Exploring Spacetime Curvature: A Review of Non-Euclidean Geometry, Relativistic Limits, and Cosmic Expansion

This is a review, not a report of new experiments. It draws together a few connected ideas — Poincaré's thought experiment on scale invariance [2], Einstein's General Relativity [1], and the shift from flat, Euclidean space to the curved geometry needed to describe gravity [4] — along with the standard relativistic relations for time dilation and the light-speed limit. One piece is original: as a check on the framework, we work through Einstein's light-deflection formula for the Sun and get roughly 1.75″, matching the value associated with the 1919 eclipse observations [3]. We also go over the corrected OPERA neutrino result [7], [8], time-dilation tests with rubidium clocks [9], and touch briefly on higher-dimensional extensions of the theory [5], [6]. Rather than claiming an answer, the paper ends by pointing out what would actually have to be shown before faster-than-light travel in some alternative geometry could be taken seriously.