B Twin paradox explained for laymen

[martinbn]

Even then the acceleration profiles were different. Both twins agree which twin is the early accelerating and which is the late accelerating twin. The acceleration still eliminates the symmetry.

Again, (proper) acceleration does not cause the differential aging, but it does break the symmetry.

Of course there is no symmetry, but the experienced acceleration is the same for the two. It is not the acceleration that makes it asymmetric.

 

[FactChecker]

The twin who slingshots around a large mass is lower in a potential well. So he ages more slowly. And he is moving faster. So he ages more slowly.

Both of these are coordinate-dependent heuristics for time dilation, but the result should hold for differential aging: Shorter elapsed time for the slingshotting twin.

Sorry. I stand corrected.

 

[FactChecker]

Of course there is no symmetry, but the experienced acceleration is the same for the two. It is not the acceleration that makes it asymmetric.

The relative distances when the acceleration occurs is different. So it is the acceleration and the timing/relative distance of it that breaks the symmetry.

 

[PeroK]

The relative distances when the acceleration occurs is different. So it is the acceleration and the timing/relative distance of it that breaks the symmetry.

That's a remarkably uncovariant explanation.

 

[martinbn]

The relative distances when the acceleration occurs is different. So it is the acceleration and the timing/relative distance of it that breaks the symmetry.

What do you mean the relative distences are different? Distances to what?

 

[Dale]

Of course there is no symmetry, but the experienced acceleration is the same for the two. It is not the acceleration that makes it asymmetric.

I disagree that the acceleration is the same. Acceleration is a vector valued function of time and those functions are not the same. The acceleration is indeed asymmetric.

Again, my claim isn’t that the acceleration causes the aging, just that it breaks the symmetry. With no other information besides their accelerometer readings the twins agree who accelerated early and who accelerated late. So the acceleration itself does break the symmetry.

 

[FactChecker]

What do you mean the relative distences are different? Distances to what?

The relative distance between two points is the distance from one point to the other.

 

[PAllen]

One thing is true: there is only a paradox for those who do not understand SR. Mostly the general public. You would be struggling to find a physicist who thinks there is a paradox that needs an elaborate, pseudo-gravitational explanation.

Einstein was a strong proponent of the pseudogravity explanation for most of his life. He never considered there was paradox, but his preferred explanation was pseudogravity. It could be made to work for every smooth (continuous second derivative, specifically) variant in SR, and generalizes to GR.

I do not like this approach, but I find it nonsensical to claim it is invalid.

 

[jbriggs444]

The relative distance between two points is the distance from one point to the other.

A "point" in four dimensional space time is stretched into a chain of "events" known as a "world line".

Measuring the distance between two events is unambiguous. Measuring the distance between two world lines is ambiguous -- it depends on which pair of events you pick out.

The process of picking a particular world line through an event to be the "point" associated with that event is also a potential source of ambiguity.

 

[vanhees71]

Huh?

The twin who slingshots around a large mass is lower in a potential well. So he ages more slowly. And he is moving faster. So he ages more slowly.

Both of these are coordinate-dependent heuristics for time dilation, but the result should hold for differential aging: Shorter elapsed time for the slingshotting twin.

Now, I'm confused. The geodesic equation is derived from the free-particle action principle with the Lagrangian
$$L=-m c \sqrt{g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}},$$
i.e., it's the curve between fixed initial and final points which minimizes
$$A=\int_{\lambda_1}^{\lambda_2} \mathrm{d} \lambda L=-mc \tau.$$
So $\tau$ is maximal for the geodesic (free-fall worldline) connecting two points, right?

 

[PeroK]

Einstein was a strong proponent of the pseudogravity explanation for most of his life. He never considered there was paradox, but his preferred explanation was pseudogravity. It could be made to work for every smooth (continuous second derivative, specifically) variant in SR, and generalizes to GR.

I do not like this approach, but I find it nonsensical to claim it is invalid.

Touche. Obviously the great man was a law unto himself!

 

[PeterDonis]

It seems to me that slingshotting around a distant star brings an entirely different geometry into the situation. I don't think that SR applies in that situation.

