nLab
first-order formulation of gravity

Context

Gravity

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

The action functional of gravity was originally conceived as a functional on the space of pseudo-Riemannian metrics of a manifold XX. 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 Poincaré Lie algebra – essentially the Levi-Civita connection – subject to the constraint that the component in the translation Lie algebra defines a vielbein field. Mathematically this means that the field of gravity is modeled as a Cartan connection for the Lorentz group inside the Poincaré group. In physics this is known as the first order formalism or the Palatini formalism for gravity.

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.

Promoting this perspective form the Poincaré group to the super Poincaré group yields supergravity. Promoting it further to the Lie n-algebra extensions of the super Poincaré group (from the brane scan/brane bouquet) yields type II supergravity, heterotic supergravity and 11-dimensional supergravity in higher Cartan geometry-formulation (D'Auria-Fré formulation of supergravity).

local model spaceglobal geometrydifferential cohomologyfirst order formulation of gravity
generalKlein geometryCartan geometryCartan connection
examplesEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
Lorentzian supergeometysupergeometrysuperconnectionsupergravity
generalKlein 2-geometryCartan 2-geometry
higher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

References

A decent introduction is in sections 4 and 5 of

A detailed account is in section I.4.1 of

Revised on October 14, 2013 13:03:50 by Toby Bartels (98.19.41.253)