Gravitation in Flat Euclidean Spacetime Frame: Unified Electrogravity and Magnetogravity Forces

An effective description of physics requires an appropriate geometrical frame. Three-dimensional Euclidean space provides the geometrical frame for non-relativistic physics. A derivation of an imaginary temporal axis −icˆq the speed, ˆq the unit wave-vector of light, extends the standard Euclidean space into a well-defined four-dimensional Euclidean spacetime frame, which provides the natural mathematical framework for relativistic physics. The basic elements of the Euclidean spacetime frame are fully specified four-component complex vectors satisfying standard vector operations and vector identities. In developing a theory of gravitation in the Euclidean spacetime frame, we have used the Lense-Thirring spacetime metric of linearized general relativity to derive an appropriate complex four-component gravitational field potential vector. Taking the curl of the field potential vector provides a unified complex gravitational field strength composed of electric-type and magnetic-type components. Taking the cross-product of the complex four-component velocity and the field strength provides a unified complex gravitational force intensity composed of gravitoelectric and gravitomagnetic components. Application to the motion of a gyroscope in the gravitational field of the earth provides the standard results of frame-dragging and geodetic effects as determined in linearized general relativity theory.