A.5.6 (HL)—Lorentz transformations

Set the relativistic factor

For two inertial frames with relative speed $v$, use $\gamma=1/\sqrt{1-v^2/c^2}$. At ordinary speeds $\gamma\approx1$; as $v$ approaches $c$, the difference from Galilean coordinates becomes significant.

Transform one event

For an event with coordinates $(x,t)$ in S, the syllabus equations are $x′=\gamma(x-vt)$ and $t′=\gamma(t-vx/c^2)$ in S′. Use the same signed direction for $x$ and $v$, and calculate both coordinates from the same event.

Show the coordinate method

Write $\gamma$ first, substitute the given $x,t,v$, then report $x′$ and $t′$ with units. If the question provides a space–time diagram, reading the coordinates is an alternative only when the axes and scale are correctly interpreted.

Worked example from local Question Bank rows 31630–31631

For $v=0.745c$, $\gamma=1/\sqrt{1-0.745^2}=1.499$. An event at $x=1.0\,\mathrm{m}$ and $t=0$ transforms to

$$x'=\gamma(x-vt)=1.499(1.0)=1.50\,\mathrm{m}$$
$$ct'=\gamma(ct-\tfrac{v}{c}x)=1.499(0-0.745\times1.0)=-1.12\,\mathrm{m}$$

Thus $t'=-1.12/c\,\mathrm{s}$; the negative coordinate is valid and reflects the chosen origins.

Respect the syllabus boundary

Know and apply the transformation equations; their derivation is not required. The frames must be inertial, and the relative speed must satisfy $v<c$ so that $\gamma$ is real.