I have noticed that in the literature on causality in general relativity one sees apparent counterexamples to the cosmic censorship hypothesis (somehow you have models for gravitational collapse which assume spherical symmetry and things like this so that naked singularities can in fact arise). Hawking conceded that these were counterexamples, but then re-instated the hypothesis because these examples are in some sense unrealistic or unphysical.


I was wondering if the Penrose conjecture is also likely to have 'unphysical' or 'unrealistic' violations (so somehow make some special assumptions and then cook up a black hole spacetime which violates the Penrose inequality), or whether the conjecture is that one can simply never create a counterexample at all to the inequality?


**Edit:** I am aware of the counterexample of Carrasco and Mars to a stronger version of the conjecture. In that paper they find slices of the Kruskal spacetime for which the outermost *generalized* apparent horizon has area strictly greater than $16 \pi M^2$, and so this is not a counterexample to the true Penrose inequality as far as I am aware.


Jarosław Kopiński has mentioned to me in private communication that there is in fact already a counterexample to the Penrose inequality with '[apparent horizon](https://en.wikipedia.org/wiki/Apparent_horizon)':


* Ishai Ben-Dov, *The Penrose inequality and apparent horizons*, Phys.Rev. D **70** (2004) 124031, doi:[10.1103/PhysRevD.70.124031](https://doi.org/10.1103/PhysRevD.70.124031), arXiv:[gr-qc/0408066](https://arxiv.org/abs/gr-qc/0408066),


and so that it is not that surprising that one can construct counterexamples when the inner boundary is even more general.



Counterexamples to the Penrose Conjecture
Having thought about this more and discussed it with others, the answer seems to be that there are likely no counterexamples to the Penrose inequality, even if one allows for unphysical violations.


For recent numerical evidence of this, [a paper](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.103.064025) of Kulczycki and Malec investigates the Penrose inequality and other modifications, finding counterexamples to the modified versions but no violations of the original inequality.


