# Corrections to precession of the perihelion of Mercury

Consider one of the classical observables of General Relativity, the precession of the perihelion of an elliptical orbit: $$\delta\varphi=\frac{6\pi G^2M^2m^2}{c^2L^2}$$

I am interested in known corrections to this formula, either classical or quantum-mechanical, coming from more general descriptions of General Relativity. More precisely, what is the value of $\delta\varphi$ in

• $f(R)$ gravity?
• naïve quantum gravity (i.e., from the one-loop correction to the Hilbert Lagrangian, in the form $c_1 R^{abcd}R_{abcd}+c_2 R^{ab}R_{ab}+c_3 R^2$)?
• Supergravity and low-energy effective String Theory (e.g., the effect of a non-trivial dilaton, etc.)?

Corrections to other classical observables, such as the deflection of light by a massive body, or the gravitational redshift of light are of interest as well.

• $f(R)$ are more of a family of theories but if you accept series in $R$, then this paper should qualify: arxiv.org/abs/1104.0819 – user154997 Oct 11 '17 at 15:29
• Wouldn't quantum gravity effects be too small to observe by many, many orders of magnitude? – Ben Crowell Oct 11 '17 at 15:33
• @LucJ.Bourhis I guess I was hoping that a result of $\delta\varphi$ for arbitrary $f$ was known, but now I realise I was being far too optimistic. A series expansion it is then. Thanks! – AccidentalFourierTransform Oct 11 '17 at 15:34
• @BenCrowell yes, of course. From the theoretical point of view, the prediction is very interesting anyway, even if untestable in practice. – AccidentalFourierTransform Oct 11 '17 at 15:35
• The main result I had taken from that paper I cited is that it predicts that light deflection is not modified, and that is depressing! – user154997 Oct 11 '17 at 15:48