This Quantum World/Appendix/Relativity/4-vectors

4-vectors

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

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 ${\displaystyle {\vec {a}}=(a_{t},\mathbf {a} )=(a_{0},a_{1},a_{2},a_{3})}$  and ${\displaystyle {\vec {b}}=(b_{t},\mathbf {b} )=(b_{0},b_{1},b_{2},b_{3}),}$  defined by

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

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

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

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

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

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

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