nLab
first-order formulation of gravity

Idea

The action functional of gravity was originally conceived as a functional on the space of pseudo-Riemannian metrics of a manifold $X$. Later on it was realized that it may alternatively naturally be thought of as a functional on the space of connections with values in the Poincare Lie algebra – essentially the Levi-Civita connection. This is known as the first order formalism or the Palatini formalism. The field strength of gravity – the Riemann tensor – is the curvature of Levi-Civita connection. Typically this is referred to a spin connection in this context.

The first order formalism is also the formalism in which supergravity has to be formulated.

Related concepts

• supergravity

• D'Auria-Fre formulation of supergravity

• gravity as a BF-theory.

• teleparallel gravity

References

A decent introduction is in sections 4 and 5 of

• Jorge Zanelli, Lecture notes on Chern-Simons (super-)gravities. Second edition (February 2008) (arXiv:hep-th/0502193)

A detailed account is in section I.4.1 of

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Supergravity and Superstrings - A Geometric Perspective