A particle moves with speed along the -axis in frame

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Special relativity: spacetime and kinematics · Relativistic dynamics and fields in flat spacetime

Invariance, Ch General bases, the metric and tensors in spacetime

Problem

A particle moves with speed

v = 0.60c

along the

x

-axis in frame

S

, where

c

is the speed of light. ((a)) Write the four-velocity

U^\mu = (\gamma c, \gamma v, 0, 0)

in units of

c

, and verify

U^\mu U_\mu = -c^2

using

\eta_{\mu\nu} = \mathrm{diag}(-1,1,1,1)

. ((b)) Frame

S'

moves at

v' = 0.80c

along the same axis relative to

S

. Using the boost

U'^0 = \gamma'(U^0 - (v'/c)U^1)

U'^1 = \gamma'(U^1 - (v'/c)U^0)

with

\gamma' = (1-v'^2/c^2)^{-1/2}

, find

U'^\mu

in units of

c

. ((c)) Read off the particle's speed in

S'

from

U'^\mu

, as a multiple of

c

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