How much Gravity is required to stop time?

Question

Clocks free of gravitational influence run faster than those experiencing gravity. Is it possible for gravitational influence to bring time to a stop? Additionally can acceleration affect clocks in the same manner as gravity with time running fastest with no acceleration. In a hypothetical case where a clock in deep space is not subjected to gravitational influence and is therefore not moving, does the rate of time reach maximum?

Answer 1

Time stops on the event horizon of a black hole according to a distant observer. One might consider Einstein'r original insight into relativity, where he realized on taking a street car that if the car were moving the speed of light he would never see a clock he was moving away from tick off its next increment of time. Something similar happens as one observes a body fall towards a black hole. The object is time dilated and red shifted arbitrarily as it approaches the event horizon.

More formally this can be see with the Schwarzschild metric for a nonrotating black hole $$ \text{d}s^2 = \left(1 - \frac{2GM}{rc^2}\right)\text{d}t^2 - \left(1 - \frac{2GM}{rc^2}\right)^{-1}\text{d}r^2 - r^2\text{d}\Omega^2. $$ We look at the situation for light rays leaving a body close to the black hole hole with the horizon at $r = 2GM/c^2$. The interval for null rays is zero $ds = 0$ and we integrate the time $$ \int^T \text{d}t = \int^R\left(1 - \frac{2GM}{rc^2}\right)^{-1} \text{d}r = R - \frac{2GM}{c^2}\ln\left(R - \frac{2GM}{c^2}\right). $$ This diverges as $R \rightarrow \frac{2GM}{c^2}$, which means you have to wait an infinite time to see the clock tick off the time it crossed the horizon.

Answer 2

A clock running faster or slower must always be considered with respect to some other clock. As mentioned in the comments, an observer will always see a clock they are holding ticking at the very same rate, but they might see some other observer's clock ticking faster or slower. For an example concerning the movie Interstellar, see this answer I've wrote for another post. I'll try to give the basics needed in here in any way.

One situation in which one could say "time stops" in a very specific and technical sense is at the event horizon, which is the boundary of a black hole. Suppose, for example, the situation in Interstellar: there is a planet really close to the black hole and a spaceship further away. Some people from the spaceship went to the planet to explore it. They are closer to the black hole than the people on the spaceship.

If the crew close to the black hole check their own watches, they won't see anything weird. However, suppose now they pick a telescope and take a look at a clock on the spaceship: they would see the clock ticking much faster than their own watches. Hence, in their point of view, time on the spaceship is passing faster. Similarly, from the point of view of the spaceship, time on the planet is passing by at an incredibly slow rate.

As you get closer to the black hole, the effect gets stronger. For example, suppose that instead of landing on a planet, some astronaut decided they wanted to jump straight into the black hole while holding a walkie talkie, so they could tell the people over at the spaceship what they are seeing. In addition, the walkie talkie sends a signal every second (according to the falling astronaut's watch) to the spaceship.

At first, the people on the spaceship wouldn't notice anything weird. They would be getting the signals roughly one second apart and so on. However, as the astronaut falls further down, the signals the crew receives star to take longer to come. Eventually, there is a two-second delay between signals, instead of one. As the astronaut falls further, the delay increases, until eventually it is infinite. As the delay increases, the people at the spaceship look through the window and see the astronaut getting closer to the black hole and redder (due to gravitational redshift). Eventually, the redshift gets so large they can't even see the astronaut anymore, but they never saw them falling into the black hole. In this very specific sense, the people at the spaceship see the astronaut's watch stopping at the event horizon of the black hole.

Why did I bother to write "in this very specific sense"? Because the astronaut can't notice anything about this. They will enter the black hole in finite time with respect to their watch and, if the black hole is large enough, the gravitational field at the horizon will be so weak they won't feel a thing until they are way down the black hole. Not only their time didn't stop, but they didn't feel anything weird when crossing the horizon and (assuming, for simplicity, an uncharged, non-rotating black hole) they will reach the singularity at "the core" of the black hole in finite time with respect to their watch.

Let's summarize our conclusions. Can strong gravitational fields stop time? Only if by "stop time" you mean "an external observer will see the watch of the observer in a strong gravitational field ticking slower and slower until it stops", but the observer who is in a strong gravitational field will not feel like their own time is slowing down or anything similar.