Some Consequences of the Lorentz Equations — Question 8

Question 8

A hyperbolic trajectory has subluminal tangents but no single rest frame

A worldline in one spatial dimension is $x(t)=\sqrt{b^2+c^2t^2}$, with $b>0$. Consider $t\ge0$.

1. Find its instantaneous velocity and show its magnitude stays below c.

2. At any event, boost with $\beta=ct/x$. Find that event coordinates and explain why this does not make the entire worldline stationary.

Question 8 - Solution

A hyperbolic trajectory has subluminal tangents but no single rest frame

(a) Method and calculation

$$v=\frac{dx}{dt}=\frac{c^2t}{x},\quad\boxed{\beta=ct/x<1}.$$$$1-\beta^2=\frac{x^2-c^2t^2}{x^2}=b^2/x^2,\quad\gamma=x/b.$$

(b) Interpretation and further calculation

$$ct\prime=\gamma(ct-\beta x)=\boxed{0},\quad x\prime=\gamma(x-\beta ct)=\boxed{b}.$$$$\frac{d\beta}{dt}=\frac{cb^2}{(b^2+c^2t^2)^{3/2}}>0.$$

Physical check

The required boost varies from event to event. A momentarily comoving inertial frame is not a global rest frame for accelerated motion.