Does time stop in black holes?

Question

I was going through Hawking's incantations on black holes, where black holes get defined as follows:

Black Hole: A region of spacetime from which the escape velocity exceeds the velocity of light.

I am told that nothing in this universe can have a speed greater than that of light. By Einstein's theory, if black holes actually do have escape velocity greater than the speed of light, this should mean that time in the black hole has stopped completely. In the center of a black hole is a gravitational singularity, a one-dimensional point which contains a huge mass in an infinitely small space, where density and gravity become infinite and space-time curves infinitely, and where the laws of physics as we know them cease to operate. Although supermassive black holes do not usually indulge in the process of moving about the universe, smaller black holes do. I have read from various sources that these black holes move at an appreciable speed comparing with light. So my question here is about the increase in mass and density of the black hole further.

By Einstein's equation, the density of the point where the gravitational singularity has been developed and the mass of the black hole should constantly be increasing. Does this imply that at one time, the black hole should grow vast enough to swallow the entire universe and after the "spaghettification" all the mass of the universe gets stored at the infinitely small point with infinite density?

What I further do not understand is, is that how do black holes grow in mass? In my opinion, by swallowing stars, asteroids, etc., the size of black holes should remain constant and the mass of the point where the singularity is developed should be increased. I have also read that when two black holes crash into each other, their gravitational fields combine to create an even stronger gravitational field.

In this case, are the black holes superposing on each other by pure constructive interference?

Answer 1

... this should mean that time in the blackhole has stopped completely

Not sure how you reach this conclusion. If we assume that general relativity still applies beyond the event horizon of a black hole (although we cannot observe this, we have no reason to think otherwise) then the time co-ordinate of objects that have fallen within the event horizon still changes. The key thing is that all timelike paths inside the future light cone of an object within the event horizon end up at the centre of the black hole. So all objects within the event horizon will reach the centre of the black hole in a finite amount of proper time. What happens to them then we do not know - GR predicts a singularity at the centre of a black hole, which we believe is not physically realistic. But to go beyond this we need a theory of quantum gravity, which we do not yet have.

Answer 2

"if black holes actually do have escape velocity greater than the speed of light, this should mean that time in the black hole has stopped completely"

What it means is that the time axis has twisted round so far that the entire future of an event on the event horizon points entirely into the black hole. You can't escape from the black hole for the same reason you can't escape from the future.

Time doesn't stop at the event horizon. If you fall into a black hole, time continues as normal for you as you pass the event horizon. From the point of view of somebody outside, if you try to join up the time and space coordinates valid a long way away from point to point, as you get closer the flow of time diverges and it looks like time 'stops'. But this is just a peculiarity of the choice of coordinates. It's the same as saying time 'stops' for a ray of light - but we commonly cross the paths of light rays without anything odd happening.

In fact, you can get event horizons without black holes. If a rocket accelerates in a straight line at a constant rate, it gets closer and closer to the path of a particular ray of light. An observer on the rocket can never see anything that happens at the origin after this particular ray of light, as the light from any such event can never catch up to the accelerating observer - it is an event horizon. The diagram here shows the paths of observers with increasing accelerations, and how they all approach a particular diagonal line representing a light ray from the origin. Time for the rocket-propelled observer is shown by the fan of lines through the origin, that go to infinity as the diagonal is approached. An observer on the rocket looking back sees the stationary observer's time apparently 'stop' as their own time continues on to infinity. Of course, time doesn't stop for the stationary observer - they pass through the event horizon without noticing a thing. It's just a peculiarity of the coordinate system of the accelerated observer.

This is an expected consequence of the equivalence principle - that gravity and acceleration look locally the same. The accelerated rocket observers correspond to stationary observers hovering a fixed distance outside a black hole. The stationary observers watching the rocket correspond to a freefall observer falling into the black hole. The spacetime diagram for the Rindler coordinates of uniform acceleration look almost exactly like the Kruskal-Szekeres diagram for a black hole. (Note for the picky, the spacing of the lines in the two diagrams is not quite the same. Rindler 'artificial gravity' goes like $1/r$, not $1/r^2$.) Static observers hovering outside the hole follow the hyperbolic paths in the right-hand quadrant. The event horizon of the black hole is the diagonal lines crossing in the middle. A freefalling observer crosses the event horizon of the black hole like the static observer crosses the event horizon of a uniformly accelerating rocket - simply passing from the past into the future. Once past, he can never move fast enough to recross the line and meet up with the observer on the hyperbola, without exceeding c. Event horizons criss-cross every point in space and time - they're just light rays.

However time does stop at the singularity at the centre of the black hole. The history of particles falling in just stop when they hit it. General relativity predicts the curvature goes to infinity, and can no longer say what happens next.

Answer 3

Escape velocity is just the velocity required to escape completely from the gravitational pull of a body. For example if you're on earth's surface and you are not able to reach escape velocity you will be stuck on earth. Similarly if you are near the event horizon of a black hole and are not able to reach escape velocity you won't be able to escape. Since it's physically impossible to reach the speed of light it is impossible for matter to ever escape a black hole. Light can't escape either but the argument is slightly more complicated.

Time does stop in a sense near a black hole but this statement depends on the observer. If an outside observer $O$ sees another observer $F$ falling into a black hole then $O$ will see the watch of $F$ ticking slower and slower. For $O$ it will take infinite time for $F$ to cross the event horizon. This is not an optical illusion: from $O$'s perspective the watch of $F$ actually slows down. For the falling observer $F$ nothing peculiar happens. For him the event horizon is just an arbitrary distance. So when I say that time slows I mean for an outside observer.

So finally I should stress, like in m4r35n357's comment, that escape velocity is just a theoretical velocity needed to escape from some body. It is not related to time slowing down.