Time Dilation & Superluminal Travel: 2 Questions

[jkbhagatio]

Two quick questions on time dilation and superluminal travel for a situation in which two people are in motion with respect to each other.

1) If Jack is moving with respect to Jill, Jill sees Jack's clock move more slowly relative to her clock, so Jill perceives Jack aging more slowly relative to her.

But, couldn't Jack treat this situation as Jill moving with respect to him, and so Jack would see Jill's clock moving more slowly relative to his?

Isn't this a paradox?2) Jack is traveling towards Jill at more than half the speed of light (say 0.51c), and fires a light beam towards Jill, and at the same time fires a baseball towards Jill at 0.51c.

From Jill's perspective, wouldn't the baseball travel towards her at more than the speed of the light, and reach her before the light beam reaches her? Why is this impossible?

 

[Ibix]

But, couldn't Jack treat this situation as Jill moving with respect to him, and so Jack would see Jill's clock moving more slowly relative to his?

Yes (or, to be precise, both would see the other's clock ticking fast due to their changing distance, but would calculate that the other's clock was ticking slow once they corrected for the changing distance).

Isn't this a paradox?

No. Look up "relativity of simultaneity". In short, for there to be a paradox they would have to start their clocks at the same time and stop their clocks at the same time and have conflicting readings. But their frames have different notions of what "at the same time" means for events that aren't in the same place, and these differences will always account for any clock differences.

2) Jack is traveling towards Jill at more than half the speed of light (say 0.51c), and fires a light beam towards Jill, and at the same time fires a baseball towards Jill at 0.51c.

0.51c relative to whom? 99% of confusion over relativity is people not understanding the relativity of simultaneity. The other 1% is people failing to specify who is measuring what.

From Jill's perspective, wouldn't the baseball travel towards her at more than the speed of the light, and reach her before the light beam reaches her? Why is this impossible?

Assuming you meant that Jack measures the speed of the baseball to be 0.51c relative to him and Jill measures Jack to be doing 0.51c relative to her, look up "relativistic velocity addition". Velocities $u$ and $v$ combine to be$$\frac{u+v}{1+uv/c^2}$$(some sources may have minus signs, depending on their choice of definition of direction of $v$). Note that when $u$ and $v$ are both much less than $c$ this is practically indistinguishable from the familiar Newtonian notion that velocities add linearly.

 

[jkbhagatio]

Hi @Ibix, thanks for the reply! Yes, I have read about relativity of simultaneity, though I guess I am still not understanding it clearly : )

I will try to slightly reframe my first question to hopefully make more clear what I was trying to get at: if Jack and Jill are at a space station together on earth, and Jack flies around the Earth and comes back to Earth to see Jill, then from Jill's perspective he will have aged less relative to her, but from Jack's perspective Jill could have been moving relative to him, so he could see Jill as having aged less relative to him.

However, when I've read about examples like this, when Jack and Jill meet again, it's always Jack who has aged less relative to Jill. Why is this necessarily true, i.e. when they meet again, why can't Jack see Jill as having aged less?For my second question:

Assuming you meant that Jack measures the speed of the baseball to be 0.51c relative to him and Jill measures Jack to be doing 0.51c relative to her

Yes, this is what I meant. And ah yes, relativistic velocity addition answers my second question, thanks!

 

[mitochan]

Hi @jkbhagatio
I understand you that Jill is at still in a inertial frame of reference (IFR) and Jack goes round and come back to see Jill again. When they meet Jack ages less.
Jill stays still in IFR, Jack stays still in rotating frame of reference, not IFR. This difference breaks the assumed symmetry.

 

[FactChecker]

Each is observing the other's clock and processes at different positions as they move along. So the disagreement in time passage is due to how the clocks are synchronized within their own reference frame along their relative paths. Since each can understand what is going on with the other's clocks and processes using Special Relativity, there is no paradox.

