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 v ≤ c 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.