# On-shell versus Off-shell Equivalence in 3D Gravity

## TL;DR

This paper investigates the distinction between on-shell and off-shell equivalence in 3D gravity theories, highlighting potential misconceptions and clarifying the conditions under which different actions are truly equivalent.

## Contribution

It demonstrates that off-shell equivalence is not guaranteed by on-shell equivalence, especially in linearized 3D massive gravity models, correcting recent misconceptions.

## Key findings

- Off-shell and on-shell equivalences are distinct concepts.
- Misinterpretations can lead to incorrect conclusions in gravity models.
- Clarifies the conditions for equivalence in 3D massive gravity.

## Abstract

A given field theory action determines a set of field equations but other actions may yield equivalent field equations; if so they are on-shell equivalent. They may also be off-shell equivalent, being related by the elimination of auxiliary fields or by local field redefinitions, but this is not guaranteed, as we demonstrate by consideration of the linearized limit of 3D massive gravity models. Failure to appreciate this subtlety has led to incorrect conclusions in recent studies of the "Minimal Massive Gravity" model.