[Paper Review] Teleparallel Geometry with Spherical Symmetry: The diagonal and proper frames

R. J. van den Hoogen, H. Forance|arXiv (Cornell University)|Aug 23, 2024

Advanced Numerical Analysis Techniques4 citations

TL;DR

This paper derives the diagonal and proper orthonormal co-frame and spin connection for spherically symmetric teleparallel geometries in $f(T)$ gravity, demonstrating that the diagonal frame with non-trivial spin connection simplifies field equation analysis compared to the proper frame with trivial spin connection. It shows that the Lorentz transformation linking the two frames reveals the two physical degrees of freedom in the spin connection and confirms the consistency of Kantowski-Sachs geometries in $f(T)$ gravity with power-law solutions.

ABSTRACT

We present the proper co-frame and its corresponding (diagonal) co-frame/spin connection pair for spherically symmetric geometries which can be used as an initial ansatz in any theory of teleparallel gravity. The Lorentz transformation facilitating the move from one co-frame to the other is also presented in factored form. The factored form also illustrates the nature of the two degrees of freedom found in the spin connection. The choice of coordinates in restricting the number of arbitrary functions is also presented. Beginning with a thorough presentation of teleparallel gravity using the metric affine gauge theory approach and concentrating on f(T) teleparallel gravity, we express the field equations in the diagonal co-frame. We argue that the choice of diagonal co-frame may be more advantageous over the proper co-frame choice. Finally, assuming one additional symmetry, we restrict ourselves to the Kantowski-Sachs tele-parallel geometries, and determine some solutions.

Motivation & Objective

• To construct a consistent orthonormal co-frame and spin connection pair for spherically symmetric teleparallel geometries in $f(T)$ gravity.
• To clarify the role of the spin connection in preserving Lorentz invariance and separating inertial from gravitational effects.
• To compare the diagonal co-frame (with non-trivial spin connection) with the proper co-frame (with trivial spin connection) in terms of field equation complexity and physical interpretation.
• To determine whether Kantowski-Sachs geometries are consistent with $f(T)$ teleparallel gravity and to derive explicit solutions under additional symmetry assumptions.

Proposed method

• Uses the metric affine gauge (MAG) approach to formulate a fully covariant framework for teleparallel gravity, enforcing zero curvature and zero non-metricity via Lagrange multipliers.
• Derives the diagonal orthonormal co-frame and its corresponding spin connection for spherically symmetric spacetimes, identifying five arbitrary functions of time and radial coordinates.
• Constructs the proper orthonormal co-frame via a Lorentz transformation from the diagonal frame, explicitly factoring the transformation into boosts and rotations dependent on two functions from the spin connection.
• Applies the field equations in the diagonal co-frame gauge and analyzes the antisymmetric part of the $f(T)$ field equations to verify consistency with the spin connection structure.
• Imposes additional symmetry (spatial homogeneity and anisotropy) to restrict the geometry to Kantowski-Sachs type, solving the field equations for power-law solutions.
• Compares the torsion tensor and field equations in both diagonal and proper frames, confirming equivalence under Lorentz transformation.

Experimental results

Research questions

• RQ1 Can a diagonal orthonormal co-frame with non-trivial spin connection be consistently constructed for spherically symmetric teleparallel geometries in $f(T)$ gravity?
• RQ2 How does the Lorentz transformation between the diagonal and proper co-frames reveal the physical degrees of freedom in the spin connection?
• RQ3 Is the Kantowski-Sachs geometry consistent with $f(T)$ teleparallel gravity, and what are the resulting field equations and solutions?
• RQ4 Does the diagonal co-frame gauge offer computational advantages over the proper co-frame gauge in solving $f(T)$ field equations?
• RQ5 What constraints on the parameters $\gamma$ and $\alpha_2$ yield real, non-trivial power-law solutions in the Kantowski-Sachs model?

Key findings

• The diagonal orthonormal co-frame and its corresponding spin connection provide a simpler ansatz with five arbitrary functions, four of which are in the metric and one in the spin connection.
• The proper orthonormal co-frame, obtained via a Lorentz transformation from the diagonal frame, has a trivial spin connection and eliminates inertial effects, but requires a non-diagonal frame structure.
• The Lorentz transformation linking the diagonal and proper frames is factored into boosts and rotations, explicitly revealing the two physical degrees of freedom in the spin connection.
• A two-parameter family of power-law solutions exists for the Kantowski-Sachs geometry in $f(T)$ gravity, parameterized by $\gamma$ and $\alpha_2$, with the condition $0 < c_1^2 < 1 + 2\alpha_2$ restricting valid parameter space.
• The antisymmetric part of the $f(T)$ field equations is satisfied by the spin connection in the Kantowski-Sachs geometry, confirming consistency of this class of anisotropic, spatially homogeneous models with $f(T)$ teleparallel gravity.
• The diagonal frame is shown to be more advantageous for solving field equations due to the explicit structure of the spin connection, despite the proper frame's conceptual clarity in isolating gravitational effects.