Is this a mistake in the derivation of the spherically symmetric field equations in $f(R)$ gravity?

Question

Abstract:

An f(R) gravitation for galactic environments.

We propose an action-based f(R) modification of Einstein's gravity which admits of a modified Schwarzschild-deSitter metric. In the weak field limit this amounts to adding a small logarithmic correction to the newtonian potential. A test star moving in such a spacetime acquires a constant asymptotic speed at large distances. This speed turns out to be proportional to the fourth root of the mass of the central body in compliance with the Tully-Fisher relation. A variance of MOND's gravity emerges as an inevitable consequence of the proposed formalism

In the paper associated with the above abstract arxiv.org/abs/astro-ph/0603302, the authors have derived field equations for $f(R)$ gravity in the case of a static, spherically symmetric metric (Eq. 3). It appears that they wrote $f(R(r))$ as $f(r)$ for some $f$ where $R$ is the Ricci scalar which only depends on the $r$ coordinate. To me this looks suspicious: the field equations are $$f'(R)R_{\mu \nu} - \tfrac12 f(R) g_{\mu \nu} + g_{\mu \nu} \Box f'(R) - \nabla_\mu \nabla_\nu f'(R) = 0 $$ so surely in the replacement $f(R(r)) \to f(r)$ the equation would become
$$\frac{f'(r)}{R'(r)}R_{\mu \nu} - \tfrac12 f(r) g_{\mu \nu} + g_{\mu \nu} \Box \left (\frac{f'(r)}{R'(r)} \right) - \nabla_\mu \nabla_\nu \left (\frac{f'(r)}{R'(r)} \right) = 0 $$

however the authors seem to have written it as $$f'(r)R_{\mu \nu} - \tfrac12 f(r) g_{\mu \nu} + g_{\mu \nu} \Box f'(r) - \nabla_\mu \nabla_\nu f'(r) = 0. $$

I do not believe they and others could make a mistake like this so it must be something wrong in my thinking, but I do not know what.

I cannot confirm this is precisely what they have done but I have implemented these equations in Mathematica and assuming the code is correct (which I was able to check against other sources along with reading over the $f(R)$ field equation function multiple times), I am able to obtain equations very similar, though not identical, to the ones they give with the latter prescription I have described.

Tellingly, instead of an arbitrary $f$, a function $f(r) = r$ does not reduce to Einstein's equations. I would appreciate any help in this matter.

Answer 1

The third equation can be dismissed immediately by dimensional analysis. Note that the curvature tensor has mass dimension $[R]=2$, and so $f(R)$ also has mass dimension $[f]=2$. However, $r$ has mass dimension $[r]=-1$. Thus, in the third equation, the first term has mass dimension $[f'(r)R_{\mu\nu}]=5$, while the second term has mass dimension $[f(r)g_{\mu\nu}]=2$. Thus, this equation makes no sense from the start.

I suspect your derivation of the Einstein equations per component may have some errors if you're getting a solution such that the third equation you wrote matches what is written in the paper, since all of their expressions are fine on dimensional grounds.