I Three inertial particles: Separating the twins from the paradox

[PeterDonis]

I have a Ph.D. in pure mathematics

Your credentials do not automatically mean everything you post is correct. @PeroK correctly pointed out a claim of yours that was wrong, the same one I did. If you have a Ph.D. in math, you should have no trouble doing the required math to either see that @PeroK and I are right and retract your claim, or show us the math that proves your claim is correct.

There is a big difference between derivatives of velocities in a non-inertial reference frame and physical accelerations.

Yes, agreed. But I don't think that is an important confusion in the OP of this thread.

IMO, that is a major problem for people who are struggling with the twin paradox.

That's not the sense I have gotten from other PF threads on the topic. Do you have any particular examples in mind?

I have read your article and do not think it adequately addresses that issue.

The article only uses proper acceleration and doesn't even mention coordinate acceleration, that's true. That's because I don't think coordinate acceleration even plays any part at all in analyzing the scenario, whereas proper acceleration does play a role at least in the standard scenario since it is a physical observable that is asymmetric between the twins.

With that being said, I feel that I should leave this thread to others.

Not until you either retract your claim or back it up with math. Drive-bys are not appreciated here.

 

[PeroK]

To give another explicit example. Suppose an alien spaceshift explored our part of the galaxy. They approach Earth at $0.6c$ and proceed to Alpha Centauri. In the local reference frame this journey is about 4.4 light years and takes about 7 years. The Aliens know this and can calculate this. The onboard journey time, however, is about 5.5 years.

Note that there is an asymmetry between the reference frames, as we have a single spaceship and two fixed points in the local frame.

We can also assume that we have a space station orbiting Alpha Centauri and this knows the local Earth time (using the Einstein synchronization convention).

In any case when the Aliens pass the space station, they are not surprised that 7 years Earth time has passed. What they cannot do is imagine that the time dilation of the Earth clock implies that only about 4 years has passed in the local reference frame during the journey.

In learning SR, resolution of this scenario should precede tackling the full twin paradox, which essentially puts two of these Alien journeys back to back - with some sort of turnaround, potentially requiring an acceleration phase.

In this way, the full twin paradox is resolved. Moreover, the acceleration phase is seen to be not critical to the argument, but merely a physical constraint in the process of changing direction.

 

[vanhees71]

It has nothing to do with acceleration. The asymmetry is more fundamental. The turnaround times differ.

Let's try my simple way for the standard scenario. We use the inertial frame of the resting twin as "calcluational frame", because that's the most simple one to calculate invariant (!!!) quantities.

The worldline of the twin at rest, parametrized with the "coordinate time" (which is this twin's proper time of course) is simply
$$x_A=(t , 0), \quad t \in [0,T].$$
The world line of the traveling twin is
$$x_B=\begin{cases} (t,vt) & \text{for} \quad t \in [0,T/2], \\
(t, v(T-t))&\text{for} \quad t \in [T/2,T]. \end{cases}$$
Then you get
$$\tau_A=\int_0^T \mathrm{d} t \sqrt{\dot{x}_A(t) \cdot \dot{x}_A(t)}= T$$
and
$$\tau_B=\int_0^T \mathrm{d} t \sqrt{\dot{x}_B(t) \cdot \dot{x}_B(t)}=\int_0^T \mathrm{d} t \sqrt{1-v^2}=T \sqrt{1-v^2}.$$
Case closed.

 

[Peter Strohmayer]

You're wrong about acceleration being the only resolution to twin paradox.

The world line of the traveling twin is not straight. So how can one claim that acceleration is not relevant to understanding the "twin paradox"?

 

[vanhees71]

I'd say it rather has to do with time dilation than with acceleration. It's about the dependence of the time between two events on the clock that's used which measures them. All you can say is that the measured time is maximal for a clock moving along a geodesic in spacetime. In Minkowski spacetime that's uniform rectilinear motion.

 

[Nugatory]

I wouldn't call it my analysis, since I am trying to explain the flaw in the analysis of those who are confused by the Twins Paradox.

You could improve your explanation some by rephrasing this part:

As long as both twins remain in their own inertial reference frame...

There's no such thing as "remaining in" in a reference frame (in SR everything is always in all frames everywhere) and "their own reference frame" is a misleading way of saying “using a frame in which they are at rest”. Without this sloppy terminology it is difficult to even state the paradox - we find ourself saying something like “using the inertial frame in which the traveling twin is at rest throughout the journey” and the flaw becomes obvious.

 

[vanhees71]

Just once more: According to the clock postulate, what's measured is the proper time of (ideal) clocks. This is an invariant, independent on the choice of reference frame. It doesn't even need to be an inertial frame of reference and the corresponding "Lorentzian" coordinates to calculate it. As also in Newtonian physics, the description in terms of coordinates referring to a non-inertial frame is perfectly valid also in SR. In GR there are only local inertial frames anyway.

 

[Trysse]

Although I have enjoyed the discussion, I find, that it is beside the point (of the OP). The Idea of the OP was to present the scenario without any narrative tweaks and turns that make the description paradoxical.

The paradox arises in the way the story is told. If I tell the story as "the twin paradox", I (intentionally or unintentionally) tell a story that is misleading. What is misleading in that story are the words that are used. "Innocent" words such as seeing, measuring, and calculating are prone to hide far more complex processes that involve many steps. Other more technical words that are prone to hide complex concepts are coordinate system, frame of reference, etc.

Unless complex processes are described and concepts are explained in detail, the meaning of those words remains ambiguous. This can lead to a feeling of intellectual unease. Depending on the inclination of the audience, this unease leads to the notion, that there is a paradoxical situation. However, the situation is not paradoxical, only the description of the situation is wrong/incomplete/misleading. However, the "paradoxical story" is not the topic of the OP.

It would be a paradoxical situation if both twins had aged identically!

I think the point that is worth discussing is what @PeterDonis pointed out. His point directly addresses the geometric model.

Unfortunately, your attempt obfuscates the actual spacetime geometry because it uses Euclidean geometry, whereas the geometry of spacetime is Minkowskian. The straightedge for making lines is all right, but the compass is wrong; for Minkowskian geometry it should be a tool that draws hyperbolas about a given center, not circles. For this particular scenario, you can get away with it because moving a term from one side of the equation to the other makes it look like the ordinary Pythagorean theorem. But this method will not generalize to other scenarios; whereas the "spacetime geometry" viewpoint described in the Insights article works for any scenario whatever, in full generality.

 

[PeroK]

The world line of the traveling twin is not straight. So how can one claim that acceleration is not relevant to understanding the "twin paradox"?

Read my posts or check resources online.

 

[A.T.]

He can use an arbitrary (accelerating) rest frame. Then of course he has to use the adequate components of the fundamental form, $g_{\mu \nu}$.

And that makes the analysis in an accelerating frame different from the analysis in an inertial frame. So the different accelerations of the reference frames break the symmetry between the analyses.

This has per se nothing to do with the accelerations of some clocks that are being analysed, it just coincides in some cases.

 

[vanhees71]

The important point is that it does not make the analysis different, if you take the (little) effort to formulate everything in a manifestly covariant way. The proper time of a clock along an arbitrary time-like worldline, expressed in arbitrary "generalized spacetime coordinates", $q^{\mu}$, in an arbitrary parametrization simply reads
$$\tau=\int_{\lambda_1}^{\lambda_2} \mathrm{d} \lambda \sqrt{g_{\mu \nu} \dot{q}^{\mu} \dot{q}^{\nu}}.$$