# Quasi-local energy (in gravity) Energy is a tricky concept in General Relativity. Unlike in Newtonian physics, energy is usually considered as a global property of (a time slice of) the spacetime. For example, the [[0487 ADM mass|ADM energy]] is defined using data at spatial infinity and measures the total mass of the spacetime. It is certainly useful to have a notion of energy for a finite region. For example, we would like to talk about the mass of a black hole in astrophysics. If the only notion of energy is the total mass of the universe, its utility would be very limited. But it turned out to be quite challenging. More rigorously, *quasi-local* quantities are quantities associated with open subsets of spacetime whose closure is compact. ## Refs - reviews - [[2004#Szabados (Review)]] ## Misner-Sharp energy The Misner-Sharp energy is a [[0595 Quasi-local energy|quasi-local]] definition of [[0592 Gravitational energy|gravitational energy]] for spacetimes with spherical symmetry. Using coordinates $\mathrm{d} s^2=r^2 \mathrm{~d} \Omega^2-2 \mathrm{e}^{-f} \mathrm{~d} \xi^{+} \mathrm{d} \xi^{-},$it is defined to be$E=\frac{1}{2} r\left(1-g^{-1}(\mathrm{~d} r, \mathrm{~d} r)\right)=\frac{1}{2} r+\mathrm{e}^f r \partial_{+} r \partial_{-} r=\frac{1}{2} r+\frac{1}{4} \mathrm{e}^f r^3 \theta_{+} \theta_{-}.$ ### Refs - original - [[1964#Misner, Sharp]] - general 2-surfaces - [[1993#Hayward (Sean)]]: generalises to any 2-surface (in 4D) - higher-derivative - [[0425 Gauss-Bonnet gravity|GB gravity]]: [[2006#Maeda]] and [[2007#Maeda, Nozawa]] - applications - [[1994#Hayward]]: studies various properties and application of the Misner-Sharp energy including a proof of [[0476 Penrose inequality|Penrose inequality]] ## Hayward energy The Hayward energy is a quasilocal [[0592 Gravitational energy|gravitational energy]], defined on any spacelike 2-surface (in a 4D spacetime) to be$E=\frac{1}{8 \pi} \sqrt{\frac{A}{16 \pi}} \int_S \mu\left(\mathcal{R}+\theta \tilde{\theta}-\frac{1}{2} \sigma_{a b} \tilde{\sigma}^{a b}-2 \omega_a \omega^a\right)$where $\mu$ is area density, $\mathcal{R}$ is the intrinsic Ricci scalar, $\theta$, $\sigma$, and $\omega$ are the expansion, shear, and twist, respectively. ### Refs - original: [[1993#Hayward (Sean)]] ### Properties - it reduces to the [[0595 Quasi-local energy#Misner-Sharp energy|Misner-Sharp energy]] in spherical symmetry - it vanishes in flat space ### Issues - the $\omega^2$ term is not gauge invariant - it tends to Newman-Unti rather than Bondi-Sachs at null infinity ## Wang-Yau energy Works for a general 2-surface. ### Refs - origin: [[2008#Wang, Yau]] ### Applications - binary BH: [[2023#Pook-Kolb, Zhao, Andersson, Krishnan, Yau]]