Is there a "square root" version of the Einstein field equation?

Question

It is well known that the Klein-Gordon equation have a kind of "square root" version :  the Dirac equation.
The Maxwell equations can also be formulated in a Dirac way.
It is also well known that the metric of general relativity have a kind of "square root" version :  the tetrad field (or vierbein) of components $e_{mu}^a(x)$ :
begin{equation}tag{1}
g_{mu nu}(x) = eta_{ab} , e_{mu}^a(x) , e_{nu}^b(x).
end{equation}
Now, a natural question to ask is if the full Einstein equations :
begin{equation}tag{2}
G_{mu nu} + Lambda , g_{mu nu} = -, kappa , T_{mu nu},
end{equation}
could be reformulated for the tetrad field only (or other variables ?), as a kind of a "Dirac version" of it ?  In other words :  is there a "square root" version of equation (2) ?

Qmechanic · Answer

Since Nature has fermionic matter we are anyway ultimately forced to rewrite the metric in GR in terms of a vielbein (and introduce a spin connection). See e.g. my Phys answer here. The fermionic matter obeys a Dirac equation in curved spacetime. This however would not amount to a square root of EFE.

There exist supersymmetric extensions of GR, such as, SUGRA.

Another idea is to consider YM-type theories as a square root of GR, or GR as a double copy of YM. See e.g. the Ashtekar formulation or the KLT relations.

Alex Nelson · Answer

By taking the "Dirac square root" of the Hamiltonian constraint for GR, you naturally end up with Supergravity...so in some appropriate sense, SUGRA "is" a "square root" of GR. For more on this, see:

Romualdo Tabensky, Claudio Teitelboim, "The square root of general relativity". Physics Letters B 69 no.4 (1977) pp 453-456. Eprint
Claudio Teitelboim, "Supergravity and Square Roots of Constraints". Phys. Rev. Lett. 38 (1977) 1106. Eprint

Is there a "square root" version of the Einstein field equation?

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