General relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

 and supergravity

Supergravity

In theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry...

 in all dimensions meet each other at a common assumption:

Any configuration space

Configuration space

- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...

 can be coordinatized by gauge fields , where the index is a Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

 index and is a spatial manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 index.

Using these assumptions one can construct an effective field theory

Effective field theory

In physics, an effective field theory is, as any effective theory, an approximate theory, that includes appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, while ignoring substructure and degrees of freedom at shorter distances .-The renormalization...

 in low energies for both. In this form the action of general relativity can be written in the form of the Plebanski action which can be constructed using the Palatini action to derive Einstein's field equations of general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

.

The form of the action introduced by Plebanski

Jerzy Plebanski

Jerzy Franciszek Plebański Rosiński was a Polish theoretical physicist best known for his extensive research into general relativity and supergravity.-Sources:*...

 is:


where


are internal indices,


is a curvature on the orthogonal group and the connection

Connection (mathematics)

In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...

 variables (the gauge fields) are denoted by
.

The symbol


is the Lagrangian multiplier and


is the antisymmetric symbol valued over .