# JKM ambiguity ## Refs - [[1993#Jacobson, Kang, Myers]] not only mentions but also show that they vanish when evaluated a Killing horizon - Also mentioned in [[1993#Wald]] and [[1994#Iyer, Wald]] but not all three are mentioned ## Ambiguities 1. boundary term $L \rightarrow L+d \mu$ - then $\Theta \rightarrow \Theta +\delta \mu$ - extra terms $d i_{\xi} \alpha$ and $i_{\xi} \alpha$ in $\mathbf{j}$ and $\mathbf{Q}$ respectively - at bifurcation surface $\xi=0$ so they vanish on the bifurcation surface - in fact everywhere on the Killing horizon 2. addition of exact form to the symplectic potential $\Theta \rightarrow \Theta + dY$ - systematic treatment in [[2019#Harlow, Wu]] - $d \gamma\left(\mathcal{L}_{\xi} \psi\right)$ and $\gamma\left(\mathcal{L}_{\xi} \psi\right)$ in $\mathbf{j}$ and $\mathbf{Q}$ respectively - vanish because $\mathcal{L}_{\tilde{\chi}} \psi=0$ for background fields in a stationary solution 3. $J=d Q$ so can add a closed form to $Q$ - not important as horizon section is a closed surface ## Resolutions Unfortunately, most methods of computing the [[0004 Black hole entropy|black hole entropy]] does not fix this ambiguity, except two: - [[0145 Generalised area|LM method]] - see [[2013#Dong]] - [[0005 Black hole second law|BH second law method]] - see [[2015#Wall (Essay)]]