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| In [[general relativity]], the '''positive energy theorem''' (more commonly known as the '''positive mass conjecture''' in [[differential geometry]]) states that, assuming the [[energy conditions#Mathematical statement|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 (mathematics)|rigidity]].
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| 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 [[spinor]]s, inspired by positive energy theorems in the context of [[supergravity]]. An extension of the theorem for the [[Bondi mass]] was given by [[Malcolm Ludvigsen|Ludvigsen]] and James Vickers, Gary Horowitz and [[Malcolm Perry (physicist)|Malcolm Perry]], and Schoen and Yau.
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| [[Gary Gibbons]], [[Stephen Hawking]], Horowitz and Perry proved extensions of the theorem to asymptotically [[anti-de Sitter spacetime]]s and to [[Einstein field equations#Einstein–Maxwell equations|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 <math>Q</math> and magnetic charge <math>P</math>, the mass of the spacetime satisfies
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| :<math>M \geq \sqrt{Q^2 + P^2},</math>
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| with equality for the Majumdar–Papapetrou [[extremal black hole]] solutions.
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| == See also ==
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| * In 1984 Schoen used the positive mass theorem in his work which completed the solution of the [[Yamabe problem]].
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| * The positive mass theorem was used in [[Hubert Bray]]'s proof of the [[Riemannian Penrose inequality]].
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| == References ==
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| * R. Schoen and S.-T. Yau, "On the proof of the positive mass conjecture in general relativity", ''Commun. Math. Phys.'' '''65''', 45 (1979).
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| * R. Schoen and S.-T. Yau, "Proof of the positive mass theorem. II", ''Commun. Math. Phys.'' '''79''', 231 (1981).
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| * E. Witten, "A new proof of the positive energy theorem", ''Commun. Math. Phys.'' '''80''', 381 (1981).
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| * [http://www.iop.org/EJ/abstract/0305-4470/14/10/002/ M. Ludvigsen and J. A. G. Vickers, "The positivity of the Bondi mass", ''J. Phys. A'' '''14''', L389 (1981).]
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| * [http://prola.aps.org/abstract/PRL/v48/i6/p371_1 G. T. Horowitz and M. J. Perry, "Gravitational mass cannot become negative", ''Phys. Rev. Lett.'' '''48''', 371 (1982).]
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| * [http://prola.aps.org/abstract/PRL/v48/i6/p369_1 R. Schoen and S. T. Yau, "Proof that the Bondi mass is positive", ''Phys. Rev. Lett.'' '''48''', 369 (1982).]
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| * [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103922377 G. W. Gibbons, S. W. Hawking, G. T. Horowitz and M. J. Perry, "Positive mass theorems for black holes", ''Commun. Math. Phys.'' '''88''', 295 (1983).]
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| {{relativity-stub}}
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| [[Category:Mathematical methods in general relativity]]
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| [[Category:Theorems in general relativity]]
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