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

So far, the four-dimensional Minkowski spacetime frame has provided the geometrical framework for describing the more accurate relativistic dynamics. But a serious, though unnoticed, drawback of the Minkowski spacetime frame is that it does not specify a unit vector in the temporal direction, yet all the three associated spatial coordinate axes, namely the x-axis, y-axis and z-axis are specified by their respective unit vectors

In recent work [

In the specification of the Euclidean

The Euclidean spacetime frame is characterized by an imaginary temporal axis specified by the unit wave vector

A general Euclidean four-vector

The basic mathematical operations such as addition and subtraction, dot and cross products, divergence and curl of Euclidean four-vectors, and the Euclidean gradient of a scalar, are defined and evaluated in the standard vector operation forms:

In detailed calculations presented in [

The basic algebraic operations presented in

In a systematic formulation of physics in the Euclidean spacetime frame, we apply the mathematical property that the basic elements of the Euclidean spacetime frame are Euclidean four-vectors, which as defined above, are complex four-component vectors. The basic dynamical elements, the mechanical elements and the electromagnetic elements, of a physical system are defined as Euclidean four-vectors. As usual, the fundamental physical properties of matter are mass and electric charge, which generate appropriately defined force fields. We interpret a force field, with the generating mass or electric charge at the center (origin), as a bounded four-dimensional Euclidean spacetime frame. A mechanical (mass-generated) or electromagnetic (electric charge-generated) force field is characterized by a Euclidean field potential four-vector. In general, mathematical operations with the physical Euclidean four-vectors have algebraic properties which reveal fundamental features of dynamics within a Euclidean spacetime frame.

Consistently with standard descriptions of physics in various geometrical frames, we identify the basic dynamical properties of physics in the Euclidean

Other physical quantities such as orbital and spin angular momenta, together with the associated orbital and spin magnetic moments, etc., can be defined as desired.

In general, physics in the Euclidean spacetime frame naturally satisfies the basic conservation laws and invariance under Lorentz transformation. Application of the conservation laws and invariance under the Lorentz transformation easily provide the fundamental relativistic properties of time dilation and mass increase with speed, which have been obtained together with the corresponding

In the special case where the temporal unit vector

In the Euclidean spacetime frame in our case the gravitational field, a force field is characterized by a Euclidean field potential four-vector

The Euclidean field potential four-vector, with the associated field strength and force intensity in

An elegant property which has emerged is that the mathematical operations with Euclidean four-vectors provide a natural transition from non-relativistic physics in three-dimensional Euclidean space to relativistic physics in four-dimensional Euclidean spacetime frame. This property means that the Euclidean

We now introduce the Euclidean gravitational force field. To determine the appropriate form of the gravitational Euclidean field potential four-vector, we begin by observing that in Einstein’s general theory of relativity, the gravitational field potential is defined by a rank-2 symmetric metric tensor in Riemann geometry. The general form of the metric tensor characterizes gravitation in a non-inertial spacetime frame. Noting that in a weak gravitational field, linearization of Einstein’s general theory of relativity reduces to

In GEM, the standard Lense-Thirring spacetime metric is obtained in the form [

We have thus used the Lense-Thirring spacetime metric to determine the appropriate form of the GEM field potential four-vector

The GEM field potential four-vector

With the Euclidean gravitational field potential four-vector determined as in

The gravitational field strength

The gravitational force intensity

We identify

Noting that

We observe that, apart from a missing factor

The other component can be interpreted as the rate of change of spin

It is the magnetic-type forces that cause a change in the spin of a gyroscope moving in the gravitational field of the earth. Relativistic effects are therefore obtained by taking the cross-product of the spin-vector and the magnetic-type forces.

Finally, we have the frame-dragging and geodetic effects

From Astronomical point of view, reference frames serve as the observational perspectives from which we perceive the motion of objects in spacetime. When a massive object, such as a rotating black hole, drags spacetime along with its rotation, the effect is observable from different reference frames. From the perspective of an observer on the site to the rotating mass, the dragging of spacetime appears minimal or non-existent. However, for an observer distant from the rotating mass, the frame-dragging effect becomes more pronounced, influencing the motion of nearby objects and altering their trajectories. Therefore, reference frames provide the framework for understanding relativistic effects that depend on the observer’s motion relative to the rotating mass [

Similarly, geodetic effects are intimately connected to reference frames in general relativity. These effects demonstrate the idea of curvature of

In astrophysics, frame dragging influences the behaviour of massive rotating objects such as black holes and neutron stars. This effect affects the orbits of nearby objects and the emission of gravitational waves, providing valuable insights into the dynamics of these systems [

Understanding relativistic effects is essential for precise navigation systems, particularly in space missions where accurate positioning is critical. Corrections based on general relativity, including geodetic effects, are necessary for achieving the high levels of accuracy required for GPS applications on Earth and in space. For instance, the Global Positioning System relies on corrections derived from general relativity to ensure accurate positioning and timing information for navigation purposes.

As it is, we have demonstrated that it is possible to handle

We thank the two institutions, Maseno University and Meru University of Science and Technology for providing facilities and a conducive work environment during the preparation of the manuscript.