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

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.