I recently asked this question on the physics stack exchange. After some discussion in the comments, I didn't really arrive at a satisfactory conclusion, so I thought it might be worth it to ask it here as well. I'm new to posting questions here, so if it goes against guidelines, then apologies and a mod can close this. Anyway, here is the question.
For context, I am reading this paper. The authors speak of the radius of curvature of a spacetime at a particular point. However, they don't give an explicit definition of this term. So more precisely, let $(M,g)$ be a $D$-dimensional Lorentzian manifold with Riemann tensor $R$, which is non-vanishing, and consider an arbitrary point $p\in M$. Then how is the radius of curvature at $p$ defined? In the comments to my original post, user Prahar provided the following definition: \begin{equation} \tag{1} r^2(x)=\frac{D(D-1)}{R(x)} \end{equation} where $r(x)$ is the radius of curvature at $x$ and $R(x)$ is the Ricci scalar at $x$. The problem is, the authors of the article mentioned above, have to be using a definition where the radius of curvature is finite, even in a Ricci-flat region of $M$. This is clearly not the case for this definition, since the Ricci scalar would be identically zero in a Ricci flat region and hence $r$ would diverge.
My question can thus be summarized: Is there a definition of the radius of curvature at a point on a Lorentzian manifold such that the radius of curvature is finite even in a Ricci-flat region of the manifold? One candidate I could think of would be to replace $R(x)$ with the square-root of the Kretschmann scalar in $(1)$. However I don't know if this even makes sense as a definition of radius of curvature.
Any answers or reference suggestions are greatly appreciated.