[Paper Review] Totally Asymmetric Torsion on Riemann-Cartan Manifold

Yuyiu Lam|ArXiv.org|Nov 2, 2002

Advanced Differential Geometry Research1 references3 citations

TL;DR

This paper proposes a relativistic theory on a Riemann-Cartan manifold where torsion is totally antisymmetric, naturally arising from the vanishing of the symmetric part of the affine connection in local inertial frames. The theory ensures coincidence between autoparallel and metric geodesics, leading to a scalar field $φ$ that emerges from the torsion structure and modifies the Einstein tensor via a conformal-like term, yielding a theory conformally related to Brans-Dicke gravity with $ω = 0$ in vacuum.

ABSTRACT

A relativistic theory constructed on Riemann-Cartan manifold with a derived totally antisymmetric torsion is proposed. It follows the coincidence of the autoparallel curve and metric geodesic. The totally antisymmetric torsion naturally appears in the theory without any ad hoc imposed on.

Motivation & Objective

• To resolve the G-A problem (discrepancy between autoparallel and metric geodesics) in Einstein-Cartan theory by enforcing total antisymmetry of torsion.
• To show that totally antisymmetric torsion arises naturally from the vanishing of the symmetric affine connection in normal coordinates, without ad hoc assumptions.
• To derive a field equation where the totally antisymmetric torsion leads to a scalar field $φ$ that couples to gravity but not to matter in the variational principle.
• To establish a conformal equivalence between the proposed theory and Brans-Dicke theory with $ω = 0$ in vacuum, suggesting new avenues for quantum gravity.

Proposed method

• Utilizes normal (Gaussian) coordinates around a point $p$ in a $U_4$ manifold to show that the symmetric part of the affine connection vanishes, implying $\Gamma^{\lambda}_{\mu\nu}(0) = 0$.
• Applies the exponential map and geodesic structure to derive $\Gamma^{\lambda}_{\mu\nu} + \Gamma^{\lambda}_{\nu\mu} = 0$ at $p$, leading to $S^{\lambda}_{\mu\nu} = S^{\lambda}_{[\mu\nu]}$.
• Uses Cartan's structure equations with orthonormal frames to express torsion one-form $\Theta^\lambda = \frac{1}{2} S_{\mu\nu}^\lambda \theta^\mu \wedge \theta^\nu$ and curvature $\Omega^\mu_\nu = \frac{1}{2} R^\mu_{\nu\lambda\gamma}(\Gamma) \theta^\lambda \wedge \theta^\gamma$, linking torsion to curvature.
• Derives the Ricci scalar $R(\Gamma)$ in terms of the standard Ricci tensor and the scalar field $\phi$, showing $R_{\mu\nu}(\Gamma) = R_{\mu\nu} + 2(g_{\mu\nu} \partial_\lambda \phi \partial^\lambda \phi - \partial_\mu \phi \partial_\nu \phi)$.
• Applies Hamilton's principle to the vacuum action $\int R(\Gamma) \sqrt{-g} \, d^4x$, varying the metric and deriving the field equation $G_{\mu\nu} = -6(\partial_\mu \phi \partial_\nu \phi - \frac{1}{2} g_{\mu\nu} \partial_\alpha \phi \partial^\alpha \phi)$.
• Imposes the condition $\partial_\lambda \partial^\lambda \phi = 0$ to simplify boundary terms and ensure vanishing of divergence terms in the variation.

Experimental results

Research questions

• RQ1 Can totally antisymmetric torsion emerge naturally from the geometric structure of a Riemann-Cartan manifold without external imposition?
• RQ2 Does the vanishing of the symmetric part of the affine connection in normal coordinates imply that torsion must be totally antisymmetric?
• RQ3 How does the resulting scalar field $\phi$ affect the gravitational field equations in vacuum?
• RQ4 What is the conformal relationship between this theory and Brans-Dicke theory, particularly in the limit $\omega = 0$?
• RQ5 Can the scalar field $\phi$ be consistently derived from the torsion tensor without coupling to matter in the variational principle?

Key findings

• The totally antisymmetric torsion $S_{\lambda\mu\nu} = S_{[\lambda\mu\nu]}$ arises naturally from the vanishing of the symmetric part of the affine connection in normal coordinates, without ad hoc assumptions.
• The coincidence of autoparallel and metric geodesics is achieved because $\Gamma^{\lambda}_{\mu\nu} = \{\}_{\mu}^{\lambda}_{\nu} + S_{\mu\nu}^{\lambda}$, with $S^{\lambda}_{\mu\nu} = S^{\lambda}_{[\mu\nu]}$, implying $\Gamma^{\lambda}_{(\mu\nu)} = 0$ at $p$.
• The scalar field $\phi$ emerges from the torsion structure and appears in the Ricci tensor as $R_{\mu\nu}(\Gamma) = R_{\mu\nu} + 2(g_{\mu\nu} \partial_\lambda \phi \partial^\lambda \phi - \partial_\mu \phi \partial_\nu \phi)$.
• The vacuum field equation is $G_{\mu\nu} = -6(\partial_\mu \phi \partial_\nu \phi - \frac{1}{2} g_{\mu\nu} \partial_\alpha \phi \partial^\alpha \phi)$, showing a conformal coupling to gravity with a negative sign.
• The theory is conformally equivalent to Brans-Dicke theory with $\omega = 0$ in vacuum, as the scalar field $\phi$ appears with a coupling strength that matches the $\omega = 0$ limit.
• The scalar field $\phi$ does not couple to matter in the variational principle, consistent with test particles moving along geodesics, and the theory reduces to standard GR when $P^\lambda = 0$ and $T^{[\mu\nu]} = 0$.