This Quantum World/Appendix/Relativity/4-vectors

4-vectorsEdit

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

 

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

 

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.