Special relativistic notation Minkowski space-time. Lorentz transformation

Special relativistic notation Minkowski space-time. Lorentz transformations 

An event in Minkowski space-time is defined, relative to a coordinate frame 5, by a 4-vector jc = (/t = 0,1,2,3) where Jt = ctis the time coordinate 

Special relativity asserts that, in the absence of gravitational fields, the speed of light is the same for all observers in free fall, the so-called inertial observers. Let S,S be inertial frames of two such observers the invariance of the speed of light implies that the coordinates of the same event, x in S and x = hx m S, must satisfy s = (x,x) = xf,x/). Since for every pair of vectors x,y. 

Equation (4) implies that (detA) = 1, detA = 1. It follows that A is a non-singular matrix, with A = g Ag. Along with contravariant vectors we associate covariant vectors (covectors) a such that 

Covectors transform under Lorentz transformations according to 

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