Constraints on BMS Transformations via Energy Conditions and implications on black hole geometry

Abstract

We investigate whether the formally infinite-dimensional supertranslation sector of the Bondi-Metzner-Sachs (BMS) group remains fully physically admissible once classical energy conditions are enforced. Working in a perturbative framework $g_{ab}\rightarrow g_{ab}+h_{ab}$, we first develop a general toolkit by expanding the curvature tensors and the Ricci scalar in powers of the perturbation $h_{ab}$ and recast the strong, weak, null and dominant energy conditions (SEC, WEC, NEC and DEC, respectively) as explicit inequalities on $h_{ab}$ following from the Raychaudhuri equation. The formalism is general, but to obtain concrete constraints we specialize to the standard BMS form on a Schwarzschild background and parametrize $h_{ab}=\mathcal{L}_\eta g_{ab} $ by a supertranslation function $f(\theta,\phi)$. We find that the SEC and WEC impose nontrivial angular restrictions on $f$ already at next-to-leading order (NLO) in the perturbation, whereas the NEC and DEC are preserved at linear order and acquire their first nontrivial contributions only at next-to-next-to-leading order (NNLO). Notably, the NNLO NEC reduces to a purely angular condition (independent of the radial coordinate), providing the strongest constraint on admissible supertranslations. Thus, imposing energy conditions substantially reduces the space of physically admissible supertranslations; the allowed sector, although remains infinite-dimensional in principle, is substantially constrained in practice.