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

$({\vec {a}},{\vec {b}})=a_{0}b_{0}-a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3},$

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 ${\vec {c}}={\vec {a}}+{\vec {b}}$ 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.