Abstract
We investigate the coupling of matter to geometry in conformal quadratic Weyl gravity, by assuming a coupling term of the form L_m{tilde{R}}^2, where L_m is the ordinary matter Lagrangian, and {tilde{R}} is the Weyl scalar. The coupling explicitly satisfies the conformal invariance of the theory. By expressing {tilde{R}}^2 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 fleft( R,L_mright) 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 zapprox 3.
Highlights
The birth of general relativity through the work by Einstein [1] and Hilbert [2] did have a deep impact on physics, and on mathematics, leading to several extensions of the Riemannian geometry
In a Weyl geometric framework the parallel transport does not keep the length of a vector constant. This feature of the Weyl geometry led to Einstein’s severe criticism of its initially proposed physical interpretation, based on the claim that since the behavior of the atomic clocks would depend on their past history, the existence of sharp spectral lines in the presence of an electromagnetic field would be impossible
The Weyl geometry is defined by classes of equivalence gαβ, ωμ of the metric gαβ and of the Weyl vector gauge field ωμ, related by the Weyl gauge transformations [70], gμν =
Summary
The birth of general relativity through the work by Einstein [1] and Hilbert [2] did have a deep impact on physics, and on mathematics, leading to several extensions of the Riemannian geometry. For extensive reviews and discussions of theories with geometry-matter coupling see [80–85] Such a geometrical-physical approach leads to gravitational models more complicated than standard general relativity, and they represent an interesting possibility for explaining the accelerating expansion of the Universe, dark energy, and dark matter, respectively. These types of theories raise a number of extremely difficult physical and mathematical questions. It is the goal of the present paper to investigate an extension of the conformally invariant Weyl geometric gravity, as introduced in [64–66], by allowing the possibility of a conformally invariant coupling between matter and curvature in a Weyl geometric framework. In this study we use the Landau–Lifshitz [86] sign conventions, and definitions of the geometric quantities
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