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<p>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 <i>g</i><sub><i>ab</i></sub> → <i>g</i><sub><i>ab</i></sub> + <i>h</i><sub><i>ab</i></sub>, we first develop a general toolkit by expanding the curvature tensors and the Ricci scalar in powers of the perturbation <i>h</i><sub><i>ab</i></sub> and recast the strong, weak, null and dominant energy conditions (SEC, WEC, NEC and DEC, respectively) as explicit inequalities on <i>h</i><sub><i>ab</i></sub> 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 <i>h</i><sub><i>ab</i></sub> = <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <msub> <mi mathvariant="script">L</mi> <mi>η</mi> </msub> <msub> <mi>g</mi> <mi mathvariant="italic">ab</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">\( {\mathcal{L}}_{\eta }{g}_{ab} \)</EquationSource> </InlineEquation> by a supertranslation function <i>f</i> (<i>θ</i>, <i>ϕ</i>). We find that the SEC and WEC impose nontrivial angular restrictions on <i>f</i> 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.</p>

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Constraints on BMS transformations via energy conditions and implications on black hole geometry

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  • 发表日期：

    2026年5月26日

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  • Nihar Ranjan Ghosh,
  • Malay K. Nandy

摘要

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 gab → gab + hab, we first develop a general toolkit by expanding the curvature tensors and the Ricci scalar in powers of the perturbation hab and recast the strong, weak, null and dominant energy conditions (SEC, WEC, NEC and DEC, respectively) as explicit inequalities on hab 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 hab = L η g ab \( {\mathcal{L}}_{\eta }{g}_{ab} \) by a supertranslation function f (θ, ϕ). 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.