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Abstract

Following a brief review of perturbative Einstein-Hilbert gravity, we extract the Feynman rules from the flat-space expanded action. Computing diagrams order by order in the gravitational coupling G_N, we construct the small rₛ/r expansion of the perturbed Schwarzschild metric to 𝒪(1/r²), where r is radial coordinate and rₛ=2G_Nm is the Schwarzschild radius for an object of mass m. In applying the same treatment to the Einstein-Maxwell action, we compute an additional graviton-photon interaction vertex and the 𝒪(1/r²) correction to the photon 1-point function. Our result is then verified against the small G_N expansion of the Reissner-Nordström (RN) metric. For a pair of interacting RN black holes, each with electric charge Q, we show that the force vanishes at each order in G_N provided the black holes are extremal (Q=√G_Nm). Based on this result, we propose a novel perturbative expansion in the relative velocity v² for multi-centre black hole solutions, to be explored and further developed in future work.