Positive energy theorem

<h2 id="historical-overview">Historical overview</h2>
The original proof of the theorem for <a href="/facts/ADM_mass/48tVaQfr">ADM mass</a> was provided by <a href="/facts/Richard_Schoen/RP34L6ND">Richard Schoen</a> and <a href="/facts/Shing-Tung_Yau/qIQclz7g">Shing-Tung Yau</a> in 1979 using <a href="/facts/Variational_methods/Ya5Lsma3">variational methods</a> and <a href="/facts/Minimal_surfaces/Fy7VkrLQ">minimal surfaces</a>. <a href="/facts/Edward_Witten/lPUylPmj">Edward Witten</a> gave another proof in 1981 based on the use of <a href="/facts/Spinor/sKFNnEwP">spinors</a>, inspired by positive energy theorems in the context of <a href="/facts/Supergravity/jeNk3aAB">supergravity</a>. An extension of the theorem for the <a href="/facts/Bondi_mass/7IycC0jq">Bondi mass</a> was given by <a href="/facts/Malcolm_Ludvigsen/ypwCfB2L">Ludvigsen</a> and James Vickers, Gary Horowitz and <a href="/facts/Malcolm_Perry_(physicist)/dz2pJg1n">Malcolm Perry</a>, and Schoen and Yau.
<a href="/facts/Gary_Gibbons/PuWYkNcf">Gary Gibbons</a>, <a href="/facts/Stephen_Hawking/b7dcce7n">Stephen Hawking</a>, Horowitz and Perry proved extensions of the theorem to asymptotically <a href="/facts/Anti-de_Sitter_spacetime/xoDubsIe">anti-de Sitter spacetimes</a> and to <a href="/facts/Einstein_field_equations/j2rJTzW9">Einstein–Maxwell theory</a>. 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 <a href="/facts/Electric_charge/xxsBfmlB">electric charge</a> 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 and <a href="/facts/Magnetic_charge/R1TSbfMY">magnetic charge</a> 
 
 
 
 P
 
 
 {\displaystyle P}
 
, the mass of the spacetime satisfies (in <a href="/facts/Gaussian_units/1mtqI96c">Gaussian units</a>)

M
        ≥
        
          
            
              Q
              
                2
              
            
            +
            
              P
              
                2
              
            
          
        
        ,
      
    
    {\displaystyle M\geq {\sqrt {Q^{2}+P^{2}}},}

with equality for the <a href="/facts/Sudhansu_Datta_Majumdar/rscyuo35">Majumdar</a>–<a href="/facts/Achilles_Papapetrou/RJeCfaf9">Papapetrou</a> <a href="/facts/Extremal_black_hole/F5SYaqc3">extremal black hole</a> solutions.

<h2 id="initial-data-sets">Initial data sets</h2>
An initial data set consists of a <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> (M, g) and a symmetric 2-tensor field k on M. One says that an initial data set (M, g, k):

<ul><li>is time-symmetric if k is zero</li>
<li>is maximal if trgk = 0 <a class="footnote-ref" id="fnref:1" href="#fn:1">1</a></li>
<li>satisfies the dominant energy condition if</li></ul>

R
          
            g
          
        
        −
        
          |
        
        k
        
          
            |
          
          
            g
          
          
            2
          
        
        +
        (
        
          tr
          
            g
          
        
        ⁡
        k
        
          )
          
            2
          
        
        ≥
        2
        
          
            |
          
        
        
          div
          
            g
          
        
        ⁡
        k
        −
        d
        (
        
          tr
          
            g
          
        
        ⁡
        k
        )
        
