TITLE:
Embedding the Einstein Tensor in the Klein-Gordon Equation Using Geometric Algebra Cl3,0
AUTHORS:
Jesús Sánchez
KEYWORDS:
Geometric Algebra, Einstein Tensor, Klein-Gordon Equation, Bra-Ket Product, Non-Euclidean Metric
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.13 No.10,
October
31,
2025
ABSTRACT: In this paper, we will use Geometric Algebra to be able to embed the Klein-Gordon equation for a particle in a non-Euclidean field (gravitational field). This way, we will obtain an expression similar to the Dirac equation, but with a slight change in one of the terms. This variation is produced and depends on the curvature of the space where the particle lies in (the Ricci scalar). In a similar manner, we will find variations in the equation for the energy of a particle and in the Einstein gravitational equation that will depend again on the value of the Ricci scalar (the curvature of the space where the particle lies in). An important outcome will be an equation that limits the value of the Ricci scalar depending on the value of the mass that provokes it (the value of the mass, not the mass density), highly reducing the possibilities of arriving at singularities. In fact, the value of this R has been found to be equal to the cosmological constant (both in the order of 1E-52), making it a perfect candidate for the dark energy. Also, the magnetic-like effects of gravitation coming from the equations are sufficient to explain the rotation of the galaxies (NGC 1560, NGC 3198 and NGC 3115) without the need for dark matter.