Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 354 / 2 / b / i / Solution

Let $R=R_{\mathrm{AdS}}$ and $r_c=\pi /2-ϵ$. For ${\mathrm{AdS}}_3$, $\mathcal{R}=-6/R^2$ and $\Lambda =-1/R^2$, whileThe bulk term isWith the inward normal $n^r=-\cos r/R$, the induced metric has $\sqrt{-h}=R^2\sin r/{\cos }^{2}r$ andThe Gibbons–Hawking–York boundary term is consequentlyThus, with the orientation and Lorentzian signs displayed in the question,