# 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 $\vec{x}=(ct,x,y,z).$

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 $\vec{a}=(a_t,\mathbf{a})=(a_0,a_1,a_2,a_3)$ and $\vec{b}= (b_t,\mathbf{b})=(b_0,b_1,b_2,b_3),$ defined by

$(\vec{a},\vec{b})=a_0b_0-a_1b_1-a_2b_2-a_3b_3,$

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 $\vec{c}=\vec{a}+\vec{b}$ and calculate

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

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

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

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

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