# Is special relativity immune to the paradox of Achilles?

According to the entry "Proper Time" in Wikipedia, for an object in a SR spacetime traveling with velocity v for a time interval Δ T

c2Δ T 2 = c2 Δτ2 + v 2 Δ T2,

where Δ T is the coordinate time interval and Δ τ proper time interval experienced for the object. Furthermore,

• Δ x =v Δ T is the distance traveled by the object in the coordinate space at speed v during Δ T,
• c Δ τ = (c 2 - v2) 1/2 Δ T is the distance traveled by light during Δ τ (for example in the object's internal clock) which may also be interpreted as the distance traveled by the object, say in the direction orthogonal to the coordinate space, at speed (c2 - v 2)1/2 during Δ T [as if the object did so],
• c Δ T may similarly be interpreted as the distance an stationary (with respect to the coordinate space) observer travels at speed c in the direction orthogonal to the coordinate space,
• The coordinate (3-dimensional) space and the axis orthogonal to it which is just that of coordinate time together form a (4-dimensional) coordinate spacetime.
• Therefore, the distance that the observer (that does not travel in the coordinate space) travels in the coordinate spacetime is the the same as the distance the object travels in the direction at angle arcsin(v/c) with respect to the observer, both at the same speed c, during the same coordinate time interval.

The implication of the interpretation is that all objects (as well as observers) travel with the same speed c (the resultant of the two velocity vectors one with the speed vc in the coordinate space and the other with the speed (c2 - v 2)1/2 in the coordinate time).

If Zeno's paradox of Achilles and the tortoise is indeed sound, it only rules out the cases where objects move at different speeds but not the cases where objects move at the same speed but different directions, since in such cases the objects travel the same distance during the same time interval.

It appears that there is an interpretation of Special Relativity which is not vulnerable to Zeno's argument for the paradox of Achilles: all objects and observers move at the speed of light in a 4-dimensional coordinate spacetime -- there is only one speed at which any entity can move. Projecting the motion of such objects onto any 3-dimensional space displays them as having different speeds and traveling different distances.

• Zeno's arguments "prohibit" any motion whatsoever (in the "true" sense), or none at all (as a sensory illusion). And only objects with zero rest mass, like photons, can move at the speed of light in SR. That excludes a lot of objects. Commented Apr 15, 2023 at 4:55
• Special relativity comes about from the constancy of the speed of light, not from the "3-dimensional space" alone. Commented Apr 15, 2023 at 15:41
• @Conifold the point is that perhaps those two option do not exhaust al the possible interpretations of this particular paradox (the Achilles and the tortoise). Also, when we say no object with mass can reach the speed of light, we mean it in the sense that their projected motion onto a lower-dimensional (3-D) space cannot reach the speed of light. Commented Apr 15, 2023 at 16:24
• @Frank What this interpretation implies is that another underlying phenomenon gives rise to the constancy of the speed of light: that there can only be one speed (possible for any motion). I am not sure I understand the second part of your comment. Commented Apr 15, 2023 at 16:28
• It is an interesting visualization. I once visualized gravity as being like a floor that slants down, at a greater or lesser angle according to its strength. But this floor slants down in every direction. Commented Apr 20, 2023 at 2:29

Yes, there are n things wrong with your argument, where I estimate n to have an upper bound of around 10. Here are some of them...

None of the versions of Zeno's paradox about Achilles that I have ever read prohibit two particles moving at different speeds.

In Special Relativity, temporal axes in spacetime cannot be set in any direction.

Your third sentence needs to be tightened up. You should say that in A's rest frame, assuming A is coasting inertially, A moves parallel with the time axis. That will eliminate the potential for confusion should your reader imagine that A considers themself to be driving towards Reno, say.

Your fourth sentence is also ambiguous. It suggests that B is moving orthogonally to A, in which case it is not moving in A's direction, regardless of how much it moves, contrary to what you say.

Your fifth sentence is incorrect. There will be a frame in which A and B are moving with the same speed, albeit in different directions.

Your sixth sentence is incorrect, as the phenomenon you describe in the preceding sentence does not give rise to the formulation of special relativity.

Nitpicker as I am, I would suggest there are two questionable aspects to your seventh sentence. The first is the opening word 'Hence' which you seem to be using as a shorthand for some unimaginable (to me at least) leap of logic. The second is all the words that follow, which are entirely unjustified by any of the preceding six sentences taken individually or together.

Apart from that, and the possibility of one or two other slips I have missed, I agree with everything you say.

• I don't think there are several "temporal axes" in SR to start with. Commented Apr 15, 2023 at 15:42
• @frank there are! I sharpened all three of mine just yesterday. You can't beat a newly sharpened temporal axe for cutting short any boring experience. Commented Apr 15, 2023 at 16:04
• Ah yes, I had forgotten about that - you are right :-) Commented Apr 15, 2023 at 16:25
• If I had n hours to cut something short, I would spend the first n-2 sharpening the axes. Commented Apr 20, 2023 at 2:25

A paradox is something which seems to follow from a model that seems to be sound, yet is false, indicating a hidden problem with the argument; or it is something true, yet seems like it should be false, indicating a problem with the reasoner's intuition or comprehension. Since anyone who can catch a tortoise can demonstrate that the Achilles argument has a false conclusion, and likewise for the arrow and stadium, they must be of the former type. Something was wrong with the argument, although it would take a few thousand years to formally pinpoint exactly what.

Zeno's paradoxes, expressed in terms of formal logic, were solved using formal logic - the calculus of infinite series sums - in the 19th century. No physics was needed beyond the mathematical relationship between distance, time, and velocity in order to translate Zeno's words into symbolic logic.

• Even if the "Standard Solution" is completely satisfactory, it does not block other solutions. The proposed interpretation of the this particular paradox (Achilles and the tortoise) provides something like a validation or support for SR, the same way it might be said that the Standard Solution provides support for the theory of calculus. Commented Apr 15, 2023 at 16:59