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As a newbie I was reading the book "The Theory of Everything", and came about these two paragraphs which I just don't understand:

I had already discussed with Roger Penrose the idea of defining a black hole as the set of events from which it was not possible to escape to a large distance. This is now the generally accepted definition. It means that the boundary of the black hole, the event horizon, is formed by rays of light that just fail to get away from the black hole. Instead, they stay forever, hovering on the edge of the black hole. It is like running away from the police and managing to keep one step ahead but not being able to get clear away.

Suddenly I realized that the paths of these light rays could not be approaching one another, because if they were, they must eventually run into each other. It would be like 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 light rays in the event horizon had to be moving parallel to, or away from, each other.

The first paragraph is for context, the second para is what I don't understand at all.

The paths of these light rays could not be approaching one another.

Why?

It would be great if you explain the analogy given after this in a more detailed yet layman manner.

Qmechanic
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Gautam
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  • See also this question: https://physics.stackexchange.com/questions/451884/how-do-light-rays-move-parallel-at-the-event-horizon-and-why-is-this-necessary – Nadav Har'El Sep 19 '23 at 08:49
  • It means that the photons need to be on a purely radial path (transverse paths already fall in below the photon sphere which is far above the horizon) – Yukterez Sep 26 '23 at 03:32

2 Answers2

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It's like running on a treadmill at maximum speed. It's such treadmill that it's wide, and it's curved longitudinally, so that if you slow down, you drift backwards, were your maximum speed is lower because of the curvature.

Now unless you run straight ahead, you fall from the the treadmill, right?

Oh yes, there are many people running at their maximum speed on the wide treadmill. So unless you run straight ahead, you move towards an other runner, and you cause the other runner to became gravitationally time dilated by your mass, which will lead to the other runner falling off from the treadmill too.

In this analogy runners are light rays. And non-parallel light rays have a slowing down effect on each other. Like in this case: https://en.wikipedia.org/wiki/Kugelblitz_(astrophysics)

stuffu
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The idea of this has to do with what is referred to as the "expansion" of null (light) rays. When the expansion $\theta $ is zero, the two light rays will just be parallel to each other, whereas if it is positive (or negative), the would move away from (respectively towards) each other. If the rays were to meet, the expansion must become zero after finite affine time, i.e. $\theta =0$, after which they would emerge with a positive expansion $\theta \to +\infty $. This is evident from the Raychaudari equation, which, under the assumption of the null energy condition (which states that if the initial value of the expansion is negative, it must be so until a finite affine time before emerging with a positive expansion (this point is referred to as a caustic or a focal point)) requires that this be the case. I believe this was from A Brief History of Time (?), on the discussion on the singularity theorems, following Penrose's 1965 theorem.

VaibhavK
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