Extensions of GR using Projective-Invariance

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 ($Γ_{μν}^λ\neqΓ_{νμ}^λ$), 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 $Γ_{μν}^λ$, 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 $Γ_{[μν]}^ν$ 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.

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