Extensions of GR using Projective-Invariance

TL;DR

This paper proposes a geometric unification of electromagnetism and gravity through a non-symmetric affine connection and projective symmetry, offering new insights into classical and quantum field interpretations within affine gravity theories.

Contribution

It introduces a novel approach unifying electromagnetism and gravity via projective invariance in affine geometry, linking electromagnetic fields to Ricci tensors and exploring implications for f(R) gravity.

Findings

Classical Einstein-Maxwell equations derived from affine geometry.

Electromagnetic field interpreted as preserving projective invariance.

Discussion on the role of projective invariance in f(R) gravity theories.

Abstract

We show that the unification of electromagnetism and gravity into a single geometrical entity can be beautifully accomplished in a theory with non-symmetric affine connection (${\Gamma}_{\mu\nu}^{\lambda}\neq{\Gamma}_{\nu\mu}^{\lambda}$), and the unifying symmetry being projective symmetry. In addition, we show that in a purely-affine theory where there are no constrains on the symmetry of ${\Gamma}_{\mu\nu}^{\lambda}$, the electromagnetic field can be interpreted as the field that preserves projective-invariance. The matter Lagrangian breaks the projective-invariance, generating classical relativistic gravity and quantum electromagnetism. We notice that, if we associate the electromagnetic field tensor with the second Ricci tensor and ${\Gamma}_{[\mu\nu]}^{\nu}$ with the vector potential, then the classical Einstein-Maxwell equation can be obtained. In addition, we explain the geometrical interpretation of projective transformations. Finally, we discuss the importance of the role of projective-invariance in f(R) gravity theories.