Exploring Length Contraction and Light Rays in Frame O: Analysis and Discrepancy

[ijustlost]

A rod of length 2l is at rest in frame O' with co-ords (x',y',z')=(±l,λ,0)

Observer O moves at speed u along the x axis.

The first part of the question is just to derive the length contraction - fine, O measures the length as 2l / γ

The next part has me stuck:

"Show that, at t = 0, the observer O sees the light rays from the two ends of the rod
coming in at angles that lead to an apparent length of 2lγ (making the assumption
that the rod is oriented parallel to the x-axis with a y separation of λ from the origin).
Comment briefly on the discrepancy between the two results. "

I've tried drawing space-time diagrams, and just got in a bit of a mess. I can work out equations for the worldlines of the front and back of the pole (e.g. for one end (ct,x,y,z) = (ct'γ - lγ u/c, γ t'-lγ ,λ,0) etc), but setting t=0 for front and back just gives the 2l / γ. result

 

[SergioPL]

The light rays are sent in the same time at the rod system or in the same time at the observer's system. In the first case, the ray sent from the beginning of the rod will be advanced in time gamma*l*u^2 seconds and the ray sent from the end will be delayed the same quantity.

The result will be that it looks like the rays have been sent from a distance that looks (gamma*l). You can also transform the vector 2*l x+0 t from the Rod system to the observer system, it will become 2*l*gamma.

 

[SergioPL]

The light rays are sent in the same time at the rod system or in the same time at the observer's system. In the first case, the ray sent from the beginning of the rod will be advanced in time gamma*l*u^2 seconds and the ray sent from the end will be delayed the same quantity.

The result will be that it looks like the rays have been sent from a distance that looks (gamma*l). You can also transform the vector 2*l x+0 t from the Rod system to the observer system, it will become 2*l*gamma.