Combining metric tensors/curvature tensors

I was thinking about the following scenario:

Consider a particle which causes a metric $g_{\mu\nu}$ on an otherwise Minkowski spacetime (or any manifold). Now, consider another particle, somewhere in the vicinity of the first particle, which causes a metric $h_{\mu\nu}$ on a spacetime which would have been Minkowski if not for these two particles.

Then, what what would the metric in the vicinity of these two points be? I am guessing that it is: $$(g_{\mu\nu}-\eta_{\mu\nu})+(h_{\mu\nu}-\eta_{\mu\nu}) + \eta_{\mu\nu} = g_{\mu\nu}+h_{\mu\nu} - \eta_{\mu\nu}$$

Also, does the Riemann Curvature Tensor $R_{\mu\nu\rho}^\sigma$add up directly? I don't think it should because the Einstein tensor $G_{\mu\nu}$ does (I think) and it is dependent on the Ricci Curvature AND the spacetime metric tensor.

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You can do this addition at weak field, so long as your coordinates make the metric nonsingular (rectangular coordinates as the unperturbed metric), and add the usually negligible corrections perturbatively. –  Ron Maimon Aug 22 '13 at 23:57

Unlike classical electromagnetism, General Relativity is highly nonlinear--this means that the gravitational field can serve as its own source. A consequence of this fact is that fields decidedly do not superpose, and you can get all sorts of effects even from vacuum relativity. The most notable of these effects are things such as Brill waves and Geons, where gravitational waves collide or collapse to form black holes. You can work out solutions where this happens even when the spaces initially are empty outside of the two regions before overlap occurs.

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