Positive energy theorem

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. Suppose that it has one end, and it is asymptotically Schwarzschild in the following sense:

Suppose that K is an open precompact subset of M such that there is a diffeomorphism Φ : ℝ³ − B₁(0) → M − K, and suppose that there is a number m such that the symmetric 2-tensor

${\displaystyle h_{ij}=(\Phi ^{\ast }g)_{ij}-\delta _{ij}-{\frac {m}{2|x|}}\delta _{ij}}$

on ℝ³ − B₁(0) is such that for any i, j, p, q, the functions ${\displaystyle |x|^{2}h_{ij}(x),}$ ${\displaystyle |x|^{3}\partial _{p}h_{ij}(x),}$ and ${\displaystyle |x|^{4}\partial _{p}\partial _{q}h_{ij}(x)}$ are all bounded.

Schoen and Yau's theorem asserts that m must be nonnegative. If, in addition, the functions ${\displaystyle |x|^{5}\partial _{p}\partial _{q}\partial _{r}h_{ij}(x),}$ ${\displaystyle |x|^{5}\partial _{p}\partial _{q}\partial _{r}\partial _{s}h_{ij}(x),}$ and ${\displaystyle |x|^{5}\partial _{p}\partial _{q}\partial _{r}\partial _{s}\partial _{t}h_{ij}(x)}$ are bounded for any ${\displaystyle i,j,p,q,r,s,t,}$ then m must be positive unless the boundary of M is empty and (M, g) is isometric to ℝ³ with its standard Riemannian metric.

Note that the conditions on h are asserting that h, together with some of its derivatives, are small when x is large. Since h is measuring the defect between g in the coordinates Φ and the standard representation of the t = constant slice of the Schwarzschild metric, these conditions are a quantification of the term "asymptotically Schwarzschild". 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. If (M, g) is a complete Riemannian manifold with multiple ends, then the above result applies to any single end, provided that there is a positive mean curvature sphere in every other end. This is guaranteed, for instance, if each end is asymptotically flat in the above sense; one can choose a large coordinate sphere as a boundary, and remove the corresponding remainder of each end until one has a Riemannian manifold-with-boundary with a single end.

Let (M, g, k) be an initial data set satisfying the dominant energy condition. Suppose that (M, g) is an oriented three-dimensional smooth complete Riemannian manifold (without boundary); suppose that it has finitely many ends, each of which is asymptotically flat in the following sense.

Suppose that ${\displaystyle K\subset M}$ is an open precompact subset such that ${\displaystyle M\smallsetminus K}$ has finitely many connected components ${\displaystyle M_{1},\ldots ,M_{n},}$ and for each ${\displaystyle i=1,\ldots ,n}$ there is a diffeomorphism ${\displaystyle \Phi _{i}:\mathbb {R} ^{3}\smallsetminus B_{1}(0)\to M_{i}}$ such that the symmetric 2-tensor ${\displaystyle h_{ij}=(\Phi ^{\ast }g)_{ij}-\delta _{ij}}$ satisfies the following conditions:

• ${\displaystyle |x|h_{ij}(x),}$ ${\displaystyle |x|^{2}\partial _{p}h_{ij}(x),}$ and ${\displaystyle |x|^{3}\partial _{p}\partial _{q}h_{ij}(x)}$ are bounded for all ${\displaystyle i,j,p,q.}$

Also suppose that

• ${\displaystyle |x|^{4}R^{\Phi _{i}^{\ast }g}}$ and ${\displaystyle |x|^{5}\partial _{p}R^{\Phi _{i}^{\ast }g}}$ are bounded for any ${\displaystyle p}$
• ${\displaystyle |x|^{2}(\Phi _{i}^{\ast }k)_{ij}(x),}$ ${\displaystyle |x|^{3}\partial _{p}(\Phi _{i}^{\ast }k)_{ij}(x),}$ and ${\displaystyle |x|^{4}\partial _{p}\partial _{q}(\Phi _{i}^{\ast }k)_{ij}(x)}$ for any ${\displaystyle p,q,i,j}$
• ${\displaystyle |x|^{3}((\Phi _{i}^{\ast }k)_{11}(x)+(\Phi ^{\ast }k)_{22}(x)+(\Phi _{i}^{\ast }k)_{33}(x))}$ is bounded.

The conclusion is that the ADM energy of each ${\displaystyle M_{1},\ldots ,M_{n},}$ defined as

${\displaystyle {\text{E}}(M_{i})={\frac {1}{16\pi }}\lim _{r\to \infty }\int _{|x|=r}\sum _{p=1}^{3}\sum _{q=1}^{3}{\big (}\partial _{q}(\Phi _{i}^{\ast }g)_{pq}-\partial _{p}(\Phi _{i}^{\ast }g)_{qq}{\big )}{\frac {x^{p}}{|x|}}\,d{\mathcal {H}}^{2}(x),}$

is nonnegative. Furthermore, supposing in addition that

• ${\displaystyle |x|^{4}\partial _{p}\partial _{q}\partial _{r}h_{ij}(x)}$ and ${\displaystyle |x|^{4}\partial _{p}\partial _{r}\partial _{s}\partial _{t}h_{ij}(x)}$ are bounded for any ${\displaystyle i,j,p,q,r,s,}$

the assumption that ${\displaystyle {\text{E}}(M_{i})=0}$ for some ${\displaystyle i\in \{1,\ldots ,n\}}$ implies that n = 1, that M is diffeomorphic to ℝ³, and that Minkowski space ℝ^3,1 is a development of the initial data set (M, g, k).

