# Positive energy theorem (aka positive mass theorem) A positive energy theorem states that the energy of a certain gravitational theory is bounded from below and takes the minimum value only for a unique vacuum. This is interesting because it is obviously untrue in Newtonian physics: gravity is attractive, so the potential energy can be made arbitrarily negative if you put the particles closer and closer together. On the other hand, it is obviously true for non-gravitational theories: as long as the time-time component of the stress tensor is non-negative, the total energy, which is just the integral of it, is immediate non-negative. In General Relativity, even for $3+1$ dimensional asymptotically Minkowski spacetimes, it is a difficult task to show that the energy is bounded from below. In $2+1$ dimensions, it is easy to show that the positive energy theorem follows from the Gauss-Bonnet theorem. The original proof by Schoen and Yau uses it to prove the positive energy theorem in $3+1$ dimensions by showing that the $2+1$ dimensional one would be violated if the $3+1$ dimensional one is false. This generalises to higher dimensional asymptotically Minkowski spacetimes, $M^{4+n}$ to at least 7 dimensions. But if the higher dimensional spacetime is asymptotically $M^4\times B_n$ where $B_n$ is some $n$-dimensional compact space, then it might be true or false, depending on what $B_n$ is. Famously, the positive energy theorem is false when $B_n$ is $S^1$, as in [[0169 Kaluza-Klein|Kaluza-Klein theory]]. It is also possible to formulate a positive energy theorem for the Bondi mass. The Bondi mass $m(u)$ is defined at some retarded time $u$ and does not include the energy of the radiation that escaped to null infinity before time $u$. Positivity of the Bondi mass at $u\to\infty$ would imply that the energy lost in the radiation after time $u$ cannot be more than $m(u)$. ## Proofs in flat space - using minimal surface: - [[SchoenYau1979]] (letter) and [[SchoenYau1981]] (full proof) - extension to all dimensions: [[SchoenYau2017]][](https://arxiv.org/abs/1704.05490) - supersymmetry-inspired proof using spinorial techniques: [[1981#Witten]] - works for all manifolds admitting a spin structure - using the inverse mean curvature flow: [[Geroch1973]] and [[HuiskenIlmanen1997]] - focusing of null geodesics near conformal infinity: [[PenroseSorkinWoolgar1993]] ## Proofs in AdS - spinorial proof for asymptotically AdS spacetimes without a horizon: [[AbbottDeser1982]] - extends spinorial proof to AdS with a black hole (in 4D): [[1983#Gibbons, Hawking, Horowitz, Perry]] - proof based on null focusing techniques: [[1994#Woolgar]] - using holography: [[2002#Page, Surya, Woolgar]] ## Positive energy theorems for the Bondi mass - [[LudvigsenVickers1981]][](https://iopscience.iop.org/article/10.1088/0305-4470/14/10/002) - [[HorowitzPerry1982]][](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.48.371) and [[SchoenYau1982]][](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.48.369) (simultaneous) ## Other theories - [[0332 Supergravity|SUGRA]]: - an argument given in [[DeserTeitelboim1977]][](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.39.249) - a comparison of this argument with the spinor method given in [[Hull1983]][](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-90/issue-4/The-positivity-of-gravitational-energy-and-global-supersymmetry/cmp/1103940417.full) - with EM charge: - [[1983#Gibbons, Hawking, Horowitz, Perry]] - [[0006 Higher-derivative gravity|HDG]] - $R+R^2$: [[Strominger1984]][](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.30.2257) - Einstein-Aether and Horava: [[GarfinkleJacobson2011]][](https://arxiv.org/abs/1108.1835) ## Related - [[0476 Penrose inequality]] - [[0168 Bubble of nothing]] ## Refs - a short and quite readable introduction by [Witten](https://www.ias.edu/sites/default/files/sns/%5B50%5DParticles_and_Fields2_Proc_1981_BanffSummerSch_-1981.pdf) - another short and quite readable introduction by [Horowitz](https://arxiv.org/pdf/1508.04230.pdf) - modern review with many refs: [[Zhang2015]][](https://arxiv.org/pdf/1508.04230.pdf)