Dirac equation for massive neutrinos in a Schwarzschild-de Sitter spacetime from a 5D vacuum

Starting from a Dirac equation for massless neutrino in a 5D Ricci-flat background metric, we obtain the effective 4D equation for massive neutrino in a Schwarzschild-de Sitter (SdS) background metric from an extended SdS 5D Ricci-flat metric. We use the fact that the spin connection is defined to an accuracy of a vector, so that the covariant derivative of the spinor field is strongly dependent of the background geometry. We show that the mass of the neutrino can be induced from the extra space-like dimension.

💡 Research Summary

The paper investigates how a neutrino that is massless in a five‑dimensional (5D) Ricci‑flat background can acquire an effective mass when the theory is reduced to a four‑dimensional (4D) Schwarzschild‑de Sitter (SdS) spacetime. The authors begin with the 5D Dirac equation for a massless spin‑½ field, (\Gamma^{A}\nabla_{A}\Psi=0), where (\Gamma^{A}) are the 5D gamma matrices and (\nabla_{A}) includes the spin connection (\omega_{A}^{;BC}). They emphasize that the spin connection is defined only up to a vectorial freedom; this ambiguity is encoded in an additional term (\Lambda_{A}^{;BC}) that does not affect the torsion‑free condition but does influence the covariant derivative of the spinor.

The 5D metric employed is an “extended” SdS metric: \