          
            
              |
            
          
          
            g
          
        
        ,
      
    
    {\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},}

where Rg denotes the <a href="/facts/Scalar_curvature/tfgm3AuK">scalar curvature</a> of g.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
Note that a time-symmetric initial data set (M, g, 0) satisfies the dominant energy condition if and only if the scalar curvature of g is nonnegative. One says that a Lorentzian manifold (M, g) is a development of an initial data set (M, g, k) if there is a (necessarily spacelike) hypersurface embedding of M into M, together with a continuous unit normal vector field, such that the induced metric is g and the <a href="/facts/Second_fundamental_form/3Ihz5tEc">second fundamental form</a> with respect to the given unit normal is k.
This definition is motivated from <a href="/facts/Mathematics_of_general_relativity/5RktLkom">Lorentzian geometry</a>. Given a Lorentzian manifold (M, g) of dimension n + 1 and a spacelike immersion f from a connected n-dimensional manifold M into M which has a trivial normal bundle, one may consider the induced Riemannian metric g = f *g as well as the <a href="/facts/Second_fundamental_form/3Ihz5tEc">second fundamental form</a> k of f with respect to either of the two choices of continuous unit normal vector field along f. The triple (M, g, k) is an initial data set. According to the <a href="/facts/Gauss-Codazzi_equations/PxQSgtNn">Gauss-Codazzi equations</a>, one has

G
                    ¯
                  
                
                (
                ν
                ,
                ν
                )
              
              
                
                =
                
                  
                    1
                    2
                  
                
                
                  
                    (
                  
                
                
                  R
                  
                    g
                  
                
                −
                
                  |
                
                k
                
                  
                    |
                  
                  
                    g
                  
                  
                    2
                  
                
                +
                (
                
                  tr
                  
                    g
                  
                
                ⁡
                k
                
                  )
                  
                    2
                  
                
                
                  
                    )
                  
                
              
            
            
              
                
                  
                    G
                    ¯
                  
                
                (
                ν
                ,
                ⋅
                )
              
              
                
                =
                d
                (
                
                  tr
                  
                    g
                  
                
                ⁡
                k
                )
                −
                
                  div
                  
                    g
                  
                
                ⁡
                k
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\overline {G}}(\nu ,\nu )&={\frac {1}{2}}{\Big (}R^{g}-|k|_{g}^{2}+(\operatorname {tr} ^{g}k)^{2}{\Big )}\\{\overline {G}}(\nu ,\cdot )&=d(\operatorname {tr} ^{g}k)-\operatorname {div} ^{g}k.\end{aligned}}}

where G denotes the <a href="/facts/Einstein_tensor/ejyGdaFz">Einstein tensor</a> Ricg - ⁠1/2⁠Rgg of g and ν denotes the continuous unit normal vector field along f used to define k. So the dominant energy condition as given above is, in this Lorentzian context, identical to the assertion that G(ν, ⋅), when viewed as a vector field along f, is timelike or null and is oriented in the same direction as ν.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="the-ends-of-asymptotically-flat-initial-data-sets">The ends of asymptotically flat initial data sets</h2>
In the literature there are several different notions of "asymptotically flat" which are not mutually equivalent. Usually it is defined in terms of weighted Hölder spaces or weighted Sobolev spaces.
However, there are some features which are common to virtually all approaches. One considers an initial data set (M, g, k) which may or may not have a boundary; let n denote its dimension. One requires that there is a compact subset K of M such that each connected component of the complement M − K is diffeomorphic to the complement of a closed ball in Euclidean space ℝn. Such connected components are called the <a href="/facts/End_(topology)/Rtwywetf">ends</a> of M.

<h2 id="formal-statements">Formal statements</h2>
<h3>Schoen and Yau (1979)</h3>
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:

<blockquote>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

 
 
 
 
 h
 
 i
 j
 
 
 =
 (
 
 Φ
 
 ∗
 
 
 g
 
 )
 
 i
 j
 
 
 −
 
 δ
 
 i
 j
 
 
 −
 
 
 m
 
 2
 
 |
 
 x
 
 |
 
 
 
 
 
 δ
 
 i
 j
 
 
 
 
 {\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 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 2
 
 
 
 h
 
 i
 j
 
 
 (
 x
 )
 ,
 
 
 {\displaystyle |x|^{2}h_{ij}(x),}
 
 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 3
 
 
 
 ∂
 
 p
 
 
 
 h
 
 i
 j
 
 
 (
 x
 )
 ,
 
 
 {\displaystyle |x|^{3}\partial _{p}h_{ij}(x),}
 
 and 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 4
 
 
 
 ∂
 
 p
 
 
 
 ∂
 
 q
 
 
 
 h
 
 i
 j
 
 
 (
 x
 )
 
