Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 354 / 2 / c

The two-sided eternal black hole is dual to the thermofield double stateTracing out the left CFT givesThus the geometric period $\beta$ is the boundary inverse temperature, and $S_{\mathrm{BH}}$ is the Von Neumann entropy $S_R=-\mathrm{Tr}({\rho }_R\log {\rho }_R)$, equivalently the entanglement entropy between the two CFTs.

The modular Hamiltonian of this thermal state isFor any first-order state variation with $\mathrm{Tr}\delta \rho =0$,This is the first law of entanglement entropy.

The shell is created by a unitary $U_R$ acting only on the right CFT. Therefore ${\rho }_R\mapsto U_R{\rho }_R{U}_R^{†}$ and its eigenvalues, hence the exact left--right entanglement entropy, do not change. The Hubeny–Rangamani–Takayanagi surface homologous to the complete right boundary remains the old extremal bifurcation surface in the portion of the bulk preceding the shell, behind the enlarged late-time event horizon. Its entropy isnot $A({\mu }^{\prime })/(4G_N)$.

The larger late-time horizon area $A({\mu }^{\prime })$ instead gives the coarse-grained thermodynamic entropy of the final equilibrium black hole. It counts the entropy obtained after discarding detailed information about the coherent unitary excitation. The distinction between the unchanged HRT area and the increased final horizon area is the bulk counterpart of fine-grained entropy conservation under unitary evolution alongside thermodynamic entropy production after coarse graining.