[Paper Review] Isotropic cosmology in metric-affine gauge theory of gravity

А. В. Минкевич, A. S. Garkun|ArXiv.org|May 4, 1998

Cosmology and Gravitation TheoriesPhysics and Astronomy16 citations

TL;DR

This paper investigates isotropic cosmological models within metric-affine gauge theory of gravity (MAGT), incorporating curvature, torsion, and nonmetricity tensors via a generalized gravitational Lagrangian. It derives a generalized Friedmann equation and shows that regular cosmological solutions—previously found in Poincaré gauge theory—also emerge in MAGT, particularly in Weyl-Cartan space-time, under specific constraints on Lagrangian parameters, preserving key physical consequences like finite energy density limits.

ABSTRACT

Geometrical structure of homogeneous isotropic models in the frame of the metric-affine gauge theory of gravity (MAGT) is analyzed. By using general form of gravitational Lagrangian including both a scalar curvature and various invariants quadratic in the curvature, torsion and nonmetricity tensors, gravitational equations of MAGT for homogeneous isotropic models are deduced. It is shown, that obtained gravitational equations lead to generalized cosmological Friedmann equation for the metrics by certain restrictions on indefinite parameters of gravitational Lagrangian. Isotropic models in the Weyl-cartan space-time are discussed.

Motivation & Objective

• To analyze the geometrical structure of homogeneous isotropic models in metric-affine gauge theory of gravity (MAGT), including curvature, torsion, and nonmetricity tensors.
• To derive gravitational field equations in MAGT using a general gravitational Lagrangian containing quadratic invariants of curvature, torsion, and nonmetricity tensors.
• To investigate whether regular cosmological solutions—previously found in Poincaré gauge theory—also exist in MAGT, particularly in Weyl-Cartan space-time.
• To determine the conditions under which isotropic models with nonvanishing torsion and nonmetricity (Weyl-Cartan) emerge, and to derive the corresponding generalized Friedmann equation.

Proposed method

• The study employs a general gravitational Lagrangian in MAGT that includes scalar curvature and quadratic invariants of curvature, torsion, and nonmetricity tensors.
• The field equations are derived from the variational principle applied to this Lagrangian, leading to a system of equations involving the scale factor R(t), torsion S(t), and nonmetricity Q_i(t).
• The analysis is restricted to Robertson-Walker metrics in comoving coordinates, with the tetrad formalism used to express anholonomic components of torsion and nonmetricity.
• The system of field equations is reduced to a set of five equations (47)–(52), which are solved under specific symmetry and invariance assumptions, including space inversion invariance.
• Solutions are classified based on constraints on Lagrangian parameters (e.g., a, m, k, k_i, m_i), distinguishing Riemann-Cartan, Weyl, and Weyl-Cartan space-time models.
• The generalized Friedmann equation is derived and shown to be consistent across different geometric structures, with solutions depending on the vanishing or non-vanishing of torsion and nonmetricity.

Experimental results

Research questions

• RQ1 Under what conditions do isotropic cosmological models with nonvanishing torsion and nonmetricity emerge in metric-affine gauge theory of gravity?
• RQ2 How does the inclusion of nonmetricity in the gravitational Lagrangian affect the cosmological dynamics compared to standard Poincaré gauge theory?
• RQ3 What are the necessary and sufficient conditions on the Lagrangian parameters for the existence of regular cosmological solutions in Weyl-Cartan space-time?
• RQ4 How does the generalized Friedmann equation in MAGT compare to the standard Friedmann equation in general relativity and other gauge gravity models?
• RQ5 Can the physical consequences of Poincaré gauge theory—such as finite energy density limits and gravitational repulsion—be preserved in the broader framework of MAGT?

Key findings

• The generalized Friedmann equation in MAGT is derived and shown to reduce to the standard form under specific parameter constraints, preserving the dynamics of regular cosmological models.
• Isotropic models in Weyl-Cartan space-time exist when 3m² = 4ak and additional constraints on Lagrangian parameters (e.g., k₂+k₄+2k₅ + m/8 = 0) are satisfied, allowing nonvanishing torsion and nonmetricity.
• Torsion and nonmetricity are expressed as derivatives of ln|1 - β(ρ - 3p)|, with explicit relations Q₁ = [a/(a - 2m)] d/dt ln|1 - β(ρ - 3p)| and S = [m/(2(a - 2m))] d/dt ln|1 - β(ρ - 3p)| under these conditions.
• When m = a = 0 and k ≠ 0, the solution reduces to Riemann-Cartan space-time with S ≠ 0 and Q₁ = 0, corresponding to models previously found in Poincaré gauge theory.
• When k = m = 0 and a ≠ 0, the solution corresponds to Weyl space-time with S = 0 and Q₁ ≠ 0, again consistent with earlier results in PGT.
• All derived solutions satisfy the generalized cosmological field equations (GCFE), confirming that regular cosmological solutions from PGT are also valid in MAGT, including key physical effects like finite energy density limits and gravitational repulsion.