Bell’s twin rockets non-inertial length enigma resolved by real geometry

Abstract

A priori uniformity and monotonicity of the ‘non-inertial length’ expansion of a uniformly co-accelerating medium, uniquely yield an unfamiliar ‘hemicoid’ real-values metric surface ϒ in R3. ϒ(τ,l) hosts congruent helicoidally distributed fixed-l ‘hemix world-lines’ tracing medium increments’ clock times τ and crossed by fixed-τ medium helices of parameterized length λ sharing comoving ‘non-inertial frames’. Radar intervals and expansion factor ∂λ/∂l=√(1+v2/c2) conform to requirements established in Coleman, Results in Physics, 6, 2016—Minkowski spacetime does not apply to a homogeneously accelerating medium. Co-directional radar paths on ϒ mapped from home frame chart diagonals crossing hyperbolic world-lines, surf ‘horizon’ increment hemices, whereas counter-directional radar paths tend to ‘overlap’ horizon medium helices. They also traverse each medium expansion helix at respectively identical angles and geodesic curvatures, independently of differing rocket emission times. Surface ϒ’s real metric is: ds2=dτ2+dλ2+2tanhτ.(tanhτ-1/coshτ)/√(1+tanh2τ)dτ.dλ. Keywords: Homogeneous acceleration, Non-inertial length, Radar intervals, Real-values metric, Minkowski metric, Photon crossing rate, Hemix, Hemicoid