifolds.