Hawking energy (also called the Hawking mass) is a proposed quasi-local mass in general relativity associated with a closed spacelike 2-surface in spacetime. It was introduced by Stephen Hawking in 1968 as a simple geometric quantity intended to measure the mass or energy contained within a finite region, using only geometric data defined on the bounding surface rather than at infinity.
In general relativity, a quasi-local energy aims to assign an energy (or mass) to a finite spacetime region bounded by a closed surface. Unlike global notions such as the ADM mass, quasi-local quantities depend on the geometry of the chosen surface and generally require additional conditions to exhibit physically desirable properties.
Definition Let Σ {\displaystyle \Sigma } be a smooth closed spacelike 2-surface in a four-dimensional spacetime. At each point of Σ {\displaystyle \Sigma } , there exist two future-directed null vector fields orthogonal to the surface, one outgoing and one ingoing. The corresponding null expansions θ + {\displaystyle \theta _{+}} and θ − {\displaystyle \theta _{-}} measure the divergence of these families of null geodesics as they emanate orthogonally from Σ {\displaystyle \Sigma } . The null vector fields are chosen so that their inner product satisfies ⟨ ℓ + , ℓ − ⟩ = − 2 {\displaystyle \langle \ell _{+},\ell _{-}\rangle =-2} , which fixes the normalization of the null expansions. With this convention, the Hawking energy is defined by
E H ( Σ ) = | Σ | 16 π ( 1 + 1 16 π ∫ Σ θ + θ − d μ ) , {\displaystyle E_{H}(\Sigma )={\sqrt {\frac {|\Sigma |}{16\pi }}}\left(1+{\frac {1}{16\pi }}\int _{\Sigma }\theta _{+}\theta _{-}\,d\mu \right),}
where | Σ | {\displaystyle |\Sigma |} denotes the area of Σ {\displaystyle \Sigma } and d μ {\displaystyle d\mu } is its induced area measure.
Physical interpretation The null expansions θ + {\displaystyle \theta _{+}} and θ − {\displaystyle \theta _{-}} measure the divergence of outgoing and ingoing families of light rays orthogonal to the surface Σ {\displaystyle \Sigma } . Their product therefore encodes how bundles of light rays are focused or defocused by the spacetime geometry. From this perspective, the Hawking energy can be interpreted as a measure of the gravitational focusing of light caused by the matter content and curvature enclosed by Σ {\displaystyle \Sigma } .
Expression in a spacelike hypersurface If Σ {\displaystyle \Sigma } lies in a spacelike hypersurface represented by an initial data set ( M , g , k ) {\displaystyle (M,g,k)} , where ( M , g ) {\displaystyle (M,g)} is a three-dimensional Riemannian manifold with metric g {\displaystyle g} and k {\displaystyle k} is the second fundamental form of M {\displaystyle M} as embedded in the ambient spacetime, the product of null expansions can be expressed in terms of the geometry of Σ {\displaystyle \Sigma } . In this setting, it can be written using the mean curvature H {\displaystyle H} of Σ ⊂ ( M , g ) {\displaystyle \Sigma \subset (M,g)} (the trace of the second fundamental form of Σ {\displaystyle \Sigma } in ( M , g ) {\displaystyle (M,g)} ) and the trace P = t r Σ k {\displaystyle P=\mathrm {tr} _{\Sigma }k} of k {\displaystyle k} restricted to Σ {\displaystyle \Sigma } . In this case, the Hawking energy takes the form
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