Penrose inequality and apparent horizons

A spherically symmetric spacetime is presented with an initial data set that is asymptotically flat, satisfies the dominant energy condition, and such that on this initial data $M<\sqrt{A/16\pi }$, where $M$ is the total mass and $A$ is the area of the apparent horizon. This provides a counterexample to a commonly stated version of the Penrose inequality, though it does not contradict the true Penrose inequality.

Figure 1

A spacetime diagram of the Kruskal extension of a Schwarzschild spacetime. Two angular dimensions are suppressed. Each point therefore represents a 2-sphere.

Figure 2

A spacetime diagram of a closed, dust, RW spacetime.

Figure 3

A spacetime diagram of the original OS model.

Figure 4

A spacetime diagram of a generalized OS spacetime with ${\chi }_0>\frac{\pi }{2}$.

Figure 5

Embedding diagrams of OS-like spacetimes sliced at the moment of time symmetry. One angular dimension is suppressed by setting $\theta =\pi /2$. Left panel: a normal OS spacetime. Right panel: a generalized OS spacetime with ${\chi }_0>\frac{\pi }{2}$.

Figure 6

A spacetime diagram of a counterexample to the apparent horizon Penrose inequality. Notice how the slice passes just below the event horizon in portion 1. Therefore, the apparent horizon is located where the initial data slicing intersects the marginally outer trapped 2-spheres in the second Schwarzschild region, portion 3.