 
 {\displaystyle |x|^{4}\partial _{p}\partial _{q}h_{ij}(x)}
 
 are all bounded.</blockquote>
Schoen and Yau's theorem asserts that m must be nonnegative. If, in addition, the functions 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 5
 
 
 
 ∂
 
 p
 
 
 
 ∂
 
 q
 
 
 
 ∂
 
 r
 
 
 
 h
 
 i
 j
 
 
 (
 x
 )
 ,
 
 
 {\displaystyle |x|^{5}\partial _{p}\partial _{q}\partial _{r}h_{ij}(x),}
 
 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 5
 
 
 
 ∂
 
 p
 
 
 
 ∂
 
 q
 
 
 
 ∂
 
 r
 
 
 
 ∂
 
 s
 
 
 
 h
 
 i
 j
 
 
 (
 x
 )
 ,
 
 
 {\displaystyle |x|^{5}\partial _{p}\partial _{q}\partial _{r}\partial _{s}h_{ij}(x),}
 
 and 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 5
 
 
 
 ∂
 
 p
 
 
 
 ∂
 
 q
 
 
 
 ∂
 
 r
 
 
 
 ∂
 
 s
 
 
 
 ∂
 
 t
 
 
 
 h
 
 i
 j
 
 
 (
 x
 )
 
 
 {\displaystyle |x|^{5}\partial _{p}\partial _{q}\partial _{r}\partial _{s}\partial _{t}h_{ij}(x)}
 
 are bounded for any 
 
 
 
 i
 ,
 j
 ,
 p
 ,
 q
 ,
 r
 ,
 s
 ,
 t
 ,
 
 
 {\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 <a href="/facts/Schwarzschild_metric/gkm25ScM">Schwarzschild metric</a>, 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.

<h3>Schoen and Yau (1981)</h3>
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 
 
 
 
 K
 ⊂
 M
 
 
 {\displaystyle K\subset M}
 
 is an open precompact subset such that 
 
 
 
 M
 ∖
 K
 
 
 {\displaystyle M\smallsetminus K}
 
 has finitely many connected components 
 
 
 
 
 M
 
 1
 
 
 ,
 …
 ,
 
 M
 
 n
 
 
 ,
 
 
 {\displaystyle M_{1},\ldots ,M_{n},}
 
 and for each 
 
 
 
 i
 =
 1
 ,
 …
 ,
 n
 
 
 {\displaystyle i=1,\ldots ,n}
 
 there is a diffeomorphism 
 
 
 
 
 Φ
 
 i
 
 
 :
 
 
 R
 
 
 3
 
 
 ∖
 
 B
 
 1
 
 
 (
 0
 )
 →
 
 M
 
 i
 
 
 
 
 {\displaystyle \Phi _{i}:\mathbb {R} ^{3}\smallsetminus B_{1}(0)\to M_{i}}
 
 such that the symmetric 2-tensor 
 
 
 
 
 h
 
 i
 j
 
 
 =
 (
 
 Φ
 
 ∗
 
 
 g
 
 )
 
 i
 j
 
 
 −
 
 δ
 
 i
 j
 
 
 
 
 {\displaystyle h_{ij}=(\Phi ^{\ast }g)_{ij}-\delta _{ij}}
 
 satisfies the following conditions:

<ul><li>
 
 
 
 
 |
 
 x
 
 |
 
 
 h
 
 i
 j
 
 
 (
 x
 )
 ,
 
 
 {\displaystyle |x|h_{ij}(x),}
 
 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 2
 
 
 
 ∂
 
 p
 
 
 
 h
 
 i
 j
 
 
 (
 x
 )
 ,
 
 
 {\displaystyle |x|^{2}\partial _{p}h_{ij}(x),}
 
 and 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 3
 
 
 
 ∂
 
 p
 
 
 
 ∂
 
 q
 
 
 
 h
 
 i
 j
 
 
 (
 x
 )
 
 
 {\displaystyle |x|^{3}\partial _{p}\partial _{q}h_{ij}(x)}
 
 are bounded for all 
 
 
 
 i
 ,
 j
 ,
 p
 ,
 q
 .
 
