 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-354/2/a/ii/solution

Regulate the surface at

z = ϵ

, so

r_{m a x} = R^{2} - ϵ^{2}

. On the hemisphere,

1 + z^{'2} = R / z

, and its area is

Area (γ_{D}) = 2 π R \int_{0}^{r_{m a x}} \frac{r d r}{( R ^{2} - r ^{2} ) ^{3/2}} = 2 π (\frac{R}{ϵ} - 1) .

S_{D} = \frac{π}{2 G} (\frac{R}{ϵ} - 1) .

The 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.

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