Positive-Energy Theorem  

In General > s.a. energy conditions.
* Idea: If Σ is an asymptotically flat non-singular spacelike hypersurface, and the dominant energy condition is satisfied, then the total energy-momentum p is a future-directed timelike or null vector; Furthermore, p = 0 iff the spacetime is flat in a neighborhood of Σ.
* Motivation: Originally the motivation for a positive-energy theorem came from the Newtonian physics observation that only positive masses are seen; A result of this type is also required by the stability of the theory; Otherwise, for example, in thermodynamics one could get a perpetual motion machine by pumping energy out of a black hole (? : Nester).
* Rem: We do not have positive-energy theorems for spacetimes with a positive cosmological constant.
> Online resources: see Wikipedia page.

References > s.a. asymptotic flatness at null infinity [positivity of the Trautman-Bondi mass]; Penrose Inequality.
@ Introductions and reviews: Dain in(14)-a1302; Zhang IJMPA(15)-a1508-proc [spacetimes with non-zero cosmological constant].
@ Conjecture: Geroch in(78); Geroch & Horowitz AP(79); Brill & Jang in(80).
@ Schoen & Yau proof: Schoen & Yau CMP(79), PRL(79), CMP(81), CMP(81); Choquet-Bruhat in(83).
@ Witten proof: Witten CMP(81); Nester PLA(81); Parker & Taubes CMP(82); Reula JMP(82); Bizoń & Malec CQG(86); Dimakis & Müller-Hoissen CQG(90).
@ Other proofs: Grisaru PLB(78); Kijowski in(86); Jezierski & Kijowski PRD(87); Nester PLA(89); Bergqvist CQG(92); Penrose et al gq/93; Nester et al CQG(94), Nester & Tung PRD(94)gq; Pelykh JMP(00) [equivalence Witten-Nester]; Chruściel & Galloway CQG(04)gq [and Lorentzian splitting theorem]; Chee et al PRD(05)gq [spinorial variables]; Sarıoğlu & Tekin a0709 [using asymptotic symmetries]; Bäckdahl & Valiente Kroon CQG(11) [invarianrt deviation from Minkowski space, approximate twistors]; Jezierski & Waluk BCP(16)-a1508 [using spacetime foliations].
@ Proofs for special cases: Jang JMP(76) [flat initial hypersurface], JMP(78); Chruściel CQG(04)gq [using null rigidity]; a1010 [Riemannian, manifolds that are graphs of smooth functions]; Grant & Tassotti a1205, a1408 [low-regularity metrics]; Chruściel & Paetz CQG(14) & CQG+ [positivity of the Trautman-Bondi mass for spacetimes containing complete smooth light cones, elementary proof]; Shyam a1408-wd [maximally sliced, asymptotically flat spacetimes].
@ Related topics: Jang & Wald JMP(77) [and cosmic censorship]; Bekenstein PRD(13)-a1310 [why is mass always positive, if vacuum energy can be negative?]; Carlotto et al IM(16)-a1503 [effective versions].

Extensions > s.a. de sitter spacetime [asymptotically de Sitter]; energy [asymptotically FLRW spacetimes]; quasilocal energy; teleparallel.
@ For black holes: Gibbons, Hawking, Horowitz & Perry CMP(83); Herzlich JGP(98); Rogatko CQG(00) [in Einstein-Maxwell-dilaton].
@ Spacetimes with horizons and couplings: Finster et al JMP(00)gq; Khuri & Weinstein JMP(13) [charged Dirac fields].
@ Asymptotically (locally) AdS spacetimes: Page et al PRL(02)ht [from boundary causality]; Cheng & Skenderis JHEP(05)ht; Luo et al NPB(10).
@ Higher dimensions: Witten NPB(82) [no, instability]; Lee & Sorkin CMP(88) [5D Kaluza-Klein theory]; Zhang JMP(99) [5D version of Witten's proof]; Ding JMP(08); Choquet-Bruhat a1107 [arbitrary space dimension]; Shiromizu & Soligon a2007 [d+1 dimensional Kaluza-Klein]; Cameron a2010 [Penrose et al proof]; Nguyen a2102 [explicit negative-mass solutions].
@ 2+1 dimensions: Ashtekar & Varadarajan PRD(94)gq [and upper bound on energy]; Menotti & Seminara AP(95); Wong a1202; Barnich & Oblak CQG(14)-a1403 [holographic].
@ Manifolds with corners: Miao ATMP(02)mp; McFeron & Székelyhidi CMP(12).
@ Lower bounds, Bogomolny-type inequalities: Gibbons & Hull PLB(82); Gibbons & Wells gq/93, Gibbons CQG(99)ht/98 [asymptotically AdS, black holes].
@ Quantum version: Thiemann CQG(98)gq/97; Smolin PRD(14)-a1406.
@ Other theories: Rogatko CQG(02) [Einstein-Maxwell-axion-dilaton]; Deser CQG(09)-a0907 [topologically massive gravity]; Garfinkle & Jacobson PRL(11)-a1108 [Einstein-aether and Hořava gravity]; Nozawa & Shiromizu NPB(14)-a1407 [extended supergravities]; > s.a. hořava gravity.
@ Conformal: Simon LMP(99)gq/00; Tam & Wang a1607 [on asymptotically hyperbolic manifolds].
@ Manifolds with boundary: Kim DG&A(04) [with inner boundary]; Bryden et al JMP(19)-a1806 [in terms of angular momentum and charge]; Hirsch & Miao PJM-a1812; Almaraz et al a1907 [initial data sets with non-compact boundary]; Galloway & Lee a2105.
@ Related topics: Chruściel CQG(86); Yip CMP(87); Zhang & Zhang CMP(00) [time symmetric]; Zhang et al PRD(05) [with angular momentum contribution]; Okikiolu CMP(09) [negative-mass theorem for surfaces of positive genus]; Lee & LeFloch a1408, Shibuya a1803 [manifolds with distributional curvature].

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