The Lorentz transformations

The three ingredients developed so far — time dilation for cospatial events, length contraction for simultaneous events, the formula for the relativity of simultaneity — are enough to derive the Lorentz transformations, i.e. the compact way to rewrite the coordinates of any event when passing from one frame to another. They are the mathematical heart of special relativity: once obtained, every kinematic effect follows from them as a special case.

Setup. Two events $E_1$ and $E_2$ have coordinates $(t_{1(A)}, \vv{x}_{1(A)})$ and $(t_{2(A)}, \vv{x}_{2(A)})$ in SRA. We want to find $\Delta t_{21(B)}$ and $\Delta \vv{x}_{21(B)}$ in SRB, which moves with velocity $\vv{v}$ relative to SRA. For brevity we set $\Delta t = \Delta t_{21(A)}$ and $\Delta \vv{x} = \Delta \vv{x}_{21(A)}$.

The auxiliary-event trick. We introduce a third event $E_3$ constructed ad hoc, so that it is at the same time:

• cospatial with $E_1$ in SRA: $\vv{x}_{3(A)} = \vv{x}_{1(A)}$;
• simultaneous with $E_2$ in SRA: $t_{3(A)} = t_{2(A)}$.

The choice is clever because to $E_3$ we can apply separately time dilation (pairing it with $E_1$) and the relativity of simultaneity (pairing it with $E_2$), two formulae we already know.

Step 1 (time of $E_3$ relative to $E_1$). $E_1$ and $E_3$ are cospatial in SRA, so in SRB their times are dilated:

$t_{3(B)} - t_{1(B)} = \gamma\,(t_{3(A)} - t_{1(A)}) = \gamma\,\Delta t$

Step 2 (time of $E_3$ relative to $E_2$). $E_3$ and $E_2$ are simultaneous in SRA, with $\vv{L}_{32(A)} = \vv{x}_{3(A)} - \vv{x}_{2(A)} = -\Delta \vv{x}$. By the relativity of simultaneity:

$t_{3(B)} - t_{2(B)} = -\gamma\,\frac{\vv{\beta}\cdot(-\Delta\vv{x})}{c} = \gamma\,\frac{\vv{\beta}\cdot\Delta\vv{x}}{c}$

Step 3 (transformation of time). Subtracting the two previous equations eliminates $t_{3(B)}$ and leaves the transformation of time:

$\ev{\Delta t_{21(B)} = \gamma\left[\Delta t - \frac{\vv{v}\cdot\Delta\vv{x}}{c^2}\right]}$

Step 4 (position of $E_3$ relative to $E_1$). In SRA, $E_1$ and $E_3$ are at the same point; in SRB, however, in the meantime time $\gamma\Delta t$ has elapsed and SRA has shifted by $-\vv{v}\,\gamma\Delta t$ relative to SRB. Hence:

$\vv{x}_{3(B)} - \vv{x}_{1(B)} = -\vv{v}\,\gamma\,\Delta t$

Step 5 (position of $E_3$ relative to $E_2$). $E_3$ and $E_2$ are simultaneous in SRA, so their distance in SRA is a proper length measured in SRA; seen from SRB (where the two events are not simultaneous) that distance appears dilated by the factor $\gamma$ — the opposite of length contraction, because here it is SRA performing the measurement:

$\vv{x}_{3(B)} - \vv{x}_{2(B)} = \gamma\,(\vv{x}_{3(A)} - \vv{x}_{2(A)}) = -\gamma\,\Delta\vv{x}$

Step 6 (transformation of position). Subtracting the equations of steps 4 and 5 eliminates $\vv{x}_{3(B)}$ and leaves the transformation of position:

$\ev{\Delta\vv{x}_{21(B)} = \gamma\left[\Delta\vv{x} - \vv{v}\,\Delta t\right]}$

Putting the two boxed results together we obtain the full transformation law, in the most common case of motion along a single axis.

Lorentz transformations (1D, along $x$)

If SRB moves with velocity $v$ along the $x$ axis of SRA and the origins coincide as the synchronisation event, then for every pair of events: $\Delta t_{(B)} = \gamma\left(\Delta t_{(A)} - \frac{v\,\Delta x_{(A)}}{c^2}\right), \qquad \Delta x_{(B)} = \gamma\left(\Delta x_{(A)} - v\,\Delta t_{(A)}\right)$

The inverse transformation is obtained with the more intuitive rule of relativity: if SRB moves at $+v$ relative to SRA, then SRA moves at $-v$ relative to SRB. It suffices to swap the subscripts $(A)\leftrightarrow(B)$ and substitute $v \to -v$:

$\Delta t_{(A)} = \gamma\left(\Delta t_{(B)} + \frac{v\,\Delta x_{(B)}}{c^2}\right), \qquad \Delta x_{(A)} = \gamma\left(\Delta x_{(B)} + v\,\Delta t_{(B)}\right)$

The derivation with the “auxiliary event” is longer than others, but has the merit of using only already-known tools: dilation, contraction and relative simultaneity. We did not have to postulate anything new.

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Topics: Special relativity Concepts: Lorentz transformations Skills: Changing reference frame Methods: Minkowski diagram

Related exercises: Galileo versus Einstein · Space-type or time-type · Problem — Tunnel at 0.8c