Gravitational Time Dilation Formula?

[Ethan O'Farrell]

 I am curious about the formula for gravitational time dilation. I have scoured the internet and can't seem to find any explanation for the variables in the formula. I am aware of velocity time dilation and that formula, but gravitational time dilation stumps me. It should be noted I am only a sophomore in high school so i might not fully grasp all of the mathematics yet however, any help is much appreciated. 

[swansont]

It's the change in the gravitational potential, divided by c^2. So near the surface of the earth, gh/c^2

[Ethan O'Farrell]

How would I use the formula, let's say, to calculate the gravitational time dilation on the moon?

[adsar2]

On 8/10/2017 at 9:44 AM, Ethan O'Farrell said:

 I am curious about the formula for gravitational time dilation. I have scoured the internet and can't seem to find any explanation for the variables in the formula. I am aware of velocity time dilation and that formula, but gravitational time dilation stumps me. It should be noted I am only a sophomore in high school so i might not fully grasp all of the mathematics yet however, any help is much appreciated. 

The variables stem from using the [url=https://en.wikipedia.org/wiki/Schwarzschild_metric#The_Schwarzschild_metric]Schwarzschild solution[/url] to the Einstein Field Equations.

[adsar2]

On 8/10/2017 at 10:00 AM, Ethan O'Farrell said:

How would I use the formula, let's say, to calculate the gravitational time dilation on the moon?

This may be an easy or a very difficult question, depending how you looked at it:

 

1. The easy way - ignore the gravitational mass of the Moon. Apply the Schwarzschild solution

 

[math](d\tau/dt)^2=(1-r_s/r) -(v/c)^2[/math]

 

Where r is the distance between the Earth and Moon centers, r_s=9mm is the Earth Schwarschild radius, v is the tangential speed of the Moon wrt the Earth

 

2. The very difficult way, use the "two body solution" for the EFE's. This solution has become available only recently. The same steps apply as above.

[ravell]

The formula for the gravitational time dilation can be easily deduced by yourself, using VETER program available on the link:

https://www.dropbox.com/s/9ljwu5wwsi0v9up/VerificationTheoryRelativity.xlsx?dl=0.

Based on the sheet "GPS2" of the above program, the derived formula will be as follows:

  TR1 = TR2*c/ [c^2 - 2GM*(R2 - R1)/(R1*R2)]^0,5

Where:
TR1 - clock tick at a lower level (R1), TR2 - clock tick at a higher level (R2),
  G - gravitational constant, M - mass of the object (eg moon), R1 and R2 – radius  of the orbits between which the time dilation is calculated, c - speed of light.

From the above formula we can see that the ticks of the light clock at a lower level (R1) will be longer than the clock ticks at a higher level (R2). That means that the light clocks on a higher altitude run faster than the clocks on a lower altitude.