[Paper Review] Nonexistence of a parent theory for general relativity and unimodular gravity

Gerardo García-Moreno, Alejandro Jiménez-Cano|arXiv (Cornell University)|Sep 13, 2023

Cosmology and Gravitation TheoriesPhysics and Astronomy13 references3 citations

TL;DR

This paper investigates whether a parent theory unifying General Relativity (GR) and Unimodular Gravity (UG) exists by introducing Stueckelberg fields to restore full diffeomorphism and Weyl symmetry in both theories. Despite their dynamical equivalence, the presence of the cosmological constant as a global degree of freedom in UG obstructs the construction of a consistent parent theory, proving such a unification impossible under standard gauge completion procedures.

ABSTRACT

General Relativity (GR) and Unimodular Gravity (UG) provide two equivalent descriptions of gravity that differ in the nature of the cosmological constant. While GR is based on the group of diffeomorphisms that permits the cosmological constant in the action, UG is based on the subgroup of volume-preserving diffeomorphisms together with Weyl transformations which forbid the presence of the cosmological constant. However, the cosmological constant reappears in UG as an integration constant so it arises as a global degree of freedom. Since gauge symmetries are simply redundancies in our description of physical systems, a natural question is whether there exists a "parent theory" with the full diffeomorphisms and Weyl transformations as gauge symmetries so that it reduces to GR and UG respectively by performing suitable (partial) gauge fixings. We will explore this question by introducing Stueckelberg fields both in GR and UG to complete the gauge symmetries in each theory to that of the would-be parent theory. Despite the dynamical equivalence of the two theories, we find that precisely the additional global degree of freedom provided by the cosmological constant in UG obstructs the construction of the parent theory.

Motivation & Objective

• To determine whether a single parent theory can unify General Relativity (GR) and Unimodular Gravity (UG) via shared gauge symmetries.
• To investigate if GR and UG can be derived as distinct gauge fixings of a common theory with full diffeomorphism and Weyl invariance.
• To analyze the role of the cosmological constant as a global degree of freedom in UG and its implications for gauge completion.
• To assess whether Stueckelberg field methods can reconcile the gauge structures of GR and UG into a unified framework.
• To clarify the fundamental incompatibility between the global nature of the cosmological constant in UG and the requirements for a parent gauge theory.

Proposed method

• Introduce Stueckelberg fields into General Relativity to restore Weyl invariance, transforming it into a theory with extended gauge symmetry.
• Apply the same Stueckelberg procedure to Unimodular Gravity to restore full diffeomorphism invariance, completing its gauge structure.
• Compare the resulting Stueckelberg-modified theories of GR and UG to test whether they can be derived from a common parent theory.
• Analyze the linearized WTDiff (Weyl + transverse diffeomorphisms) and Fierz-Pauli (GR linearized) theories to identify structural differences.
• Examine the nonlinear versions of GR and UG with Stueckelberg fields to assess the viability of a parent theory at the full nonlinear level.
• Use the trinity formulation of UG (metric, teleparallel, symmetric teleparallel) to compare its structure with GR analogs and confirm inequivalence.

Experimental results

Research questions

• RQ1 Can a single parent theory exist that reduces to both General Relativity and Unimodular Gravity via different gauge fixings?
• RQ2 Does the cosmological constant in Unimodular Gravity, appearing as a global integration constant, prevent the existence of a unified gauge-theoretic parent theory?
• RQ3 Are the Stueckelberg-modified versions of GR and UG dynamically equivalent, and can they be unified under a common symmetry structure?
• RQ4 What is the role of Weyl symmetry and diffeomorphism invariance in the construction of a parent theory for GR and UG?
• RQ5 Why do the Stueckelberg completions of GR and UG lead to inequivalent theories despite the original dynamical equivalence of GR and UG?

Key findings

• The Stueckelberg completion of General Relativity restores Weyl invariance but does not yield a theory equivalent to the Stueckelberg-completed Unimodular Gravity.
• The Stueckelberg-completed versions of GR and UG are inequivalent, indicating that no common parent theory can consistently reduce to both via gauge fixing.
• The cosmological constant in Unimodular Gravity arises as a global degree of freedom, which is not present in the same form in GR, breaking the symmetry structure needed for a parent theory.
• The presence of this global degree of freedom in UG prevents the construction of a unified gauge theory that includes both full diffeomorphisms and Weyl symmetry as fundamental symmetries.
• The linearized analysis of WTDiff and Fierz-Pauli theories confirms that the two theories differ fundamentally due to the additional global mode in WTDiff, which corresponds to the cosmological constant.
• The trinity formulation of Unimodular Gravity (metric, teleparallel, symmetric teleparallel) shows that none of its variants are equivalent to their GR counterparts, reinforcing the fundamental distinction between the two theories.