That's correct. The post I was responding to there was, IIRC, not limiting itself to SR scenarios.

we know that the twin who slingshots around a large mass would age faster

First, as @jbriggs444 points out, the twin is lower in a gravitational potential well, which makes him age slower, not faster.

Second, the turnaround in this case, just as in the usual SR twin paradox scenario, can be made to occupy a negligibly short portion of the traveling twin's entire worldline, so the aging during it can be ignored. Its primary effect is simply to allow the traveling twin to turn around, i.e., to make his worldline such that he can come back to meet up with the stay-at-home twin again.

 

[PeterDonis]

in GR the maximum proper time among all time-like curves connecting to points is reached for a geodesic connecting the two points.

Not always. The case I described is a counterexample, if we have the stay-at-home twin floating in free space instead of sitting on Earth's surface. Then the stay-at-home twin's worldline is a geodesic, and so is the traveling twin's (since his turnaround is accomplished by a free-fall slingshot maneuver around a distant planet or star), but only the former's worldline is a worldline of maximal proper time between the starting and ending events.

Of course there are simpler examples possible: for example, an astronaut in a spaceship orbiting Earth launches a probe radially outward, with just the right velocity such that the probe returns to the ship after the ship has made exactly one orbit. The ship's and the probe's worldlines are both geodesics, but only the latter's worldline is one of maximal proper time between the starting and ending events.

 

[PeterDonis]

The relative distances when the acceleration occurs is different.

A better way to state this would be that the time elapsed on the traveling twin's clock between the starting event (where he leaves the stay-at-home twin) and when the acceleration occurs is different. That makes it clear that the difference is invariant.

 

[PeroK]

I do not like this approach, but I find it nonsensical to claim it is invalid.

I'm not sure whether I'd say it's invalid, but here are the two things I dislike most:

1) It avoids using the simple geometry of flat spacetime - that explanation should be a lightbulb moment for any student of SR.

2) It fails to dispel the myth that GR is required to explain the twin paradox. Reading the 100+ posts in this thread you may come to the conclusion that whether GR and gravity are required to explain the twin paradox is still an open question.

 

[FactChecker]

That's correct. The post I was responding to there was, IIRC, not limiting itself to SR scenarios.
First, as @jbriggs444 points out, the twin is lower in a gravitational potential well, which makes him age slower, not faster.

Second, the turnaround in this case, just as in the usual SR twin paradox scenario, can be made to occupy a negligibly short portion of the traveling twin's entire worldline, so the aging during it can be ignored. Its primary effect is simply to allow the traveling twin to turn around, i.e., to make his worldline such that he can come back to meet up with the stay-at-home twin again.

Yes. I misspoke in the earlier post when I said that the traveling twin would age faster during the slingshot. He would age slower. The twin on Earth would age more. But that is just substituting actual gravity for acceleration (pseudo-gravity). Either one makes the situation of the twins non-symmetric and alleviates the "paradox" complaint.

 

[vanhees71]

Not always. The case I described is a counterexample, if we have the stay-at-home twin floating in free space instead of sitting on Earth's surface. Then the stay-at-home twin's worldline is a geodesic, and so is the traveling twin's (since his turnaround is accomplished by a free-fall slingshot maneuver around a distant planet or star), but only the former's worldline is a worldline of maximal proper time between the starting and ending events.

Of course there are simpler examples possible: for example, an astronaut in a spaceship orbiting Earth launches a probe radially outward, with just the right velocity such that the probe returns to the ship after the ship has made exactly one orbit. The ship's and the probe's worldlines are both geodesics, but only the latter's worldline is one of maximal proper time between the starting and ending events.

You are right. It's not so clear, for general geodesics connecting the same spacetime points. So one has to do the calculation. Perhaps your latter example is a nice exercise for testbodies in a Schwarzschild spacetime (e.g., one in a circular orbit, the other being shot radially out in the and falling back as described. I'll try that!

 

[jbriggs444]

Not always. The case I described is a counterexample, if we have the stay-at-home twin floating in free space instead of sitting on Earth's surface.

