# Time dilation and relative velocity

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## Main Question or Discussion Point

https://www.physicsforums.com/showthread.php?p=3577768#post3577768" has caused me to question my assumptions.

If a target of observation is moving at relativistic velocities but its distance does not change wrt the observer, is it observed to be experiencing time dilation?

Alice stands at North Pole. Bob zooms around equator in a tight circle at .9c. Do they observe dilation in each other? I guess this gets into the non-inertial frame of reference aspect of SR.

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ghwellsjr
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This is just like what happens with gravitational time dilation which is non-repciprocal. Alice at the North Pole always sees Bob's clock as running slower than her own and Bob see's Alice's clock as running faster than his, neglecting additional influence from gravity. It would be better to concoct the scenario in space, far removed from any gravity.

By the way, Einstein mentioned this exact same scenario in his 1905 paper.

Dale
Mentor
Alice stands at North Pole. Bob zooms around equator in a tight circle at .9c. Do they observe dilation in each other? I guess this gets into the non-inertial frame of reference aspect of SR.
Your guess is correct. Bob's frame is non inertial, so although Alice is going at .9 c in circles in Bobs frame the time dilation formula in Bobs frame is not the usual one. If you go through all of the math to calculate the time dilation formula in Bobs frame you will find that Bob agrees that Alice ages faster.

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It would be better to concoct the scenario in space, far removed from any gravity.
Agreed. I phrased it the way I did for clarity of model. Harder to describe the setup in space.

Could have used a low mass asteroid...

Isn't this similar to the situation where you have a rotating circle ?

So Alice is in the center and Bob circles around at relativistic velocity.
I saw this situation explained before, and all relativistic effects do occur for Bob.

I believe this link will help:

Please, take into account that in order to observe Bob all the time, Alice should rotate on her axis too, and her radial speed of rotation would be less than Bob's. When you compare their speeds, it's like Bob is receding from Alice all the time, but without success (it happens with real humans too ). I am talking about recession, because Dave's question was caused by my allegations about the influence of approaching/receding of the objects on time dilation (see https://www.physicsforums.com/showthread.php?t=543550").

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Please, take into account that in order to observe Bob all the time, Alice should rotate on her axis too, and her radial speed of rotation would be less than Bob's. When you compare their speeds, it's like Bob is receding from Alice all the time, but without success (it happens with real humans too ). I am talking about recession, because Dave's question was caused by my allegations about the influence of approaching/receding of the objects on time dilation (see https://www.physicsforums.com/showthread.php?t=543550").
No, Alice doesn't have to rotate, nor move in any other way.
Just consider Alice has an omnidirectional detector.

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Dale
Mentor
Please, take into account that in order to observe Bob all the time, Alice should rotate on her axis too, and her radial speed of rotation would be less than Bob's. When you compare their speeds, it's like Bob is receding from Alice all the time, but without success (it happens with real humans too ). I am talking about recession, because Dave's question was caused by my allegations about the influence of approaching/receding of the objects on time dilation (see https://www.physicsforums.com/showthread.php?t=543550").
This is not correct. Bob is not receding from Alice at any time in either Bob's or Alice's frame. This does not depend on whether or not Alice is rotating at the center, or even whether or not Alice is exactly at the center.

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No, Alice doesn't have to rotate, nor move in any other way.
Just consider Alice has an omnidirectional detector.
Even if Alice is an omnidirectional detector, she has to rely on light for her observations, and Bob's light does not come to her in circles - it comes in straight lines moving in circles. The results of the observation of a cycle of Bob's movement are represented not by a single point, but by a circle as well, with the radius less than the radius of Bob's orbit and more than zero (one point can not represent more than one point). Consider a projection of Bob's orbit on the omnidirectional detector: if this projection is just a point, your observation does not make sense. And Alice would also have to reflect the direction of Bob's movement in the results of her observations, so in observational terms she has to follow Bob's movement in a smaller orbit one way of another. For observational purposes it does not matter, if Alice follows Bob physically or not.

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Even if Alice is an omnidirectional detector, she has to rely on light for her observations, and Bob's light does not come to her in circles - it comes in straight lines moving in circles. The results of the observation of a cycle of Bob's movement are represented not by a single point, but by a circle as well, with the radius less than the radius of Bob's orbit and more than zero (one point can not represent more than one point). Consider a projection of Bob's orbit on the omnidirectional detector: if this projection is just a point, your observation does not make sense. And Alice would also have to reflect the direction of Bob's movement in the results of her observations, so in observational terms she has to follow Bob's movement in a smaller orbit one way of another. For observational purposes it does not matter, if Alice follows Bob physically or not.
How does any of this mean anyone is receding? They are at the same distance from each other at all times.

Alexroma, none of this makes any sense. I think this thread needs to close. There's no science being done here.

