Convergent sum of EFT corrections to the Schwarzschild metric requires UV locality

Corrections to vacuum black hole solutions of general relativity (GR) are considered in an effective field theory (EFT) framework, perturbatively in EFT coefficients, focusing on the Schwarzschild solution of GR. We find dominant corrections to the Schwarzschild metric in all orders in the derivative expansion far away from the horizon. These corrections can be summed up in a closed form through EFT coefficients up to all orders in derivatives and to the second order in curvature. It occurs that such a summation is convergent only for localizable theories, making a direct connection between the graviton scattering amplitudes properties and the applicability of a perturbative treatment of an EFT of gravity. We further apply our results to logarithmic form factors which appear in the one-loop effective action for GR in four dimensions. We find out that the corresponding corrections to the Schwarzschild metric are stronger than those from the tree-level EFT operators. The developed framework can be extended to account for the corrections to the other black hole solutions in GR, such as the Kerr metric.