Twin paradox explained for laymen

[PeroK]

There is a physical rationalization (within GR?) for the traveling twin seeing the stationary twin aging very fast during the turn-around. The traveling twin experiences acceleration during the turn-around which is equivalent to there being a gravitational field pulling toward the stationary twin. The farther away the stationary twin is, the greater the speedup of his aging. Therefore, the turn-around has a much greater aging effect than the opposite acceleration/deceleration effects when the trip starts and ends with the twins near each other.

It's none of these things. It's simple flat spacetime geometry. It's really no more mysterious than Pythagoras' theorem.

 

[Sagittarius A-Star]

which is equivalent to there being a gravitational field pulling toward the stationary twin.

I think it should say: "which is equivalent to in this frame everywhere being a pseudo-gravitational field pulling toward the traveling twin."

The "stationary" twin is in free fall towards the traveling twin (= only coordinate-acceleration of the stationary twin).

 

[FactChecker]

It's none of these things. It's simple flat spacetime geometry. It's really no more mysterious than Pythagoras' theorem.

When something is true, there may be many ways to intuit it. There may be a simple mathematical calculation or it may just be compatible with other things.

 

[PeroK]

When something is true, there may be many ways to intuit it. There may be a simple mathematical calculation or it may just be compatible with other things.

The problem with "something weird happens during acceleration" is that you can remove all of the acceleration from the scenario.

 

[Sagittarius A-Star]

that you can remove all of the acceleration from the scenario.

You can also keep all of the acceleration in the scenario. I see no problem with this. SR is fine dealing with accelerated reference frames.

 

[PeroK]

You can also keep all of the acceleration in the scenario. I see no problem with this. SR is fine dealing with accelerated reference frames.

That's not the point. If acceleration were the "cause", then you couldn't remove it. You can always keep extraneous factors in the scenario.

 

[FactChecker]

That's not the point. If acceleration were the "cause", then you couldn't remove it. You can always keep extraneous factors in the scenario.

It is good to know that things are compatible, regardless of what you use for a proof.

(FYI. I do not think that there is a purely mathematical definition of an inertial reference frame that would allow one to distinguish between the two twins. There would need to be some way of identifying which was doing the traveling. That requires a (mathematically) arbitrary reference point. Otherwise, a purely mathematical approach makes the situation of the twins completely symmetric. )

 

[Sagittarius A-Star]

That's not the point. If acceleration were the "cause", then you couldn't remove it. You can always keep extraneous factors in the scenario.

1) In the rest frame of the stationary twin, acceleration of the traveling twin is no direct "cause". "Gamma" depends only directly on velocity.

2) In the rest frame of the traveling twin, pseudo-gravitational time-dilation is part of the "cause", besides the "Gamma" of the "stationary" twin.

Often, the turn-around is described as instantaneously. Then in the rest frame of the traveling twin, the wristwatch of the stationary twin makes a jump to a later time instead of a continuous change to that time. The acceleration is not left out, but only hidden in a "dirac delta function".

 

[Vanadium 50]

None of that is correct. You can set this up entirely with people looking at clocks through windows without anyone accelerating.

 

[Sagittarius A-Star]

None of that is correct. You can set this up entirely with people looking at clocks through windows without anyone accelerating.

You can always redefine the scenario, for example to a triplet scenario. But you don't have to.

 

[PeterDonis]

I do not think that there is a purely mathematical definition of an inertial reference frame that would allow one to distinguish between the two twins.

Certainly there is. In any scenario where the traveling twin has to undergo nonzero proper acceleration to turn around, his worldline has nonzero path curvature. The mathematical definition of an inertial frame specifies that worldlines with constant spatial coordinates have zero path curvature (that's what "inertial" means), so it is impossible to find an inertial frame in which the traveling twin has constant spatial coordinates for the entire trip.

 

[PeterDonis]

1) In the rest frame of the stationary twin, acceleration of the traveling twin is no direct "cause". "Gamma" depends only directly on velocity.

2) In the rest frame of the traveling twin, pseudo-gravitational time-dilation is part of the "cause", besides the "Gamma" of the "stationary" twin.

Note that these two statements are compatible with different definitions of what a "cause" can be.

In 1), the acceleration of the traveling twin is an invariant, so this statement is compatible with a definition of "cause" which requires any possible "cause" to be an invariant.

In 2), the pseudo-gravitational time dilation is not an invariant (it vanishes in the stay at home twin's rest frame), so this statement is not compatible with a definition of "cause" which requires any possible "cause" to be an invariant.

