Remarks on Pure Spin Connection Formulations of Gravity

TL;DR

This paper compares two methods for deriving pure spin connection actions in gravity, showing they produce identical results for pure gravity cases with or without a cosmological constant.

Contribution

It demonstrates the equivalence of first order and Hamiltonian approaches in deriving pure spin connection formulations of gravity.

Findings

Both methods yield identical pure spin connection actions.

The results apply to pure gravity with or without a cosmological constant.

The equivalence simplifies understanding of gravity's spin connection formulations.

Abstract

In the derivation of a pure spin connection action functional for gravity two methods have been proposed. The first starts from a first order lagrangian formulation, the second from a hamiltonian formulation. In this note we show that they lead to identical results for the specific cases of pure gravity with or without a cosmological constant.