A bunch of protons travels at the constant speed along the axis of the laboratory 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 bunch of protons travels at the constant speed

0.500c

along the

+x

axis of the laboratory frame

S

, where

c = 2.998\times 10^{8}\ \mathrm{m\,s^{-1}}

. The bunch's rest frame

S'

is in standard configuration with

S

. Gravity is ignored and the light travels in vacuum. ((a)) Find the rapidity

\psi

S'

relative to

S

. ((b)) Write the four-velocity

\mathbf{U}

of the bunch in

S

, each component as a multiple of

c

, and evaluate

\mathbf{U}\cdot\mathbf{U}

. ((c)) An atom travelling with the bunch emits light of frequency

\nu_{\rm em} = 4.00\times 10^{14}\ \mathrm{Hz}

in the bunch's rest frame. A detector at rest in

S

sits far downstream behind the bunch, so the source recedes directly along the line of sight. What frequency

\nu_{\rm rec}

does it receive?

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