Relativity says that no experiment can tell a frame at rest from one moving uniformly. Its transformations, the Lorentz boosts, mix time and space, keeping the speed of light the same for everybody. Time and space become four components of one four-vector, $x^{\mu}=(ct,\mathbf{r})$, and what every observer agrees on is not the length of $\mathbf{r}$ or the time $t$ but the interval $c^{2}t^{2}-\mathbf{r}^{2}$. Energy and momentum make a four-vector too, $p^{\mu}=(E/c,\mathbf{p})$, and its invariant square is $m^{2}c^{2}$: that is the energy–momentum relation, $E^{2}=p^{2}c^{2}+m^{2}c^{4}$. In quantum mechanics the four-momentum becomes the operators $i\hbar\partial/\partial t$ and $-i\hbar\nabla$, together $i\hbar\partial_{\mu}$. A relativistic wave equation must keep its form when the coordinates are boosted: that single demand shapes the Klein–Gordon and Dirac equations of the next lessons.

Walking diagonally across a square room, you can describe your path by steps east and north or by steps along the room’s diagonals: the steps change, the distance walked does not. A boost is such a change of description for spacetime, and the interval is the distance it keeps.

Time up, $ct$, and one direction of space across. The dots are one event, at interval 1 from the origin, as seen from frames moving at different speeds: boosts move it along the hyperbola $c^{2}t^{2}-x^{2}=1$, never across the light cone, dashed.

With $x^{\mu}=(ct,\mathbf{r})$, the metric $g_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$ and , a boost along is

• The interval $x_{\mu}x^{\mu}=c^{2}t^{2}-\mathbf{r}^{2}$ is the same in every frame; so is for any two four-vectors.

Put the boosted coordinates into the interval and expand. The cross terms in $ctx$ cancel, and what is left is the old interval times $\gamma^{2}(1-\beta^{2})$, which is 1.

The boost.

The terms $\mp2\beta\,ctx$ cancel.

The definition of $\gamma$.

The interval is the same in both frames; so, the same way, is $E^{2}/c^{2}-p^{2}=m^{2}c^{2}$.

A boost at speed $v=\beta c$ mixes time and space, with the factor $\gamma=1/\sqrt{1-\beta^{2}}$.

Every observer agrees on $c^{2}t^{2}-\mathbf{r}^{2}$, though not on $t$ or separately.

Energy and momentum form one four-vector, whose invariant square is $m^{2}c^{2}$; energy alone changes from frame to frame.

Great work! You now know how to:

• write a boost and show that the interval is invariant
• combine energy and momentum into a four-vector
• turn the four-momentum into operators