Phys 481 Theory of Relativity I Homework 1B. Ozgur Sarıoglu due: Nov 4, 2013; Monday lecture

This is a one time reminder: You are NOT allowed to talk to anyone – and especially to each other– about the questions. It is very important that you write your solutions on your own, and youbetter do this neatly and carefully. It is also important that you explain your reasoning in eachquestion!

1-a) Consider two inertial frames S and S′ in standard configuration. In S′ the standard latticeclocks all emit a ‘flash’ at noon. Prove that in S this flash occurs on a plane orthogonal to thex-axis and traveling in the positive x-direction at speed c2/v.b) Prove that at any instant there is just one plane in S on which the clocks of S agree with theclocks of S′, and that this plane moves with velocity (c2/v)(1− 1/γ).c) Illustrate on a Minkowski diagram a situation similar to that of part a) above: A flash occurseverywhere at once on the spatial x′-axis of the frame S′ at some instant t′ = t′0. Show that inthe usual second frame S, this flash is a bright spot traveling forward along the spatial x-axis atsuperluminal speed c2/v.

2-a) Establish the approximation γ(v) ≈ 10n when v/c = 0.99 . . . 995 (with 2n nines). Show alsothat

i) γv = c(γ2 − 1)1/2 , ii) c2dγ = γ3vdv , iii) d(γv) = γ3dv .

We will use the latter identities later in our lectures, so these are important!b) A spaceship travels to a star 5 light-years away, in a time its crew considers to be 5 years. Whatis the ship’s speed? [This is a former GRE question and was used in an earlier “PhD qualifying”exam in our department!]

3- Consider the ‘Cin Ali observing two planes’ example we had in class. Take the proper length ofeach plane to be L. i) Deduce that the relative velocity between the two planes is given by v = L/3in the time units the clocks indicated in class. ii) If these units are 10−7 s, prove that v = 2

√2c/3

and L = 85 m. iii) What is the velocity of each plane relative to the ground?

4-a) Prove the following additional relations between the ‘rapidity’ φ and the velocity v:

coshφ = γ , sinhφ =v

cγ , tanhφ =

v

c.

b) Three inertial frames S, S′, S′′ are in standard configuration with each other; S′ has velocity v1relative to S, and S′′ has velocity v2 relative to S′. Prove that in this case S′′ must have velocity(v1 + v2)/(1 + v1v2/c

2) relative to S.

5- In an inertial frame S a train of plane light waves of wavelength λ travels in the negative x-direction towards the observer at the origin. The loci of the wavecrests then satisfy an equationof the form x = −ct + nλ , with n an integer. Sketch a few of such loci on a Minkowski diagramand then apply an active Lorentz transformation to deduce that in the usual second frame S′, thewavelength will be given by

λ′ = λ

√c− vc+ v

.

This is a relativistic Doppler formula, which we will later have more to say about.

6- A stellar object at some known large distance ejects a ‘jet’ at speed v towards an observerobliquely, making an angle θ with the line of sight. To the observer the jet appears to be ejectedsideways at speed V . Prove that V = c sin θ/(c/v−cos θ), and show that V can exceed c, e.g. whenθ = π/4.