How to use Ashtekar's variables in classical gravitational physics? I have often heard of Ashtekar's variables in General Relativity, because of the naturalness with which they would allow a canonical formulation of gravity, useful for a hypothetical quantum gravity theory.
But what I do not understand is that use has been done so far in classical gravitational physics.
Example: 


*

*How is it possible to express gravitational waves through Ashtekar variables? Are there any papers, or reviews in which the subject is addressed?

*Similar question, but for the case of a Schwarzschild black hole?
I know the formulation of these variables, what is not clear to me is how to implement them at the classical level, and unfortunately I did not find any examples.
 A: First, some general info on Ashtekar variables.
Ashtekar-Barbero connection is an $\mathfrak{su}_2$ valued spatial connection
$$ A = A_a^i \tau_i dx^a, $$
which is by definition related to the $\mathfrak{so}_{3,1} \sim \mathfrak{sl}_{2,\mathbb{C}}$ valued spin connection $\omega$ by
$$ A_a^{ij} = \varepsilon^{ijk} \omega_a^{0k} + \gamma^{-1} \omega_a^{ij}, $$
where $\gamma$ is a dimensionless constant called the Immirzi parameter.
The other part of Ashtekar's variables is given by the densitized triad field
$$ E^a_i = \frac{1}{2} \varepsilon^{abc} \varepsilon_{ijk} e^j_b e^k_c = |e| \cdot e^a_i. $$
There's two exceptional properties which make this objects so useful in canonical General Relativity:


*

*It transforms as a $\mathfrak{su}_2$ valued spatial connection, e.g. under coordinate-dependent spatial rotations $\Omega(x) \in SU(2)$,
$$ A \rightarrow \Omega A \Omega^{-1} + \Omega d \Omega^{-1}. $$
Note that this property is non-trivial. The second summand of $A$ indeed transforms like a connection, but the first summand transforms like a 3-vector. But the transformation law of the gauge connection allows us to add an arbitrary vector to it. So $A$ ends up being a $\mathfrak{su}_2$ connection on the spatial slice.

*$A$ and $E$ come naturally as a canonical conjugate pair for the Holst action
$$ S_{\text{Holst}}[\omega, e] = \intop_M e^I \wedge e^J \left( \star + \gamma^{-1} \right) F_{IJ}, $$
where $\star$ is an internal automorphism of $\mathfrak{so}_{3,1}$ (aka the electro-magnetic duality operator) defined as
$$ \star F_{IJ} = \frac{1}{2} \varepsilon_{IJKL} F^{KL}, $$
and $F = d\omega + \omega \wedge \omega$ is the curvature of the spin connection.
It is straightforward to check that the canonical conjugate of $A$ is $E$.
Since Holst action is classically equivalent to General Relativity (because the equations of motion coming from it are equivalent to Einstein's equations), this allows for re-interpretation of the phase space variables of General Relativity in terms of a $\mathfrak{su}_2$ connection $A$ and its canonical conjugate densitized triad field $E$.
You might be wondering why this is important for Quantum Gravity. Well, in the quantum theory states are naturally functions over $1/2$ of the phase space variables (this is sometimes called polarization in geometric quantization). This opens an interesting possibility of using certain functionals called cylindrical functions over the space of connections as describing states of the Quantum Gravity theory. These functionals can only be built over the space of connections – which is true for the Ashtekar connection $A$. After defining the constraints as quantum operators on this space, the formal definition of a Quantum Gravity theory is complete. This is in fact exactly the outline of the Canonical Loop Quantum Gravity programme.
You might be also wondering if it is possible to have a simpler connection describing gravity. It turns out that it is not possible to satisfy both (1) and (2) with just one summand from the definition of $A$: it is the first summand that makes $A$ canonically conjugate to the densitized triad, and it is the second summand that makes it transform as a $\mathfrak{su}_2$ connection.
Now to finally answer your question.
You don't have a resource-recommendation tag, so I assume that you are willing to do some work and derive the expressions, given that I provide the general algorithm. So here goes.


*

*Take the spacetime metric $g_{\mu \nu}$ of interest. This could either be a gravitational wave or a Schwarzschild black hole.

*Choose a compatible tetrad and spin connection. Note that you have $SO(3,1)$ freedom here, so there's no unique solution to this problem. You might want to choose the simplest form, or just any form. But your spin connection has to be compatible with the tetrad and give the correct Riemann tensor, whereas tour tetrad should give the correct spacetime metric $g_{\mu \nu}$.

*Calculate $A$ and $E$ using the definitions given in the previous section of my answer.
A: This equation only used when waves come to the picture. Where there are no wave in classical physics. Only in quantum gravity space time warps waves.
