Geometrodynamics based on geodesic equation with Cartan structural equation on Riemannian manifolds

TL;DR

This paper develops a geometrodynamical framework on Riemannian manifolds using Cartan structural equations and a novel geospin matrix, linking geometric identities to dynamical equations.

Contribution

It introduces a new dynamical formulation of Cartan structural equations on Riemannian manifolds using four variables, connecting geometric identities with physical dynamics.

Findings

The geodesic equation is well explained by the first dynamical equation.

The second equation corresponds to the first Bianchi identity.

The last equation reveals the dynamical aspect of the second Bianchi identity.

Abstract

Motivated by the geospin matrix $W$ as a new variable in Riemannian geometry. Then, we use four real dynamical variables $\left\{ a,\alpha ,v,W \right\}$ to show the dynamical essence of Cartan structural equation, we obtain the geometrodynamics on Riemannian manifolds that can be expressed below \begin{align} & \Theta /d{{t}^{2}}=a-v\wedge W,\nonumber & d\Theta /d{{t}^{3}}=v\wedge \alpha -a\wedge W,\nonumber & \Omega /d{{t}^{2}}=\alpha -W\wedge W, \nonumber & d\Omega /d{{t}^{3}}=W\wedge \alpha -\alpha \wedge W\nonumber \end{align} that is valid more generally for any connection in a principal bundle. We can see that the first equation explains the geodesic equation very well, the second formula means the first Bianchi identity, while the last equation reveals the dynamical nature of the second Bianchi identity. It implies that Cartan structural equations on Riemannian manifolds can be rewritten in a real geometrodynamical form based on the geospin matrix.