 
 {\displaystyle i,j,p,q.}
 
</li></ul>
Also suppose that

<ul><li>
 
 
 
 
 |
 
 x
 
 
 |
 
 
 4
 
 
 
 R
 
 
 Φ
 
 i
 
 
 ∗
 
 
 g
 
 
 
 
 {\displaystyle |x|^{4}R^{\Phi _{i}^{\ast }g}}
 
 and 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 5
 
 
 
 ∂
 
 p
 
 
 
 R
 
 
 Φ
 
 i
 
 
 ∗
 
 
 g
 
 
 
 
 {\displaystyle |x|^{5}\partial _{p}R^{\Phi _{i}^{\ast }g}}
 
 are bounded for any 
 
 
 
 p
 
 
 {\displaystyle p}
 
</li>
<li>
 
 
 
 
 |
 
 x
 
 
 |
 
 
 2
 
 
 (
 
 Φ
 
 i
 
 
 ∗
 
 
 k
 
 )
 
 i
 j
 
 
 (
 x
 )
 ,
 
 
 {\displaystyle |x|^{2}(\Phi _{i}^{\ast }k)_{ij}(x),}
 
 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 3
 
 
 
 ∂
 
 p
 
 
 (
 
 Φ
 
 i
 
 
 ∗
 
 
 k
 
 )
 
 i
 j
 
 
 (
 x
 )
 ,
 
 
 {\displaystyle |x|^{3}\partial _{p}(\Phi _{i}^{\ast }k)_{ij}(x),}
 
 and 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 4
 
 
 
 ∂
 
 p
 
 
 
 ∂
 
 q
 
 
 (
 
 Φ
 
 i
 
 
 ∗
 
 
 k
 
 )
 
 i
 j
 
 
 (
 x
 )
 
 
 {\displaystyle |x|^{4}\partial _{p}\partial _{q}(\Phi _{i}^{\ast }k)_{ij}(x)}
 
 for any 
 
 
 
 p
 ,
 q
 ,
 i
 ,
 j
 
 
 {\displaystyle p,q,i,j}
 
</li>
<li>
 
 
 
 
 |
 
 x
 
 
 |
 
 
 3
 
 
 (
 (
 
 Φ
 
 i
 
 
 ∗
 
 
 k
 
 )
 
 11
 
 
 (
 x
 )
 +
 (
 
 Φ
 
 ∗
 
 
 k
 
 )
 
 22
 
 
 (
 x
 )
 +
 (
 
 Φ
 
 i
 
 
 ∗
 
 
 k
 
 )
 
 33
 
 
 (
 x
 )
 )
 
 
 {\displaystyle |x|^{3}((\Phi _{i}^{\ast }k)_{11}(x)+(\Phi ^{\ast }k)_{22}(x)+(\Phi _{i}^{\ast }k)_{33}(x))}
 
 is bounded.</li></ul>
The conclusion is that the ADM energy of each 
 
 
 
 
 M
 
 1
 
 
 ,
 …
 ,
 
 M
 
 n
 
 
 ,
 
 
 {\displaystyle M_{1},\ldots ,M_{n},}
 
 defined as

E
        
        (
        
          M
          
            i
          
        
        )
        =
        
          
            1
            
              16
              π
            
          
        
        
          lim
          
            r
            →
            ∞
          
        
        
          ∫
          
            
              |
            
            x
            
              |
            
            =
            r
          
        
        
          ∑
          
            p
            =
            1
          
          
            3
          
        
        
          ∑
          
            q
            =
            1
          
          
            3
          
        
        
          
            (
          
        
        
          ∂
          
            q
          
        
        (
        
          Φ
          
            i
          
          
            ∗
          
        
        g
        
          )
          
            p
            q
          
        
        −
        
          ∂
          
            p
          
        
        (
        
          Φ
          
            i
          
          
            ∗
          
        
        g
        
          )
          
            q
            q
          
        
        
          
            )
          
        
        
          
            
              x
              
                p
              
            
            
              
                |
              
              x
              
                |
              
            
          
        
        
        d
        
          
            
              H
            
          
          