Physically, only invariants correspond to observable quantities, and it seems like anything that could be a "cause" should be an observable quantity, so only the first statement would be compatible with what seems like a physically reasonable definition of "cause".

 

[Sagittarius A-Star]

Physically, only invariants correspond to observable quantities, and it seems like anything that could be a "cause" should be an observable quantity, so only the first statement would be compatible with what seems like a physically reasonable definition of "cause".

The nonzero path curvature of the worldline of the traveling twin is invariant. In the rest frame of the stationary twin it appears as an acceleration (not influencing time-dilation). In the rest frame of the traveling twin, it appears as pseudo-gravity (influencing time-dilation).

 

[FactChecker]

Certainly there is. In any scenario where the traveling twin has to undergo nonzero proper acceleration to turn around, his worldline has nonzero path curvature. The mathematical definition of an inertial frame specifies that worldlines with constant spatial coordinates have zero path curvature (that's what "inertial" means), so it is impossible to find an inertial frame in which the traveling twin has constant spatial coordinates for the entire trip.

Using only the relative positions of the twins, one can not mathematically define which is moving, which is accelerating, etc. There must be a definition of "inertial" or "stationary" using some external or mathematically arbitrary reference. Which reference frame is defined as "inertial" must be based on physics and has associated consequences.

 

[PeterDonis]

The nonzero path curvature of the worldline of the traveling twin is invariant.

Yes.

In the rest frame of the stationary twin it appears as an acceleration

As coordinate acceleration, yes.

(not influencing time-dilation)

Not directly, but indirectly it does. The traveling twin's coordinate acceleration changes his velocity in the stationary twin's rest frame, which does affect the traveling twin's time dilation. See below.

In the rest frame of the traveling twin, it appears as pseudo-gravity

Basically, yes. But see below.

(influencing time-dilation)

The stationary twin's pseudo-gravitational time dilation in the traveling twin's rest frame is due to the stationary twin's position (at much higher "altitude" than the traveling twin). But if you are going to call that an influence of path curvature, manifesting as pseudo-gravity, on time dilation, it is not direct, only indirect. So in both cases (traveling twin in stationary twin's rest frame, or stationary twin in traveling twin's rest frame), the path curvature of the traveling twin's worldline does affect time dilation, but only indirectly.

 

[Sagittarius A-Star]

The stationary twin's pseudo-gravitational time dilation in the traveling twin's rest frame is due to the stationary twin's position (at much higher "altitude" than the traveling twin). But if you are going to call that an influence of path curvature, manifesting as pseudo-gravity, on time dilation, it is not direct, only indirect. So in both cases (traveling twin in stationary twin's rest frame, or stationary twin in traveling twin's rest frame), the path curvature of the traveling twin's worldline does affect time dilation, but only indirectly.

I think, the influence of pseudo-gravitational potential-difference on time dilation is directly. But that's no problem, because time dilation (tickrate-ratios at a certain instance of time) is frame-dependent. Only the end-result (age-difference when meeting the 2nd time) is absolute. The influence of pseudo-gravitational potential-differences on that is only indirectly. The age difference depends, amoung others, on the pseudo-gravitational time-dilation, integrated over the turnaround-time.

 

[PeterDonis]

I think, the influence of pseudo-gravitational potential-difference on time dilation is directly.

"Pseudo-gravitational potential difference" is not the same as "path curvature". The latter is the invariant, and, as I said, it affects time dilation indirectly, and does so in both frames. So there is no need to point to any frame-dependent quantity like "pseudo-gravitational field" as a cause of time dilation; the invariant, path curvature, works fine as a cause in any frame.

time dilation (tickrate-ratios at a certain instance of time) is frame-dependent

Under this definition of "time dilation" (which I have no issue with, but not everyone is careful about using the term only with this definition), it doesn't need to have a cause, because frame-dependent things don't need causes. Only invariant things, like the age difference of the twins when they meet again, need to have causes.

 

[Sagittarius A-Star]

Under this definition of "time dilation" (which I have no issue with, but not everyone is careful about using the term only with this definition)

That's right. For example in the English Wikipedia they define time dilation as a difference in elapsed time, measured by two clocks (due to velocity or gravitational potential difference), and in the German Wikipedia that processes in a physical system tick slower relative to the observer, if the system is moving relative to the observer.

