Revisiting Coincident GR in Internal STEGR Formulation

TL;DR

This paper revisits Coincident General Relativity within the internal-space formulation of Symmetric Teleparallel Gravity, analyzing its theoretical structure, stability, and particle motion implications.

Contribution

It introduces the gauge approach to STEGR in internal space, derives its field equations, and discusses potential ghost instabilities and particle kinematics.

Findings

The theory may require a degenerate condition for stability.

Derived field equations in the internal gauge formulation.

Analyzed test particle motion in non-metricity spacetime.

Abstract

We revisit Coincident General Relativity (CGR) in the gauge approach to gravity based on Symmetric Teleparallel Equivalent to General Relativity (STEGR) in the {\it internal-space formulation}, which one of the authors recently proposed in Ref.~[J. Math. Phys. 66 (2025) 5, 052505]. First, we review the standard formulation of STEGR theories in the Palatini approach to gravity, in which formulation we impose the teleparallel and torsion-free conditions by using Lagrange multipliers. Second, we introduce the STEGR theories in the gauge approach to gravity, which is formulated in the internal space, and derive its field equations. We briefly discuss whether the Ostrogradski ghost instability exists and find that the theory may require a degenerate condition to be imposed. Finally, assuming the coincident gauge, we derive CGR in both terms of the action integral and the field equation. Discussing the possible kinematics of the STEGR theory, we formulate a motion of a test scalar particle in the spacetime with non-metricity.