This Quantum World/Appendix/Relativity/4-vectors

4-vectors

3-vectors are triplets of real numbers that transform under rotations like the coordinates $x,y,z.$ 4-vectors are quadruplets of real numbers that transform under Lorentz transformations like the coordinates of $\overrightarrow{x}=(ct,x,y,z).$

You will remember that the scalar product of two 3-vectors is invariant under rotations of the (spatial) coordinate axes; after all, this is why we call it a scalar. Similarly, the scalar product of two 4-vectors $\overrightarrow{a}=(a_t,𝐚)=(a_0,a_1,a_2,a_3)$ and $\overrightarrow{b}=(b_t,𝐛)=(b_0,b_1,b_2,b_3),$ defined by

$(\overrightarrow{a},\overrightarrow{b})=a_0{b}_0-a_1{b}_1-a_2{b}_2-a_3{b}_3,$

is invariant under Lorentz transformations (as well as translations of the coordinate origin and rotations of the spatial axes). To demonstrate this, we consider the sum of two 4-vectors $\overrightarrow{c}=\overrightarrow{a}+\overrightarrow{b}$ and calculate

$(\overrightarrow{c},\overrightarrow{c})=(\overrightarrow{a}+\overrightarrow{b},\overrightarrow{a}+\overrightarrow{b})=(\overrightarrow{a},\overrightarrow{a})+(\overrightarrow{b},\overrightarrow{b})+2(\overrightarrow{a},\overrightarrow{b}).$

The products $(\overrightarrow{a},\overrightarrow{a}),$ $(\overrightarrow{b},\overrightarrow{b}),$ and $(\overrightarrow{c},\overrightarrow{c})$ are invariant 4-scalars. But if they are invariant under Lorentz transformations, then so is the scalar product $(\overrightarrow{a},\overrightarrow{b}).$

One important 4-vector, apart from $\overrightarrow{x},$ is the 4-velocity $\overrightarrow{u}=\frac{d\overrightarrow{x}}{ds},$ which is tangent on the worldline $\overrightarrow{x}(s).$ $\overrightarrow{u}$ is a 4-vector because $\overrightarrow{x}$ is one and because $ds$ is a scalar (to be precise, a 4-scalar).

The norm or "magnitude" of a 4-vector $\overrightarrow{a}$ is defined as $\sqrt{|(\overrightarrow{a},\overrightarrow{a})|}.$ It is readily shown that the norm of $\overrightarrow{u}$ equals $c$ (exercise!).

Thus if we use natural units, the 4-velocity is a unit vector.