#### Lorentz Transformations
red.sr.Lt

To talk about coordinate transformations, it is necessary to talk about events, namely occurrences at a specific point in space and time.

We will use two inertial frames of reference: \({\cal S}\) and \(\bar{\cal S}\). We will assume that \(\bar{\cal S}\) is moving relative to \({\cal S}\) at velocity \(v\) in the positive \(x\) direction.

Imagine we have an event \(E\) occurring at \((t, x, y, z)\). We would like to know the coordinates \((\bar{t}, \bar{x}, \bar{y}, \bar{z})\) in system \(\bar{\cal S}\).

If the clock is started at the moment at which the origins \({\cal O}\) and \(\bar{\cal O}\) pass each other, then after a time \(t\), \(\bar{\cal O}\) will be at a distance \(vt\) from \({\cal O}\) and we would have \[ x = d + vt, \] where \(d\) is the distance from \(\bar{\cal O}\) to \(\bar{\cal A}\) (\(\bar{\cal A}\) being the point on the \(\bar{x}\) axis that is even with \(E\) when the event occurs).

If Galilean physics is used, then \(d = \bar{x}\) and the transformation rules are \[ \bar{t} = t, ~~\bar{x} = x - vt, ~~\bar{y} = y, ~~\bar{z} = z. \] If however we properly take into account Lorentz contraction, we must set \[ d = \frac{1}{\gamma} \bar{x} ~~\longrightarrow~~ \bar{x} = \gamma (x - vt). \]

Doing the same argument from the point of view of an observer at rest in frame \(\bar{S}\), we would write \[ \bar{x} = \bar{d} - v \bar{t} \] where \(\bar{d}\) is the distance from \({\cal O}\) to \({\cal A}\) at time \(\bar{t}\) (\({\cal A}\) being the point on the \(x\) axis that is even with \(E\) when the event occurs). Again invoking Lorentz, \[ \bar{d} = \frac{1}{\gamma} x ~~\longrightarrow~~ x = \gamma(\bar{x} + v \bar{t}). \] Solving these relations yields the dictionary for

Lorentz transformations (motion along \(x\) at velocity \(v\))

\begin{align} \bar{t} &= \gamma \left( t - \frac{v}{c^2} x \right), & \bar{y} &= y, \nonumber\\ \bar{x} &= \gamma \left( x - vt \right), & \bar{z} &= z \end{align}

Einstein's velocity addition rule: using these rules, one can show that velocities add as \[ v_{13} = \frac{v_{12} + v_{23}}{1 + v_{12}v_{23}/c^2} \]

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42