What does Hawking mean by "Light rays that form the edge of the event horizon could never approach one another"?

Question

In Hawking's famous publication "A Brief History of Time", he discusses the steps leading to his discovery of Hawking Radiation. He first argues that due to the aforementioned feature of light at the edge of the event horizon, the area of the event horizon is non-decreasing. He then goes on to say that black holes appear to violate the Second Law of Thermodynamics since you can throw particles in a black hole decreasing entropy outside the black hole. Therefore it was proposed that the area of the event horizon would be proportional to the entropy of the black hole allowing it to conform with the Second Law. This led him to deduce that black holes must have temperature and thus must emit radiation.

Unfortunately I have had difficulty grasping the first step in this sequence - i.e. how Hawking initially postulated that black holes have non-decreasing area. His exact words are:

"Suddenly I realised that the paths of these light rays could never approach one another. If they did, they must eventually run into one another. It would be like meeting someone else running away from the police in the opposite direction - you would both be caught! (Or, in this case, fall into a black hole.) But if these light rays were swallowed up by the black hole, then they could not have been on the boundary of the black hole. So the paths of light rays in the event horizon had always be moving parallel to, or away from, each other. Another way of seeing this is that the event horizon, the boundary of the black hole, is like the edge of a shadow - the shadow of impending doom. If you look at the shadow cast by a source at a great distance, such as the sun, you will see that the rays of light in the edge are not approaching each other.

If the rays of light that form the event horizon, the boundary of the black hole, can never approach each other, the area of the event horizon might stay the same or increase with time but if could never decrease because that would mean that at least some of the rays of light in the boundary would have to be approaching each other.

I guess the root of my difficulty here is my interpretation of the edge of the event horizon. I thought it was essentially the surface of a sphere or ellipsoid whereby matter has to travel at the speed of light in order to escape. I can't seem to fit in the ideas of "parallel" and "approaching each other" in the context of the edge of the event horizon.

I know I could be missing something really obvious here but any help appreciated!

Answer 1

Raychaudhuri's equation was obtained using parameters involving the propagation of null congruence (a group of light rays spaced out over space and propagating in time) and is used to calculate what are called optical scalars in General Relativity. It is best derived in the Newman Penrose formalism, and one of the scalar parameters included are the congruence 'expansion'. There are two others, one is the shear and the other the twist.

These are described and the equations covered at Wikipedia.

It was used by Penrose, Hawking and others to prove various theorems on the inevitability of the formation of Black Holes, with horizons, and some properties of any such horizon. One of the results was that to have an event horizon form as collapse proceeds, it is necessary (and I can't remember what else was required to make it also sufficient, perhaps it was the positive energy condition but I'm not claiming I am stating it all correctly) that there are null congruences that are converging, which was defined as the expansion optical scalar being negative. See the reference for the conditions. The

I have not read in full detail Hawking's "A Brief History of Time", but in GR that was used to prove some of the Black Hole (BH) theorems that once a trapped surface forms (and again see the reference for any other conditions), which requires negative expansion null congruences, a BH will inevitably form, and under some conditions lead to the singularity inside the horizon. And it was also shown that at (or near?) the horizon the congruences can ONLY diverge or stay parallel (positive or zero expansion scalar) which means the horizon expands or stays the same, never gets smaller.

That would explain Hawking's claim about the horizon never shrinks. He was just doing it in a popular book, not a scientific paper. Knowing some of the real physics you can understand his non-scientific explanations. I do not know if otherwise they are convincing, they would not convince me but titillate my imagination. That is also why this was a great question, question the popularly stated explanations if you feel you really want to understand. You will never learn, or really understand real physics from popular books, but they are a lot of fun if you know some of the physics, and can give you some intuition, but careful not to do it w/o knowing some of the physics or you'll develop the wrong intuition

See one reference at Wikipedia for the Penrose-Hawking singularity theorems, it talks about converging light rays and it also discusses some the Raychaudhuri's equation, which also involved Sachs and other physicists of the time. There's more to read to get it good enough, but it's a start. See this page.

Answer 2

In order to have a simple understanding of Hawking's statement, "light rays that form the edge of the event horizon can never approach one another," think of a plane slicing the BH diameter. The resulting circle will be a "few" photons "wide." The photons closer to the center of the BH will spiral into the center, those that are "just right" will stay at the same radius (parallel to each other), the remainder will escape (move away from the BH center) from the event horizon. Therefore, Hawking's statement is correct!