general relativity - Is GR vacuum equation unique?

The title question would be too long if I tried to specify it clearly. So let me be more clear. Consider the class of theories having the following properties:

1. The langrangian density is only dependent of scalars created from the curvature tensors (for example $R, R^{ab}R_{ab}, R^{abcd}R_{abcd}, C^{abcd}C_{abcd}, (R)^2, R^{ab}R_{bc}R^{cd}R_{da}$, etc.) + generic matter using the usual minimal coupling.


2. The field theory is generated from the lagrangian by considering variations in the metric (ie. we will not consider Palatini or torsion theories)


3. The field equation automatically provides a covariantly conserved stress energy tensor (this means like in GR, the covariant derivative of the Ricci terms are collectively zero just from geometric requirements and not an extra postulate. I've seen papers claim this follows automatically from 1, but I have not seen proof, so I'm specifying it just in case.)


4. Let's stay with 3+1 dimensions.

My question is then: GR is the theory starting from using $R$ as the lagrangian density, and the vacuum equation in GR reduces to $R^{ab}=0$. Is there another function of curvature scalars in this class which would yield the same vacuum equations?

(For clarity: Yes, for this question I don't care if the field equations differ in non-vacuum. I'm just curious if they can possibly match in vacuum.)

One may be tempted to immediately say it is unique, but consider the theories from $(R)^2$, $R^{ab}R_{ab}$, and $R^{abcd}R_{abcd}$.(1) One might intuitively expect that $C^{abcd}C_{abcd}$ would then add yet another theory, but it turns out this theory can be written as a combination of the previous three. (2)(3) And it turns out that in 4-dimensions, there are actually only two linearly independent combinations of these four theories. (4)

Playing with it to try to find if there is a theory equivalent to GR in vacuum, I was looking for a theory which in vacuum after I contract the indices, I can show the only possible result is R=0 (For an example if the contraction gave a polynomial like $R + R^3 + R^{abcd}R_{abcd}R = 0$, then the only solution with real curvature is $R=0$). And then hoping that applying this to the field equations reduces everything to $R^{ab}=0$. This hunt and peck method is failing me, as I haven't found anything yet and clearly won't let me prove GR is unique if that is the answer.

Can someone give an example which yields the GR vacuum equation or prove mathematically no such theory exists and thus the GR vacuum equation is unique in this class?

Answer

You can add a Gauss-Bonnet term $R^2 - 4R^{\mu\nu}R_{\mu\nu} + R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}$. It's purely topological.