How Do Equations Demonstrate Time Dilation Near Black Holes?

[bunsen1]

What are the equations that shows that time slows down as you near a black hole? I'm trying to prove to a non-believer that the math is there and shows this. Thanks!

 

[Bandersnatch]

Hi bunsen1,

Have you seen the wikipedia article on gravitational time dilation? It's got a good descriptive explanation together with the equations.
http://en.wikipedia.org/wiki/Gravitational_time_dilation
You'll probably want to read the "Outside a non-rotating sphere" section.

Do note that the effect is present in any gravitational field, not just that of black holes. E.g., clocks on Earth tick slower than those in ISS.

Come back if anything's unclear.

 

[PeterDonis]

Do note that the effect is present in any gravitational field, not just that of black holes. E.g., clocks on Earth tick slower than those in ISS.

No, they don't, because the ISS is in orbit about the Earth, which adds an extra effect due to the orbital velocity. Clocks on the ISS actually run slower than clocks on the surface of the Earth, because the slowdown due to orbital velocity outweighs the speedup due to higher altitude. (For objects in higher orbits, such as the GPS satellites, the altitude speedup outweighs the orbital velocity slowdown, so the "natural" rate of the GPS satellite clocks is faster than that of clocks on the Earth's surface. This has to be adjusted for to make the GPS system work.)

To isolate just the gravitational time dilation, you need to consider objects that are static in the gravitational field, i.e., not moving at all relative to the source of the field. Technically, this means objects on the Earth's surface aren't static either, because the Earth is rotating. A truly correct comparison that involves only gravitational time dilation would be an object at rest on the surface of a non-rotating, perfectly spherical planet, vs. an object "hovering" at a high altitude above the planet, using rocket engines or something equivalent to hold itself in place. In the case of a black hole, since there is no surface, you would compare the clocks on two rockets, one "hovering" close to the horizon vs. one "hovering" much higher up.

 

[Bandersnatch]

Welp, that's a brainfart. Thanks for the correction. I actually had written GPS satellites there, before chaning it to "something more iconic" without thinking of the huge difference in orbital radii.

 

[bunsen1]

"In the case of a black hole, since there is no surface, you would compare the clocks on two rockets, one "hovering" close to the horizon vs. one "hovering" much higher up."

If you are in one of the rockets nearing the black hole, does the gravitational time dilation apply to all things affected by time, including biochemical processes, as you get closer and closer to the black hole? In other words, everything slows down inside the rocket, not just relative to another object?

 

[PeterDonis]

If you are in one of the rockets nearing the black hole, does this time dilation apply to all things affected by time, including biochemical processes, as you get closer and closer to the black hole?

Yes.

 

[ChrisVer]

Can't this be seen in Oppenheimer-Snyder coords?

 

[PeterDonis]

Can't this be seen in Oppenheimer-Snyder coords?

What do you mean by Oppenheimer-Snyder coordinates?

(The general answer to your question is that gravitational time dilation is independent of coordinates; it's defined in terms of the time translation symmetry of the spacetime. But how to compute it can be more evident in some coordinates than others.)