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

TL;DR

This paper explores how classical energy conditions restrict the supertranslation transformations in the BMS group, revealing that many are physically inadmissible, thus significantly narrowing the allowed supertranslation sector in black hole spacetimes.

Contribution

It develops a perturbative framework to derive explicit inequalities from energy conditions on supertranslations, constraining the BMS group on Schwarzschild backgrounds.

Findings

Energy conditions impose angular restrictions on supertranslations.

SEC and WEC restrict supertranslations at next-to-leading order.

NEC and DEC constraints appear only at higher orders, especially NNLO.

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.