This is only my second question on mathoverflow, so my apologies if this would be more appropriate at a physics site. My question concerns a modification to the Einstein-Hilbert action. The standard action is given (in the absence of matter and with cosmological constant $\Lambda=0$) by


$$ \mathcal{S\_{EH}}(g\_{\mu\nu}) = \int\_M R \sqrt{-g}\mbox{ }d^4x$$


where $M$ is a (compact) differentiable 4-manifold, $g\_{\mu\nu}$ is a Lorentzian metric on $M$, $R$ is scalar curvature and $\sqrt{-g}\mbox{ }d^4x$ is the standard volume form. Critical points of this action (with respect to variations in $g\_{\mu\nu}$) give Lorentzian metrics which are solutions to Einstein's field equations for general relativity.



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> QUESTION: Does anyone know of work using a similar action, but where the absolute value $|R|$ appears instead of $R$? That is, I'm interested in references to previous work concerning the action $$ \mathcal{S}(g\_{\mu\nu}) = \int\_M |R| \sqrt{-g}\mbox{ }d^4x.$$
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Given the huge amount of interest in quantum gravity, I would assume that someone has examined this. However, I was unsuccessful in my searches. I'm not a physicist, so perhaps I'm missing some bit of terminology that is standard. Any help pointing me in the right direction would be greatly appreciated!



Work on an Einstein-Hilbert type action but with the *absolute value* of scalar curvature?
In the vacuum case this is not greatly different from the Einstein-Hilbert action. 


Let $(M,g)$ be a classical solution to the variational problem as you posed. Suppose $p\in M$ is such that $R(p) \neq 0$, then by continuity in a small neighborhood of $p$, the scalar curvature $R$ is signed, and hence locally in that neighborhood it is also a critical point to the Einstein-Hilbert action. But then it must be Ricci flat, contradicting the assumption that $R \neq 0$ at $p$. 


Conversely, if $(M,g)$ is a classical solution to the Einstein-Hilbert variational problem, then it is Ricci flat and hence scalar flat. And hence you have that all Einstein-vacuum solutions are also solutions to the critical point problem you posed. 


Going back forwards again, note that by definition any scalar flat 4 manifold will be a minimizer of the action. Hence you have that for the vacuum problem of your proposed action:



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> There are no critical points which do not minimise the action; the action minimisers are precisely the scalar flat Lorentzian 4 manifolds. 
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In any case, if you really are interested in this action, for literature searches the relevant keyword is [f(R) gravity theories](http://en.wikipedia.org/wiki/F%28R%29_gravity). 


