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The theory of the conservation of energy in the thin layer approximation hasbeen extended to special relativity. Four models for the density of thecircumstellar medium are analyzed, which are represented by constant, powerlaw, exponential and Emden (n=5) profile for density. The astrophysical resultsare presented in a numerical way, except for a Taylor expansion of the fourtrajectories in the surrounding of the origin. The free parameters of themodels are particularized for SN1993j, for which the radius versus time isknown. Some evaluations on the time dilation are presented.

Energy Conservation in the thin layer approximation: III. The spherical relativistic case for supernovae