            2
          
        
        (
        x
        )
        ,
      
    
    {\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

<ul><li>
 
 
 
 
 |
 
 x
 
 
 |
 
 
 4
 
 
 
 ∂
 
 p
 
 
 
 ∂
 
 q
 
 
 
 ∂
 
 r
 
 
 
 h
 
 i
 j
 
 
 (
 x
 )
 
 
 {\displaystyle |x|^{4}\partial _{p}\partial _{q}\partial _{r}h_{ij}(x)}
 
 and 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 4
 
 
 
 ∂
 
 p
 
 
 
 ∂
 
 r
 
 
 
 ∂
 
 s
 
 
 
 ∂
 
 t
 
 
 
 h
 
 i
 j
 
 
 (
 x
 )
 
 
 {\displaystyle |x|^{4}\partial _{p}\partial _{r}\partial _{s}\partial _{t}h_{ij}(x)}
 
 are bounded for any 
 
 
 
 i
 ,
 j
 ,
 p
 ,
 q
 ,
 r
 ,
 s
 ,
 
 
 {\displaystyle i,j,p,q,r,s,}
 
</li></ul>
the assumption that 
 
 
 
 
 E
 
 (
 
 M
 
 i
 
 
 )
 =
 0
 
 
 {\displaystyle {\text{E}}(M_{i})=0}
 
 for some 
 
 
 
 i
 ∈
 {
 1
 ,
 …
 ,
 n
 }
 
 
 {\displaystyle i\in \{1,\ldots ,n\}}
 
 implies that n = 1, that M is diffeomorphic to ℝ3, and that Minkowski space ℝ3,1 is a development of the initial data set (M, g, k).

<h3>Witten (1981)</h3>
Let 
 
 
 
 (
 M
 ,
 g
 )
 
 
 {\displaystyle (M,g)}
 
 be an oriented three-dimensional smooth complete Riemannian manifold (without boundary). Let 
 
 
 
 k
 
 
 {\displaystyle k}
 
 be a smooth symmetric 2-tensor on 
 
 
 
 M
 
 
 {\displaystyle M}
 
 such that

R
          
            g
          
        
        −
        
          |
        
        k
        
          
            |
          
          
            g
          
          
            2
          
        
        +
        (
        
          tr
          
            g
          
        
        ⁡
        k
        
          )
          
            2
          
        
        ≥
        2
        
          
            |
          
        
        
          div
          
            g
          
        
        ⁡
        k
        −
        d
        (
        
          tr
          
            g
          
        
        ⁡
        k
        )
        
          
            
              |
            
          
          
            g
          
        
        .
      
    
    {\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 
 
 
 
 K
 ⊂
 M
 
 
 {\displaystyle K\subset M}
 
 is an open precompact subset such that 
 
 
 
 M
 ∖
 K
 
 
 {\displaystyle M\smallsetminus K}
 
 has finitely many connected components 
 
 
 
 
 M
 
 1
 
 
 ,
 …
 ,
 
 M
 
 n
 
 
 ,
 
 
 {\displaystyle M_{1},\ldots ,M_{n},}
 
 and for each 
 
 
 
 α
 =
 1
 ,
 …
 ,
 n
 
 
 {\displaystyle \alpha =1,\ldots ,n}
 
 there is a diffeomorphism 
 
 
 
 
 Φ
 
 α
 
 
 :
 
 
 R
 
 
 3
 
 
 ∖
 
 B
 
 1
 
 
 (
 0
 )
 →
 
 M
 
 i
 
 
 
 
 {\displaystyle \Phi _{\alpha }:\mathbb {R} ^{3}\smallsetminus B_{1}(0)\to M_{i}}
 
 such that the symmetric 2-tensor 
 
 
 
 
 h
 
 i
 j
 
 
 =
 (
 
 Φ
 
 α
 
 
 ∗
 
 
 g
 
 )
 
 i
 j
 
 
 −
 
 δ
 
 i
 j
 
 
 
 
 {\displaystyle h_{ij}=(\Phi _{\alpha }^{\ast }g)_{ij}-\delta _{ij}}
 
 satisfies the following conditions:

