Time dilations on confusing situations

[Solvay]

I'm curious about time dilation sizes of below 4 situations. Please assume I'm observing the situations far away in zero gravity. And please ignore all SR effects, just focus on GR.

http://dishdev.me/data/timedilationq.png

Equivalence Principle says acceleration and gravitational fields can substitute for each other. I also learned a time dilation by GR occurs in gravitational fields and accelerating frames, and I can interpret the time dilation by any of the two concepts.

But I'm confused by situations when acceleration and gravity both exist.

The clock in A has a plain orbital motion. The clock in B also has the same orbital motion but it's caused by artificial propulsion directed to the center of this rotational motion, not by gravity of the heavy star. Do both clocks have the same time dilations because of the same accelerations they have? Or the clock in A has more time dilation by gravitational field of the heavy star?

The clock in C has no movement at all because it cancels the gravity by the same size propulsion. It also has no apparent acceleration. Does it have the same time dilation as the clock in A? Or smaller because it doesn't have acceleration?

In final D, the clock is on the center of two heavy stars, the spot where sizes of two gravity are exactly the same. Does it have no time dilation like observer in zero gravity space? Or more time dilation than a clock on gravitational field of one heavy star?

 

[PeterDonis]

Equivalence Principle says acceleration and gravitational fields can substitute for each other.

That's not really what the EP says, although a lot of people mistakenly think it is.

What the EP actually says is that being at rest in a gravitational field is equivalent, locally, to accelerating in empty space so that you feel the same weight. For example, being at rest on the surface of the Earth is equivalent, locally, to accelerating at 1 g in empty space.

However, that "locally" is important. Time dilation is not a local phenomenon; it requires you to compare elapsed time for different observers that are co-located, then separate, then are co-located again (or who have some other invariant way of getting a common time reference). So you can't use the EP to figure out time dilation problems.

Also, "acceleration" here means proper acceleration, i.e., feeling weight. It does not mean coordinate acceleration. See further comments below.

The clock in A has a plain orbital motion. The clock in B also has the same orbital motion but it's caused by artificial propulsion directed to the center of this rotational motion, not by gravity of the heavy star. Do both clocks have the same time dilations because of the same accelerations they have?

They don't have the same accelerations. Clock A is in free fall, weightless. Clock B feels weight (we don't usually call it that, but physically it's the same thing). So these two clocks are not equivalent according to the EP.

As far as time dilation is concerned, you can't really compare these clocks because they're in different spacetimes. There's no unique way to figure out any meaningful correspondence between them that will let you compare their elapsed times per orbit.

The clock in C has no movement at all because it cancels the gravity by the same size propulsion. It also has no apparent acceleration.

Clock C is accelerated in the proper sense (see above); it feels weight.

Does it have the same time dilation as the clock in A? Or smaller because it doesn't have acceleration?

Clock C has less time dilation than A--more precisely, if clock A passes clock C once per orbit, then C's elapsed time between successive meetings will be more than A's elapsed time. The difference has nothing to do with acceleration; it is because clock A is moving and clock C is not, i.e., it's due to velocity, not acceleration. (There are subtleties here as well, since strictly speaking "velocity" is relative; but in this particular case there is an invariant notion of "velocity", and A has it and C doesn't, so A has more time dilation.)

In final D, the clock is on the center of two heavy stars, the spot where sizes of two gravity are exactly the same. Does it have no time dilation like observer in zero gravity space?

No. This is a good illustration of how time dilation and "gravitational field" are not the same and don't always correspond.

Or more time dilation than a clock on gravitational field of one heavy star?

It's more complicated than that. Unlike the previous situations, there is no well-defined notion of "time dilation" in general in this one. That's because the two heavy stars can't be static; they must either be orbiting their common center of mass, or be falling into each other. Time dilation can only be defined for static situations (more precisely, for stationary situations, so that the field of a single rotating body can still qualify; but the situation with two heavy stars isn't stationary either).

In the limit where the two stars are small enough that their gravitational fields are very weak, and far enough apart so that their relative motion is small, there is an approximate notion of time dilation that is just the sum of the time dilations for each star in isolation; but this is only an approximation.

 

[harrylin]

I'm curious about time dilation sizes of below 4 situations. Please assume I'm observing the situations far away in zero gravity. And please ignore all SR effects, just focus on GR.

http://dishdev.me/data/timedilationq.png

Equivalence Principle says acceleration and gravitational fields can substitute for each other. I also learned a time dilation by GR occurs in gravitational fields and accelerating frames, and I can interpret the time dilation by any of the two concepts.

But I'm confused by situations when acceleration and gravity both exist.

The clock in A has a plain orbital motion. The clock in B also has the same orbital motion but it's caused by artificial propulsion directed to the center of this rotational motion, not by gravity of the heavy star. Do both clocks have the same time dilations because of the same accelerations they have? [..]

Welcome to Physicsforums. :)

There is a lot of confusion related to acceleration. Clocks are assumed not to be influenced by acceleration. With GR's equivalence principle you can pretend that an accelerating clock in empty space is not accelerating but in rest in a gravitational field; but in your first example that approach only complicates matters.

