Can I keep multiple clocks in sync after a ride on a roller coaster?

[metastable]

Suppose there are 6 cars on a roller coaster and I synchronize 6 clocks at rest with respect to each other in a box with no wiggle room on the space station. The bottom of the box has the same shape as the curvature of the earth, and the box is always kept "top side up" with respect to the ground. Next I bring the 6 clocks down to Earth along with the next set of returning astronauts. I place the box next to the train and load the clocks, one per car, in such a way that each clock remains the same distance from the ground during the loading process. The train starts perfectly level in the sense that the track exactly follows the curvature of the Earth (and in an idealization the Earth is a perfect sphere as well). Once the clocks are loaded (none having changed elevation during the loading process), the rollercoaster accelerates, starts turning wildly and doing loop the loops, before returning to the "level" stretch of track. Next the clocks are carefully unloaded without changing their elevations, returned to the box, the box is kept right side up and returned to the space station on the next launch. In space, the box is opened and the clocks checked for synchronization. Are they still in sync?

 

[Ibix]

Generally, elapsed time is a measure of the "length" of an object's worldline. If you sync clocks then move them in the same way then they have identical worldlines, so identical elapsed times and are obviously in sync at the end. If they don't move in the same way then they have different worldlines that may or may not have the same elapsed time so they may or may not end up in sync.

The answer for your scenario therefore depends (in principle) on details of how you load and unload the clocks, since that's where you could be treating them differently. Such differences are almost certainly negligible.

 

[A.T.]

Suppose there are 6 cars on a roller coaster and I synchronize 6 clocks at rest with respect to each other in a box with no wiggle room on the space station. The bottom of the box has the same shape as the curvature of the earth, and the box is always kept "top side up" with respect to the ground. Next I bring the 6 clocks down to Earth along with the next set of returning astronauts. I place the box next to the train and load the clocks, one per car, in such a way that each clock remains the same distance from the ground during the loading process. The train starts perfectly level in the sense that the track exactly follows the curvature of the Earth (and in an idealization the Earth is a perfect sphere as well). Once the clocks are loaded (none having changed elevation during the loading process), the rollercoaster accelerates, starts turning wildly and doing loop the loops, before returning to the "level" stretch of track. Next the clocks are carefully unloaded without changing their elevations, returned to the box, the box is kept right side up and returned to the space station on the next launch. In space, the box is opened and the clocks checked for synchronization. Are they still in sync?

Is there no simpler scenario possible to get to the point?

 

[jbriggs444]

The ride on the roller coaster will not result in anything as simple as a time shift of the same velocity profile. The clocks are treated differently during the ride. For instance, a lead clock will be moving more rapidly at the top of a hill than will a middle clock.

I would expect the result to "average out", but have no clue how to begin assessing second order effects.

One can arrange to slowly move the clocks from a common box at the start of the ride to their respective cars and slowly move the clocks back to the box after the ride. The only interesting part of the scenario is what happens during the ride.

 

[metastable]

If they don't move in the same way then they have different worldlines that may or may not have the same elapsed time so they may or may not end up in sync.

The answer for your scenario therefore depends (in principle) on details of how you load and unload the clocks, since that's where you could be treating them differently. Such differences are almost certainly negligible.

So if the clocks are brought back to the space station will they remain in sync with a 7th clock that remained on board the whole time?

 

[jbriggs444]

So if the clocks are brought back to the space station will they remain in sync with a 7th clock that remained on board the whole time?

For a space station in low Earth orbit, I believe that time dilation from orbital speed dominates over the earthbound time dilation from gravitational potential. The space station clock will be slow relative to the clocks that took a holiday at the theme park.

Barring some very high speed roller coaster rides.

 

[Ibix]

So if the clocks are brought back to the space station will they remain in sync with a 7th clock that remained on board the whole time?

No.

 

[A.T.]

...7th clock...

Why do you need 7 clocks for this?

 

[metastable]

Why do you need 7 clocks for this?

If I understand the scenario correctly, the 6 clocks are out of sync during the loop the loops, but return to synchrony at the end of the ride. But I suppose you are correct 5 or fewer clocks might do to confirm whether I have a correct understanding.

 

[A.T.]

But I suppose you are correct 5 or fewer clocks might do to confirm whether I have a correct understanding.

Maybe also ask yourself if you really need all the other stuff.

 

[metastable]

How do you propose factoring both the gravitational time dilation effects & the velocity time dilation effects, in a simpler manner?

 

[Ibix]

Why are you involving gravity at all? Why not just synchronise your clocks in the cars and drop all the to-and-from-the-station stuff?

 

[jbriggs444]

How do you propose factoring both the gravitational time dilation effects & the velocity time dilation effects, in a simpler manner?

As I understand it, it is only in the weak field limit that gravitational time dilation and velocity time dilation can be successfully modeled as independent effects that combine linearly.

GPS is a good worked example where the tick rate of clocks that are moving and are at altitude has been calculated and measured.

 

[metastable]

Why are you involving gravity at all? Why not just synchronise your clocks in the cars and drop all the to-and-from-the-station stuff?

I think it is a very simplified case compared to the study of gravitational and velocity time dilation effects in the previously discussed more complicated scenario involving canceling galactic orbital velocity with a 215km/s burn, performing Oberth maneuvers at periapsis with each passing star, followed by a final Oberth maneuver in proximity to Sagittarius A*.

 

[Ibix]

I think it is a very simplified case compared to...

That doesn't answer the question. What are you hoping to learn here? Why are you inventing difficult-to-generalise scenarios involving roller coasters and surface-to-space transfers?

 

[metastable]

I'd like to better understand how time dilation is affected by extreme velocities and extreme gravitational fields.

 

[PeterDonis]

I think it is a very simplified case

Your definition of "simplified" does not appear to be the same as anyone else's.

I'd like to better understand how time dilation is affected by extreme velocities and extreme gravitational fields.

Extreme velocity means more time dilation. Extreme gravitational fields mean more time dilation. That's about the best that can be done at a "B" level given the generality of what you're asking. Thinking up various complicated scenarios that are difficult and time consuming for other people to analyze won't change that.

If you want more detail than that, you need to learn the actual math, so that you can do the calculations yourself instead of trying to outsource them to other PF posters.

Thread closed.

 

[Nugatory]

I'd like to better understand how time dilation is affected by extreme velocities and extreme gravitational fields.

For that, you will have to follow a more disciplined path. Start by learning special relativity (in the simplest case of no gravity and no acceleration) and understanding how velocity-based time dilation is related to the relativity of simultaneity. Get comfortable with applying the Lorentz transformations. Use them to resolve the twin paradox and Bell's spaceship paradox.

Because you eventually want to understand gravitational effects, next you'll need to learn the geometrical approach to SR; this is usually an early chapter in GR textbooks, and the treatment in Taylor and Wheeler's "Spacetime Physics" is also good.

After this you'll be able to start considering problems in which gravity is relevant. Not ethat "extreme" gravitational fields implies that the weak field approximation doesn't apply, so be cautious about anything you read that doesn't show the math.