[Paper Review] Weyl Geometries and Timelike Geodesics

L. Fatibene, M. Francaviglia|arXiv (Cornell University)|Jun 10, 2011

Advanced Differential Geometry Research5 references4 citations

TL;DR

This paper establishes that any global timelike vector field in a Lorentzian spacetime uniquely determines a Weyl geometry ([g], Γ) in which the field is geodesic with respect to the Weyl connection Γ. The key result is a constructive mechanism to derive a Weyl geometry from observable fluid flow lines, enabling a geometric description of gravitational fields beyond standard general relativity via the covector A encoding the connection's deviation from Levi-Civita.

ABSTRACT

In view of Ehlers-Pirani-Schild formalism, since 1972 Weyl geometries should be considered to be the most appropriate and complete framework to represent (relativistic) gravitational fields. We shall here show that in any given Lorentzian spacetime (M,g) that admits global timelike vector fields any such vector field u determines an essentially unique Weyl geometry ([g], Γ) such that u is Γ-geodesic (i.e. parallel with respect to Γ).

Motivation & Objective

• To establish a geometric framework linking observable fluid flow lines to gravitational field geometry in relativistic theories.
• To resolve the rigidity of standard general relativity by allowing timelike congruences to define a Weyl connection Γ, not just the Levi-Civita connection.
• To provide a mechanism to construct a Weyl geometry ([g], Γ) from a physical congruence of timelike curves, such as fluid worldlines.
• To support the use of Weyl geometries in modified gravity models, particularly f(R) theories, by showing their compatibility with geodesic fluid flows.

Proposed method

• Given a conformal structure ([g]) and a [g]-timelike vector field u, the paper constructs a unique torsionless connection Γ such that u is Γ-geodesic.
• The connection Γ is derived from the Levi-Civita connection {g} of a representative metric g and a covector A via the formula: Γαβμ = {g}αβμ + (gαϵgβμ − 2δα(βδϵμ))Aϵ.
• The construction ensures compatibility between the conformal structure [g] and the projective structure [Γ], satisfying the Ehlers-Pirani-Schild (EPS) axioms.
• The method is conformally invariant: rescaling g to ˜g = Ω²g induces a transformation of A and Γ that preserves the Weyl geometry structure.
• The formalism is applied to cosmological models, showing that in Friedmann-Robertson-Walker spacetimes, the time-like vector field ∂t remains geodesic even with non-zero pressure.
• The paper demonstrates that the energy-momentum tensor conservation law in Weyl geometry requires careful choice of connection, with Γ being the physically relevant one in f(R) models.

Experimental results

Research questions

• RQ1 Can any timelike congruence in a spacetime be realized as a geodesic congruence of a Weyl connection Γ, given a fixed conformal structure [g]?
• RQ2 How is the Weyl connection Γ uniquely determined by a timelike vector field u, and what is its relation to the Levi-Civita connection of a representative metric g?
• RQ3 Under what conditions can a fluid with non-zero pressure still be described by a geodesic congruence in a Weyl geometry?
• RQ4 How does conformal rescaling of the metric affect the Weyl connection and the associated covector A?
• RQ5 In f(R) gravity models, how does the Weyl geometry framework support the physical interpretation of the connection Γ as the dynamical field for free-fall trajectories?

Key findings

• For any globally defined [g]-timelike vector field u in a Lorentzian spacetime (M, g), there exists a unique Weyl geometry ([g], Γ) such that u is a Γ-geodesic.
• The Weyl connection Γ is explicitly constructed from the Levi-Civita connection {g} and a covector A via the formula Γαβμ = {g}αβμ + (gαϵgβμ − 2δα(βδϵμ))Aϵ.
• The construction is conformally invariant: under g → Ω²g, the covector A transforms as A → A − d ln Ω, preserving the Weyl geometry.
• In Friedmann-Robertson-Walker cosmologies, the time-like vector field ∂t is a geodesic for the Weyl connection Γ even when pressure p ≠ 0, due to the symmetry of the metric.
• The energy-momentum tensor conservation law implies that ∇μp(gμν + nμnν) = 0 for geodesic n only under special conditions, which are satisfied in cosmological models.
• The Weyl geometry framework provides a natural setting for f(R) gravity, where the connection Γ is metric and automatically compatible with the EPS formalism, allowing Γ to describe free-fall motion while {g} serves as a kinematic reference.