Time dilation due to SR plus GR

[powerplayer]

So I've been trying to find an equation that will represent total time dilation.

I've looked through a couple threads and it seems the consensus of the threads I've seen on the topic say that total time dilation is the product of time dilation due to velocity and gravity. But I'm not clear on it and would like to be sure.

Here are a few example situations that I'd like to know how the total time dilation worked:

1. Your traveling straight with a very high velocity and you pass a massive object with close proximity.

2. You orbit a massive object with high velocity.

3. you travel at high velocity with no gravity effecting you.

4. you maintain 0 velocity in a strong gravitational field.

are there any equations(or derived equations) that will describe the total time dilation for all these situations?

 

[Passionflower]

So I've been trying to find an equation that will represent total time dilation.

I've looked through a couple threads and it seems the consensus of the threads I've seen on the topic say that total time dilation is the product of time dilation due to velocity and gravity. But I'm not clear on it and would like to be sure.

Here are a few example situations that I'd like to know how the total time dilation worked:

1. Your traveling straight with a very high velocity and you pass a massive object with close proximity.

2. You orbit a massive object with high velocity.

3. you travel at high velocity with no gravity effecting you.

4. you maintain 0 velocity in a strong gravitational field.

are there any equations(or derived equations) that will describe the total time dilation for all these situations?

For cases 1 and 2 you can use the Schwarzschild solution. If you are interested in the velocity with respect to static observers you can decompose the gravitational and velocity based time dilation. Thus the velocity with respect to arbitrary observers simply becomes a calculation.

For case 3 you would have to describe the velocity with respect to something else as there is no such thing as absolute velocity. Of course in all cases the gravitational part would be zero.

For case 4 you have to be more specific. In case of the Schwarzschild solution it is simple because if we take the velocity with respect to stationary observers the velocity is zero, so there is only gravitational time dilation.

For an arbitrary spacetime I think it is near hopeless to decompose the gravitational and velocity based parts.

 

[pervect]

I'd suggest focusing on round trips - computing "time dilation" requires that you specify how the clocks are going to be compared. If you do that via a round trip, it's automatically well defined. And it's also something concrete that you can measure, not something abstract.

For round trips, it's also easy (in principle) to calculate, given a metric and a path. You just integrate dtau, which you get from the metric, along the curve you travel, and that's the total elapsed time.

Given some specific coordinate system, you can think of the coefficient of g_00 as being a time dilation that makes your elapsed time for the trip less (compared to what it would be if g_00 was 1). And you can always think of space-like parts of the trip as subtracting from the Lorentz interval (so moving automatically makes your elapsed time less).

 

[powerplayer]

ok, well in all cases assume that the observer is at Earth and "you" (the traveler) are flying away from Earth to the massive object.

so it seems that the schwarzschild metric describes total time dilation for both velocity and gravitational time dilation in these cases. is this a correct statement?

 

[pervect]

The metric allows you to calculate your trip time, and the elapsed time for other observers - or pretty much any other measurement you want to make.

 

[powerplayer]

The metric allows you to calculate your trip time, and the elapsed time for other observers - or pretty much any other measurement you want to make.

so that's a yes to the question above your last post?