Ghost instabilities and strong coupling in quadratic non-metricity theories

TL;DR

This paper investigates quadratic non-metricity theories with a parity-violating term within Newer General Relativity, showing that linear perturbations around flat FLRW generically reveal ghost instabilities or extra DOFs, except for STEGR and a parity-violating TDiff subclass which are linearly healthy but become strongly coupled non-linearly due to PV breaking of non-linear TDiff. The authors perform a thorough non-linear DOF census using both the Dirac-Bergmann Hamiltonian approach and the Cartan-Kuranishi algorithm, focusing on STEGR with PV, and demonstrate that eight propagating DOFs generically arise. The two methods yield consistent results, indicating the presence of non-linear degrees of freedom beyond the usual tensor modes and signaling strong coupling that undermines perturbative reliability. These findings emphasize caution in adopting parity-violating non-metricity theories as fundamental descriptions and illustrate the utility of complementary non-linear constraint analyses. The work also highlights methodological tensions between Hamiltonian and involutive approaches in field theories with non-algebraic constraints, suggesting broader applicability of Cartan-Kuranishi techniques to metric-affine gravity.

Abstract

We revisit the framework of Newer General Relativity, defined by all independent quadratic invariants of the non-metricity tensor, including the unique quadratic parity-violating term. We analyze linear perturbations around a flat FLRW background and find that the theory generically exhibits ghost instabilities and/ or propagates more degrees of freedom than in the Minkowski limit, signalling strong coupling. There are two notable exceptions: the Symmetric Teleparallel Equivalent of General Relativity (STEGR) and the transverse-diffeomorphism-invariant gravity subclass, both of which are supplemented by the parity-violating operator. However, since the parity-violating term explicitly breaks (transverse) diffeomorphism invariance, we show, using both the Dirac-Bergmann procedure and the Cartan-Kuranishi algorithm, that the parity-violating extension of STEGR propagates eight degrees of freedom at the fully non-linear level.