 

[PeroK]

I will try to slightly reframe my first question to hopefully make more clear what I was trying to get at: if Jack and Jill are at a space station together on earth, and Jack flies around the Earth and comes back to Earth to see Jill, then from Jill's perspective he will have aged less relative to her, but from Jack's perspective Jill could have been moving relative to him, so he could see Jill as having aged less relative to him.

However, when I've read about examples like this, when Jack and Jill meet again, it's always Jack who has aged less relative to Jill. Why is this necessarily true, i.e. when they meet again, why can't Jack see Jill as having aged less?

It's important to understand that time dilation refers specifically to measurements carried out using an inertial reference frame. That is to say, in order for you to apply the rule of time dilation you must be moving inertially.

Someone who is moving inertially (i.e. not accelerating) may apply time dilation to anything that is moving relative to them. But, someone who is accelerating cannot consider themselves an inertial observer and apply time dilation to someone moving inertially.

For example, if someone is in a circular orbit (which implies acceleration) about an inertial object, then there is an asymmetry and it is the accelerating object that ages more than the central inertial object.

Likewise, if you change direction suddenly, then you cannot claim to be a totally inertial observer. Changing your inertial reference frame has the same effect as acceleration.

These cases generally come under the name of the twin paradox.

Note that if two observers remain inertial, then (in special relativity), then may only pass each other once and can never do so again. There is, therefore, no paradox in their mutually symmetric time dilation as measured by the other.

 

[Dale]

from Jack's perspective

The issue is that Jack’s perspective is a non-inertial reference frame. The rules for inertial reference frames do not apply for non-inertial reference frames. So it is not correct that “he could see Jill as having aged less”. If you work out the actual rules for Jack’s perspective then you will find that he in fact sees Jill as having aged more.

 

[hutchphd]

Let me add the piece to this that confused me for a long while (until Physics Forums in fact).
Suppose you have a bunch of synchronized clocks "attached" in your reference frame on a grid. As your frame of reference is accelerated the clocks will no longer be synchronized. Voila!
Hope that helps.

 

[jkbhagatio]

Thanks for the replies, all. I think what I am trying to get at concerns reference frames.

If you work out the actual rules for Jack’s perspective then you will find that he in fact sees Jill as having aged more.

@Dale If by chance you could refer me to a similar example problem with the worked out solution that would be really helpful!In regards to my reframed question:

if Jack and Jill are at a space station together on earth, and Jack flies around the Earth and comes back to Earth to see Jill, then from Jill's perspective he will have aged less relative to her

and this reply

Someone who is moving inertially (i.e. not accelerating) may apply time dilation to anything that is moving relative to them. But, someone who is accelerating cannot consider themselves an inertial observer and apply time dilation to someone moving inertially.

I guess what I'm asking is:
Why can't Jack see himself as maintaining the inertial reference frame, and Jill (and the rest of the planet) as accelerating around him?

 

[PeroK]

I guess what I'm asking is:
Why can't Jack see himself as maintaining the inertial reference frame, and Jill (and the rest of the planet) as accelerating around him?

Because, if you are accelerating then by Newton's second law you must feel a real force, which you can measure. $$F = ma$$

 

[Dale]

@Dale If by chance you could refer me to a similar example problem with the worked out solution that would be really helpful!

Sure. This is not the only possible approach, but it is my favorite:

https://arxiv.org/abs/gr-qc/0104077

As you will see, the whole process is rather cumbersome, so it is not commonly done. Nevertheless, when you say something like “Jack’s perspective” what you are actually saying is substantially different from an inertial frame.

Why can't Jack see himself as maintaining the inertial reference frame, and Jill (and the rest of the planet) as accelerating around him?

Because his accelerometer does not always read 0.

 

[PeroK]

If by chance you could refer me to a similar example problem with the worked out solution that would be really helpful!

I did an analysis of a circular orbit, considered as polygon motion with a large number of inertial sections, join by instantaneous changes of direction. The key is, of course, the relativity of simultaneity:

https://www.physicsforums.com/threa...ther-in-a-circular-orbit.896607/#post-5660815