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

The horizon is the largest positive root of $f(r_H)=0$. With $x=r_H^2$,Choosing the positive root gives

After Wick rotation $t=-i{t}_E$, the near-horizon metric hasRegularity at the origin of this polar plane requires the Euclidean black-hole regularity condition ${\beta }_H=4\pi /f^{\prime }(r_H)$. Sincewe obtain

Keep the smooth $\beta ={\beta }_H$ geometry fixed away from the horizon but identify Euclidean time with arbitrary period $\beta$. The horizon then has deficit angle $2\pi (1-\beta /{\beta }_H)$. Its delta-function curvature contributesand henceUsing $\log Z=-I_{\mathrm{grav}}$,

All smooth bulk terms, including the cosmological-constant volume term, are proportional to the Euclidean time period and are annihilated by $1-\beta {\partial }_{\beta }$. The asymptotic Gibbons–Hawking–York boundary term and holographic counterterms are likewise smooth and linear in $\beta$. Only the curvature singularity at the fixed point of the Euclidean time circle survives.

Varying the boundary period and filling it by the corresponding smooth Euclidean saddle computes the same canonical partition function. Because each bulk metric obeys the Einstein equation, the implicit first-order metric variation of the on-shell action reduces to boundary terms; regularity relates the varied horizon radius to the varied period. The thermodynamic identitythen yields the same Bekenstein-Hawking entropy. The conical method is an off-shell way to isolate the local horizon term, while the smooth-saddle method packages that term into the variation of the entire 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.

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.