<ul><li>
 
 
 
 
 |
 
 x
 
 |
 
 
 h
 
 i
 j
 
 
 (
 x
 )
 ,
 
 
 {\displaystyle |x|h_{ij}(x),}
 
 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 2
 
 
 
 ∂
 
 p
 
 
 
 h
 
 i
 j
 
 
 (
 x
 )
 ,
 
 
 {\displaystyle |x|^{2}\partial _{p}h_{ij}(x),}
 
 and 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 3
 
 
 
 ∂
 
 p
 
 
 
 ∂
 
 q
 
 
 
 h
 
 i
 j
 
 
 (
 x
 )
 
 
 {\displaystyle |x|^{3}\partial _{p}\partial _{q}h_{ij}(x)}
 
 are bounded for all 
 
 
 
 i
 ,
 j
 ,
 p
 ,
 q
 .
 
 
 {\displaystyle i,j,p,q.}
 
</li>
<li>
 
 
 
 
 |
 
 x
 
 
 |
 
 
 2
 
 
 (
 
 Φ
 
 α
 
 
 ∗
 
 
 k
 
 )
 
 i
 j
 
 
 (
 x
 )
 
 
 {\displaystyle |x|^{2}(\Phi _{\alpha }^{\ast }k)_{ij}(x)}
 
 and 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 3
 
 
 
 ∂
 
 p
 
 
 (
 
 Φ
 
 α
 
 
 ∗
 
 
 k
 
 )
 
 i
 j
 
 
 (
 x
 )
 ,
 
 
 {\displaystyle |x|^{3}\partial _{p}(\Phi _{\alpha }^{\ast }k)_{ij}(x),}
 
 are bounded for all 
 
 
 
 i
 ,
 j
 ,
 p
 .
 
 
 {\displaystyle i,j,p.}
 
</li></ul>
For each 
 
 
 
 α
 =
 1
 ,
 …
 ,
 n
 ,
 
 
 {\displaystyle \alpha =1,\ldots ,n,}
 
 define the ADM energy and linear momentum by

E
        
        (
        
          M
          
            α
          
        
        )
        =
        
          
            1
            
              16
              π
            
          
        
        
          lim
          
            r
            →
            ∞
          
        
        
          ∫
          
            
              |
            
            x
            
              |
            
            =
            r
          
        
        
          ∑
          
            p
            =
            1
          
          
            3
          
        
        
          ∑
          
            q
            =
            1
          
          
            3
          
        
        
          
            (
          
        
        
          ∂
          
            q
          
        
        (
        
          Φ
          
            α
          
          
            ∗
          
        
        g
        
          )
          
            p
            q
          
        
        −
        
          ∂
          
            p
          
        
        (
        
          Φ
          
            α
          
          
            ∗
          
        
        g
        
          )
          
            q
            q
          
        
        
          
            )
          
        
        
          
            
              x
              
                p
              
            
            
              
                |
              
              x
              
                |
              
            
          
        
        
        d
        
          
            
              H
            
          
          
            2
          
        
        (
        x
        )
        ,
      
    
    {\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),}

P
        
        (
        
          M
          
            α
          
        
        
          )
          
            p
          
        
        =
        
          
            1
            
              8
              π
            
          
        
        
          lim
          
            r
            →
            ∞
          
        
        
          ∫
          
            
              |
            
            x
            
              |
            
            =
            r
          
        
        
          ∑
          
            q
            =
            1
          
          
            3
          
        
        
          
            (
          
        
        (
        
          Φ
          
            α
          
          
            ∗
          
        
        k
        
          )
          
            p
            q
          
        
        −
        
          
            (
          
        
        (
        
          Φ
          
            α
          
          
            ∗
          
        
        k
        
          )
          
            11
          
        
        +
        (
        
          Φ
          
            α
          
          
            ∗
          
        
        k
        
          )
          
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    {\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 
 
 
 
 α
 =
 1
 ,
 …
 ,
 n
 ,
 
 
 {\displaystyle \alpha =1,\ldots ,n,}
 
 consider this as a vector 
 
 
 
 (
 
 P
 
 (
 
 M
 
 α
 
 
 
 )
 