Let ${\displaystyle (M,g)}$ be an oriented three-dimensional smooth complete Riemannian manifold (without boundary). Let ${\displaystyle k}$ be a smooth symmetric 2-tensor on ${\displaystyle M}$ such that

${\displaystyle R^{g}-|k|_{g}^{2}+(\operatorname {tr} _{g}k)^{2}\geq 2{\big |}\operatorname {div} ^{g}k-d(\operatorname {tr} _{g}k){\big |}_{g}.}$

Suppose that ${\displaystyle K\subset M}$ is an open precompact subset such that ${\displaystyle M\smallsetminus K}$ has finitely many connected components ${\displaystyle M_{1},\ldots ,M_{n},}$ and for each ${\displaystyle \alpha =1,\ldots ,n}$ there is a diffeomorphism ${\displaystyle \Phi _{\alpha }:\mathbb {R} ^{3}\smallsetminus B_{1}(0)\to M_{i}}$ such that the symmetric 2-tensor ${\displaystyle h_{ij}=(\Phi _{\alpha }^{\ast }g)_{ij}-\delta _{ij}}$ satisfies the following conditions:

• ${\displaystyle |x|h_{ij}(x),}$ ${\displaystyle |x|^{2}\partial _{p}h_{ij}(x),}$ and ${\displaystyle |x|^{3}\partial _{p}\partial _{q}h_{ij}(x)}$ are bounded for all ${\displaystyle i,j,p,q.}$
• ${\displaystyle |x|^{2}(\Phi _{\alpha }^{\ast }k)_{ij}(x)}$ and ${\displaystyle |x|^{3}\partial _{p}(\Phi _{\alpha }^{\ast }k)_{ij}(x),}$ are bounded for all ${\displaystyle i,j,p.}$

For each ${\displaystyle \alpha =1,\ldots ,n,}$ define the ADM energy and linear momentum by

${\displaystyle {\text{E}}(M_{\alpha })={\frac {1}{16\pi }}\lim _{r\to \infty }\int _{|x|=r}\sum _{p=1}^{3}\sum _{q=1}^{3}{\big (}\partial _{q}(\Phi _{\alpha }^{\ast }g)_{pq}-\partial _{p}(\Phi _{\alpha }^{\ast }g)_{qq}{\big )}{\frac {x^{p}}{|x|}}\,d{\mathcal {H}}^{2}(x),}$

${\displaystyle {\text{P}}(M_{\alpha })_{p}={\frac {1}{8\pi }}\lim _{r\to \infty }\int _{|x|=r}\sum _{q=1}^{3}{\big (}(\Phi _{\alpha }^{\ast }k)_{pq}-{\big (}(\Phi _{\alpha }^{\ast }k)_{11}+(\Phi _{\alpha }^{\ast }k)_{22}+(\Phi _{\alpha }^{\ast }k)_{33}{\big )}\delta _{pq}{\big )}{\frac {x^{q}}{|x|}}\,d{\mathcal {H}}^{2}(x).}$

For each ${\displaystyle \alpha =1,\ldots ,n,}$ consider this as a vector ${\displaystyle ({\text{P}}(M_{\alpha })_{1},{\text{P}}(M_{\alpha })_{2},{\text{P}}(M_{\alpha })_{3},{\text{E}}(M_{\alpha }))}$ in Minkowski space. Witten's conclusion is that for each ${\displaystyle \alpha }$ it is necessarily a future-pointing non-spacelike vector. If this vector is zero for any ${\displaystyle \alpha ,}$ then ${\displaystyle n=1,}$ ${\displaystyle M}$ is diffeomorphic to ${\displaystyle \mathbb {R} ^{3},}$ and the maximal globally hyperbolic development of the initial data set ${\displaystyle (M,g,k)}$ has zero curvature.

According to the above statements, Witten's conclusion is stronger than Schoen and Yau's. However, a third paper by Schoen and Yau[4] shows that their 1981 result implies Witten's, retaining only the extra assumption that ${\displaystyle |x|^{4}R^{\Phi _{i}^{\ast }g}}$ and ${\displaystyle |x|^{5}\partial _{p}R^{\Phi _{i}^{\ast }g}}$ are bounded for any ${\displaystyle p.}$ It also must be noted that Schoen and Yau's 1981 result relies on their 1979 result, which is proved by contradiction; therefore their extension of their 1981 result is also by contradiction. By contrast, Witten's proof is logically direct, exhibiting the ADM energy directly as a nonnegative quantity. Furthermore, Witten's proof in the case ${\displaystyle \operatorname {tr} _{g}k=0}$ can be extended without much effort to higher-dimensional manifolds, under the topological condition that the manifold admits a spin structure.[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.

As of April 2017, Schoen and Yau have released a preprint which proves the general higher-dimensional case in the special case ${\displaystyle \operatorname {tr} _{g}k=0,}$ without any restriction on dimension or topology. However, it has not yet (as of May 2020) appeared in an academic journal.