Hawking energy

The Hawking energy or Hawking mass is one of the possible definitions of mass in general relativity. It is a measure of the bending of ingoing and outgoing rays of light that are orthogonal to a 2-sphere surrounding the region of space whose mass is to be defined.

Let ${\displaystyle ({\mathcal {M}}^{3},g_{ab})}$ be a 3-dimensional sub-manifold of a relativistic spacetime, and let ${\displaystyle \Sigma \subset {\mathcal {M}}^{3}}$ be a closed 2-surface. Then the Hawking mass ${\displaystyle m_{H}(\Sigma )}$ of ${\displaystyle \Sigma }$ is defined to be

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