While responding to a question on ResearchGate, I thought about black holes and event horizons.

When you study general relativity, you learn that a star that is dense enough and massive enough will undergo gravitational collapse. The result will be a black hole, an object from which nothing, not even light, can escape. A black hole is surrounded by a spherical surface, its event horizon. It is not a physical surface, but a region that is characterized by the fact that the geometric distortions of spacetime due to gravity become extreme here. Once you cross the horizon, there is no turning back. It acts as a one-way membrane. Anything inside will unavoidably fall into the so-called singularity at the center of the black hole. What actually happens there, no-one really knows; gravity becomes so strong that quantum effects cannot be ignored, but since we don’t have a working quantum theory of gravity, we can’t really tell what happens.

That said, when you study general relativity, you also learn that a distant observer (such as you) can never see the horizon form. The horizon will forever remain in the distant observer’s infinite future. Similarly, we never see an object (or even a ray of light) cross the horizon. For a distant observer, any information coming from that infalling object (or ray of light) will become dramatically redshifted, so much so that the object will appear to crawl to a halt, essentially remaining frozen near the horizon. But you won’t actually get a chance to see even that; that’s because due to the redshift, rays of light from the object will become ever longer wavelength radio waves, until they become unobservable. So why do we bother even thinking about something that provably never happens in a finite amount of time?

For one thing, we know that even though a distant observer cannot see a horizon form, an infalling observer can. So purely as a speculative exercise, we would like to know what this infalling observer might experience.

And then there is the surface of last influence. We may not see an object cross the horizon, but there is a point in time beyond which we can no longer influence an infalling object. That is because any influence from us, even a beam of light, will not reach the object before the object crosses the horizon.

This is best illustrated in a so-called Penrose diagram (named after mathematician Roger Penrose, but also known as a conformal spacetime diagram.) In this diagram, spacetime is represented using only two dimensions on a sheet of paper; two spatial dimensions are suppressed. Furthermore, the remaining two dimensions are grossly distorted, so much so that even the “point at infinity” is drawn at a finite distance from the origin. However, the distortion is not random; it is done in such a way that light rays are always represented by 45° lines. (Such angle-preserving transformations are called “conformal”; hence the name.)

So here is the conformal spacetime diagram for a black hole, showing also an infalling object and a distant observer trying to communicate with this object:

Time, in this diagram, passes from bottom to top. The world line of an object is a (not necessary straight) line that also moves from bottom to top, and is never more then 45° away from the vertical (as that would represent faster-than-light motion).

In this diagram, a piece of infalling matter crosses the horizon. It is clear from the diagram that once that happens, there is nothing that can be done to avoid hitting the singularity near the top of the diagram. To escape, the object would need to move faster than light, in order to cross, from the inside to the outside, the 45° line representing the horizon.

An observer traveling along with the infalling object can bounce, e.g., radar waves off that object. However, that cannot go on forever. Once the observer’s world line crosses the line drawn to represent the surface of last influence, his radar waves will no longer reach the infalling object outside the horizon. Any echo from the object, therefore, will not be seen outside the horizon; it will remain within the horizon and eventually be swallowed by the singularity.

So does the existence of this surface of last influence mean that the event horizon exists for real, even though we cannot see it? This was an argument made in the famous textbook on relativity, Gravitation by Misner, Thorne and Wheeler. However, I tend to disagree. Sure, once you cross the surface of last influence, you can no longer influence an infalling object. Nonetheless, you still won’t see the object actually cross the horizon. Moreover, if the object happens to be, say, a very powerful rocket, its pilot may still change his mind and turn around, eventually re-emerging from the vicinity of the black hole. The surface of last influence remains purely hypothetical in this case; it is defined by the intersection of the infalling object and the event horizon, something that never actually happens.