A space station defines the inertial frame ; a courier ship defines the inertial frame

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Special relativity: spacetime and kinematics · Special relativity: spacetime and kinematics

Relativity and Gravitation, Ch Minkowski spacetime, four-vectors and causal structure

Problem

A space station defines the inertial frame SS; a courier ship defines the inertial frame S′S', which moves at v=0.600cv = 0.600c along the common +x+x axis in the standard configuration (axes parallel, origins coinciding at t=t′=0t = t' = 0). Gravity is ignored; both frames are inertial. Two events are marked on the station's wall, with coordinates given in SS: event 1 at t1=0t_1 = 0, x1=0x_1 = 0; event 2 at t2=5.00 μst_2 = 5.00\ \mu\mathrm{s}, x2=1200 mx_2 = 1200\ \mathrm{m} (with y=z=0y = z = 0 for both). ((a)) Starting from the Lorentz transformation Δt′=γ(Δt−vΔx/c2)\Delta t' = \gamma(\Delta t - v\Delta x/c^2), Δx′=γ(Δx−vΔt)\Delta x' = \gamma(\Delta x - v\Delta t), show by direct substitution that −c2Δt′2+Δx′2=−c2Δt2+Δx2-c^2\Delta t'^2 + \Delta x'^2 = -c^2\Delta t^2 + \Delta x^2 for any pair of events. ((b)) Evaluate Δt′\Delta t' and Δx′\Delta x' for this pair, and evaluate Δs2\Delta s^2 in both frames to four significant figures. ((c)) The courier ship moves at 0.600c0.600c in SS. Write its four-velocity components U=(γc,γu⃗)\mathbf{U} = (\gamma c, \gamma \vec u) in SS and compute U⋅U=−(U0)2+(U1)2\mathbf{U}\cdot\mathbf{U} = -(U^0)^2 + (U^1)^2. Take c=2.998×108 m s−1c = 2.998\times 10^{8}\ \mathrm{m\,s^{-1}}.

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