Superposition of Ricci scalars Suppose I have two point/line singularities in spacetime (what is important to me is that they are localized). Also suppose I have some fields in spacetime and that the two singularities interact with each other via the fields (fields are non-minimally coupled to the curvatures). Now if I denote the Ricci scalar of the entire system by $R$ and for each of the singularities by $R_1$ and $R_2$, is $R=R_1+R_2$? I am not able to make up my mind. I don't think so and I think only to some order in perturbation this is true. But your help/comments will be greatly appreciated.
 A: A few things:
1) If you have a spacetime containing an isolated object like a black hole or a cosmic string, or any matter distribution whatsoever, $R$ is zero outside the matter distribution, since $T_{ab}$ is zero.
2) Also, consider the Schwarzschild solution in asymptotically Cartesian coordinates.  Place the $r=0$ singularity at $(x_{1},y_{1},z_{1})$.  If you try and write the metric according to the ansatz $g_{ab}={}^{1}g_{ab}+{}^{2}g_{ab}$, where ${}^{1}g_{ab}$ is the Schwarzschild solution with the hole at $(x_{1},y_{1},z_{1})$, you will find that the solution you get is only an approximation of a vacuum solution of Einstein's equation.  
3) There are terms in $R_{abcd}$ that have to do with things like the boundary charges and initial conditions in the spacetime.  In particular, if you two singularities are moving with respect to each other, or have appreciable gravitational binding energy, or emit gravitational radiation, then you will have boundary terms that don't satisfy $M_{adm}=M_{1}+M_{2}$, and so, you wouldn't expect that $R_{abcd}={}^{1}R_{abcd}+{}^{2}R_{abcd}$
