Area Theorem rederived

Lecture XXVIII: Horizons and the area theorem

We begin by reconsidering the nature of the horizon. As we found in the Schwarzschild case, when properly considered in terms of a set of coordinates that are nonsingular at the horizon (ũ, ṽ, θ, φ), the horizon (in that case, ũ = 0) is actually a 3-dimensional submanifold H of the overall 4D manifold M. This is not surprising since it represents a boundary between the region observable by an o...

The Pythagorean Theorem and Area: Postulates into Theorems

An area theorem for the same-different experiment.

If observers in a same-different experiment base their decisions on the absolute difference between observations on a trial, the area under the receiver operating characteristic equals the maximum proportion of correct decisions that an unbiased independent-observations observer could attain. Even though the differencing strategy is suboptimal, the area measure yields an index of optimal perfor...

Cosmic Censorship, Area Theorem, and Self-Energy of Particles

The (zeroth-order) energy of a particle in the background of a black hole is given by Carter’s integrals. However, exact calculations of a particle’s selfenergy (first-order corrections) are still beyond our present reach in many situations. In this paper we use Hawking’s area theorem in order to derive bounds on the self-energy of a particle in the vicinity of a black hole. Furthermore, we sho...

Is There a General Area Theorem for Black Holes?

The general validity of the area law for black holes is still an open problem. We first show in detail how to complete the usually incompletely stated text-book proofs under the assumption of piecewise C-smoothness for the surface of the black hole. Then we prove that a black hole surface necessarily contains points where it is not C (called “cusps”) at any time before caustics of the horizon g...