Minimal Lines and Geodesics within Matter: a Fundamental Difficulty of Einstein's Theory

Abstract

ACCORDING to Einstein's relativity theory the minimal lines, ds2 = gixdxidxx = 0, represent, in any circumstances (that is, in space-time upon which any metrical tensor gix satisfying the field-equations has been impressed), light-lines, and thus the laws of propagation of light. Now, in vacuo, the representation is of course correct, giving—apart from minor refinements—uniform, isotropic propagation with the velocity c, as pre-arranged. But inside matter, considered as a continuous medium characterized by the material tensor Tix, the minimal lines manifestly cannot represent light propagation, even to a rough approximation. For in such a medium, supposed isotropic, the light velocity is, essentially, c/ (where is the refractive index, say, for light of a fixed frequency), whereas Tix, determining the gix, contains no trace of, in fact, no properly optical feature of the medium in question. Thus, even without detailed mathematical deduction, one can see that, within matter, minimal lines are not light-lines; that is, ds = 0 does not represent light propagation.