3-vectors are triplets of real numbers that transform under rotations like the coordinates
4-vectors are quadruplets of real numbers that transform under Lorentz transformations like the coordinates of
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
and
defined by
![{\displaystyle ({\vec {a}},{\vec {b}})=a_{0}b_{0}-a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96031e13734e451fd6783fc47a4d70302dcb4f41)
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
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}}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e84a17de417af222f350903c873da5ec836e0b2)
The products
and
are invariant 4-scalars. But if they are invariant under Lorentz transformations, then so is the scalar product
One important 4-vector, apart from
is the 4-velocity
which is tangent on the worldline
is a 4-vector because
is one and because
is a scalar (to be precise, a 4-scalar).
The norm or "magnitude" of a 4-vector
is defined as
It is readily shown that the norm of
equals
(exercise!).
Thus if we use natural units, the 4-velocity is a unit vector.