Dynamical entropy on local causal horizons: the Einstein equation of state and canonical-energy flux

Abstract

For a twist-free null hypersurface, the Hollands-Wald-Zhang (HWZ) representative $S_{\mathrm{loc}}=(4G{)}^{-1}\int (1-V\theta )dA$, local on each cross-section (cut), obeys an exact off-shell Raychaudhuri identity with no term linear in the expansion, so no final equilibrium cut is needed. Given an entropy density $1/4G$, the Unruh temperature $\kappa /2\pi$ of an approximate local boost, local equilibrium cuts, and stress-tensor conservation, equating the leading equilibrium change of $S_{\mathrm{loc}}$ to matter boost heat yields $R_{kk}=8\pi G{T}_{kk}$ for every null direction, hence the Einstein equation with undetermined cosmological constant, an equation of state in Jacobson's conditional sense. On a fixed caustic-free generator bundle with $dA=Jd{A}_{\mathrm{ref}}$, the functional $S_{\mathrm{L}}=(4G{)}^{-1}\int (1+\ln J-V\theta )d{A}_{\mathrm{ref}}$ obeys an exact balance per reference cell with curvature, shear, and expansion terms all positive, so under null convergence $S_{\mathrm{L}}$ is nondecreasing for $V\ge 0$ until the first caustic. Conversely, requiring this balance for positive affine parameter on every small bundle and null direction is equivalent to the Einstein equation, given conservation and the stated normalization. Cut-locality, positivity, and first-order HWZ normalization select the density $1+\ln J-V\theta$ uniquely within a two-function class. Both functionals depend on the affine origin; we give their shift laws, identify the anchoring-invariant content, and show the origin is physical data fixed by the equilibrium cut. Near equilibrium, $S_{\mathrm{L}}$ agrees with the HWZ dynamical entropy at first order, and the temperature-weighted change of its Hessian equals the HWZ modified canonical-energy flux for stationary-vacuum perturbations. At finite amplitude, $S_{\mathrm{L}}$ is only a monotone geometric functional; reading it as entropy requires a coarse-graining postulate, stated but not derived.

arXiv comment26 pages

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