# 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.