Black hole equations question

In summary: In Oppenheimer-Snyder coordinates, each point in spacetime is specified by its position and its momentum. The time translation symmetry of spacetime means that the same coordinate system can be used to describe the motion of all particles in the system, regardless of their relative motion. That is, given two points in spacetime, you can always find a third point that they intersect, and which corresponds to their relative momenta. This is useful for calculating things like the deflection of light by a massive object, or the gravitational waves that are emitted by a black hole.In summary, gravitational time dilation is the phenomenon whereby the rate at which time passes appears to slow down as an object approaches a black hole.
  • #1
bunsen1
2
0
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!
 
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  • #2
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.
 
  • #3
Bandersnatch said:
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.
 
  • #4
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.
 
  • #5
"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?
 
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  • #6
bunsen1 said:
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.
 
  • #7
Can't this be seen in Oppenheimer-Snyder coords?
 
  • #8
ChrisVer said:
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.)
 

1. What are black hole equations and how are they calculated?

Black hole equations are mathematical formulas used to describe the behavior and properties of black holes. They are calculated using principles from Einstein's theory of general relativity.

2. Can black hole equations be solved analytically?

No, black hole equations are highly complex and cannot be solved analytically. They require sophisticated computational techniques and numerical methods to approximate solutions.

3. How do black hole equations relate to the event horizon?

The event horizon is a key concept in black hole equations. It is the boundary surrounding a black hole from which nothing, including light, can escape. The equations describe the properties of the event horizon, such as its size and shape.

4. What is the Schwarzschild radius and how is it calculated?

The Schwarzschild radius is the distance from the center of a black hole at which the event horizon is located. It is calculated using the mass of the black hole and the speed of light. The formula is 2GM/c^2, where G is the gravitational constant and c is the speed of light.

5. Can black hole equations be used to understand the behavior of other objects in space?

Yes, black hole equations can be applied to a wide range of astronomical objects, including neutron stars and galaxies. They can also be used to study the effects of gravity on space and time in the presence of massive objects.

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