# Three inertial particles: Separating the twins from the paradox

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In summary, the conversation discusses a response to an insight article about the twin paradox and proposes a model using only a straightedge and a compass. The model aims to eliminate the paradox by stripping away elements such as the narrative of two twins and technical gimmicks. Instead, it focuses on three particles and three events, and calculates the proper time experienced by each particle between the events. The conversation also encourages readers to try drawing the model themselves and provides hints and step-by-step guides for those who need assistance.
FactChecker said:
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.

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FactChecker and PeroK
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.

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.

PeterDonis said:
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.

Peter Strohmayer said:
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.

vanhees71 said:
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.

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
$$\tau=\int_{\lambda_1}^{\lambda_2} \mathrm{d} \lambda \sqrt{g_{\mu \nu} \dot{q}^{\mu} \dot{q}^{\nu}}.$$

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