Real-metric spacetime own-surfaces hosting nongeodesic radar paths crossing ‘hemix’ own-lines and shared velocity helices

Abstract

Pondering Enrique Loedel’s 1948 symmetrical spacetime chart whose angle’s sine reflects scaled velocity between inertial reference frames, led in 2004 to a most curious discovery: Angles of a unit sines law ratio spherical triangle geometricise relativistic velocity composition. Overlooked in a paper by Einstein’s colleague Sommerfeld and kept under wraps until publication of a recent book, the germane criterion points to a seemingly unknown hemispherical ‘hemix’ spiral which encapsulates the Gudermannian relating a fixed thrust rocket’s velocity and clocked own-time. This opens new avenues for analysing relativistic noninertial length contexts whereby hemix-generated real-metric λ|τ ‘own-surfaces’ visualisable in R3 are corroborated by radar path attributes, oddly a strategy unexploited in relativity. The Euclidean models epitomise not only Born’s (one off) ‘rigid motion’ but also non-Minkowski extended medium acceleration scenarios such as the previously unresolved Bell’s spaceships problem where the ±[+++-] signature is misplaced, increment ‘own-lines’ and radar paths turn out to be nongeodesics, and Gauss curvature remains the same along each shared own-time likewise nongeodesic ‘medium curve’. Keywords: Spacetime metric, Own-surface, Hemix, Hemicoid, Bell’s string paradox, Rigor mortis motion, Minkowski metric, Spherical triangle, Radar paths, Geodesics