[Paper Review] A Solution to Symmetric Teleparallel Gravity

Muzaffer Adak, Özcan Sert|ArXiv.org|Dec 2, 2004

Cosmology and Gravitation Theories17 citations

TL;DR

This paper proposes a symmetric teleparallel gravity model where spacetime curvature and torsion are zero, but nonmetricity is nonvanishing, offering a novel geometric framework for gravity. By solving the field equations under spherical symmetry, it derives a solution identical to the Schwarzschild metric, revealing that while the spacetime is regular at the Schwarzschild horizon from a Riemannian perspective, the nonmetricity invariant diverges there, indicating a physical singularity in this formalism.

ABSTRACT

Teleparallel gravity models, in which the curvature and the nonmetricity of spacetime are both set zero, are widely studied in the literature. We work a different teleparallel theory, in which the curvature and the torsion of spacetime are both constrained to zero, but the nonmetricity is nonzero. After reformulating the general relativity in this spacetime we find a solution and investigate its singularity structure.

Motivation & Objective

• To explore a gravity model in spacetime with zero curvature and torsion but non-zero nonmetricity, offering an alternative to standard general relativity.
• To investigate whether such a symmetric teleparallel gravity (STPG) framework can yield physically meaningful solutions, particularly black hole-like geometries.
• To analyze the singularity structure of the solution, focusing on the behavior of curvature and nonmetricity invariants.
• To clarify the physical and geometric implications of nonmetricity in STPG, especially at horizons.

Proposed method

• The authors adopt a spacetime structure {M, g, ∇} with a Lorentzian metric g and a linear connection ∇, where curvature and torsion are constrained to zero.
• They decompose the connection into Levi-Civita, contortion, and nonmetricity parts using Cartan's structure equations.
• A spherically symmetric metric ansatz is used, with nonmetricity components derived from the field equations under the constraint that the total curvature vanishes.
• The field equations are solved by imposing consistency conditions on the nonmetricity and connection components, leading to differential equations for the metric functions F(r) and G(r).
• The Riemannian curvature invariants are computed using the Levi-Civita connection, while nonmetricity invariants are derived from the irreducible components of Qab.
• The solution is derived by solving the resulting differential equations, yielding F²(r) = 1 - C/r and G(r) = 1/F(r), which reproduces the Schwarzschild geometry.

Experimental results

Research questions

• RQ1 Can a gravity model with zero curvature and torsion but non-zero nonmetricity yield a consistent solution resembling the Schwarzschild black hole?
• RQ2 How do the curvature and nonmetricity invariants behave in this solution, particularly at the Schwarzschild radius?
• RQ3 Is the Schwarzschild horizon a regular surface in the context of symmetric teleparallel gravity, or does it exhibit physical singularities due to nonmetricity?
• RQ4 What is the geometric and physical significance of nonmetricity in this alternative gravity framework?

Key findings

• The solution derived from the symmetric teleparallel gravity model with zero curvature and torsion is mathematically equivalent to the Schwarzschild solution, with F²(r) = 1 - C/r and G(r) = 1/F(r).
• The Riemannian curvature invariant R_ab(ω) ∧ *R^ab(ω) is singular at r = 0 and regular at r = C, indicating a horizon from the Riemannian perspective.
• The nonmetricity invariant Q_ab ∧ *Q^ab diverges not only at r = 0 but also at the Schwarzschild radius r = C, signaling a physical singularity in the STPG framework.
• The horizon at r = C is regular in Riemannian geometry but singular in symmetric teleparallel gravity due to the divergence of the nonmetricity invariant.
• The quadratic curvature invariant scales as 6C²/r⁶, confirming the Schwarzschild-like behavior at the origin.
• The nonmetricity invariant exhibits a complex structure, with divergences at both r = 0 and r = C, suggesting that horizons may not be physically acceptable in this formalism.