Coupling matter and curvature in Weyl geometry: conformally invariant $f\left(R,L_m\right)$ gravity

Abstract

We investigate the coupling of matter to geometry in conformal quadratic Weyl gravity, by assuming a coupling term of the form Lm R2 , where Lm is the ordinary matter Lagrangian, and R is the Weyl scalar. The coupling explicitly satisfies the conformal invariance of the theory. By expressing R2 with the help of an auxiliary scalar field and of the Weyl scalar, the gravitational action can be linearized, leading in the Riemann space to a conformally invariant f (R, Lm) type theory, with the matter Lagrangian nonminimally coupled to the Ricci scalar. We obtain the gravitational field equations of the theory, as well as the energy-momentum balance equations. The divergence of the matter energy-momentum tensor does not vanish, and an extra force, depending on the Weyl vector, and matter Lagrangian is generated. The thermodynamic interpretation of the theory is also discussed. The generalized Poisson equation is derived, and the Newtonian limit of the equations of motion is considered in detail. The perihelion precession of a planet in the presence of an extra force is also considered, and constraints on the magnitude of the Weyl vector in the Solar System are obtained from the observational data of Mercury. The cosmological implications of the theory are also considered for the case of a flat, homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker geometry, and it is shown that the model can give a good description of the observational data for the Hubble function up to a redshift of the order of z ≈ 3.

Dark energy and accelerating cosmological evolution from osculating Barthel–Kropina geometry

“…Alternatively, one could construct the field equations in a less rigorous approach directly by substituting in the standard general relativistic equations the Riemannian quantities with their Finslerian counterparts. Modified gravity theories with curvature-matter coupling [114][115][116] could also be extended to a Finslerian geometric framework. Riemannian geometries with torsion [117,118] or non-metricity [119][120][121] can also represent a source of inspiration for obtaining their Finslerian analogues.…”

Dark energy and accelerating cosmological evolution from osculating Barthel-Kropina geometry

“…Alternatively, one could construct the field equations in less rigorous approach directly by substituting in the standard general relativistic equations the Riemannian quantities with their Finslerian counterparts. Modified gravity theories with curvature-matter coupling [114][115][116] could also be extended to a Finslerian geometric framework. Riemannian geometries with torsion [117,118] or nonmetricity [119][120][121] can also represent a source of inspiration for obtaining their Finslerian analogues.…”