A.5.3 (HL)—Galilean transformations

Transform event coordinates

For frames S and S′ that are coincident at $t=t′=0$, with S′ moving at velocity $v$ in the positive $x$-direction relative to S, the Galilean transformations are $x′=x-vt$ and $t′=t$.

Use the sign convention

Start with the event coordinates $(x,t)$ in S. Substitute the frame velocity with its signed value, calculate $x′$, and carry the unchanged time $t′=t$ into S′. A positive $x′$ means the event is on the positive side of S′’s origin.

Check the assumptions

These equations describe the classical model: inertial frames, common synchronized time, and an origin coincidence at $t=0$. They are not the Lorentz transformations and do not preserve the speed of light between frames.

Symbolic example from local Question Bank row 31622

An event has $x'=L$ and $t'=L/c$ in S′. Galilean time is absolute, so $t=t'=L/c$. Using the inverse position transformation,

$$x=x'+vt=L+v\frac{L}{c}$$

The extra $vL/c$ is the distance travelled by the S′ origin during the shared time interval.

Common trap

Do not change $t$ using $t′=t-vx/c^2$; that belongs to the relativistic transformation. In a Galilean transformation, time is shared: $t′=t$.