Stationary first law from the covariant boundary identity

Differentiate $J_{\xi }$ at fixed $\xi$. Since ${\mathcal{L}}_{\xi }\varphi =0$ on the stationary background and $\theta$ is linear in its variation argument, $\delta J_{\xi }=\theta (\varphi ;\delta ({\mathcal{L}}_{\xi }\varphi ))-{\iota }_{\xi }\delta \mathcal{L}=\theta (\varphi ;{\mathcal{L}}_{\xi }\delta \varphi )-{\iota }_{\xi }(E_{\varphi }[\delta \varphi ]+d\theta (\varphi ;\delta \varphi ))\mathrm{.}$ The first equality uses that the coefficient variation of $\theta (\varphi ;0)$ is zero; the second uses $\delta ({\mathcal{L}}_{\xi }\varphi )={\mathcal{L}}_{\xi }\delta \varphi$ because $\xi$ is fixed. Covariance of $\theta$ identifies $\theta (\varphi ;{\mathcal{L}}_{\xi }\delta \varphi )={\mathcal{L}}_{\xi }\theta (\varphi ;\delta \varphi )$. Since $E_{\varphi }=0$, Cartan's identity [F1] yields $\delta J_{\xi }=d({\iota }_{\xi }\theta )$.

By [A1], $C_{\xi }=J_{\xi }-d{Q}_{\xi }$ and $\delta C_{\xi }=0$. Differentiating this identity gives $\delta J_{\xi }=d(\delta Q_{\xi })$ because $d$ is independent of the field variation and the spacetime coordinates. Combining with step 1.1 gives $d(\delta Q_{\xi }-{\iota }_{\xi }\theta )=0$. This conclusion uses the explicit constraint-variation premise; it is not inferred from the closure of a presymplectic current.

Apply [F2] to $\alpha =\delta Q_{\xi }-{\iota }_{\xi }\theta$ on $C_R$. By step 2.1 and $\mathrm{\partial }{C}_R=S_R-B$, $0={\int }_{C_R}d\alpha ={\int }_{S_R}(\delta Q_{\xi }-{\iota }_{\xi }\theta )-{\int }_B(\delta Q_{\xi }-{\iota }_{\xi }\theta )\mathrm{.}$ On $B$, $\xi =0$, so ${\iota }_{\xi }\theta =0$. Taking the limit in the first term is permitted by the boundary convergence assumption in [A1], and its value is $\delta H_{\xi }$ by the specified definition of the Hamiltonian variation. Hence $\delta H_{\xi }={\int }_B\delta Q_{\xi }$.

By [F3], the charge used here is in the representative class required by [L1]. Apply [L1] to the right-hand side of step 3.1. This gives $\delta H_{\xi }=(\mathrm{\hslash }\kappa \mathrm{/}(2\pi k_B))\delta S_W$, with all field variations taken at fixed generator and fixed background $\kappa$.

For ${\mathrm{\ell }}_{\mathrm{E}\mathrm{H}}=R\mathrm{/}(16\pi G)$, the curvature convention in the preceding entropy-variation theorem gives $R=g^{ac}{g}^{bd}{R}_{abcd}$. Holding $g$ fixed and using $\delta R_{abcd}=-\delta R_{abdc}$, the Riemann-symmetric derivative is $P^{abcd}=\frac{1}{32\pi G}(g^{ac}{g}^{bd}-g^{ad}{g}^{bc})\mathrm{.}$ Its binormal contraction is $P^{abcd}{ϵ}_{ab}{ϵ}_{cd}=(-2-2)\mathrm{/}(32\pi G)=-1\mathrm{/}(8\pi G)$, because ${ϵ}_{ab}{ϵ}^{ab}=-2$ and ${ϵ}_{ab}{ϵ}^{ba}=2$. Substitution in the definition of $S_W$ yields $S_W=k_B{\int }_{B}d{A}_B\mathrm{/}(4G\mathrm{\hslash })=k_B{A}_B\mathrm{/}(4G\mathrm{\hslash })$.