Notes on the Nonextensive-Entropy Derived Evolution and Fundamental Metric Tensors in the Continuum Mechanics of Media and of Special and General Relativity

Many successes of special and general relativity in describing the universe have occurred as a result of the derivation of flat and curved spacetime metrics compatible with the Einstein equation of general relativity. These metrics were written on, or rather multiplied linear and nonlinear scalar coefficient functions to the background of a flat extensive spacetime (‘special relativity’) metric. In this letter we present a new perspective complementary to the one of fractional derivatives, this where fractional deformations of spacetime are supposed, which in this letter is obtained from a q-deformed non-extensive entropy derived type of metric that reduces to the flat Minkowski or Euclidean metric depending on application dimension and signature. This q-deformed metric then, is a parametrically deviated-from flat spacetime metric that encodes parametrically a scalar 'curvature' which tends to zero (flat) as q->1 for several instances and under minimal assumptions. Its derivation & some of its qualities will be presented. Also presented are calculations of the Ricci tensors, and scalar curvatures and therefore the stress-energy tensors for some of these cases, with at least one case closely corresponding to a q-deformed ADS metric.