Straight lines in K and K’

Let S1 and S2 be two planes in K perpendicular to the x-axis, and let Δx = x2 – xdenote the signed distance between them, i.e. the difference between their respective x-coordinates. In K', let S1' and S2' be the corresponding planes perpendicular to the x'-axis, marked out at a given t in K.

Using a similar argument to that when λ was introduced, it can be easily seen that for the corresponding signed distance Δx' in K':

(Lemma) Δx' = μ · Δx, where μ > 0 is a constant.

Notes: (a) the value of μ can only depend on the choice of K and K', or rather only on v due to symmetry reasons, (b) in classical mechanics, μ = 1.

Since distances on planes perpendicular to the x-axis don't change, it follows immediately that (yet again, meaning "location" in the general sense):

(Theorem) If l is a straight line (segment) in K, and l' is the corresponding location in K' marked out at time t in K, then l' is a straight line (segment) too.

(Proof) It's due to the proportionality between each of Δx and Δx', Δy and Δy', Δz and Δz'.

But how is l perceived in K'? There is no guarantee that at time t in K, all points of l have the very same corresponding t' values in K'. (In classical mechanics it is guaranteed that they all do, since λ = 1.)

(Theorem) l is perceived in K' as a straight line (segment).

(Proof) Let P be a fixed point on l in K, and P' the corresponding point in K' at a given time t', marked out at an appropriate t in K. Now, let Q be an another point on l in K, with Δx = xQ – xP being the difference between the x-coordinates of Q and P. It was shown earlier that for the corresponding time value t'Q in K', read off at t in K, the deviation from t' is proportional to Δx, i.e. t'Q – t' ~ Δx. This also entails t'Q – t' ~ Δx', as Δx = (1 / μ) · Δx'. Thus, to find out where Q was (or will be) in K' at t', we have to shift it away from l', along v' by the (signed) amount of -(t' – t'Q) · v'. Since this amount is proportional to Δx', shifting all points on l' accordingly will result in a straight line (segment) in K'.

Note: it will result in a "proportional" straight line (segment) in K', meaning that e.g. middle points in K are perceived as middle points in K' too.

Making use of v' = v, the following relationship can be derived:

(Theorem) μ = λ.

(Proof) An observer in K, located on S2 sees that Δt = Δx / v time elapsed between meeting S2' and S1'. In addition, the observer also sees a corresponding time Δt' = λ · Δt elapsed in K'. This means that in K', S2 travelled Δt' = λ · Δx / v time from S2' to S1'. And since S2 is moving at v relative to K', the distance between S1' and S2' is Δx' = v · Δt' = λ · Δx.

Consequently, if there was a rod r' in K', parallel to the x'-axis and reaching from S1' to S2', its length would be Δx' in K' but an observer in K would see it Δx.

(Corollary) In K, the perceived length of r' is 1 / λ times its rest length in K'.

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