Holographic argument for the Penrose inequality in AdS spacetimes

We give a holographic argument in favor of the AdS Penrose inequality, which conjectures a lower bound on the total mass in terms of the area of apparent horizons. This inequality is often viewed as a test of cosmic censorship. We further find a connection between the area law for apparent horizons and the Penrose inequality. Finally, we show that the argument also applies to solutions with charge, resulting in a charged Penrose inequality in AdS.

Figure 1

Adapted from [15]. An apparent horizon $\sigma$ (purple) and its outer wedge $O_W[\sigma ]$ (shaded gray). By assumption there exists a Cauchy slice $H$ of $O_W[\sigma ]$ on which $\sigma$ is the minimal area surface homologous to the boundary.

Figure 2

Adapted from [15]. The initial data slice $\mathrm{\Sigma }$ (green) used to prepare the doubled spacetime with an HRT surface $X$ whose area is identical to the area of $\sigma$. Note that $O_W[\sigma ]$ is fixed by taking the component of $\mathrm{\Sigma }$ in $O_W[\sigma ]$ to be identical in the doubled spacetime and in the original spacetime. The entanglement wedge bounded by $X$ and the right boundary is dual to ${\rho }_0$.

Figure 3

The area increases in a spacelike direction along a future holographic screen, which corresponds to time increase on the boundary. Allowing time-dependent sources on the boundary results in an increase in $E$. We maximize $S_{vN}$ subject to $E$ at the past boundary of $O_W[\sigma ]$, so the mass in the Penrose inequality increases correspondingly with the apparent horizon area increase.