# Penrose inequality In asymptotically flat spacetime, the Penrose inequality states that $M_{i} \geq M \geq \sqrt{A / 16 \pi} \geq \sqrt{A_{i} / 16 \pi}.$So the inequality compares the initial mass with the area of an apparent horizon at some initial hypersurface. The first sign follows from the assumption that the mass does not increase; the second comes from assuming the final state is a Kerr solution; the last one comes from the [[0005 Black hole second law|BH second law]]. It is a generalisation of [[0116 Positive energy theorem|the positive energy theorem]] and is implied by (and therefore a test of) [[0221 Weak cosmic censorship|the cosmic censorship]]. In higher dimensions, weak cosmic censorship can be violated, but those naked singularities are expected to be mild; so a weaker form of WCC can still imply the Penrose inequality. The AdS$_{d+1}$ version is given by$M-M_{0} \geq \frac{(d-1) \Omega_{d-1, k}}{16 \pi}\left[q^{2}\left(\frac{\Omega_{d-1, k}}{A}\right)^{\frac{d-2}{d-1}}+k\left(\frac{A}{\Omega_{d-1, k}}\right)^{\frac{d-2}{d-1}}+\frac{1}{l^{2}}\left(\frac{A}{\Omega_{d-1, k}}\right)^{\frac{d}{d-1}}\right]$ where $M_{0}=\frac{1}{8 \pi}(-k)^{d / 2} \frac{(d-1) !^{2}}{d !} l^{d-2} \Omega_{d-1, k}$for $d$ even and zero for $d$ odd. Here $k$ depends on the topology of the asymptotic boundary and $q$ is the electric charge. More generally, beyond Einstein gravity (say with matter or [[0006 Higher-derivative gravity|higher-derivative corrections]]), the Penrose inequality in asymptotically AdS spacetimes states that the area of any apparent horizon $\sigma$ in a spacetime with mass $M$ satisfies$A[\sigma] \leq A_{\mathrm{BH}}(M),$where $A_{\mathrm{BH}}(M)$ is the area of the stationary black hole with mass $M$ in the theory. If the theory has more than one stationary black hole with mass $M$, choose the one with the largest area. ## Refs - original: [[1973#Penrose]] - reviews - [[2003#Bray, Chrusciel (Review)]] - [[2009#Mars (Review)]] - proofs - for maximal hypersurfaces and with spherical symmetry: - [[1994#Malec, Murchadha]] - with spherical symmetry: - [[1994#Hayward]] - with time-symmetry but no other symmetry: - [[2001#Huisken, Ilmanen]] - [[1999#Bray]] - cohomogeneity-one initial data: - [[2024#Khuri, Kunduri]] - in AdS - [[ItkinOz2011]][](https://arxiv.org/pdf/1106.2683.pdf): simple derivation of the expression - [[HusainSingh2017]][](https://arxiv.org/abs/1709.02395): spherically symmetric, but time asymmetric - [[Ambrozio2014]][](https://arxiv.org/abs/1402.4317): time-symmetric perturbations - [[2022#Khuri, Kopinski]]: some perturbations of Schwarzschild-AdS - [[2022#Folkestad]]: scalar field with various potentials - with higher-genius boundary topology - [[2013#Lee, Neves]] - [[2022#Alaee, Hung, Khuri]] - holographic proof - [[2019#Engelhardt, Horowitz]] - quantum version - [[2019#Bousso, Shahbazi-Moghaddam, Tomasevic (Aug, Letter)]] - with two angular momenta (for bi-axisymmetric initial data in 5D spacetime) - [[AlaeeKunderi2022]][](https://arxiv.org/abs/2205.09737) - with rotations - [[2023#Feng, Yan, Gao, Lau, Yau]] - applications - [[2022#Folkestad]]: use it as a swampland condition - [[2025#Folkestad]]: all known examples violating Penrose inequality also violates positive mass theorem ## Relations to other concepts - [[0487 ADM mass|ADM mass]]: [[YangLiCai2022]][](https://arxiv.org/abs/2205.08246)