Time dilation of an orbiting ship

[Buckethead]

I'm sure this has become a tedious question, but wasn't able to ween out an exact answer through searching.

A ship orbits an asteroid using retro rockets to maintain an orbit. The gravity is insignificant and can be ignored. A clock on the ship will tick more slowly than a clock on the asteroid. Even though the ship feels acceleration due to the retro rockets, time dilation in this case is not due to acceleration by virtue of the Clock Hypothesis so I'm guessing we can ignore the forces felt on the ship as having anything to do with the time dilation. The ship is experiencing time dilation strictly because of its velocity relative to the asteroid. Now if we switch our observation from the asteroid to the ship then an observer on the ship would instead see the asteroid orbiting the ship at the same relative velocity. This appears to be a symmetric situation with the exception of the retro rockets. So is this situation the same as just two ships passing each other (each seeing the others clock move more slowly) or is this an asymmetric situation (similar to the twin paradox or a curved timeline in a world line diagram) where the ship has to be considered accelerating while the asteroid is not.

 

[russ_watters]

A ship orbits an asteroid using retro rockets to maintain an orbit...

This appears to be a symmetric situation with the exception of the retro rockets.

That's a big exception.

 

[Nugatory]

where the ship has to be considered accelerating while the asteroid is not.

If an accelerometer on the ship records acceleration - and it will in this case, because of the rockets - then the ship is accelerating, by definition.

The general technique for solving these problems is to calculate the proper time along the worldlines of the two objects. In this case the gravitational effects are negligible so we have flat Minkowski spacetime, which makes the problem quite a bit easier. We only need two spatial dimensions because the asteroid and the ship lie in the same plane. If we put the origin of our coordinate system at the center of the asteroid, then the worldline of the asteroid is just the time axis; the worldine of the spaceship is a helix coiling around the time axis.

 

[Buckethead]

If an accelerometer on the ship records acceleration - and it will in this case, because of the rockets - then the ship is accelerating, by definition.

The general technique for solving these problems is to calculate the proper time along the worldlines of the two objects. In this case the gravitational effects are negligible so we have flat Minkowski spacetime, which makes the problem quite a bit easier. We only need two spatial dimensions because the asteroid and the ship lie in the same plane. If we put the origin of our coordinate system at the center of the asteroid, then the worldline of the asteroid is just the time axis; the worldine of the spaceship is a helix coiling around the time axis.

OK, but can't we put the ship at the origin instead or does the fact that it feels acceleration prohibit that. If we can do that, then doesn't that make it a symmetrical situation like two passing ships?

 

[Orodruin]

OK, but can't we put the ship at the origin instead or does the fact that it feels acceleration prohibit that. If we can do that, then doesn't that make it a symmetrical situation like two passing ships?

You could find a coordinate system where this would be true. However, that would not be an inertial frame and so you could not apply the expressions derived for inertial frames without thinking. You would need to redo the entire computation and in the end you would get the same result as that in the asteroid frame.

 

[Janus]

I'm sure this has become a tedious question, but wasn't able to ween out an exact answer through searching.

A ship orbits an asteroid using retro rockets to maintain an orbit. The gravity is insignificant and can be ignored. A clock on the ship will tick more slowly than a clock on the asteroid. Even though the ship feels acceleration due to the retro rockets, time dilation in this case is not due to acceleration by virtue of the Clock Hypothesis so I'm guessing we can ignore the forces felt on the ship as having anything to do with the time dilation. The ship is experiencing time dilation strictly because of its velocity relative to the asteroid.

Nothing ever "experiences" time dilation. Time dilation is something you measure as happening to clocks either with a relative motion with respect to you or at a different gravitational potential, or in a different position to you relative to the acceleration if you are in an accelerated frame. The clock postulate simply means that an accelerating clock won't exhibit any additional time dilation due to its acceleration as measured from any other frame. It does not apply to what time dilation an observer in the accelerated frame would measure in other clocks. The fact that the circling spaceship is in an accelerating frame does effect how it measures what is happening to the asteroid clock. Since it is constantly accelerating towards that clock, it will measure it as running fast compared to its own. You can't swap an accelerated frame's observations for an inertial frame's observations.

