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ω − s · k , consider the case when the vectors s and k are collinear.

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1 Using the formula for the Doppler shift derived in the lecture ω

0

= (1 − v

2

)

−1/2

ω − s · k , consider the case when the vectors s and k are collinear.

2 Consider a relativistic particle moving with constant acceleration w the (t, x) plane.

Assuming that x(0) = 0 and dx

dτ (0) = 0 , show that the wordline is given by t(τ ) = a

−1

sinh(aτ ), x(τ ) = a

−1

(cosh(aτ ) − 1), where a = √

w · w .

3 Find the Lorentz transformations corresponding to the following SL(2, C) matrices U = ±  e

iφ/2

0

0 e

−iφ/2



and U = ±  e

ψ/2

0 0 e

−ψ/2

 ,

where φ and ψ are real parameters.

4 A photon with momentum k bounces (perpendicularly) o a mirror that moves with velocity u. Find the momentum of the photon after reection.

5 A pion moves in an accelerator on the circular orbit with radius R and period T (as

measured in the lab frame). What is the proper period (as measured in the pion's frame)?

Cytaty

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