In the ADM formulation of general relativity, spacetime is split into spatial slices and a time axis. The basic variables are taken to be the induced metric ${\displaystyle q_{ab}(x)}$ on the spatial slice and the metric's conjugate momentum ${\displaystyle K^{ab}(x)}$, which is related to the extrinsic curvature and is a measure of how the induced metric evolves in time.^[1] These are the metric canonical coordinates.

In 1986 Abhay Ashtekar introduced a new set of canonical variables, Ashtekar (new) variables to represent an unusual way of rewriting the metric canonical variables on the three-dimensional spatial slices in terms of an SU(2) gauge field and its complementary variable.^[2]

Overview

Ashtekar variables provide what is called the connection representation of canonical general relativity, which led to the loop representation of quantum general relativity^[3] and in turn loop quantum gravity and quantum holonomy theory.^[4]

Let us introduce a set of three vector fields ${\displaystyle \ E_{j}^{a}\ ,}$ ${\displaystyle \ j=1,2,3\ }$ that are orthogonal, that is,

${\displaystyle \delta _{jk}=q_{ab}\ E_{j}^{a}\ E_{k}^{b}~.}$

The ${\displaystyle \ E_{i}^{a}\ }$ are called a triad or drei-bein (German literal translation, "three-leg"). There are now two different types of indices, "space" indices ${\displaystyle \ a,b,c\ }$ that behave like regular indices in a curved space, and "internal" indices ${\displaystyle \ j,k,\ell \ }$ which behave like indices of flat-space (the corresponding "metric" which raises and lowers internal indices is simply ${\displaystyle \ \delta _{jk}\ }$). Define the dual drei-bein ${\displaystyle \ E_{a}^{j}\ }$ as

${\displaystyle \ E_{a}^{j}=q_{ab}\ E_{j}^{b}~.}$

We then have the two orthogonality relationships

${\displaystyle \ \delta ^{jk}=q^{ab}\ E_{a}^{j}\ E_{b}^{k}\ ,}$

where ${\displaystyle q^{ab}}$ is the inverse matrix of the metric ${\displaystyle \ q_{ab}\ }$ (this comes from substituting the formula for the dual drei-bein in terms of the drei-bein into ${\displaystyle \ q^{ab}\ E_{a}^{j}\ E_{b}^{k}\ }$ and using the orthogonality of the drei-beins).

and

${\displaystyle \ E_{j}^{a}\ E_{b}^{k}\ =\delta _{b}^{a}\ }$

(this comes about from contracting ${\displaystyle \ \delta _{jk}=q_{ab}\ E_{k}^{b}\ E_{j}^{a}\ }$ with ${\displaystyle \ E_{c}^{j}\ }$ and using the linear independence of the ${\displaystyle \ E_{a}^{k}\ }$). It is then easy to verify from the first orthogonality relation, employing ${\displaystyle \ E_{j}^{a}\ E_{b}^{j}=\delta _{b}^{a}\ ,}$ that

${\displaystyle \ q^{ab}~=~\sum _{j,\ k=1}^{3}\;\delta _{jk}\ E_{j}^{a}\ E_{k}^{b}~=~\sum _{j=1}^{3}\;E_{j}^{a}\ E_{j}^{b}\ ,}$

we have obtained a formula for the inverse metric in terms of the drei-beins. The drei-beins can be thought of as the 'square-root' of the metric (the physical meaning to this is that the metric ${\displaystyle \ q^{ab}\ ,}$ when written in terms of a basis ${\displaystyle \ E_{j}^{a}\ ,}$ is locally flat). Actually what is really considered is

${\displaystyle \ \left(\mathrm {det} (q)\right)\ q^{ab}~=~\sum _{j=1}^{3}\;{\tilde {E}}_{j}^{a}\ {\tilde {E}}_{j}^{b}\ ,}$

which involves the "densitized" drei-bein ${\displaystyle {\tilde {E}}_{i}^{a}}$ instead (densitized as ${\textstyle \ {\tilde {E}}_{j}^{a}={\sqrt {\det(q)\ }}\ E_{j}^{a}\ }$). One recovers from ${\displaystyle \ {\tilde {E}}_{j}^{a}\ }$ the metric times a factor given by its determinant. It is clear that ${\displaystyle {\stackrel{~<!--\; ~\; -->}{E}}_j^a}$Zdroj:https://en.wikipedia.org?pojem=Ashtekar_variables
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