[Paper Review] Scale covariant gravity and equilibrium cosmologies

Erhard Scholz|ArXiv.org|Mar 20, 2007

Cosmology and Gravitation Theories37 references3 citations

TL;DR

This paper proposes a scale-covariant gravity framework using integrable Weyl geometry to derive equilibrium cosmological solutions—Weyl universes—that avoid the singularities and expansion dynamics of standard cosmology. By coupling the Hilbert-Einstein action to a scale-covariant scalar field and introducing a natural vacuum gauge, the model yields stable, isotropic Robertson-Walker solutions with a constant Hubble-like connection, matching supernova, quasar, and Pioneer anomaly data while offering a geometric realization of Mach's principle.

ABSTRACT

Causal structure, inertial path structure and compatibility with quantum mechanics demand no full Lorentz metric, but only an integrable Weyl geometry for space time (Ehlers/Pirani/Schild 1972, Audretsch e.a. 1984). A proposal of (Tann 1998,,Drechsler/Tann 1999) for a minimal coupling of the Hilbert-Einstein action to a scale covariant scalar vacuum field $ϕ$ (weight -1) plus (among others) a Klein-Gordon action term opens the access to a scale covariant formulation of gravity. The ensuing scale covariant K-G equation specifies a natural scale gauge vacuum gauge). Adding other natural assumptions for gauge conditions (in particular Newton gauge, with unchanging Newton constant) the chosen Ansatz leads to a class of Weyl geometric Robertson-Walker solutions of the Einstein equation, satisfying $a''a+a'^2= const$, analogous to the Friedmann-Lemaitre equation but with completely different dynamical properties ($a$ the warp function in Riemann gauge). The class has an asymptotically attracting 1-parameter subfamily of extremely simple space-time geometries with an isotropic Robertson-Walker fluid as source of the Einstein equation, discussed as Weyl universes elsewhere (Scholz 2005). Under the assumption of a heuristic gravitational self energy binding Ansatz for the fluid, equilibrium solutions arise, in stark contrast to classical (semi-Riemannian) cosmology. Weyl universes agree very well with a variety of empirical data from observational cosmology, in particular supernovae luminosities and quasar data.

Motivation & Objective

• To explore the viability of equilibrium cosmologies in a generalized geometric framework beyond standard semi-Riemannian relativity.
• To address the instability and singularities of classical static cosmological models by extending gravity to Weyl geometry with scale covariance.
• To construct a physically consistent, scale-invariant formulation of gravity that naturally incorporates a Higgs-like mechanism via vacuum scalar fields.
• To test whether such models can reproduce key observational data, including supernova luminosities, quasar magnitudes, and the Pioneer anomaly.
• To re-evaluate the role of equilibrium solutions in cosmology by showing their geometric and physical plausibility under Weyl geometry.

Proposed method

• Adopts integrable Weyl geometry as the fundamental spacetime structure, replacing the full Lorentz metric with a scale-covariant affine connection.
• Introduces a scale-covariant scalar vacuum field φ (weight -1) coupled to gravity via a scale-invariant Lagrangian, leading to a scale-covariant Klein-Gordon equation.
• Imposes a vacuum gauge condition that fixes the scale of the scalar field, enabling a natural definition of mass via comparison with vacuum boson mass.
• Derives a class of Weyl geometric Robertson-Walker solutions satisfying a′′a + a′² = const, distinct from Friedmann-Lemaitre dynamics.
• Introduces a negative self-binding energy term (case 2 fluid) to stabilize the fluid source and achieve equilibrium in the Einstein equation.
• Applies a heuristic gravitational self-energy binding ansatz to derive stable, isotropic equilibrium solutions with cosmological redshift encoded in the scale connection.

Experimental results

Research questions

• RQ1 Can equilibrium cosmologies exist in a geometric framework that relaxes the requirement of a full Lorentz metric?
• RQ2 How does scale covariance in Weyl geometry enable the emergence of mass for vacuum bosons without explicit mass terms?
• RQ3 Do Weyl geometric cosmologies with stable, isotropic solutions agree with observational data such as supernova luminosities and quasar magnitudes?
• RQ4 Can a self-binding energy term in the fluid source lead to stable equilibrium solutions in a scale-covariant gravity framework?
• RQ5 What is the empirical viability of Weyl universes compared to standard expanding space cosmologies?

Key findings

• The model yields a 1-parameter subfamily of Weyl geometric Robertson-Walker solutions with a′′a + a′² = const, exhibiting asymptotic stability and equilibrium behavior.
• These solutions, termed Weyl universes, are derived from static Robertson-Walker manifolds by adding a constant scale connection (Hubble connection), encoding cosmological redshift.
• The case 2 fluid model with negative self-binding energy leads to stable equilibrium solutions, in stark contrast to classical cosmology’s instability of static models.
• Empirical fits to supernova data suggest ζ ≈ 2.6, implying Ωm ≈ 2.4 and ΩΛ ≈ 1.2, indicating a high classical matter density consistent with untraceable intercluster matter.
• The model explains the Pioneer anomaly and quasar data more naturally than standard cosmology, with no need for dark energy or exotic matter.
• The vacuum gauge condition allows a natural Higgs-like mechanism via spontaneous mass generation for vacuum bosons, linking gravity to quantum field theory in a geometrically consistent way.