 

[robphy]

From a geometric viewpoint,
time-dilation compares
the measurer's t-component of the 4-velocity of the measurer [which is 1, since 4-velocities are unit vectors]
with
the measurer's t-component of the 4-velocity of an astronaut [which is what the measurer measures as the elapsed time between two ticks (marked by events) of the astronaut's clock].

These are equal only for the Galilean spacetime used in PHY 101: measurer measures $\Delta t = 1$.
In special relativity, measurer measures $\Delta t \geq 1$ (since $\gamma=\cosh\theta \geq 1$).. this is time dilation.
In Euclidean geometry (by analogy), $\Delta t \leq 1$ (since $\cos\phi \leq 1$.
See the diagrams below.

From this viewpoint, the key idea is that:
the general situation is that $\Delta t \ne 1$
and that the Galilean case is the exceptional case [not typical case].

It seems to me that the root of time-dilation is that the real spacetime we live in isn't Galilean...
but is pseudoriemannian with a Minkowski signature [for dilation].
Once this is realized, there's no need to include gravity or acceleration etc...
unless you want to calculate specific elapsed times.
Including them probably clouds the real issue above.

Our low-relative-speed lifestyles have led to our "common sense" Galilean notions of time.
Using https://www.desmos.com/calculator/wm9jmrqnw2
for Special Relativity (E=1), Galilean (E=0), and Euclidean (E=-1)...

$\quad$

$\quad$

The bottom line is to calculate the elapsed time along different worldlines from event O to event Z.
The general case is that they are different.

 

[Sagittarius A-Star]

but is pseudoriemannian with a Minkowski signature [for dilation].
Once this is realized, there's no need to include gravity or acceleration etc...
unless you want to calculate specific elapsed times.
Including them probably clouds the real issue above.

That's all fine, if you teach relativity to physics students. But the OP asks for a laymen explanation of the typical "twin paradox" scenario and states wrongly, that a scenario without real gravity would be symmetrically.

I think, for laymen an explanation with 4-vectors and 4D-spacetime may be too difficult to understand. An explanation comparing the empirical viewpoint of both twins maybe didactically better for an explanation to laymen.

And the asymmetry, related to tick-rates, can be explaind by the fact, that in the inertial frame of the "stationary" twin the tick-rate of the other twin's wristwatch can be calculated as only velocity-dependent effect, while in the accelerated frame of the "travelling" twin also pseudo-gravitational potential difference influences the tick-rate of the other twin's wristwatch.

The OP came to the wrong idea, that real gravitation is needed to break the symmetry of the typical "twin paradox" scenario. I can understand why, because at gravitational time-delation, both twins agree on, whose wristwatch ticks slower and whose faster. But in reality in this case, the symmetry breaking of tick-rates does not come from real gravitation, but from pseudo-gravitational potential-difference in one of the rest frames. So in some sense, the OP was close to this real explanation of the needed asymmetry.

 

[Nugatory]

because at gravitational time-delation, both twins agree on, whose wristwatch ticks slower and whose faster.

However, the amount by which the ages differ is a function of the amount of time the traveller spends coasting and is independent of the duration and magnitude of the pseudo-gravitational time dilation at turnaround.

 

[Sagittarius A-Star]

However, the amount by which the ages differ is a function of the amount of time the traveller spends coasting and is independent of the duration and magnitude of the pseudo-gravitational time dilation at turnaround.

That's right, because the pseudo-gravitational time dilation relates to the fast tick-rate of the stationary twin's wrist watch while the turnaround und not to the tick-rate of the traveling twin's wrist watch, which stays "normal" in his frame. And the integral of the pseudo-gravitationally time dilated tick-rate over the turnaround-time is pre-defined in the scenarion, as the acceleration of the traveller's frame shall change his velocity in the stationary twin's frame from +v to -v in a certain approximate distance.

 

[PeroK]

That's right, because the pseudo-gravitational time dilation relates to the fast tick-rate of the stationary twin's wrist watch while the turnaround und not to the tick-rate of the traveling twin's wrist watch, which stays "normal" in his frame. And the integral of the pseudo-gravitationally time dilated tick-rate over the turnaround-time is pre-defined in the scenarion, as the acceleration of the traveller's frame shall change his velocity in the stationary twin's frame from +v to -v in a certain approximate distance.

Suppose the traveller takes two stopwatches on the journey. The first watch stays on for the whole journey (including the several acceleration phases). The second watch is switched on only during the inertial phases of the motion, and is stopped during the acceleration phases.

