[Paper Review] Proper time reparametrization in cosmology: Mobius symmetry and Kodama charges

Jibril Ben Achour|SPIRE - Sciences Po Institutional REpository|Mar 19, 2021

Cosmology and Gravitation Theories54 references4 citations

TL;DR

This paper establishes a unified framework for Möbius reparametrization symmetry in homogeneous and isotropic cosmological models, showing that proper time reparametrization leaves a residual SL(2,ℝ) Noether symmetry despite gauge fixing. It derives the condition linking proper time choice and scale factor conformal weight, identifies conserved Kodama-like charges, and reveals a deeper connection between cosmological Möbius symmetry and volume-preserving diffeomorphisms in spherically symmetric gravity.

ABSTRACT

It has been noticed that for a large class of cosmological models, the gauge fixing of the time-reparametrization invariance does not completely fix the clock. Instead, the system enjoys a surprising residual Noether symmetry under a Möbius reparametrization of the proper time, which maps gauge-inequivalent solutions to the Friedman equations onto each other. In this work, we provide a unified treatment of this hidden conformal symmetry and its realization in the homogeneous and isotropic sector of the Einstein-Scalar-$Λ$ system. We consider the flat Friedmann-Robertson-Walker (FRW) model, the (A)dS cosmology and provide a first treatment of the model with spatial constant curvature. We derive the general condition relating the choice of proper time and the conformal weight of the scale factor, and give a detailed analysis of the conserved Noether charges generating this physical symmetry. Our approach allows us to identify new realizations of this symmetry while recovering previous results in a unified manner. We also present the general mapping onto the conformal particle and discuss the solution-generating nature of the transformations beyond the Möbius symmetry. Finally, we show that, at least in a restricted context, this hidden conformal symmetry is intimately related to the Kodama charges of spherically symmetric gravity. This new connection suggests that the Möbius invariance of cosmology is only the corner of a larger symmetry structure which could be relevant beyond cosmological models.

Motivation & Objective

• To provide a complete and unified treatment of the residual Möbius symmetry in homogeneous and isotropic cosmological models after gauge fixing of time-reparametrization invariance.
• To clarify how the choice of proper time clock affects the realization of this hidden conformal symmetry and derive the general condition linking clock choice and scale factor conformal weight.
• To identify and analyze the conserved Noether charges generating the Möbius symmetry across flat, (A)dS, and closed FRW models.
• To establish a novel connection between this cosmological Möbius symmetry and the Kodama charges of spherically symmetric gravity, suggesting a larger underlying symmetry structure.
• To explore the solution-generating nature of Möbius and extended Diff(S¹) transformations beyond the standard conformal group.

Proposed method

• Derives the reduced action for the Einstein-scalar-Λ system in the homogeneous and isotropic sector, focusing on proper time reparametrization invariance.
• Applies field-dependent reparametrizations to the proper time, identifying the residual SL(2,ℝ) symmetry as a Möbius transformation of the time coordinate.
• Uses the Schwarzian derivative to characterize Möbius invariance and solve the associated differential equation for reparametrization functions.
• Constructs the Hamiltonian formulation and identifies conserved Noether charges associated with the Möbius symmetry for flat, (A)dS, and closed FRW models.
• Maps the cosmological system onto the conformal particle model via the Niederer transformation, linking it to SO(2,1) invariance.
• Establishes a correspondence between the Möbius symmetry and the Kodama charge generators in spherically symmetric gravity through the conformal bridge formalism.

Experimental results

Research questions

• RQ1 How does the choice of proper time clock affect the realization of the residual Möbius symmetry in cosmological models?
• RQ2 What is the general condition relating the conformal weight of the scale factor and the choice of proper time in the presence of residual SL(2,ℝ) symmetry?
• RQ3 How are the conserved Noether charges generating the Möbius symmetry constructed and what is their physical interpretation in flat, (A)dS, and closed FRW models?
• RQ4 What is the precise connection between the cosmological Möbius symmetry and the Kodama charges in spherically symmetric gravity?
• RQ5 Can the Möbius symmetry be extended to a larger group, such as Diff(S¹), and how does this relate to solution-generating maps in cosmology?

Key findings

• The paper derives a general condition linking the conformal weight of the scale factor to the choice of proper time, ensuring the existence of residual Möbius symmetry.
• For the flat FRW model, the conserved Noether charges generating the Möbius symmetry are explicitly computed and shown to close under the SL(2,ℝ) algebra.
• In the (A)dS cosmology, the Möbius symmetry is realized with a modified conformal structure, and the associated charges are derived using the conformal particle mapping.
• For the closed universe with positive spatial curvature, the reparametrization function is constructed using the tangent function with a Möbius transformation, satisfying the Schwarzian equation with constant curvature.
• The paper establishes a direct link between the cosmological Möbius symmetry and the Kodama charges of spherically symmetric gravity, suggesting a deeper unifying symmetry beyond cosmology.
• The solution-generating nature of Möbius and extended Diff(S¹) transformations is demonstrated via the Schwarzian derivative, allowing reconstruction of (A)dS solutions from flat ones.