[Paper Review] Locally inertial coordinates with totally antisymmetric torsion

M. Socolovsky|arXiv (Cornell University)|Sep 21, 2010

Advanced Differential Geometry Research14 references3 citations

TL;DR

This paper demonstrates that the equivalence principle holds at a point in spacetime with totally antisymmetric torsion if and only if the symmetric part of the contortion tensor vanishes, enabling locally inertial coordinates without requiring torsion to vanish. The geodesic deviation equation and Newtonian limit are derived, confirming compatibility with standard gravity in the weak-field regime.

ABSTRACT

We show that the necessary and sufficient condition for erecting locally inertial coordinates at a point $p$ of a $U^4$-space, and therefore assuring the validity of the equivalence principle at that point, is the vanishing at $p$ of the symmetric part of the contortion tensor. This fact does not demand a vanishing torsion, but only a totally antisymmetric one. As an application, we derive the geodesic deviation equation; and prove the compatibility with the Newtonian limit.

Motivation & Objective

• To establish the necessary and sufficient condition for the existence of locally inertial coordinates in a spacetime with torsion.
• To investigate whether the equivalence principle remains valid when torsion is non-zero but totally antisymmetric.
• To derive the geodesic deviation equation in a Riemann-Cartan spacetime with totally antisymmetric torsion.
• To verify compatibility of totally antisymmetric torsion with the Newtonian limit of geodesic motion.
• To clarify the role of the contortion tensor's symmetric part in defining inertial frames under torsion.

Proposed method

• Derives a coordinate transformation that eliminates the symmetric part of the connection, using only the contortion tensor's symmetric component.
• Applies the metric compatibility condition and tensor transformation rules to show that first derivatives of the metric vanish at a point if the symmetric part of the contortion tensor vanishes.
• Uses the fact that only the symmetric part of the connection contributes to the geodesic equation, allowing torsion to remain non-zero but totally antisymmetric.
• Derives the geodesic deviation equation in a U⁴-space with totally antisymmetric torsion, separating curvature and torsion contributions to acceleration.
• Performs a weak-field expansion of the geodesic equation in static, weak gravitational fields to analyze the Newtonian limit.
• Imposes time-reversal invariance and small perturbations to the Minkowski metric, leading to constraints on torsion components.

Experimental results

Research questions

• RQ1 What is the necessary and sufficient condition for the existence of locally inertial coordinates in a spacetime with totally antisymmetric torsion?
• RQ2 How does the geodesic equation behave when only the symmetric part of the contortion tensor vanishes, despite non-zero torsion?
• RQ3 Can the geodesic deviation equation be consistently derived in a Riemann-Cartan spacetime with totally antisymmetric torsion?
• RQ4 Is a totally antisymmetric torsion compatible with the Newtonian limit of general relativity?
• RQ5 What constraints does the Newtonian limit impose on the components of the torsion tensor?

Key findings

• The necessary and sufficient condition for locally inertial coordinates at a point is the vanishing of the symmetric part of the contortion tensor, not the vanishing of torsion.
• A totally antisymmetric torsion tensor is compatible with the existence of locally inertial frames, as the symmetric part of the connection—governing geodesics—can still vanish.
• The geodesic deviation equation in a U⁴-space with totally antisymmetric torsion separates into curvature and torsion contributions, with the torsion part given by $ A^{ u}_{T} = 2g^{ ueta}T^{ ho}(S^{ heta}T_{ ho hetaeta})_{; ho}T^{ ho} $.
• In the Newtonian limit, the condition $ T^{0}_{k0} = 0 $ for $ k=1,2,3 $ is required to avoid unphysical forces, and this is compatible with a totally antisymmetric torsion tensor.
• The Newtonian limit leads to $ rac{d^2oldsymbol{x}}{dt^2} = - abla ilde{ ho} + oldsymbol{H}oldsymbol{T} $, and consistency requires $ oldsymbol{T} = 0 $ unless $ oldsymbol{H} $ is singular, which is ruled out by perturbation theory.
• The independent components of the totally antisymmetric torsion tensor in 4D are $ T_{120}, T_{230}, T_{310}, T_{231} $, and they fully determine the torsion contribution to geodesic deviation.