Time dilation around a planet.

In summary, gravitational time dilation and motion time dilation are different concepts. In order to accurately calculate gravitational time dilation, both the effects of curvature and motion must be taken into account. Using the SR doppler formula and the parallel transport method can provide a correct figure for gravitational redshift, also known as gravitational time dilation. Additionally, the ratio of the kinetic time dilations of a freefalling particle can be used to calculate the gravitational time dilation.
  • #1
cragar
2,552
3
If I wanted to figure out the gravitational time dilation at different points in a the field around earth, could I just use v=gt. and find the speed that I would be moving if I went into free-fall from point x to point y. To figure out the time difference between x and y?
and the t in v=gt would be the time in my free-fall frame. Then once I knew v , I could use the standard time dilation equation from SR.
 
Physics news on Phys.org
  • #2
cragar said:
If I wanted to figure out the gravitational time dilation at different points in a the field around earth, could I just use v=gt. and find the speed that I would be moving if I went into free-fall from point x to point y. To figure out the time difference between x and y?
No. Gravitational time dilation is different from movement time dilation. Here is the formula:
http://en.wikipedia.org/wiki/Gravitational_time_dilation#Outside_a_non-rotating_sphere
 
  • #3
I understand that its different I am just wondering why I can't use this math approach to figure out the time dilation between these 2 frames.
 
  • #4
cragar said:
I understand that its different I am just wondering why I can't use this math approach to figure out the time dilation between these 2 frames.

Because both curvature and motion contribute to clock difference between a source and receiver. Imagine a receiver on hovering high above earth. Imagine a bunch of clocks at some place on earth: sitting on the ground, thrown upwards, kicked sidway, etc. The receiver will see a different rate for each of the source clocks, influenced by their difference in motion and by the curvature of spacetime between them and the receiver.

In a situation like this (essentially static gravity), we call the difference observed for the clock sitting on the ground gravitational time dilation. Then the different clocks moving relative to this can be analyzed as an additional motion effect.

There is a way to treat all the cases in a uniform way a as a function of source and target world line and the intervening geometry. However, I don't know if you have the background for it: do you know what parallel transport of 4-vectors in curved spacetime is?
 
  • #5
ok thanks for your response. In my scenario I wasn't trying to figure out the time dilation for a moving observer in a G field. I was just wondering if I Could find the speed one would be at if he was in free-fall until he got to the other point, And use that speed to find the time dilation using SR equations. Is there a way to figure out how much light would be red shifted from one frame to another. And these frames are stationary in the gravitational field. Then once I knew how much the light was red shifted I could figure out what speed an observer would need to move to cancel the redshift and use this v in the SR equation to figure out the time dilation in the Gravitational field. I am not saying the observers are moving, I am just using this as a way to figure out the effect.
 
  • #6
cragar said:
ok thanks for your response. In my scenario I wasn't trying to figure out the time dilation for a moving observer in a G field. I was just wondering if I Could find the speed one would be at if he was in free-fall until he got to the other point, And use that speed to find the time dilation using SR equations. Is there a way to figure out how much light would be red shifted from one frame to another. And these frames are stationary in the gravitational field. Then once I knew how much the light was red shifted I could figure out what speed an observer would need to move to cancel the redshift and use this v in the SR equation to figure out the time dilation in the Gravitational field. I am not saying the observers are moving, I am just using this as a way to figure out the effect.

If you want to use something from SR for gravitational time dilation, what you should use is SR doppler rather than SR time dilation. Then, the parallel transport method I alluded to applies - if you compute the tangent vector (4-velocity) of a ground stationary observer, and parallel transport along a light path (null geodesic) to a higher stationary observer world line, you find the two vectors (transported tangent; and tangent of higher observer world line) are not parallel, implying a local relative motion that is equivalent to the gravitational effect. Then, if you apply the SR doppler formula, using this local relative velocity of the source along with the local light propagation vector, you would get a correct figure for gravitational redshift (= what is called gravitational time dilation).
 
  • #7
cragar said:
I was just wondering if I Could find the speed one would be at if he was in free-fall until he got to the other point, And use that speed to find the time dilation using SR equations.

Actually, if you start with a scalar speed v of a particle as measured by a local static observer at r and freefall, regardless of the path, to radius s with resulting locally measured scalar speed u, the ratio of the kinetic time dilations will equal the ratio of the gravitational time dilations, so that

sqrt(1 - (v/c)^2) / sqrt(1 - (u/c)^2) = sqrt(1 - 2 m / r) / sqrt(1 - 2 m / s)

This comes from conservation of energy, whereby the quantity E_r z_r = m c^2 sqrt(1 - 2 m / r) / sqrt(1 - (v/c)^2) is conserved for a particular freefalling particle, where z_r is the local gravitational time dilation. The same is true for frequencies of light, although massless, giving E_r z_r = h f_r z_r. So the smaller the gravitational time dilation, the greater the locally measured frequency of light.
 
Last edited:

1. What is time dilation around a planet?

Time dilation around a planet is a phenomenon where time moves slower for objects that are closer to a massive body, such as a planet, due to the curvature of space-time caused by the planet's gravitational pull.

2. How does time dilation around a planet occur?

Time dilation around a planet occurs because objects closer to the planet experience a stronger gravitational pull, which causes the fabric of space-time to warp and slow down time for those objects.

3. What is the formula for calculating time dilation around a planet?

The formula for calculating time dilation around a planet is t = t0√(1-2GM/rc^2), where t is the time experienced by an object close to the planet, t0 is the time experienced by an object far from the planet, G is the gravitational constant, M is the mass of the planet, r is the distance from the planet, and c is the speed of light.

4. How does time dilation around a planet affect space travel?

Time dilation around a planet can significantly affect space travel, as it means that time will move slower for objects closer to the planet. This can lead to discrepancies in time between two objects, causing issues with synchronization and communication.

5. Is time dilation around a planet the same as time dilation in space?

No, time dilation around a planet is different from time dilation in space. Time dilation in space occurs due to high speeds, while time dilation around a planet is caused by the curvature of space-time due to the planet's gravitational pull.

Similar threads

  • Special and General Relativity
Replies
16
Views
597
  • Special and General Relativity
2
Replies
58
Views
3K
  • Special and General Relativity
Replies
7
Views
1K
  • Special and General Relativity
Replies
10
Views
403
  • Special and General Relativity
Replies
1
Views
135
  • Special and General Relativity
2
Replies
58
Views
2K
  • Special and General Relativity
Replies
2
Views
315
  • Special and General Relativity
2
Replies
37
Views
3K
  • Special and General Relativity
Replies
21
Views
380
  • Special and General Relativity
Replies
11
Views
941
Back
Top