Penrose inequality in holography

The recent holographic deduction of the Penrose inequality only assumes the null energy condition, while a weak or dominant energy condition is required in the usual geometric proof. We take a step toward filling the gap between these two approaches. For planar or spherically symmetric asymptotically Schwarzschild anti-de Sitter (AdS) black holes, we give a purely geometric proof for the Penrose inequality by assuming the null energy condition. We also point out that two naive generalizations of the charged Penrose inequality are generally not true, and we propose two new candidates. When the spacetime is asymptotically AdS but not Schwarzschild-AdS, the total mass is defined according to holographic renormalization and depends on the scheme of quantization. In this case, the holographic argument implies that the Penrose inequality should still be valid, but we use a concrete example to show that whether the Penrose inequality holds or not will depend on what kind of quantization scheme we employ.

Figure 1

Values of $f(r)e^{-\chi (r)}/r^2$ and $\chi (r)/\chi (r_h)$. We see that $f(r)e^{-\chi (r)}/r^2$ and $\chi (r)$ have the expected asymptotic behavior, but the inequality is still broken.

Figure 2

Schematic diagram of the function $W(r)$ for $k=1$. The value of the function $W(r)$ at the horizon $r_h$ must be non-negative.

Figure 3

Mass parameter ${\tilde{f}}_0^3/2$ with two renormalization schemes.