A correspondence between $1^{st}$ and $2^{nd}$ order formalism by a metricity constraint

TL;DR

This paper introduces a method to connect first and second order formalisms in gravity by enforcing a metricity constraint with a Lagrange multiplier, enabling new theories beyond traditional formulations.

Contribution

It presents a novel approach using a Lagrange multiplier to transform first order formalism into second order, allowing exploration of new gravity theories.

Findings

Transforming first order to second order formalism via metricity constraint.

Higher derivatives appear as derivatives of the Lagrange multiplier.

Method applicable to conformal invariant theories and beyond.

Abstract

A way to obtain a correspondence between the first order and second order formalism is studied. By introducing a Lagrange multiplier coupled to the covariant derivative of the metric, a metricity constraint is implemented. The new contributions which comes from the variation of the Lagrange multiplier transforms the field equations from the first order to the second order formalism, yet the action is formulated in the first order. In this way all the higher derivatives terms in the second order formalism appear as derivatives of the Lagrange multiplier. Using the same method for breaking metricity condition and building conformal invariant theory is briefly discussed, so the method goes beyond just the study of first order or second formulations of gravity, in fact vast new possible theories of gravity are envisioned this way.