If intuition serves, a geodesic will be a path that yields a local maximum for elapsed time among a set of "similar" paths.

 

[PeterDonis]

that is just substituting actual gravity for acceleration

Not really. The traveling twin in the slingshot scenario has zero proper acceleration (which is of course the point of the scenario, to have a turnaround with zero proper acceleration). The phrase you use here is usually used to describe, for example, standing on Earth's surface as compared to standing in a rocket accelerating at 1 g in free space, i.e., the proper acceleration is the same in both cases, only the spacetime geometry, and hence the reason for the proper acceleration being present, is different.

 

[PeterDonis]

a geodesic will be a path that yields a local maximum for elapsed time among a set of "similar" paths

First, we should be restricting attention to timelike geodesics, since those are the only ones for which the "maximum" heuristic is valid in spacetime anyway.

Second, I don't think even the "local maximum" rule always works; a timelike geodesic will always be a local extremum, but I think there are cases where the extremum is a local saddle point, not a local maximum.

 

[Sagittarius A-Star]

One thing is true: there is only a paradox for those who do not understand SR. Mostly the general public. You would be struggling to find a physicist who thinks there is a paradox that needs an elaborate, pseudo-gravitational explanation.

The connection between time-dilation and the artificial gravity, that an apple-tree in an uniformly accelerated rocket experiences, can be visualized by this video (A reference to GR or the principle of equivalence is not required):

 

[PAllen]

First, we should be restricting attention to timelike geodesics, since those are the only ones for which the "maximum" heuristic is valid in spacetime anyway.

Second, I don't think even the "local maximum" rule always works; a timelike geodesic will always be a local extremum, but I think there are cases where the extremum is a local saddle point, not a local maximum.

I believe there is a theorem I’ve seen quoted in a number of GR texts and papers, which says, to the best of my memory, that for a sufficiently small causal diamond, for any two points in it that can be connected by a timelike path, there is a unique timelike geodesic contained in the causal diamond, and that this geodesic maximizes proper time among all paths contained in the diamond.This, among other things, rules out saddle points over small scales. This theorem is also what makes rigorous Synge’s notion of a World function.

This theorem also implies that for any timelike geodesic, for any sufficiently small neighborhood of any event on it, the geodesic is the maximizing path between two points on it contained in that neighborhood, among paths contained in that neighborhood.

 

[PAllen]

That's a remarkably uncovariant explanation.

Not really. General covariance means that despite different coordinate descriptions, calculation of invariants come out the same. And for physics, that means all direct observables must be invariants. However, it also entails the notion that any coordinate description is as good as any other.

Thus, for example, for muons created in the upper atmosphere reaching the ground, the Earth frame explanation of time dilation is neither more nor less valid than the muon frame explanation of the short distance between the upper atmosphere and ground.

In the twin scenario in SR, we have that in any inertial coordinates, all differences in clock rates are purely dependent on velocity, and this will be sufficient to explain any differences in proper time along paths (which are, of course invariant). However, in non-inertial coordinates, e.g. in which a non-geodesic path is a stationary position coordinate, coordinate velocity alone does not explain the relation of proper time to coordinate time. The difference can be described as pseudogravity. In any such noninertial coordinates, there is coherent explanation of proper time differences along paths that is due to a mix of velocity and the effects of pseudogravity (manifested mathematically as a non-trivial metric and nonvanishing connection). General covariance states that an explanation in any such general coordinates is equally as valid as that in inertial coordinates. None of this has anything to do with GR as it currently understood. It simply uses methods that became common with GR.

Of course I agree with you that all of this is a wildly overcomplicated way of understanding the twin scenarios. However, I also don't like the notion of "changing frames", because in my view a frame is something used by people to describe physics, and particles or bodies do not have frames - they simply may be described in any frame or coordinates.

