Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2013

(a) Let SS with coordinates (ct,x,y)(c t, x, y) and SS^{\prime} with coordinates (ct,x,y)\left(c t^{\prime}, x^{\prime}, y^{\prime}\right) be inertial frames in Minkowski space with two spatial dimensions. SS^{\prime} moves with velocity vv along the xx-axis of SS and they are related by the standard Lorentz transformation:

(ctxy)=(γγv/c0γv/cγ0001)(ctxy), where γ=11v2/c2.\left(\begin{array}{c} c t \\ x \\ y \end{array}\right)=\left(\begin{array}{ccc} \gamma & \gamma v / c & 0 \\ \gamma v / c & \gamma & 0 \\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} c t^{\prime} \\ x^{\prime} \\ y^{\prime} \end{array}\right), \quad \text { where } \gamma=\frac{1}{\sqrt{1-v^{2} / c^{2}}} .

A photon is emitted at the spacetime origin. In SS^{\prime} it has frequency ν\nu^{\prime} and propagates at angle θ\theta^{\prime} to the xx^{\prime}-axis.

Write down the 4 -momentum of the photon in the frame SS^{\prime}.

Hence or otherwise find the frequency of the photon as seen in SS. Show that it propagates at angle θ\theta to the xx-axis in SS, where

tanθ=tanθγ(1+vcsecθ)\tan \theta=\frac{\tan \theta^{\prime}}{\gamma\left(1+\frac{v}{c} \sec \theta^{\prime}\right)}

A light source in SS^{\prime} emits photons uniformly in all directions in the xyx^{\prime} y^{\prime}-plane. Show that for large vv, in SS half of the light is concentrated into a narrow cone whose semi-angle α\alpha is given by cosα=v/c\cos \alpha=v / c.

(b) The centre-of-mass frame for a system of relativistic particles in Minkowski space is the frame in which the total relativistic 3-momentum is zero.

Two particles A1A_{1} and A2A_{2} of rest masses m1m_{1} and m2m_{2} move collinearly with uniform velocities u1u_{1} and u2u_{2} respectively, along the xx-axis of a frame SS. They collide, coalescing to form a single particle A3A_{3}.

Determine the velocity of the centre-of-mass frame of the system comprising A1A_{1} and A2A_{2}.

Find the speed of A3A_{3} in SS and show that its rest mass m3m_{3} is given by

m32=m12+m22+2m1m2γ1γ2(1u1u2c2),m_{3}^{2}=m_{1}^{2}+m_{2}^{2}+2 m_{1} m_{2} \gamma_{1} \gamma_{2}\left(1-\frac{u_{1} u_{2}}{c^{2}}\right),

where γi=(1ui2/c2)1/2\gamma_{i}=\left(1-u_{i}^{2} / c^{2}\right)^{-1 / 2}

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