# Why length contraction and not apparent difference in relative velocity?

The Lorentz Transformation takes the fact relative velocities must be the same in both reference frames for an axiom, from which length contraction is derived. However, why does velocity 'take privilege' over space? Why is it the velocity that must be the same in both frames and not the distance between two objects?

• "relative velocities must be the same in both reference frames" What do you mean by that?
– user154997
Jun 5, 2017 at 20:11
• The reciprocity principle is taken to be an 'obvious' axiom. See, for example, Time and motion in physics: the Reciprocity Principle, relativistic invariance of the lengths of rulers and time dilatation: "If the velocity of O' relative to O is $\vec v$, the velocity of O relative to O' is $-\vec v$. In many discussions of special relativity, the reciprocity principle is taken as 'obvious' and is not even declared as a separate axiom" Jun 5, 2017 at 20:11
• @AlfredCentauri Yes, but why is it taken to be an obvious axiom? What's so obvious about it?
– Max
Jun 5, 2017 at 20:40
• @LucJ.Bourhis "If the velocity of O' relative to O is v, the velocity of O relative to O' is −v", basically
– Max
Jun 5, 2017 at 20:41
• Max, it seems obvious enough to me but I can't actually explain why that is the case. If it isn't obvious to you then rest assured that the principle of relativity together with homogeneity and isotropy imply reciprocity. Do these seem obvious to you? Jun 5, 2017 at 20:50

You are talking about different things. Relative velocity is the velocity that O thinks O' has. What would be the equivalent for length? It would be the distance d that O measures between himself and O'. O' will also measure the same distance d'=d. Length contraction, instead, is the difference between two distances that O, or O', measure. O will see that O' length is different if O' is moving or not. It will also measure a different velocity for O' if O' is moving or not. So no privilege of velocity over distance.

• In that regard, yes. However, if O were to measure its velocity relative to O' and calculate what it thinks O' will measure it to be, based on time dilation, it will obtain the same velocity. This is only possible through length contraction. Therefore, my question is: why do we take the velocity to take such properties and not space? For example, we could have said that the distance that O would measure between itself and O' was the same as it would think O' would measure, and then we would get that O will calculate its relative velocity to be different from what O' would measure it to be.
– Max
Jun 6, 2017 at 12:40

> Why is it the velocity that must be the same in both frames and not the distance between two objects?

It is an experimental fact that the speed of light is the same in any reference frame: see experiment of Michelson and Morley.

Contraction of lengths and times directly follows from that assumption.

• Care to explain the downvote? May 29, 2019 at 13:09
• Sure. Speed of light being constant has nothing to do with relative velocity between two objects being the same in each of those object's stationary reference frames. Hence your answer does not answer my question.
– Max
Jun 21, 2019 at 21:51
• @Max Except it does, because it is exactly the reason why. Did you read the experiment of Michelson and Morley at all and did you read how Lorentz transformations are derived at all? Apparently not :). There is no additional assumption in anything in the Lorentz transformation rather than the speed of light being constant and the reference frames being inertial, everything else follows automatically. Jun 22, 2019 at 8:54