New Area Law in General Relativity

We report a new area law in general relativity. A future holographic screen is a hypersurface foliated by marginally trapped surfaces. We show that their area increases monotonically along the foliation. Future holographic screens can easily be found in collapsing stars and near a big crunch. Past holographic screens exist in any expanding universe and obey a similar theorem, yielding the first rigorous area law in big bang cosmology. Unlike event horizons, these objects can be identified at finite time and without reference to an asymptotic boundary. The Bousso bound is not used, but it naturally suggests a thermodynamic interpretation of our result.

Figure 1

To construct a past or future holographic screen (blue solid line), one chooses a null foliation of the spacetime (green diagonal lines) and finds the maximum area surface $\sigma$ on each null slice (green dots) [12]. We prove that the area of the leaves $\sigma$ grows monotonically (in the direction of the blue triangle). (a) Black hole formed by collapse of a dust ball (gray shaded), with a null foliation by future light cones of a worldline at $r=0$. Marginally trapped spheres (dots) form a future holographic screen. This is not a dynamical horizon: the screen is timelike inside the ball and becomes spacelike only where low-density dust falls in at late times (white region). (b) Expanding matter-dominated Friedmann-Robertson-Walker universe, with a foliation by past light cones of a comoving worldline. Marginally antitrapped spheres form a past holographic screen, which is everywhere timelike.

Figure 2

There are four types of signature changes on a future holographic screen $H$ that would violate the monotonicity of the area. The dot indicates a reference leaf $\sigma (0)$, near which the area increases in the direction indicated by the arrow. On the far side of the “bend” it decreases. Note that $H$ is tangent to the marginal null direction ($\theta =0$) at the transition. Thus, the marginal null hypersurface $N$ orthogonal to a nearby leaf intersects $H$ twice (diagonal line). In the classical generic regime we consider, this is impossible.