 1
 
 
 ,
 
 P
 
 (
 
 M
 
 α
 
 
 
 )
 
 2
 
 
 ,
 
 P
 
 (
 
 M
 
 α
 
 
 
 )
 
 3
 
 
 ,
 
 E
 
 (
 
 M
 
 α
 
 
 )
 )
 
 
 {\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 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 it is necessarily a future-pointing non-spacelike vector. If this vector is zero for any 
 
 
 
 α
 ,
 
 
 {\displaystyle \alpha ,}
 
 then 
 
 
 
 n
 =
 1
 ,
 
 
 {\displaystyle n=1,}
 
 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is diffeomorphic to 
 
 
 
 
 
 R
 
 
 3
 
 
 ,
 
 
 {\displaystyle \mathbb {R} ^{3},}
 
 and the maximal globally hyperbolic development of the initial data set 
 
 
 
 (
 M
 ,
 g
 ,
 k
 )
 
 
 {\displaystyle (M,g,k)}
 
 has zero curvature.

<h3>Extensions and remarks</h3>
According to the above statements, Witten's conclusion is stronger than Schoen and Yau's. However, a third paper by Schoen and Yau<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> shows that their 1981 result implies Witten's, retaining only the extra assumption that 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 4
 
 
 
 R
 
 
 Φ
 
 i
 
 
 ∗
 
 
 g
 
 
 
 
 {\displaystyle |x|^{4}R^{\Phi _{i}^{\ast }g}}
 
 and 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 5
 
 
 
 ∂
 
 p
 
 
 
 R
 
 
 Φ
 
 i
 
 
 ∗
 
 
 g
 
 
 
 
 {\displaystyle |x|^{5}\partial _{p}R^{\Phi _{i}^{\ast }g}}
 
 are bounded for any 
 
 
 
 p
 .
 
 
 {\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 
 
 
 
 
 tr
 
 g
 
 
 ⁡
 k
 =
 0
 
 
 {\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.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> Schoen and Yau's 1979 result and proof can be extended to the case of any dimension less than eight.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> More recently, Witten's result, using Schoen and Yau (1981)'s methods, has been extended to the same context.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> 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 
 
 
 
 
 tr
 
 g
 
 
 ⁡
 k
 =
 0
 ,
 
 
 {\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.

<h2 id="applications">Applications</h2>
<ul><li>In 1984 Schoen used the positive mass theorem in his work which completed the solution of the <a href="/facts/Yamabe_problem/TekKSICb">Yamabe problem</a>.</li>
<li>The positive mass theorem was used in <a href="/facts/Hubert_Bray/8UxFXW1v">Hubert Bray</a>'s proof of the <a href="/facts/Riemannian_Penrose_inequality/692M9VUD">Riemannian Penrose inequality</a>.</li></ul>

