# Geometrical Unification of Gravitation and Electromagnetism

## TL;DR

The paper introduces GUGE, a geometrical unification theory of gravitation and electromagnetism that naturally derives Einstein-Maxwell equations without extra scalar fields, differing fundamentally from Kaluza-Klein theory.

## Contribution

GUGE provides a new geometrical framework unifying gravity and electromagnetism, deriving the metric and field equations without additional assumptions or scalar fields, and explaining charge quantization and conservation.

## Key findings

- GUGE metric is derived from proper time redefinition.
- GUGE field equations are equivalent to Einstein-Maxwell equations.
- GUGE predicts charge quantization and conservation.

## Abstract

A theory which achieves a complete geometrical unification of gravitation and electromagnetism (GUGE) is presented. This new theory is based on a recent proposal of proper time redefinition that leads to the construction of a Riemann metric, which naturally unifies gravity and electromagnetism. The 5--dimensional Riemann metric which arises looks exactly the same as the one postulated in the Kaluza-Klein (KK) 5-dimensional theory. Nevertheless, there are deep differences between GUGE and KK. The GUGE metric is deduced while the KK metric is postulated. In the GUGE field theory there is no need to impose either of the so called "cylinder" or "curling of coordinates" conditions, because they emerge as direct consequences of the construction of the GUGE metric. The GUGE field equations are fully equivalent to Einstein--Maxwell equations, while KK field equations are not. The GUGE 5--dimensional (geodesic) equations are equivalent to the 4--dimensional (non--geodesic) equations for a charged particle moving in the presence of gravitational and electromagnetic fields, unlike the KK 5--dimensional (geodesic) equations which are not and, in addition, yield non--gauge covariant geodesic equations. No extra scalar field appears in GUGE. The physical interpretation of the fifth dimension and of the role of the extra field in KK (internal coordinate in GUGE) are totally different in both approaches. Finally, GUGE results include electric charge conservation, electric charge quantization and electric charge contribution to the energy of charged particles even in the absence of electromagnetic fields, which implies (the observed fact) that there are no massless electrically charged particles in Nature, unlike the prevailing treatments of KK theories.