Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 354 / 2 / a

On the $t=0$ slice, rotational symmetry lets the surface be written $z=z(r)$. The inversion symmetry fixing the boundary circle extends to an isometry of the Poincare half-space whose fixed set isThe unique extremal surface anchored at $r=R,z=0$ must be invariant under that isometry. Hence the Ryu–Takayanagi surface for a disk is the hemisphereOne may verify directly that it extremizes the area functional $2\pi \int r{z}^{-2}\sqrt{1+z^{\prime 2}}dr$.

Regulate the surface at $z=ϵ$, so $r_{\max }=\sqrt{R^2-{ϵ}^2}$. On the hemisphere, $\sqrt{1+z^{\prime 2}}=R/z$, and its area isThe Ryu–Takayanagi formula therefore givesThe first term is the ultraviolet area-law divergence proportional to the boundary circle's length; the finite constant is universal for the vacuum of a three-dimensional conformal field theory.