Can't Solve the Relativity Problem: Seeking Help

[Jilvin]

I was reading C. Schiller's text on relativity and he gives an expression for the ratio frequencies of two observers (one a sender and one a receiver). *[Oh, and if you think that my description of the problem is unclear, I am talking about the expression on page 30 of the Motion Mountain volume II text]*

Observer S is moving at a velocity v with respect to observer R, and sends a light signal at an angle \theta_{s}. Observer R receives the signal at an angle \theta_{r}. The expression given for the ratio of the two wavelengths is:

\lambda_{r}/\lambda_{s}=\gamma(1-(v/c)cos \theta_{r})

I have tried to reproduce this assertion independently, but can't! Someone more skilled please help.

 

[bcrowell]

You can find a derivation in my book:
http://www.lightandmatter.com/genrel/
See section 4.2.3 for the 1-dimensional case, and the solution in the back of the book for problem 11 in ch. 4 for the generalization to more than one spatial dimensions. This uses four-vectors. If you aren't comfortable with four-vectors, I'm sure it can be done, but it probably becomes a more cumbersome calculation.

 

[Jilvin]

Thank you. I am slightly familiar with four-vectors but i'll need some touching up. However, I am willing to do this so this reference is satisfactory for now.

OFF TOPIC: I used you calculus book for self-study last year by the way. I only looked at it for 2 months prior to the AP Calc AB test and I managed to score a 4. So thanks for your material!