Covariant stationary Noether charge and Wald entropy — Examples

In units with $c=1$, let $\mathcal{L}=f(R){\mathit{ϵ}}_g\mathrm{/}(16\pi G)$, where $G>0$ is the Newton coupling in the given $n$-dimensional action normalization, with units $L^{n-2}\mathrm{/}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$. Choose an open interval $J\subseteq \mathbb{R}$ and $f\in C^{\mathrm{\infty }}(J)$, and admit only smooth background fields and finite-dimensional smooth field families for which $R_{{\varphi }_p}(x)\in J$ for every parameter $p$ and spacetime point $x\in M$. Thus this composed Lagrangian is smooth throughout the spacetime and variation domain required by the preceding first-law theorem; regularity only at curvature values on $B$ is not assumed. The scalar curvature $R$ has units $L^{-2}$, so $f$ has those units and $f^{\mathrm{\prime }}(R)$ is dimensionless. Under the same smooth stationary horizon assumptions as the preceding theorem, $S_W=\frac{k_B}{4G\mathrm{\hslash }}{\int }_B{f}^{\mathrm{\prime }}(R)d{A}_B\mathrm{.}$ The example computes the Noether-charge functional only; it asserts no monotonicity along a dynamical horizon.