Minkowski Space-Time Metric

A matrix A is a Lorentz transformation if it is compatible with the scalar product in Minkowski space. The Minkowski space is the set of space-time points X = (xo,x) = (ct,x), equipped with the scalar product (metric)... 

Theories like those of Lemaitre or Friedmann, which predict an expanding universe, are all based on forcing an affine metric, such as the Robertson-Walker metric, on the projective geometry of space-time. This has the effect of splitting local Minkowski space into separate space and time coordinates, without the natural complex relationship that ties space and time together. 

To specify the directions of two different vectors at nearby points it is necessary to define tangent vectors at these points. Stated in different terms, at each point of space-time, known as the contact point, there is an associated tangent Minkowski space. The theory of these spaces together with the underlying space becomes a Riemannian geometry if a Euclidean metric is introduced in each tangent space by means of a differential quadratic form. ... 

We may remind ourselves that according to conventional wisdom, Poincare failed to derive an early relativity theory mainly as a result of his stubborn adherence to the nineteenth century concept of the ether as an utter void while Einstein, just few years later, succeeded because he stayed behind the assertion of a new kind of ether that was the superfluous space-time substratum. His relativity postulate was called an absolute world by Minkowski. Fifteen years later, however, Einstein used Mach s ideas on rational movement in order to cancel the action-at-a-distance and refreshed the idea of the ether because his relativity space is endowed with physical qualities. The ether should be regarded as a primary thing (similarly to the prematurely Aristotelian plenum) assuming that matter is derived from it [40], citing the metrical qualities of the continuum of space-time differ in the environment of different points of space-time being conditioned by the matter existing outside of the territory under consideration . 

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