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

$u^{\mu}=(\gamma,\gamma v,0,0)$

with the constraint

$u^{\mu}u_{\mu}=-1$

By Newton's law

$F^{\mu}=M_0\frac{\mathrm{d} u^{\mu}}{\mathrm{d} \tau}$

Also proper time is related to coordinate time by

$\gamma d\tau=dt$

where

$\gamma=\frac{1}{\sqrt{1-v^2}}$.

Put everything together

$\gamma\frac{\mathrm{d} E}{\mathrm{d} t}=M_0\gamma\frac{\mathrm{d} \gamma}{\mathrm{d} t}$

which gives

$E=\gamma M_0$

The other equation we have is

$f_x=M_0\gamma\frac{\mathrm{d} (\gamma v)}{\mathrm{d} t}$

The calculation is a little complicated

$\frac{2f_xt}{M_0}=\frac{v}{1-v^2}+\ln \sqrt{\frac{1+v}{1-v}}$

Clearly, this relation can't be inverted.