Gravitational time dilation

In summary: If you have a "uniform gravitational field" in mind, you will need to specify a "force-free" box.In summary, the conversation discusses the formula for time dilation in a uniform gravitational field and how it can be derived using the binomial theorem. The conversation also touches on the concept of a "uniform gravitational field" in general relativity and the need for a "force-free" box in such scenarios.
  • #1
actionintegral
305
5
Hello,

I am trying to learn about Gravitational Time Dilation. I came across the following formula for time dilation in a uniform gravitational field:

[tex]T_d = {1-} \frac{gh}{c^{2}}}[/tex]

but I cannot find any derivation for this. Can someone point me in the right direction?

Thanks
 
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  • #2
Take a look at http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html

Now, the time dilation factor you state above is [tex]\sqrt{1-\frac{2gh}{c^2}[/tex] (It's equal to T0/T, in the above link).

Using the binomial theorem [tex]\sqrt{1+x}=1+\frac{1}{2x}+\cdots[/tex]

And hence, we obtain the result above [tex]T_d=1-\frac{gh}{c^2}[/tex]
 
  • #3
cristo should have explained why we are supposed to take [itex]2m/r = 2gh[/itex] (in relativistic units in which G=c=1) in the Schwarzschild vacuum.

This takes some explaining! The idea is obvious enough: in a small box near the surface [itex]r=r_0[/tex] of a spherical massive isolated object, with the exterior gravitational field modeled by the Schwarzschild vacuum, right down to the surface, the "gravitational acceleration" is the acceleration of static observers sitting on the surface, which is radially outward with magnitude [itex]m/r^2[/tex], the same expression as in Newtonian gravitation, where needless to say I am using the exterior Schwarzschild coordinate chart. So the needed explanation concerns the "small box". I'll let readers think about that...

The notion of a "uniform gravitational field" in gtr is actually rather tricky. Since actionintegral didn't cite his source, I don't know if he misread something, or if the author was too careless to mention caveats (or wanted to avoid confusing his students), or if the source is a bad one (very possible for websites by someone who is not an acknowledged expert in gtr).
 

1. What is gravitational time dilation?

Gravitational time dilation is a phenomenon in which time passes more slowly in the presence of a strong gravitational field. This is due to the curvature of spacetime caused by massive objects, which causes time to move at different rates depending on the strength of the gravitational field.

2. How does gravitational time dilation occur?

Gravitational time dilation occurs because of the effects of gravity on spacetime. As a massive object, such as a planet or star, warps the fabric of spacetime, it also affects the passage of time in that region. This means that time will move more slowly for objects closer to the massive object compared to those farther away.

3. What is the equation for gravitational time dilation?

The equation for gravitational time dilation is Δt' = Δt√(1 - 2GM/rc^2), where Δt' is the time interval measured by an observer in a strong gravitational field, Δt is the time interval measured by an observer in a weak gravitational field, G is the gravitational constant, M is the mass of the massive object, r is the distance between the two observers, and c is the speed of light.

4. Can gravitational time dilation be observed in everyday life?

Yes, although the effects are very small and only noticeable in extreme conditions. For example, astronauts in orbit around Earth experience slightly slower time compared to those on the surface, and atomic clocks on satellites run slightly faster than those on Earth due to the weaker gravitational field. However, these effects are only noticeable when very precise measurements are taken.

5. How does gravitational time dilation affect the aging process?

According to the theory of general relativity, gravitational time dilation can affect the aging process by slowing down the passage of time for objects in a strong gravitational field. This means that someone living on a planet with a strong gravitational field would age slightly slower compared to someone living on a planet with a weaker gravitational field. However, this effect is so small that it is negligible for all practical purposes.

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