Mathematics Coincides with Physics

 Zbigniew A. Nowacki

 Lodz, Poland, May 2016

General relativity cannot be considered to be a correct mathematical theory, even if it was developed only for aesthetic reasons (although in my opinion the quantum reality is far more beautiful than classical approximations, and I think that physics is algebra rather than geometry). In mathematics the existence proofs, especially of solutions for differential equations, are very important, while Hawking and Penrose proved in 1970 that the solution of Einstein's equations, that is, curved space-time, in fact, does not exist. This holds because its curvature at infinitely many points would have to be infinite.

Let us note that mathematicians examine, inter alia, how to avoid infinity (since its use frequently leads to contradictions). That is why they prohibit from dividing by zero, in order not to consider functions with infinite values they have introduced the notion of a distribution, etc. Therefore, from a mathematical point of view the non-Euclidean geometry of Einstein (unlike those discovered by Bolyai, Lobachevsky, and Riemann) is simply bad-defined.

Now we can explain why this is so. Space-time, as the name suggests, should represent the whole space and time. But since we can send (superluminal) signals across the curved spacetime, we have to assume that Einstein's proposal does not represent everything, and so it is not a space-time. We see that mathematics and physics coincide exactly.