If we assume that the acceleration phases are short (they can, theoretically, be made arbitrarily short), then the two watches show approximately the same time on return the Earth. They may, for example, differ by a few minutes if the acceleration phases lasted only a few minutes.

Given that the second watch was switched off during the critical acceleration phases, how do you explain this? Using your pseudo-gravitational explanation?

 

[Sagittarius A-Star]

Given that the second watch was switched off during the critical acceleration phases, how do you explain this? Using your pseudo-gravitational explanation?

You find the answer already in my posting #57. Let me summarize it: The pseudo-gravitational time dilation relates to the stationary twin's watch tick-rate (in the rest frame of the traveling twin) and not to the tick-rate of the travelling twin's watch.

While the inertial phases, the stationary twin's watch ticks slow (1/"Gamma"), and while the short turn-around phase, the stationary twin's watch ticks very fast (over-compensating the slow tick-rates of the inertial phases), all in the rest frame of the traveling twin.

 

[Sagittarius A-Star]

@PeroK : see also the following "twin paradox" description in the middle of the linked page:

https://en.wikisource.org/wiki/Translation:Dialog_about_Objections_against_the_Theory_of_Relativity

 

[Ibix]

Given that the second watch was switched off during the critical acceleration phases, how do you explain this? Using your pseudo-gravitational explanation?

The point is to use pseudo-gravity to regard the stay-at-home twin as above the traveller during the turnaround. Thus the stay-at-home's clock runs fast and elapses more time than the traveller.

I don't think the explanation is particularly intuitive because you need to be quite careful by what you mean by "during" the turnaround. That goes double for instantaneous turnaround - what's the pseudo-gravitational potential associated with an infinite acceleration? But if you are willing to do the mucking around it ought to work.

 

[vanhees71]

Ether theory is physically distinct from special relativity. The theories agree only up to first order in $\beta=v/c$ (roughly speaking). As you point out yourself ether theory has been ruled out at for at least one century ago with many experiments (Michelson-Morley, Trouton-Noble, Doppler effect on light,...). A nice review can be found here:

http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html

 

[Vanadium 50]

This whole "pseudo-gravity" explanation seems supremely unhelpful. Telling someone who is struggling with special relativity that understanding it requires understanding general relativity, kinda-sorta, does not sound like the easiest path. The fact that thousands of people get it without GR or GR-like or pseudo-GR establishes this.

(It might possibly be usable in reverse - starting from SR to get to GR as motivation - but it's certainly not often adopted this way)

"Acceleration" cannot be the answer to "why is the aging different" because it is possible to set this up with multiple travelers looking at clocks through windows with nobody accelerating at all. The answer is that these are different paths through spacetime, and they are different just like different paths between A and B in space (say LA to San Francisco via Baltimore) have different lengths.

"Acceleration" is only an answer to the question "I don't want to do this with windows and clocks - I want to compare two clocks, one staying home, and one that goes on the trip. How do I tell them apart?"

 

[robphy]

That's all fine, if you teach relativity to physics students. But the OP asks for a laymen explanation of the typical "twin paradox" scenario and states wrongly, that a scenario without real gravity would be symmetrically.

I think, for laymen an explanation with 4-vectors and 4D-spacetime may be too difficult to understand. An explanation comparing the empirical viewpoint of both twins maybe didactically better for an explanation to laymen.

The diagrams actually are intended for the lay person (after some introduction),
but the words I used for my reply weren't necessarily addressed to the layperson.
I see now I used "pseudo-riemannian" inappropriately to make my point
since Galilean is also pseudo-riemannian.

My main point is that
the way we use the position vs time graph in PHY 101 is only an approximation to what is really going on.
In fact, it is not well-appreciated that the position-vs-time graph is already a non-euclidean geometry.

Here's are the real measurements on a position-vs-time graph.
In this graph, the red lines are perpendicular to each other (in all three geometries mentioned earlier) and the blue segment is the hypotenuse. You can introduce units to redefine variables to make the sides have the same units... but that doesn't change the underlying geometrical relationships.

In PHY 101, we treat the blue segment as having length 1 (already in violation of Euclidean geometry)
but it's really length (approximately) $1-(5\times 10^{-17})$. (Someday, we'll have a wristwatch that will measure this.)
The event that is "Length 1 along the blue segment" occurs to the right of the red vertical line.
This is time-dilation.