 

[Buckethead]

Nothing ever "experiences" time dilation. Time dilation is something you measure as happening to clocks either with a relative motion with respect to you or at a different gravitational potential, or in a different position to you relative to the acceleration if you are in an accelerated frame.

I apologize for my sloppiness. I did mean to say that the ships clock moves slower relative to the asteroid. Accuracy is everything!

 

[PeroK]

I apologize for my sloppiness. I did mean to say that the ships clock moves slower relative to the asteroid. Accuracy is everything!

If you search for a thread titled "two twins moving around each other in a circular orbit", then in post #14 there is an analysis of this situation. This approximates the motion of the rocket as a sequence of instantaneous changes of direction.

I'm on my phone and can't find a way to link to that thread.

 

[Nugatory]

I'm on my phone and can't find a way to link to that thread.

https://www.physicsforums.com/threa...owards-each-other-in-a-circular-orbit.896607/

 

[Arkalius]

By the equivalence principle, this situation is effectively identical to a uniform gravity pulling everything in the direction the ship's rockets are firing, while the ship resists it perfectly. In this scenario, the ship is at a lower gravitational potential than the asteroid, so its time is slower as a result.

 

[PeroK]

By the equivalence principle, this situation is effectively identical to a uniform gravity pulling everything in the direction the ship's rockets are firing, while the ship resists it perfectly. In this scenario, the ship is at a lower gravitational potential than the asteroid, so its time is slower as a result.

That is not correct. Acceleration does not cause time dilation. The equivalence principle concerns experiments carried out locally within an accelerating reference frame, but would not apply to observations made from outside the rocket.

Moreover, if a rocket accelerates away from you, its time dilation is precisely related to its instantaneous velocity in your frame. There is not an additional component associated with its acceleration/"gravitational equivalence".

 

[Arkalius]

That is not correct. Acceleration does not cause time dilation. The equivalence principle concerns experiments carried out locally within an accelerating reference frame, but would not apply to observations made from outside the rocket.

I never said it did. Consider the twins paradox scenario. When the twin in the ship turns around, his acceleration to return home is equivalent to him resisting a uniform gravity causing everything to move in the direction of his rockets. Since Earth is at a higher potential in this field, its time is faster during the acceleration, thus explaining how the Earth twin ends up older upon the traveling twin's return.

If you distill the scenario in the OP down to the basics, where you have two ships at the same location in an inertial reference frame, one ship moves away from the other, and then continually fires its engines so that it maintains what is essentially a circular orbit around the other ship for some time, and then returns. Both ships observe each other in relative motion, but the one doing the orbits will have passed less proper time during this maneuver. The equivalence principle and resulting gravitational time dilation can be used to explain this, though there are other ways of explaining it as well.

 

[PeroK]

I never said it did. Consider the twins paradox scenario. When the twin in the ship turns around, his acceleration to return home is equivalent to him resisting a uniform gravity causing everything to move in the direction of his rockets. Since Earth is at a higher potential in this field, its time is faster during the acceleration, thus explaining how the Earth twin ends up older upon the traveling twin's return.

If you distill the scenario in the OP down to the basics, where you have two ships at the same location in an inertial reference frame, one ship moves away from the other, and then continually fires its engines so that it maintains what is essentially a circular orbit around the other ship for some time, and then returns. Both ships observe each other in relative motion, but the one doing the orbits will have passed less proper time during this maneuver. The equivalence principle and resulting gravitational time dilation can be used to explain this, though there are other ways of explaining it as well.

The time dilation entirely and precisely depends on the relative velocity. There is no reason I can see in complicating things by involving some sort of pseudo gravity. Even if that is a valid application of the equivalence principle, which I doubt.

 

[A.T.]

Acceleration does not cause time dilation.

The proper acceleration of the ideal clock doesn't affect the clock rate. But the proper acceleration of your reference frame does imply different clock rates between clocks resting at different positions in that non-inertial frame reference frame.

The time dilation entirely and precisely depends on the relative velocity.

Only in inertial frames. But the OP asks about a non-inertial frame.

