# Wald entropy Black hole entropy is Noether charge! Introducing [[0019 Covariant phase space|covariant phase space formalism]], Wald showed that the [[0004 Black hole entropy|entropy]] of stationary black hole can be defined as the Noether charge corresponding to the boost symmetry of the spacetime. For a general [[0006 Higher-derivative gravity|higher derivative theory]], this method also gives a formula for the black hole in such theories, known as Wald entropy. ## Refs - [[1993#Wald]] ## Derivation - current: $\mathbf{j}=\mathbf{\Theta}\left(\phi, \mathcal{L}_{\xi} \phi\right)-\xi \cdot \mathbf{L}$ - can show $d \mathbf{j}=-\mathbf{E} \mathcal{L}_{\xi} \phi$ -> 0 when EoM is satisfied - so $\mathbf{j}=d \mathbf{Q}$ when evaluated on solutions of EoM - its variation $\delta \mathbf{j}=\delta\left[\mathbf{\Theta}\left(\phi, \mathcal{L}_{\xi} \phi\right)\right]-\xi \cdot \delta \mathbf{L}=\delta\left[\mathbf{\Theta}\left(\phi, \mathcal{L}_{\xi} \phi\right)\right]-\mathcal{L}_{\xi}[\mathbf{\Theta}(\phi, \delta \phi)]+d(\xi \cdot \mathbf{\Theta})$ - -> $\delta \mathbf{j}=\Omega\left(\phi, \delta \phi, \mathcal{L}_{\xi} \phi\right)+d(\xi \cdot \Theta)$ - energy and angular momentum - $\delta \mathcal{E}=\int_{\infty}(\delta \mathbf{Q}[t]-t \cdot \mathbf{\Theta})$ -> $\mathcal{E}=\int_{\infty}(\mathbf{Q}[t]-t \cdot \mathbf{B})$ with $\delta \int_{\infty} t \cdot \mathbf{B}=\int_{\infty} t \cdot \mathbf{\Theta}$ - $\delta \mathcal{J}=-\int_{\infty} \delta Q[\varphi]$ -> $\mathcal{J}=-\int_{\infty} \mathbf{Q}[\varphi]$ - choose the Killing field of the (rotating) BH - $\xi^{a}=t^{a}+\Omega_{H}^{(\mu)} \varphi_{(\mu)}^{a}$ - $\xi^a$ is the Killing field, which vanishes on the bifurcation surface $\Sigma$ - $t^a$ is the stationary Killing field with unit norm at infinity - now Let $\delta \phi$ be an arbitrary, asymptotically flat solution of the linearised equations - the current identity above becomes $d(\delta \mathbf{Q})=d(\xi \cdot \mathbf{\Theta})$ - because $\mathbf{\Omega}\left(\phi, \delta \phi, \mathcal{L}_{\xi} \phi\right)=0$ as $\mathcal{L}_{\xi} \phi=0$ (Killing field) - taking all above: - $\delta \int_{\Sigma} \mathbf{Q}=\delta \mathcal{E}-\Omega_{H}^{(\mu)} \delta \mathcal{J}_{(\mu)}$ - now identify LHS as entropy ## Related - [[0004 Black hole entropy]] - [[0018 JKM ambiguity]]