 

[martinbn]

I might be repeating something that was aleardy said, but here is a scenario that might help with the issue of the acceleration. The two twins depart together in the same direction. One of them turns back first, the second later. They both return to the original place, the one that turns first will wait there for the second one. This way they accelerate in the begining, during the turn, and at the end in exactly the same way. So the accelerations were the same for both. Just one turns later. When they meet they compare age and it turns out that they are not the same age. So acceleration cannot be the reason for the different aging.

 

[FactChecker]

I might be repeating something that was aleardy said, but here is a scenario that might help with the issue of the acceleration. The two twins depart together in the same direction. One of them turns back first, the second later. They both return to the original place, the one that turns first will wait there for the second one. This way they accelerate in the begining, during the turn, and at the end in exactly the same way. So the accelerations were the same for both. Just one turns later. When they meet they compare age and it turns out that they are not the same age. So acceleration cannot be the reason for the different aging.

The observed effect on another observer depends on the distance between the observers. The greater the relative distance, the greater the effect on the clocks. This is why the accelerations at the start and end of the twin's travel has very small effect, but the accelerations when he turns around has a large effect. Otherwise, the acceleration effects would largely cancel out. So your example does not remove the proposed effect of acceleration.

 

[vanhees71]

I might be repeating something that was aleardy said, but here is a scenario that might help with the issue of the acceleration. The two twins depart together in the same direction. One of them turns back first, the second later. They both return to the original place, the one that turns first will wait there for the second one. This way they accelerate in the begining, during the turn, and at the end in exactly the same way. So the accelerations were the same for both. Just one turns later. When they meet they compare age and it turns out that they are not the same age. So acceleration cannot be the reason for the different aging.

It's also interesting that this question has been investigated experimentally:

http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html#Clock_Hypothesis

The upshot is that indeed even at very high proper accelerations against an inertial frame "proper clocks" like lifetimes of unstable particles (muons) show the "proper time", defined by $\mathrm{d} \tau=\sqrt{1-\beta^2} \mathrm{d} t$, i.e., independent of acceleration.

 

[Sagittarius A-Star]

So acceleration cannot be the reason for the different aging.

Acceleration does not directly effect the different aging. But it has an indirect effect on it, because for example in the inertial rest frame of the stationary twin, the integral over time of the acceleration of the traveling twin equals the change of his/her velocity.

As @FactChecker stated correctly in posting #66, in the standard "twin paradox" scenario, this indirect effect is greater, if the distance of the turnaround from Earth is greater:

1) Reason, described in 4D-spacetime: If this spatial distance is greater, then the proper time of the traveling twin is smaller, because of the minus-sign in the formula for invariant spacetime-distance.

2) Reason, described in the rest frame of the stationary twin: If this spatial distance is greater, then the traveling twin travels longer with an almost constant slow tick-rate of his/her watch (1/"Gamma").

3) Reason, described in the rest frame of the traveling twin: If this spatial distance is greater, then the pseudo-gravitational potential of the remote "stationary" twin is greater, and therefore also the tick-rate of his/her watch is greater - while the short "turnaround"-time.

 

[PeroK]

Acceleration does not directly effect the different aging. But it has an indirect effect on it, because for example in the inertial rest frame of the stationary twin, the integral over time of the acceleration of the traveling twin equals the change of his/her velocity.

As @FactChecker stated correctly in posting #66, in the standard "twin paradox" scenario, this indirect effect is greater, if the distance of the turnaround from Earth is greater:

1) Reason, described in 4D-spacetime: If this spatial distance if greater, then the proper time of the traveling twin is smaller, because of the minus-sign in the formula for invariant spacetime-distance.

2) Reason, described in the rest frame of the "stationary" twin: If this spatial distance if greater, then the traveling twin travels longer with an almost constant slow tick-rate of his/her watch (1/"Gamma").

3) Reason, described in the rest frame of the traveling twin: If this spatial distance if greater, then the pseudo-gravitational potential of the remote "stationary" twin is greater, and therefore also the tick-rate of his/her watch is greater - while the short "turnaround"-time.

That's a bit like being given a speeding ticket that, instead of saying you were driving too fast, provides an acceleration profile that implies you were drving too fast.

By that argument, acceleration is an indirect cause of speeding. That may be a true statement, but it would make for a complicated law as to if and when you got a speeding ticket.

 

[FactChecker]

By that argument, acceleration is an indirect cause of speeding. That may be a true statement, but it would make for a complicated law as to if and when you got a speeding ticket.

I have to admit that this is a good point. But I think that to say that turning around causes the effect but that acceleration does not is quibbling about semantics. Turning around is acceleration.