A relay probe moves with constant velocity along the axis of the Earth 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 relay probe moves with constant velocity 0.600c0.600c along the +x+x axis of the Earth frame SS, where c=2.998×108 m s−1c = 2.998\times 10^{8}\ \mathrm{m\,s^{-1}}. The probe's own rest frame S′S' is in standard configuration with SS. Gravity is ignored throughout, and the light travels in vacuum. ((a)) Find the rapidity ψ\psi of S′S' relative to SS. ((b)) Write the four-velocity U\mathbf{U} of the probe in SS, giving each component as a multiple of cc, and evaluate the invariant square U⋅U\mathbf{U}\cdot\mathbf{U}. ((c)) The probe's transmitter emits light of frequency νem=5.00×1014 Hz\nu_{\rm em} = 5.00\times 10^{14}\ \mathrm{Hz} as measured in the probe's rest frame, and the probe is moving directly away from the Earth receiver along the line of sight. What frequency νrec\nu_{\rm rec} is received in SS?

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