Paper 4, Section II, C

Dynamics and Relativity | Part IA, 2015

Write down the Lorentz transform relating the components of a 4-vector between two inertial frames.

A particle moves along the xx-axis of an inertial frame. Its position at time tt is x(t)x(t), its velocity is u=dx/dtu=d x / d t, and its 4 -position is X=(ct,x)X=(c t, x), where cc is the speed of light. The particle's 4-velocity is given by U=dX/dτU=d X / d \tau and its 4 -acceleration is A=dU/dτA=d U / d \tau, where proper time τ\tau is defined by c2dτ2=c2dt2dx2c^{2} d \tau^{2}=c^{2} d t^{2}-d x^{2}. Show that

U=γ(c,u) and A=γ4u˙(u/c,1)U=\gamma(c, u) \quad \text { and } \quad A=\gamma^{4} \dot{u}(u / c, 1)

where γ=(1u2/c2)12\gamma=\left(1-u^{2} / c^{2}\right)^{-\frac{1}{2}} and u˙=du/dt\dot{u}=d u / d t.

The proper 3-acceleration a of the particle is defined to be the spatial component of its 4-acceleration measured in the particle's instantaneous rest frame. By transforming AA to the rest frame, or otherwise, show that

a=γ3u˙=ddt(γu)a=\gamma^{3} \dot{u}=\frac{d}{d t}(\gamma u)

Given that the particle moves with constant proper 3 -acceleration starting from rest at the origin, show that

x(t)=c2a(1+a2t2c21)x(t)=\frac{c^{2}}{a}\left(\sqrt{1+\frac{a^{2} t^{2}}{c^{2}}}-1\right)

and that, if atca t \ll c, then x12at2x \approx \frac{1}{2} a t^{2}.

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