Even if Alice is an omnidirectional detector, she has to rely on light for her observations, and Bob's light does not come to her in circles - it comes in straight lines moving in circles. The results of the observation of a cycle of Bob's movement are represented not by a single point, but by a circle as well, with the radius less than the radius of Bob's orbit and more than zero (one point can not represent more than one point). Consider a projection of Bob's orbit on the omnidirectional detector: if this projection is just a point, your observation does not make sense. And Alice would also have to reflect the direction of Bob's movement in the results of her observations, so in observational terms she has to follow Bob's movement in a smaller orbit one way of another. For observational purposes it does not matter, if Alice follows Bob physically or not.
Alice, or the detector, can be safely considered a point. The light from Bob will cross that point.
But even if you want Alice to move, as long as the speed is not relativistic (and it won't be), the fact that she moves has no relevance.

ghwellsjr
Gold Member
Even if Alice is an omnidirectional detector, she has to rely on light for her observations, and Bob's light does not come to her in circles - it comes in straight lines moving in circles. The results of the observation of a cycle of Bob's movement are represented not by a single point, but by a circle as well, with the radius less than the radius of Bob's orbit and more than zero (one point can not represent more than one point). Consider a projection of Bob's orbit on the omnidirectional detector: if this projection is just a point, your observation does not make sense. And Alice would also have to reflect the direction of Bob's movement in the results of her observations, so in observational terms she has to follow Bob's movement in a smaller orbit one way of another. For observational purposes it does not matter, if Alice follows Bob physically or not.
Why do you think Alice will see Bob's orbit having a smaller radius than his orbital radius? Remember, length contraction only applies along the direction of motion, so neither one will see the other one as being closer than they really are.

This is not correct. Bob is not receding from Alice at any time in either Bob's or Alice's frame. This does not depend on whether or not Alice is rotating at the center, or even whether or not Alice is exactly at the center.
You are right, he's not receding in terms of change in distance. That's my methodological fault: I have not defined from the beginning what I mean by "approaching" and "receding". It's a case of bad terminology. Actually, I don't mean change in distance, because distance is not a factor in time dilation, as it's calculated with √ 1 – V2/C2. I should be rather talking about the projections of vectors of movement of the objects on the vector of light coming to the observer. I believe, I already figured out, in principle, how to explain the time dilation for circling objects, I just have to formulate it well. I hope to come back with it soon.

Dale
Mentor
Even if Alice is an omnidirectional detector, she has to rely on light for her observations, and Bob's light does not come to her in circles - it comes in straight lines moving in circles. The results of the observation of a cycle of Bob's movement are represented not by a single point, but by a circle as well, with the radius less than the radius of Bob's orbit and more than zero (one point can not represent more than one point). Consider a projection of Bob's orbit on the omnidirectional detector: if this projection is just a point, your observation does not make sense. And Alice would also have to reflect the direction of Bob's movement in the results of her observations, so in observational terms she has to follow Bob's movement in a smaller orbit one way of another. For observational purposes it does not matter, if Alice follows Bob physically or not.
Irrelevant. The word "receeding" by definition means only "distance is increasing". The distance is not changing therefore Bob is not receeding. Everything else is irrelevant.

How does any of this mean anyone is receding? They are at the same distance from each other at all times.

Alexroma, none of this makes any sense. I think this thread needs to close. There's no science being done here.
OK, you are right, it's going in wrong direction. My terminology is misleading, I have to change it. Just close it. Thank you!

Wouldn't it be a spiral instead of a circle? If you use 'time coordinates' as a added description, together with the the spatial ones. That as you can't define Bob's position in SpaceTime without it? And doing that you should get a spiral motion, you can't get it any other way, as I can see?

Assuming that this is the case, how would one express the Lorentz contraction graphically?
==

Time may be relative the observer, but I don't think I can ignore it.

Dale
Mentor
Wouldn't it be a spiral instead of a circle? If you use 'time coordinates' as a added description, together with the the spatial ones. That as you can't define Bob's position in SpaceTime without it? And doing that you should get a spiral motion, you can't get it any other way, as I can see?
It would be a helix in spacetime, not a spiral.

Assuming that this is the case, how would one express the Lorentz contraction graphically?
==

Time may be relative the observer, but I don't think I can ignore it.
What is being Lorentz contracted?

A.T.
Assuming that this is the case, how would one express the Lorentz contraction graphically?
In the inertial frame the orbiting objects are contracted along the circumferential direction.

In the co-rotating frame the orbiting objects are not contracted, and yet they have to fill the same fraction of the circumference. This leads to the conclusion that you can interpret the space as observed in the rotating frame as non-Euclidean: There is more circumference at radius R than 2*Pi*R.

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I believe, I finally solved the case of circling objects, as it applies to my calculation of time dilation, which caused Dave to question his assumptions. See my last post in the initial thread: