Black Hole Formation

Question

If you plot a space-time diagram of an object falling through the event horizon of a black hole, and draw the past and future "light cones" of the object at every point, wouldn't the point infinitely to the event horizon have a light cone which allows light being radiated by the object to reach an observer outside the event horizon at time = infinity? (at the point when the object touches the event horizon, a radiated photon will never reach the observer outside the event horizon) If so, then why don't we, outside the event horizon of black holes being formed by the collapse of stars, observe light from the collapsing star (AKA the light from a supernova forming a black hole) forever? Please let me know if I'm wrong.

Answer 1

As material falls to a horizon, emitting light as it goes, there are three effects to consider: the worldline of the emitter, the redshift of the light, and the intensity of the light (headlight effect).

As the emitter sends out light signals, they get more and more red-shifted and more and more dim and more and more infrequent at a receiver somewhere outside the horizon. For signals setting out from locations close to the horizon, the frequency and the intensity fall exponentially with time at the receiver. Faced with such an exponential decay, you could say it never quite reaches zero, but we don't normally say that for other cases of exponential decay such as atoms decaying to their ground state. We just say the atom decays. So by the same logic we should say the received light from a collapsing star falls to zero intensity, and it is not necessary to wait an infinitely long time for this to be so. Therefore the black hole is indeed black, and in practice the timescale for these decays is short (some tens of microseconds for a one solar mass black hole).

And yet, according to one very natural definition of simultaneity, the falling material does indeed not quite cross the horizon in any finite amount of time registered on the distant clock, so the matter moving to form the black hole never finishes its collapse towards its own horizon. This sounds like a very odd conclusion, but it is owing to the relativity of simultaneity and a time dilation which tends to infinity. There are plenty of other reference frames, and thus definitions of simultaneity, in which the black hole does form in a finite time. And the predictions for what emitted signals do when they arrive elsewhere is independent of such details. The signals die away. The hole is black.

Answer 2

You are not wrong. If we could live forever, and if we could observe indefinitely small light energies, we would observe the light from material falling into to a forming black hole forever. Because time appears to stop at the Schwarzschild radius, an issue is raised as to whether a singularity can actually form. In 1939 Julius Robert Oppenheimer and one his students, Hartland Snyder, published the seminal paper on gravitational collapse to a black hole (Oppenheimer J. R., Snyder H., 1939, On Continued Gravitational Attraction, Phys. Rev. 56, 455). They concluded that, from the point of view of an exterior observer, “it is impossible for a singularity to form in a finite time.”

The things usually called black holes in modern terminology are stable state solutions of Einstein's equation assuming boundary conditions which never physically appear in practice. As described by Oppenheimer and Snyder, these solutions never occur in our universe.