Positive energy theorem

In general relativity, the positive energy theorem (more commonly known as the positive mass conjecture) states that, assuming the dominant energy condition, the mass of an asymptotically flat spacetime is non-negative; furthermore, the mass is zero only for Minkowski spacetime. The theorem is a scalar curvature comparison theorem, with asymptotic boundary conditions, and a corresponding statement of geometric rigidity.

Overview

The original proof of the theorem for ADM mass was provided by Richard Schoen and Shing-Tung Yau in 1979 using variational methods. 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 an asymptotically anti-de Sitter spacetime is non-negative and only equal to zero for anti-de Sitter spacetime. In Einstein–Maxwell theory, for a spacetime with electric charge $Q$ and magnetic charge $P$, the mass of the spacetime satisfies (in Gaussian units)

$M\geq {\sqrt {Q^{2}+P^{2}}},$

with equality for the Majumdar–Papapetrou extremal black hole solutions.

Applications

• In 1984 Schoen used the positive mass theorem in his work which completed the solution of the Yamabe problem.
• The positive mass theorem was used in Hubert Bray's proof of the Riemannian Penrose inequality.

References

• Schoen, Richard; Yau, Shing-Tung (1979). "On the proof of the positive mass conjecture in general relativity". Communications in Mathematical Physics. Springer Science and Business Media LLC. 65 (1): 45–76. doi:10.1007/bf01940959. ISSN 0010-3616.
• Schoen, Richard; Yau, Shing-Tung (1981). "Proof of the positive mass theorem. II". Communications in Mathematical Physics. Springer Science and Business Media LLC. 79 (2): 231–260. doi:10.1007/bf01942062. ISSN 0010-3616.
• Witten, Edward (1981). "A new proof of the positive energy theorem". Communications in Mathematical Physics. Springer Science and Business Media LLC. 80 (3): 381–402. doi:10.1007/bf01208277. ISSN 0010-3616.
• Ludvigsen, M; Vickers, J A G (1981-10-01). "The positivity of the Bondi mass". Journal of Physics A: Mathematical and General. IOP Publishing. 14 (10): L389–L391. doi:10.1088/0305-4470/14/10/002. ISSN 0305-4470.
• Horowitz, Gary T.; Perry, Malcolm J. (1982-02-08). "Gravitational Energy Cannot Become Negative". Physical Review Letters. American Physical Society (APS). 48 (6): 371–374. doi:10.1103/physrevlett.48.371. ISSN 0031-9007.
• Schoen, Richard; Yau, Shing Tung (1982-02-08). "Proof That the Bondi Mass is Positive". Physical Review Letters. American Physical Society (APS). 48 (6): 369–371. doi:10.1103/physrevlett.48.369. ISSN 0031-9007.
• Gibbons, G. W.; Hawking, S. W.; Horowitz, G. T.; Perry, M. J. (1983). "Positive mass theorems for black holes". Communications in Mathematical Physics. 88 (3): 295-308. MR 0701918.