[Paper Review] Phenomenology of effective gravity

G. E. Volovik|ArXiv.org|Apr 16, 2003

Cosmology and Gravitation Theories14 references7 citations

TL;DR

This paper proposes that the cosmological constant is not a fixed parameter but dynamically adjusts to vacuum perturbations via the Gibbs-Duhem relation, ensuring zero gravitating vacuum energy in equilibrium. It shows that in steady-state universes (Einstein, de Sitter, Gödel), the vacuum energy responds linearly to curvature, expansion, or rotation, yielding a non-zero, matter-scale cosmological constant—resolving the hierarchy problem without fine-tuning.

ABSTRACT

The cosmological constant is not an absolute constant. The gravitating part of the vacuum energy is adjusted to the energy density of matter and to other types of the perturbations of the vacuum. We discuss how the vacuum energy responds (i) to the curvature of space in the Einstein closed Universe; (ii) to the expansion rate in the de Sitter Universe; and (iii) to the rotation in the Goedel Universe. In all these steady state Universes, the gravitating vacuum energy is zero in the absence of the perturbation, and is proportional to the energy density of perturbation. This is in a full agreement with the thermodynamic Gibbs-Duhem relation applicable to any quantum vacuum. It demonstrates that (i) the cosmological constant is not huge, since according to the Gibbs-Duhem relation the contribution of zero point fluctuations to the vacuum energy is cancelled by the trans-Planckian degrees of freedom; (ii) the cosmological constant is non-zero, since the perturbations of the vacuum state induce the non-zero vacuum energy; and (iii) the gravitating vacuum energy is on the order of the energy density of matter and/or of other perturbations. We also consider the vacuum response to the non-steady-state perturbations. In this case the Einstein equations are modified to include the non-covariant corrections, which are responsible for the relaxation of the cosmological constant. The connection to the quintessence is demonstrated. The problem of the energy-momentum tensor for the gravitational field is discussed in terms of effective gravity. The difference between the momentum and pseudo-momentum of gravitational waves in general relativity is similar to that for sound waves in hydrodynamics.

Motivation & Objective

• To resolve the cosmological constant problem by treating the vacuum energy as dynamically adjusted via thermodynamic relations.
• To demonstrate that the gravitating vacuum energy vanishes in equilibrium but becomes non-zero when perturbed, avoiding the need for fine-tuning.
• To extend the Einstein equations to allow time-varying cosmological constants under non-steady-state perturbations.
• To clarify the distinction between real and pseudomomentum in gravitational waves using condensed matter analogies.
• To establish a phenomenological framework for effective gravity where the cosmological constant evolves via relaxation mechanisms.

Proposed method

• Apply the Gibbs-Duhem relation ρ = -P to quantum vacua, ensuring vacuum energy cancels when external pressure is zero.
• Model the response of the cosmological constant to steady-state perturbations (curvature, expansion, rotation) in closed, de Sitter, and Gödel universes.
• Modify the Einstein equations by introducing non-covariant, dissipative corrections to allow Λ to vary in time during non-equilibrium conditions.
• Use analogies from condensed matter physics—particularly phonons in quantum liquids—to distinguish real momentum from pseudomomentum in gravitational waves.
• Introduce two phenomenological relaxation parameters to describe the time evolution of Λ, linking it to quintessence-like dynamics.
• Derive the effective energy-momentum tensor for gravity by separating contributions from collective modes (analogous to phonons) and back-reaction effects (analogous to gravitational field response).

Experimental results

Research questions

• RQ1 How does the vacuum energy respond to steady-state perturbations such as spatial curvature, expansion, or rotation in cosmological models?
• RQ2 Can the cosmological constant be dynamically adjusted via thermodynamic relations like the Gibbs-Duhem equation, avoiding fine-tuning?
• RQ3 What modifications to the Einstein equations are required to allow the cosmological constant to evolve under time-dependent perturbations?
• RQ4 How do the concepts of real momentum and pseudomomentum in gravitational waves parallel those in hydrodynamic phonons?
• RQ5 Can the time evolution of the cosmological constant be described phenomenologically using relaxation parameters, and how does this relate to quintessence?

Key findings

• In equilibrium, the gravitating vacuum energy is zero due to the Gibbs-Duhem relation, even though zero-point fluctuations contribute significantly at the Planck scale.
• In the presence of perturbations—such as curvature, expansion, or rotation—the cosmological constant becomes non-zero and scales with the energy density of the perturbation.
• The response of the vacuum to perturbations is universal and independent of the details of trans-Planckian physics, relying only on thermodynamic consistency.
• For time-dependent perturbations, the Einstein equations must be modified to include non-covariant, dissipative terms to allow Λ to relax toward equilibrium.
• The relaxation of Λ can be described by two phenomenological parameters, linking the dynamics to quintessence-like models.
• The distinction between real momentum and pseudomomentum in gravitational waves mirrors that in condensed matter systems, where the latter arises from collective modes and the former from back-reaction, and both are measurable separately.