<ul><li>Schoen, Richard; Yau, Shing-Tung (1979). <a href="http://projecteuclid.org/euclid.cmp/1103904790">"On the proof of the positive mass conjecture in general relativity"</a>. Communications in Mathematical Physics. 65 (1): 45–76. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1979CMaPh..65...45S">1979CMaPh..65...45S</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf01940959">10.1007/bf01940959</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0010-3616">0010-3616</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:54217085">54217085</a>.</li>
<li>Schoen, Richard; Yau, Shing-Tung (1981). <a href="http://projecteuclid.org/euclid.cmp/1103908964">"Proof of the positive mass theorem. II"</a>. Communications in Mathematical Physics. 79 (2): 231–260. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1981CMaPh..79..231S">1981CMaPh..79..231S</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf01942062">10.1007/bf01942062</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0010-3616">0010-3616</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:59473203">59473203</a>.</li>
<li>Witten, Edward (1981). <a href="http://projecteuclid.org/euclid.cmp/1103919981">"A new proof of the positive energy theorem"</a>. Communications in Mathematical Physics. 80 (3): 381–402. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1981CMaPh..80..381W">1981CMaPh..80..381W</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf01208277">10.1007/bf01208277</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0010-3616">0010-3616</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:1035111">1035111</a>.</li>
<li>Ludvigsen, M; Vickers, J A G (1981-10-01). "The positivity of the Bondi mass". Journal of Physics A: Mathematical and General. 14 (10): L389 – L391. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1981JPhA...14L.389L">1981JPhA...14L.389L</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0305-4470%2F14%2F10%2F002">10.1088/0305-4470/14/10/002</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0305-4470">0305-4470</a>.</li>
<li>Horowitz, Gary T.; Perry, Malcolm J. (1982-02-08). "Gravitational Energy Cannot Become Negative". Physical Review Letters. 48 (6): 371–374. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1982PhRvL..48..371H">1982PhRvL..48..371H</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2Fphysrevlett.48.371">10.1103/physrevlett.48.371</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>.</li>
<li>Schoen, Richard; Yau, Shing Tung (1982-02-08). "Proof That the Bondi Mass is Positive". Physical Review Letters. 48 (6): 369–371. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1982PhRvL..48..369S">1982PhRvL..48..369S</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2Fphysrevlett.48.369">10.1103/physrevlett.48.369</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>.</li>
<li>Gibbons, G. W.; Hawking, S. W.; Horowitz, G. T.; Perry, M. J. (1983). <a href="https://projecteuclid.org/euclid.cmp/1103922377">"Positive mass theorems for black holes"</a>. Communications in Mathematical Physics. 88 (3): 295–308. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1983CMaPh..88..295G">1983CMaPh..88..295G</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01213209">10.1007/BF01213209</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0701918">0701918</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121580771">121580771</a>.</li></ul>
Textbooks

<ul><li>Choquet-Bruhat, Yvonne. General relativity and the Einstein equations. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2009. xxvi+785 pp. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-923072-3</li>
<li>Wald, Robert M. General relativity. University of Chicago Press, Chicago, IL, 1984. xiii+491 pp. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-226-87032-4</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">In local coordinates, this says gijkij = 0 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">In local coordinates, this says R - gikgjlkijkkl + (gijkij)2 ≥ 2(gpq(gijkpi;j - (gijkij);p)(gklkqk;l - (gklkkl);q))1/2 or, in the usual "raised and lowered index" notation, this says R - kijkij + (kii)2 ≥ 2((kpi;i - (kii);p)(kpj;j - (kjj);p))1/2 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">It is typical to assume M to be time-oriented and for ν to be then specifically defined as the future-pointing unit normal vector field along f; in this case the dominant energy condition as given above for an initial data set arising from a spacelike immersion into M is automatically true if the dominant energy condition in its usual spacetime form is assumed. <a href="/wiki/Energy_conditions#Mathematical_statement" target="_blank">/wiki/Energy_conditions#Mathematical_statement</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Schoen, Richard; Yau, Shing Tung (1981). "The energy and the linear momentum of space-times in general relativity" (PDF). Comm. Math. Phys. 79 (1): 47–51. Bibcode:1981CMaPh..79...47S. doi:10.1007/BF01208285. S2CID 120151656. <a href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-79/issue-1/The-energy-and-the-linear-momentum-of-space-times-in/cmp/1103908887.pdf" target="_blank">https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-79/issue-1/The-energy-and-the-linear-momentum-of-space-times-in/cmp/1103908887.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Bartnik, Robert (1986). "The mass of an asymptotically flat manifold". Comm. Pure Appl. Math. 39 (5): 661–693. CiteSeerX 10.1.1.625.6978. doi:10.1002/cpa.3160390505. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Schoen, Richard M. (1989). "Variational theory for the total scalar curvature functional for Riemannian metrics and related topics". Topics in calculus of variations (Montecatini Terme, 1987). Lecture Notes in Mathematics. Vol. 1365. Berlin: Springer. pp. 120–154. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Eichmair, Michael; Huang, Lan-Hsuan; Lee, Dan A.; Schoen, Richard (2016). "The spacetime positive mass theorem in dimensions less than eight". Journal of the European Mathematical Society. 18 (1): 83–121. arXiv:1110.2087. doi:10.4171/JEMS/584. S2CID 119633794. <a href="/wiki/Lan-Hsuan_Huang" target="_blank">/wiki/Lan-Hsuan_Huang</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Positive energy theorem open-in-new

Positive energy theorem