Question

A free particle, having rest mass Mo, is in vacuum and initially at rest in the lab frame S. It undergoes an acceleration under the action of a constant pure force F = (fo, fr, 0.0), where 0 c dt Find its 4-velocity U as a function of time and force. Sketch the graphs of the dependence of normalised 3-velocity 3 v/c and the Lorentz factor y(t) of the particle as a function of time t

Answer 1

Let the 4-velocity be

(, 7u, 0,0)

with the constraint

uu=-1

By Newton's law

du FHMo dr

Also proper time is related to coordinate time by

ydTdi

where

.

1 2

Put everything together

dE dt dt

which gives

$E=\gamma M_0$

The other equation we have is

|(al)p dt Mo7

The calculation is a little complicated

1 In 1 v2 2frt 1-

Clearly, this relation can't be inverted.