 

[Arkalius]

The time dilation entirely and precisely depends on the relative velocity. There is no reason I can see in complicating things by involving some sort of pseudo gravity. Even if that is a valid application of the equivalence principle, which I doubt.

The weak equivalence principle states, among other things, that the local effects of motion in a curved spacetime (gravitation) are indistinguishable from those of an accelerated observer in flat spacetime, without exception. That means our accelerated ship in flat spacetime is the same for all intents and purposes as the same ship resisting a uniform gravity field (that keeps changing to reflect his constant shift in direction of acceleration), with all of the attendant effects such as gravitational time dilation.

 

[Nugatory]

The weak equivalence principle states, among other things, that the local effects of motion in a curved spacetime (gravitation) are indistinguishable from those of an accelerated observer in flat spacetime, without exception.

Yes, but note that word "local". An observer inside the ship and studying only what's going on inside the ship (light deflected towards the floor, dropped objects fall towards the floor, clocks near the nose run fast relative to clocks near the tail, ...) will find that these observations are consistent with either acceleration or gravity; that's the WEP at work. However, when he turns his attention to non-local phenomena such as a comparison of his clock and the clock on the asteroid, he will find that the WEP no longer works.

You would be able to apply the WEP and understand the time dilation between ship and asteroid as equivalent to gravitational time dilation, if an accelerometer on the asteroid also read 1G in thevsame direction while the two remained at rest relative to one another. That's not the case here.

 

[A.T.]

You would be able to apply the WEP and understand the time dilation between ship and asteroid as equivalent to gravitational time dilation, if an accelerometer on the asteroid also read 1G in thevsame direction while the two remained at rest relative to one another.

That would be the equivalent of gravitational time dilation in an uniform gravitational filed.

But If we would use a rotating common rest frame of ship and asteroid, with a non-uniform proper acceleration along the radial line. Would that be equivalent to a non-uniform gravitational field (same proper accelerations) in terms of gravitational time dilation?

 

[PeroK]

That would be the equivalent of gravitational time dilation in an uniform gravitational filed.

But If we would use a rotating common rest frame of ship and asteroid, with a non-uniform proper acceleration along the radial line. Would that be equivalent to a non-uniform gravitational field (same proper accelerations) in terms of gravitational time dilation?

Let's assume that time dilation were related to proper acceleration. We know, however, that it is definitely related to relative velocity. But, relative velocity and proper acceleration are independent for a circular orbit. In the sense that by changing the radius you can have a different acceleration for the same speed.

 

[A.T.]

Let's assume that time dilation were related to proper acceleration. We know, however, that it is definitely related to relative velocity.

Velocity is frame dependent. In the co-rotating frame both clocks are at rest, but still tick at different rates. In this frame that is explained by their different positions in the centrifugal potential, similar to how different positions in the gravitational potential imply gravitational time dilation.

 

[Nugatory]

Velocity is frame dependent. In the co-rotating frame both clocks are at rest, but still tick at different rates. In this frame that is explained by their different positions in the centrifugal potential, similar to how different positions in the gravitational potential imply gravitational time dilation.

I'm not seeing how that works. Consider an exchange of light signals between the two clocks, with the the ship emitting its signal when it receives the signal from the asteroid. Is that different than an exchange of light signals between the asteroid and an observer moving inertially and on a path tangent to the circular path described by the accelerated observer, hence momentarily at rest in the corotating frame? In this case the time dilation will show up as a redshift, and unlike gravitational time dilation the redshift will be symmetrical because it's caused by the relative velocity.

(We can take the duration of the signal to be arbitrarily small so that the difference between the straight-line inertial path and the circular path can also be made arbitrarily small and this is an ordinary transverse-Doppler problem).

 

[PeroK]

I don't see how it works either. A high centripetal acceleration can be achieved at non relativistic speeds.

 

[Ibix]

Have the asteroid emit light pulses at intervals of ##\Delta t## in its inertial rest frame. Since the distance to the rocket isn't changing the pulses arrive at the rocket every ##\Delta t##, again in the frame of the asteroid. But the rocket is moving. The distance traveled between pulse receptions is ##R\omega\Delta t##, so the interval between successive reception events is ##c^2\Delta\tau^2=c^2\Delta t^2-R^2\omega^2\Delta t^2##, which tells us that the time dilation factor is the ##1/\gamma## factor associated with the linear velocity, as expected.

