Special Relativity (alternative Formulations) - Euclidean Relativity

Euclidean Relativity

Equivalent to the original ? No, the velocity addition formula is different.

Euclidean relativity uses a Euclidean (++++) metric as opposed to the traditional Minkowski (+---) or (-+++) metric which is derived from the Minkowski metric by rewriting into the equivalent . The roles of time t and proper time have switched so that proper time takes the role of the coordinate for the 4th spatial dimension. A universal velocity for all objects in 4D space-time appears from the regular time derivative . The approach differs from the so-called Wick rotation or complex Euclidean relativity. In Wick rotation, time is replaced by, which also leads to a positive definite metric but it maintains proper time as the Lorentz invariant value whereas in Euclidean relativity becomes a coordinate. Because implies that photons travel at the speed of light in the subspace {x, y, z} and baryonic matter that is at rest in {x, y, z} travels normal to photons along, a paradox arises on how photons can be propagated in a space-time. The possible existence of parallel space-times or parallel worlds shifted and co-moving along is the approach of Giorgio Fontana. The Euclidean geometry is consistent with classical, Minkowski based relativity in two reference frames. The hyperbolic Minkowski geometry turns into a rotation in 4D circular geometry where length contraction and time dilation result from the geometric projection of 4D properties to 3D space. In three reference frames an inconsistency appears in the velocity addition formula, also affecting other formulas that depend on the velocity addition formula. The inconsistency does so far not imply known contradictions with experimental data but compared to the classical formula it predicts small deviations (< m/s) in the addition result when both input speeds are very high (>10 km/s) and have similar magnitude.