Does time and time dilation exist in a black hole?

Since being near stronger gravitational fields causes time dilation, can this be generalized to the region beyond the event horizon? i.e , is it meaningful to say that someone outside the black hole would travel infinitely far into the future compared to someone inside the black hole?

Yes, it is certainly true that an observer in the exterior of the black hole can reach future infinity but an observer inside the black hole cannot. But this is not a result of time dilation but rather the geometry of black holes and causality.

This can be seen by the fact that once an observer crosses the event horizon, the spacelike and timelike Killing vectors get interchanged. One way to interpret this is that the the poor chap inside the event horizon will never remain still and WILL hit the singularity. The singularity in the interior of the black hole is a result of geodesic incompleteness (for most black holes) i.e. all objects which fall into the black hole will encounter the singularity in a finite proper time and get crushed.

So essentially, an observer inside a black hole has only a finite amount of proper time left while an observer in the exterior of the black hole can in principle survive till future infinity.

Look at the above diagram (which is known as a Penrose diagram): The rules are simple:

1. All events (and observers) start off from the line Past Infinity.
2. Since the events are causal, the lines that start from past infinity cannot be spacelike i.e. they must travel within a region of $\pm$ 45 degrees with respect to the vertical axis at each point.
3. Only light can travel at precisely 45 degrees. In fact, light always travels at 45 degrees.

So you see here, once you enter the even horizon, there is no way you will escape it (unless you travel at a path which makes an angle greater than 45 degrees with respect to the vertical i.e. it travels faster than the speed of light. But we know this is impossible.) You will always hit the singularity. In fact this is also why the escape velocity at the horizon is the speed of light. You can see all this from simple geometry. On the other hand, any observer who does not enter the event horizon can continue to future infinity.

I also want to acknowledge the source of the image as Viktor Toth's website (https://spinor.info/weblog/?p=5576).

• This diagram needs some labels on its axes, what is space and what is time? This needs some futher description or a link to the source. Feb 28, 2016 at 22:45
• There is no space and time distinction in a Penrose diagram. There is no space and time distinction in GR, as a matter of fact. They're both one quantity i.e. spacetime.
– user106422
Feb 28, 2016 at 22:49
• As the creator of this particular diagram, I would have appreciated attribution. (spinor.info/weblog/?p=5576) Jul 18, 2016 at 17:22
• Dear Viktor, you're right. My apologies. I will fix this asap.
– user106422
Jul 18, 2016 at 17:23
• Dear @ViktorToth, I have acknowledged the source of the image. This is something I should have done way earlier, my apologies.
– user106422
Jul 18, 2016 at 18:02