Positive energy theorem

Let ${\displaystyle (M,g)}$ be an oriented three-dimensional smooth Riemannian manifold-with-boundary with nonnegative scalar curvature and such that each boundary component has positive mean curvature.

Suppose that ${\displaystyle K\subset M}$ is an open precompact subset such that there is a diffeomorphism ${\displaystyle \Phi :\mathbb {R} ^{3}\smallsetminus B_{1}(0)\to M\smallsetminus {\overline {K}}}$. Suppose that there is a number ${\displaystyle m}$ such that the symmetric 2-tensor ${\displaystyle h_{ij}=(\Phi ^{\ast }g)_{ij}-\delta _{ij}-{\frac {m}{2|x|}}\delta _{ij}}$ on ${\displaystyle \mathbb {R} ^{3}\smallsetminus B_{1}(0)}$ is such that for any ${\displaystyle i,j,p,q}$, 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.

• Then ${\displaystyle m}$ must be nonnegative.

Suppose, as extra conditions, that 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 ${\displaystyle i,j,p,q,r,s,t.}$

• If ${\displaystyle m=0}$ then the boundary of ${\displaystyle M}$ must be empty and ${\displaystyle (M,g)}$ must be isometric to ${\displaystyle \mathbb {R} ^{3}}$ with its standard Riemannian metric.

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

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

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 ${\displaystyle n=1}$, that ${\displaystyle M}$ is diffeomorphic to ${\displaystyle \mathbb {R} ^{3},}$ and that there is a spacelike embedding ${\displaystyle F:M\to \mathbb {R} ^{3,1}}$ whose induced metric is ${\displaystyle g}$ and whose second fundamental form is ${\displaystyle k.}$

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

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

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[1] 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; 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.[2] Schoen and Yau's 1979 result and proof can be extended to the case of any dimension less than eight.[3] More recently, Witten's result, using Schoen and Yau (1981)'s methods, has been extended to the same context.[4] 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.