Time Difference for High-Speed Travel: Asymmetric by Special Relativity

[Robert Booth]

TL;DR Summary

Relativity explains that there is no universal reference frame, all observations are relative.

So a traveller departs on a return jouney on an aircraft and finds on return that his/her time has slowed down relative to a person who remained on the ground (this is tested fact).

But why?

If everything is relative, then from the point of view of the traveller, they would have been stationary while the ground moved underneath and should expect time to have slowed down on the ground.

Special Relativity explains that there is no universal reference frame, all observations are relative.

So a traveller departs on a return jouney on an aircraft and finds on return that his/her time has slowed down relative to a person who remained on the ground (this is tested fact).

But why?

If everything is relative, then from the point of view of the traveller, they could view themself as stationary while the ground moved underneath and should expect time to have slowed down on the ground.

 

[Ibix]

If everything is relative

This is an overstatement. What relativity says is that there is nothing to choose between inertial reference frames - anyone can consider itself as "at rest". But the person who goes out-and-back isn't inertial all the time while the stay-at-home is, so there is something to choose between these observers.

In fact, it turns out that elapsed time in relativity behaves very like distance in Euclidean geometry. Just as the distance you cover between two points depends on your path through space, your elapsed time depends on your path through spacetime. The traveller took a shortcut.

 

[PAllen]

Why not start with the following FAQ of this forum:

https://www.physicsforums.com/insights/geometrical-view-time-dilation-twin-paradox/

Then search for twin paradox within this forum. I bet there are a thousand threads raising the exact question you ask that have already been answered.

 

[Nugatory]

Another good resource is http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_intro.html

 

[phinds]

@Robert Booth you are confusing time dilation with differential aging. This confusion is quite common, so all you need to do is get straight on what the two are and you'll see your confusion. The links already provided are good for that.

 

[Orodruin]

Special Relativity explains that there is no universal reference frame, all observations are relative.

This is not an accurate description. One of the most important concepts in relativity (and in geometry) is invariants - quantities that take the same value regardless of the frame.

 

[Mister T]

So a traveller departs on a return jouney on an aircraft and finds on return that his/her time has slowed down relative to a person who remained on the ground (this is tested fact).

They share the same location and synchronize their clocks. Then, after one of the clocks goes on a journey around the world, they again share the same location and compare clocks. Less time has elapsed on the traveler's clock, the clock ran slow compared to the stationary clock.

The Principle of Relativity tells us that each will observe the other's clock running slow, but only if the clocks are moving in a straight line relative to each other.

 

[phinds]

They share the same location and synchronize their clocks. Then, after one of the clocks goes on a journey around the world, they again share the same location and compare clocks. Less time has elapsed on the traveler's clock, the clock ran slow compared to the stationary clock.

No, it did not. Both clocks ticked away at one second per second. What DID happen is that they took different paths through space-time and thus encountered the effect of differential aging (one clock made fewer ticks than the other), which is NOT the same as time dilation.

 

[jbriggs444]

No, it did not. Both clocks ticked away at one second per second. What DID happen is that they took different paths through space-time and thus encountered the effect of differential aging (one clock made fewer ticks than the other), which is NOT the same as time dilation.

It does entail that for any differentiable and strictly monotone increasing function (call it a simultaneity function if you like) that maps proper times on the one path to proper times on the other path there must be at least one point at which, according to that function, time on the traveling clock ran slow relative to time on the stay at home clock. [Simple consequence of the mean value theorem]

As you suggest, however, there is no single point at which all such simultaneity functions must agree that time ran slow for the one compared to the other.

 

[phinds]

It does entail that for any differentiable and strictly monotone increasing function (call it a simultaneity function if you like) that maps proper times on the one path to proper times on the other path there must be at least one point at which, according to that function, time on the traveling clock ran slow relative to time on the stay at home clock.

As you suggest, however, there is no single point at which all such simultaneity functions must agree that time ran slow for the one compared to the other.

I don't understand what you mean. Both clocks tick constantly at one second per second throughout their journeys through space-time. Do you disagree with that?

 

[jbriggs444]

I don't understand what you mean. Both clocks tick constantly at one second per second throughout their journeys through space-time. Do you disagree with that?

No, I do not disagree. It is a tautology. But if we are to compare the clock rates against one another, we need a way to associate an event on the path over here with an event on the path over there. At least a partial foliation. Given a [suitably smooth] foliation, we can talk about relative clock rates and decide which clock is ticking slow relative to the other at a particular event.

 

[Ibix]

I don't understand what you mean. Both clocks tick constantly at one second per second throughout their journeys through space-time. Do you disagree with that?

If you have a way to say "my clock reads 12.00, and at the same time the other guy's clock reads <whatever>", and you impose the condition that the other guy's clock always ticks forwards (anything else is a Bad Idea), then you do end up stating what @jbriggs444 said - the stay-at-home must conclude that the traveller's clock ticked slowly for at least some part of the journey, and the traveller must conclude that the stay-at-home's clock ticked fast. Neither has any effect on the actual function of the other's clock, as you say - it's more akin to ruler markings not matching up if they aren't parallel.

The problem is that there's no unique way of defining the words "at the same time" that I used in my first sentence. There's an obvious way to do it for the stay-at-home (and the traveller's clock always ticks slowly by this method), but it's not obligatory. There's no single obvious way to do it for the traveller, and a naive attempt (where the stay-at-home's clock ticks slowly but undergoes a massive jump) is the root of the OP's problem.

 

[PeterDonis]

If everything is relative

Not everything is relative.

In the standard "twin paradox", the traveling twin feels acceleration when they turn around. The stay at home twin never feels acceleration at all. That difference between them is not relative.

In any "twin paradox" scenario, if each twin is emitting light signals at fixed intervals by their own clock, and detecting the other's signals, the arrival times (by the receiving twin's clock) and Doppler shifts of those signals are not relative.

So reasoning from the premise that "everything is relative" is simply wrong. You need to pay careful attention to which things are relative and which are not in order to reason correctly.

 

[PeterDonis]

a traveller departs on a return jouney on an aircraft and finds on return that his/her time has slowed down relative to a person who remained on the ground (this is tested fact).

No, it's not.

What is tested fact is that a traveler who flies eastbound around the world on an airplane, finds on returning to his starting point that his clock shows slightly less elapsed time than the clock that stayed at home.

But a traveler who flies westbound around the world on an airplane finds on returning to his starting point that his clock shows more elapsed time than the clock that stayed at home.

You can't reason correctly if you don't know what the actual data is.

https://en.wikipedia.org/wiki/Hafele–Keating_experiment#Results

 

[Ibix]

But a traveler who flies westbound around the world on an airplane finds on returning to his starting point that his clock shows more elapsed time than the clock that stayed at home.

To be pedantic, this depends on the speed of the plane, does it not? If you can circumnavigate the globe in a day (I make that about Mach 1.4 at sea level at the equator) then you are hovering inertially (at rest in the Earth-centred inertial frame) while the Earth turns under you. If you go faster, your elapsed time starts to fall again.

 

[PeterDonis]

To be pedantic, this depends on the speed of the plane, does it not?

In the general case, yes. The OP talked about "tested fact", so I was using the actual tested facts from the Hafele Keating experiment. Nobody has actually tried it with westbound supersonic aircraft.

 

[Ibix]

Nobody has actually tried it with westbound supersonic aircraft.

Fair enough. As a side note, I've always loved the Hafele-Keating experiment, just for the detail that they bought tickets for the clock in the name of Mr Clock.