Question

I was solving problem 3.4 from Rindler's "Relativity - Special, General and Cosmological" (2006), which reads :

Use a Minkowski diagram to establish the following result: Given two rods of rest lengths l1 and l2 (l2 relative to the longer rod, in which the two rods have equal lengths, provided l21(c−v).

I got the answer just fine, but at no point did I use the provided constraint cited at the very end of the problem. Pondering on it a little, I noticed how it was somehow resemblant to

(l1l2)2<1+β1−β=(λrλs)2, where the right hand side of the inequality became the Doppler shift formula for emitted wavelength λs
and received wavelength λr
of some light signal.

Although, I have no idea where it fits in this problem. Is this an errata and is it an unnecessary condition for this solution to exist? Is there a reason for this specification that I'm just not seeing?

Answer 1

The constraint in Rindler's problem about the comparison of lengths using a Minkowski diagram seems unnecessary, but it may relate to the relativistic Doppler effect or act as a specific condition under special relativity principles.

Step-by-step explanation:

The question relates to special relativity and the use of the Minkowski diagram to analyze length contraction and the relativistic Doppler effect for two rods of different rest lengths moving relative to each other. The equation provided in the problem l2 < l1/(c-v) resembles the Doppler shift formula, which expresses the change in wavelength of light due to relative motion between source and observer.

The provided constraint might be unnecessary for the solution, but it also could be considered as a condition for concluding the comparison of lengths in different frames of reference, specifically when considering the scenarios at relativistic speeds. It is pertinent to the question of how the concept of proper length and length contraction are perceived by different observers moving at high velocity relative to each other.