Black Hole Event Horizon Properties and Area Theorem

Key Insight

A black hole is precisely defined as the collection of spacetime events from which escape to a large distance is impossible. Its boundary, known as the event horizon, is composed of light rays that are perpetually on the edge of escaping, hovering indefinitely without getting clear away, akin to constantly evading capture. These light rays defining the event horizon are critically observed never to approach one another; if they did, they would eventually collide and fall into the black hole, contradicting their status as boundary elements. Therefore, the paths of light rays within the event horizon must always move parallel to, or away from, each other, much like the edge of a distant shadow where light rays do not converge.

This fundamental property implies that the area of a black hole's event horizon can only remain the same or increase over time, never decrease. A decrease would necessitate some light rays in the boundary approaching each other, which is forbidden. The event horizon's area expands whenever matter or radiation falls into the black hole. Moreover, if two black holes collide and merge, the event horizon area of the resulting single black hole will be greater than or equal to the sum of the areas of the original black holes. This nondecreasing characteristic of the event horizon's area imposes a significant restriction on how black holes can behave, a realization made in November 1970.

The nondecreasing behavior of a black hole's area strikingly resembles the second law of thermodynamics, which states that the entropy (a measure of disorder) of an isolated system always increases. For example, gas molecules spread out to fill a container, or different gases mix, increasing disorder. This led to the suggestion that the event horizon's area could represent the black hole's entropy. This proposal seemed to prevent violations of the second law, as the sum of external matter entropy and horizon area would not decrease when matter fell into a black hole. However, this idea faced a crucial challenge: if a black hole possessed entropy, it must also have a temperature and, consequently, emit radiation, which directly contradicted the definition of a black hole as an object from which nothing escapes.