[Paper Review] Torsion and Problems of Standard Cosmology

M. I. Wanas, H. A. Hassan|arXiv (Cornell University)|Sep 27, 2012

Cosmology and Gravitation Theories36 references3 citations

TL;DR

This paper proposes that incorporating torsion within the Absolute Parallelism (AP) geometry framework—instead of standard Riemannian geometry—can resolve key problems in Friedmann-Robertson-Walker (FRW) cosmology, such as singularities and particle horizons. By reformulating General Relativity in AP-geometry with a homogeneous and isotropic structure, the authors show that torsion naturally enables accelerated expansion and phantom-like behavior without external assumptions, effectively turning GR into a pure geometric theory where matter and energy emerge from torsion itself.

ABSTRACT

The field equation of orthodox general relativity are written in the context of a geometry with non-vanishing torsion, the Absolute Parallelism (AP) geometry. An AP-structure, with homogeneity and isotropy, is used for cosmological applications. The resulting dynamical equations are those of FRW-Standard cosmology, which have many problems e.g. singularity, particle horizons, ...etc. We suggest a new scheme for investigating the effect of torsion on the dynamics of FRW-Cosmology, without changing the basic structure of general relativity. It is shown that some of these problems will disappear if the torsion, associated with AP-structure used, is inserted in to the dynamical equations. Diagnose shows that problems arise when GR is written in the context of a geometry with vanishing torsion, the Riemannian geometry. This reflects the importance of using more wider geometries in studying physical phenomena.

Motivation & Objective

• To investigate whether non-vanishing torsion in Absolute Parallelism (AP) geometry can resolve longstanding problems in standard FRW cosmology, such as singularities and particle horizons.
• To reformulate General Relativity in AP-geometry while preserving its dynamical structure, thereby testing whether torsion alone can generate cosmological dynamics.
• To explore whether matter and energy can be geometrically represented as manifestations of torsion, thus restoring GR as a pure geometric theory.
• To demonstrate that inflation-like behavior and negative pressure can emerge naturally from torsion without requiring a Lagrangian, potential energy dominance, or external fields.

Proposed method

• The field equations of General Relativity are rewritten in the context of Absolute Parallelism (AP) geometry, which features non-vanishing torsion and a non-symmetric linear connection.
• An AP-structure with homogeneity and isotropy is imposed, defined by a tetrad vector field $\lambda^\mu_i$, satisfying $\lambda^\mu_i \lambda^i_\nu = \delta^\mu_\nu$ and $\lambda^\mu_i \lambda^j_\mu = \delta^j_i$.
• The metric tensor $g_{\mu\nu} = \lambda_i^\mu \lambda^i_\nu$ is derived from the tetrad, and the non-symmetric connection $\Gamma^\alpha_{\mu\nu} = \lambda^\alpha_i \partial_\nu \lambda^i_\mu$ generates torsion via $\Lambda^\alpha_{\mu\nu} = \Gamma^\alpha_{\mu\nu} - \Gamma^\alpha_{\nu\mu}$.
• The resulting dynamical equations are shown to reduce to standard FRW equations when torsion is neglected, but torsion modifies the energy-momentum conservation and evolution equations.
• The authors analyze the modified FRW equations with torsion, particularly focusing on the role of torsion in enabling negative pressure and accelerated expansion.
• The matter-energy tensor $T_{\mu\nu}$ is reinterpreted as arising from the torsion field, suggesting that torsion itself acts as a source of energy and pressure.

Experimental results

Research questions

• RQ1 Can torsion in AP-geometry resolve the singularity and particle horizon problems in standard FRW cosmology?
• RQ2 How does the inclusion of non-vanishing torsion in the field equations of General Relativity affect the dynamics of the expanding universe?
• RQ3 Can the accelerated expansion of the universe be explained by torsion without introducing a cosmological constant or dark energy?
• RQ4 To what extent can matter and energy be geometrically represented as manifestations of torsion in a pure-geometric theory of gravity?
• RQ5 What is the role of torsion in modifying the energy conservation law in cosmological models?

Key findings

• The solution to the modified FRW equations in AP-geometry exhibits no singularity and no particle horizon, resolving two major problems of standard cosmology.
• Torsion is directly linked to the Hubble parameter through equation (30), indicating a fundamental geometric relationship between expansion rate and torsion.
• Negative pressure and accelerated expansion emerge naturally from torsion without requiring a Lagrangian, potential energy dominance, minimal coupling, or external fields.
• The conservation equation (26) yields $\dot{\rho}_{\mathcal{T}} = 0$ for $w = -1$, implying constant energy density in an expanding universe, suggesting torsion energy must be included in the conservation law.
• The matter-energy tensor $T_{\mu\nu}$ is geometrically constructed from the torsion field, allowing GR to be interpreted as a pure geometric theory where matter arises from torsion.
• The model achieves inflation-like dynamics without invoking scalar fields or external potentials, suggesting torsion as a geometric origin of early-universe acceleration.