The Least Action and Matter-Field Dynamics in Gravitational Field

We define the gravitation action as an integral over the four-dimensional

space of the total curvature density. Integrating by parts, we obtain the Lagrangian

density as a function of the Christoffel symbols. From a variation of this Lagrangian

density with the metric elements, we obtain the Einstein law of gravitation for

vacuum. We define the matter action as an integral over the four-dimensional space

of the mass scalar density. With the momentum four-vector, we obtain a Lagrangian

density variation with the metric elements and the time-space coordinates. From the

total variation of the matter action in a gravitational field, we obtain the Einstein law

of gravitation in matter, and the geodesic equations. According to the Maxwell

equations, we obtain the electric and magnetic fields as an electromagnetic tensor. We

define the electromagnetic action as an integral over the four-dimensional space of

the amplitude scalar of this tensor, and obtain a variation of this action with metric

elements and the electromagnetic potentials. At the same time, we consider the

electric charge action as the scalar product of the charge flux four-vector with the

electromagnetic potential four-vector, and obtain its variation with electromagnetic

potentials and the time-space coordinates. We obtain a matter-field action variation

with three terms: for the variations of the metric elements, of the time-space

coordinates, and of the electromagnetic potentials. From the first term, we obtain the

matter dynamics in a gravitational-electromagnetic field; from the second term, we

obtain the Lorentz force in a gravitational field, and from the third term, the Maxwell

equations in the gravitational field.

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DOI https://doi.org/10.2174/9789815051094122030007	Print ISSN 2542-5064
Publisher Name Bentham Science Publisher	Online ISSN 2542-5072

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