Inequivalence of First and Second Order Formulations in D=2 Gravity Models

TL;DR

This paper investigates the differences between first and second order formulations of 2D gravity models, revealing that their equivalence breaks down, especially regarding the cosmological constant and affine formulations.

Contribution

It demonstrates the inequivalence of first and second order formulations in two-dimensional Einstein gravity, highlighting the failure of the usual equivalence in this context.

Findings

First and second order formulations are not equivalent in 2D gravity.

The cosmological constant must vanish in first order formulations.

Purely affine Eddington formulation also fails in 2D.

Abstract

The usual equivalence between the Palatini and metric (or affinity and vielbein) formulations of Einstein theory fails in two spacetime dimensions for its "Kaluza--Klein" reduced (as well as for its standard) version. Among the differences is the necessary vanishing of the cosmological constant in the first order forms. The purely affine Eddington formulation of Einstein theory also fails here.