Four Vectors in Special Relativity

            There are many important quantities that transform like 4-Vectors. They are defined so that the "length" of a 4-vector is invariant under a coordinate transformation. 

 
Space-Time 4-Vector; Length Contraction and Time Dilatation

The space-time 4-Vector is defined by:

  or                 (9)

The length of the space-time 4-Vector squared is given by:

           (10)

In differential form,  or space interval squared  is:

           (11)

The Lorentz Transformation can be expressed in matrix form using the Lorentz matrix D:

      (12)

This acts on the 4-vector coordinates of a world point in S and gives their values in S’:

        (13)

The matrices product gives relations (7).

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