 

[PeroK]

In the twin scenario in SR, we have that in any inertial coordinates, all differences in clock rates are purely dependent on velocity, and this will be sufficient to explain any differences in proper time along paths (which are, of course invariant). However, in non-inertial coordinates, e.g. in which a non-geodesic path is a stationary position coordinate, coordinate velocity alone does not explain the relation of proper time to coordinate time. The difference can be described as pseudogravity. In any such noninertial coordinates, there is coherent explanation of proper time differences along paths that is due to a mix of velocity and the effects of pseudogravity (manifested mathematically as a non-trivial metric and nonvanishing connection). General covariance states that an explanation in any such general coordinates is equally as valid as that in inertial coordinates. None of this has anything to do with GR as it currently understood. It simply uses methods that became common with GR.

That is most certainly NOT the simplest way to resolve the twin paradox! We were looking for the simplest explanation.

 

[PAllen]

That is most certainly NOT the simplest way to resolve the twin paradox! We were looking for the simplest explanation.

I strongly agree.

 

[PeterDonis]

for a sufficiently small causal diamond

The key is that "sufficiently small". There is no general formula for how to pick out the sufficiently small causal diamonds in any spacetime; it depends on the particular spacetime.

 

[Sagittarius A-Star]

That is most certainly NOT the simplest way to resolve the twin paradox! We were looking for the simplest explanation.

The calculation in the rest frame of the traveling twin can be made faily easy, if the scenaio is set-up accordingly and only an approximational calculation is done.

Let's take the specific example from Wikipedia: distance d = 4 light years, at a speed v = 0.8c. The "stationary" twin gets older by 10 years, the traveling twin by 6 years, γ = 5/3.

The traveling twin approches the star after 3 years proper time and calculates, the the "stationary" twin must have aged then by 3 years /γ = 1.8 years. So while both inertial travel phases, the "stationary" twin must get older by 2 * 1.8 years = 3.6 years. The "gap" to 10 years is (10 - 3.6) years = 6.4 years. This "gap" has to be filled by pseodo-gravitational time-dilation while "turnaround":

Δt₁ = time "gap" of the "stationary" twin's aging while "turnaround"
Δt₂ = duration of "turnaround" (defined as arbitrarily short, neglectable)
Φ = pseudo-gravitational potential
a = proper acceleration of frame while "turnaround"
h = distance of twins * γ
Δv = velocity change by turnaround = 2 * v = 1.6c

Approximate calculation:
Δt₁ = Δt₂ (1 + Φ /c²) = Δt₂ (1 + a*h /c²) = Δt₂ (1 + (Δv/Δt₂)*h /c²) = Δt₂ + (1.6c * 4LY /c²)
= Δt₂ + (1.6 * 4LY /c) = Δt₂ + (1.6 * 4Y) = 6.4 Years.

PeroK told me, that Δt₂ := a few minutes. That can be neglected.

Derivation of the used time-dilation formula:
https://www.physicsforums.com/threa...idered-to-be-accelerating.991333/post-6367026

Remark:
I find the factor γ in h in several papers (linked in posting #129), but I don't yet know the reason for it.

 

[PeterDonis]

The calculation in the rest frame of the traveling twin

There are two "rest frames" involved here (if we leave out the turnaround), the outgoing one and the returning one. Both have the same magnitude of $v$ relative to the stay-at-home frame, so the time dilation factor is the same for both, but the $v$ is in opposite directions.

 

[Sagittarius A-Star]

There are two "rest frames" involved here

I would say, that the rest frame changes from one inertial frame via the turnaround to another inertial frame.

 

[PeterDonis]

I would say, that the rest frame changes from one inertial frame via the turnaround to another inertial frame.

There is no "the" rest frame for the traveling twin unless you mean a non-inertial frame. There is no single inertial frame in which the traveling twin is at rest for the entire scenario.

In fact, even with regard to non-inertial frames, the phrase "the rest frame changes", while it is indeed a common one, is IMO misleading. "The" rest frame cannot change; the whole point of defining "the" rest frame is to have a single frame in which the chosen observer is always at rest. If you're not going to do that, you might as well drop the "rest frame" idea altogether and just do the calculation in the most convenient inertial frame for the problem, which in this case is the stay-at-home twin's frame.