Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 354 / 2 / a

Solution

The Ryu–Takayanagi formula gives the leading entropy of a static boundary region:

S (D_{1}) = \frac{Area ( γ _{D_{1}} )}{4 G} .

On the

t = 0

slice of Poincaré

AdS_{4}

d s^{2} = \frac{d z ^{2} + d ρ ^{2} + ρ ^{2} d θ ^{2}}{z ^{2}} .

The minimal surface anchored on a boundary circle of radius

R_{1}

is the hemisphere

z^{2} + ρ^{2} = R_{1}^{2} .

Cutting it off at

z = ϵ

, its area is

A_{ϵ} = 2 π R_{1} \int_{0}^{R_{1}^{2} - ϵ^{2}} \frac{ρ d ρ}{( R _{1}^{2} - ρ ^{2} ) ^{3/2}} = 2 π (\frac{R _{1}}{ϵ} - 1) .

Therefore

S (D_{1}) = \frac{π R _{1}}{2 G ϵ} - \frac{π}{2 G} .

The universal requirement was the perimeter-law divergence

π R_{1} / (2 G ϵ)

; the finite constant depends on the stated pure

AdS_{4}

- π / (2 G)

here. Since

G \sim N^{- 3/2}

, both terms are of order

N^{3/2}

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