To avoid confusion it is wiser to just account for the fact that the clock in A is in a gravitational field that reduces the clock rate, while in B the clock is not in a gravitational field. As a result, (ceteris paribus): when you look from far away through a powerful telescope at both clocks, then you should see that clock A ticks slower than clock B.

BTW, that situation is well understood, because the clocks in GPS satellites need high precision. You can find interesting articles about GPS clocks on internet, complete with the equations.

 

[Solvay]

Clock C has less time dilation than A--more precisely, if clock A passes clock C once per orbit, then C's elapsed time between successive meetings will be more than A's elapsed time. The difference has nothing to do with acceleration; it is because clock A is moving and clock C is not, i.e., it's due to velocity, not acceleration. (There are subtleties here as well, since strictly speaking "velocity" is relative; but in this particular case there is an invariant notion of "velocity", and A has it and C doesn't, so A has more time dilation.)

You mean they make difference because of SR, right? In fact, I want to concentrate on GR only, here. If I ignore the part of time dilation caused by SR, then do A and C have the same time dilation? Both clocks are affected by gravity of the same size, but one is free-falling without proper acceleration while another has proper acceleration. Doesn't this fact make any difference of time dilation size?

 

[Nugatory]

You mean they make difference because of SR, right? In fact, I want to concentrate on GR only, here. If I ignore the part of time dilation caused by SR

You can't separate them in that way. There's only one time dilation going on, and the only question is whether it can be calculated using the methods of SR (flat spacetime, no gravity) or we have to use the methods of GR (curved spacetime, gravity present).

You can't say that one part of the dilation is due to SR and another part is due to GR, any more than you can look at the length of the hypotenuse of a right triangle and say that part of the length is caused by one side of the triangle and the rest is caused by the other side.

 

[harrylin]

You mean they make difference because of SR, right? In fact, I want to concentrate on GR only, here. If I ignore the part of time dilation caused by SR, then do A and C have the same time dilation? Both clocks are affected by gravity of the same size, but one is free-falling without proper acceleration while another has proper acceleration. Doesn't this fact make any difference of time dilation size?

That is correct, according to GR there is no difference in gravitational time dilation.
And once more: acceleration itself does not cause time dilation.

Practical tests of gravitational time dilation:
- https://en.wikipedia.org/wiki/Hafele–Keating_experiment
- https://en.wikipedia.org/wiki/Gravity_Probe_A

Practical test of acceleration in orbit:
- https://www.physicsforums.com/threads/time-dilation-and-radioactive-decay.75277/#post-563718

 

[A.T.]

You can't separate them in that way.

Could you not separate the gravitational time dilation, by setting all metric components aside of g_tt to 0, and then integrating along the world-line? Of course, to get a frame independent age difference, you have to bring the clocks back together.

 

[CKH]

[Concerning case D of a clock between two stars]

No. This is a good illustration of how time dilation and "gravitational field" are not the same and don't always correspond.

Unlike the previous situations, there is no well-defined notion of "time dilation" in general in this one. That's because the two heavy stars can't be static; they must either be orbiting their common center of mass, or be falling into each other. Time dilation can only be defined for static situations (more precisely, for stationary situations, so that the field of a single rotating body can still qualify; but the situation with two heavy stars isn't stationary either).

The clock between the two stars can send out light pulses at each tick. The distant observer can compare these with his own clock ticks. So whatever the stars are doing, a specific time dilation can be measured. You suggest that the amount of time dilation is dependent on the motion of the stars. But even then, by taking that motion into consideration, we should be able to calculate a "time dilation" wrt the distant observer, even if it is changing due to the motion of the stars.

Can you answer the question for the following cases or at least describe how dilation differs in these cases:
1) The "static" case where the two stars are held stationary by rockets.
2) The case in which the stars have been held stationary but are suddenly released so that they accelerate toward one another due to gravity?
3) The case in which the stars are falling toward the center at equal and opposite velocities?
4) The case in which they orbit the central clock, in the plane of the diagram?

 

[Jonathan Scott]

Can you answer the question for the following cases or at least describe how dilation differs in these cases:
1) The "static" case where the two stars are held stationary by rockets.
2) The case in which the stars have been held stationary but are suddenly released so that they accelerate toward one another due to gravity?
3) The case in which the stars are falling toward the center at equal and opposite velocities?
4) The case in which they orbit the central clock, in the plane of the diagram?

For the weak field approximation (which holds in most places except near extremely dense matter, or for relativistic speeds), the result is the same in all of these cases. The fractional time dilation is the same as the Newtonian potential expressed in dimensionless units, that is the sum of ##-Gm/rc^2## for each source.

 

[CKH]

For the weak field approximation (which holds in most places except near extremely dense matter, or for relativistic speeds), the result is the same in all of these cases. The fractional time dilation is the same as the Newtonian potential expressed in dimensionless units, that is the sum of ##-Gm/rc^2## for each source.

Thanks. The opposite fields cancel out in the middle but the potentials add.