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Recently I have discovered the method of constructing of GR from massless field with helicity 2 theory. It is considered here, in an article "Self-Interaction and Gauge Invariance" written by Deser S.

By the few words, the idea of the method is following. When starting from massless equations for field with helicity 2 we note that it doesn't provide stress-energy momentum tensor conservation:

$$ \tag 1 G_{\mu \nu}(\partial h , \partial^{2}h) = T_{\mu \nu} \Rightarrow \partial^{\mu}T_{\mu \nu} \neq 0 $$ (here $h_{\mu \nu}$ is symmetric tensor field).

But we may change this situation by adding some tensor to the left side of equation which provides stress-tensor conservation: $$ G \to \tilde {G}: \partial \tilde{G} = 0. $$

Deser says that it might be done by modifying the action which gives $(2)$ by the following way: $$ \eta_{\mu \nu} \to \psi_{\mu \nu}, \quad \partial \Gamma \to D \Gamma . $$ Here $\eta $ is just Minkowski spacetime metric, $\Gamma$ is the Christoffel symbol with respect to $h$, $\psi^{\mu \nu}$ is some fictive field without geometrical interpretation, and $\partial \to D$ means replacing usual derivative to a covariant one with respect to $\psi $ (this means appearance of Christoffel symbols $C^{\alpha}_{\beta \gamma}$ in terms of $\psi$). By varying an action on $\psi$ we can get the expression for correction of $(1)$ which leads to stress-energy conservation.

Here is the question: I don't understand the idea of this method. Why do we admit that we must to introduce some fictive field $\psi_{\mu \nu}$ for providing conservation law? Why do we replace partial derivatives by covariant ones using $\psi $? How to "guess" this substitution? I don't understand the explanation given in an article.

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    $\begingroup$ You can freely download Thomas Ortin's Gravity and Strings book, where you find a nice section on deser's argument and how to reconstruct gravity. Just the description by Deser himself is not the best one:-) $\endgroup$ – John Aug 14 '14 at 15:44
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    $\begingroup$ Possibly helpful (or not): gr-qc/0409089, which critiques various attempts to derive GR this way, including Deser's. $\endgroup$ – benrg Aug 14 '14 at 18:33

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