#### Proper Time and Proper Velocity
red.rm.pt

If you move at velocity \(u\), then, as compared to a ground clock, your time will go slower. For a ground clock time interval \(dt\), your proper time interval is \[ d\tau = \sqrt{1 - u^2/c^2}~dt. \] For the observer on the ground, your velocity is the rate of change of your position \({\boldsymbol l}\) with respect to ground time: \[ {\boldsymbol u} = \frac{d {\boldsymbol l}}{dt} \] and this is called the ordinary velocity. The proper velocity is defined as the hybrid-frame quantity (distance as measured on the ground) divided by (proper time interval):

Proper velocity \[ {\boldsymbol \eta} \equiv \frac{d {\boldsymbol l}}{d\tau} \]

Proper and ordinary velocity are thus related by \[ {\boldsymbol \eta} = \frac{1}{\sqrt{1 - u^2/c^2}} {\boldsymbol u}. \] The nice thing about proper velocity (as compared to ordinary velocity) is that it transforms simply from one inertial system to another. By adding the zeroth component \[ \eta^0 = \frac{dx^0}{d\tau} = c \frac{dt}{d\tau} = \frac{c}{\sqrt{1-u^2/c^2}} \] we can define the

four-velocity or proper velocity four-vector \[ \eta^\mu \equiv \frac{dx^\mu}{d\tau} \] which transforms as \[ \bar{\eta}^\mu = \Lambda^\mu{}_\nu \eta^\nu \]

(the transformation is simple because \(d\tau\) in the denominator is an invariant).

By contrast, the ordinary velocities obey cumbersome transformation rules: for our usual relative frame velocity of \(v\) in the \(x\) direction, \[ \bar{u}_x = \frac{d\bar{x}}{d\bar{t}} = \frac{u_x - v}{1-v u_x/c^2}, \hspace{10mm} \bar{u}_y = \frac{d\bar{y}}{d\bar{t}} = \frac{u_y}{\gamma(1-v u_x/c^2)}, \hspace{10mm} \bar{u}_z = \frac{d\bar{z}}{d\bar{t}} = \frac{u_z}{\gamma(1-v u_x/c^2)}. \]

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42