However, the situation is not symmetrical. By the same argument as above, if the rocket emits pulses at proper time intervals of ##\Delta t'##, these must correspond to coordinate time intervals in the asteroid inertial frame that are larger than ##\Delta t'##, in fact by the ##\gamma## factor of the velocity.

So the situation is that the asteroid sees the rocket's clock ticking slow but the rocket sees the asteroid's clock ticking fast. The ratio depends on the linear speed of the rocket in the inertial frame, not its acceleration.

The asteroid's viewpoint is easy to interpret with vanilla special relativity - the rocket is always instantaneously at rest in some inertial system. But the rocket's rotating frame is different. Clocks "inside" it tick faster and clocks "below" it tick slower by the obvious ratio of ##\gamma## factors. You could interpret that as some strange gravitational field - it has the same basic behaviour wrt time dilation as long as ##0 \leq |r\omega |<c##. There'd be a lot of inertial forces and stuff to give the game away, though. Notably, the light pulses from the asteroid follow curved paths due to the Coriolis force.

So, in summary, there's a gravitation-like time dilation effect which depends on the rockets linear speed, which is related to its centripetal acceleration, but not trivially.

Does that help, or am I stating the obvious?

 

[A.T.]

A high centripetal acceleration can be achieved at non relativistic speeds.

With a small radius, which makes the potential difference low again.

 

[Arkalius]

So, in summary, there's a gravitation-like time dilation effect which depends on the rockets linear speed, which is related to its centripetal acceleration, but not trivially.

Does that help, or am I stating the obvious?

I feel like what you're talking about is what I was getting at with the equivalence principle. Using an apparent uniform (but constantly changing due to the "orbit") gravitational field with attendant time dilation effects is one way to solve the problem, but obviously not the only way.

 

[A.T.]

... the redshift will be symmetrical

You say that the spaceship will receive the asteroids signals redshifted, not blueshifted? We assume negligible asteroid mass (flat space time), right?

 

[A.T.]

Using an apparent uniform (but constantly changing due to the "orbit") gravitational field with attendant time dilation effects is one way to solve the problem, but obviously not the only way.

You can use a frame that translates on a circle with the ship, but doesn't rotate. This will produce a uniform gravitational field with a time dependent direction. However since it's time dependent, you cannot assign a potential to it. You can just argue that the asteroid is always higher in the field.

The co-rotating common rest frame has a non-uniform but time independent gravitational field, so you can assign a potential it.

 

[PeroK]

With a small radius, which makes the potential difference low again.

How are you assigning a gravitational potential?

 

[A.T.]

How are you assigning a gravitational potential?

The centrifugal potential is based on the conservative centrifugal force in the rotating frame.

 

[Janus]

While the centripetal acceleration is found by;
[tex]A_c = \omega^2 r[/tex]and
[tex] A_c = \frac{v2}{}[/tex]

How are you assigning a gravitational potential?

It is basically related to the work needed to move a test particle from one radius to another in a rotating frame.

The centripetal potential difference between a point at a given radius and the axis is found by:

[tex]\phi = -\frac{\omega^2 r^2}{2}[/tex]
(moving towards the center mean moving to a higher potential)

And since tangential velocity at r is found by:

[tex]v= \omega r[/tex]

we then get:
[tex]\phi = -\frac{v^2}{2}[/tex]

Which means that the potential difference between the Axis and r depends only on the velocity at r and is independent of r.

Contrast this to centripetal acceleration which depends on both v and r.

This is not much different than clock placed in the nose and tail of a rocket ship and undergoing an equal proper acceleration, the nose clock will tick faster than the tail clock. Now imagine instead that your rocket is continually pointing its nose towards the Asteroid as it circles under rocket power. The nose clock will now have a slightly lower proper acceleration than the tail clock. It will still run faster than the tail clock, but not by as much as it did when the nose and tail had the same proper acceleration. If you extend the nose of the rocket all the way to the axis of the forced orbit, the difference in rate between nose and tail clock will be the same as the time dilation for the tangential speed of the tail of the rocket.