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Asymptotic charges for spin-1 and spin-2 fields at the critical sets of null infinity

Mariem Magdy, Juan A. Valiente Kroon

TL;DR

The paper analyzes how massless spin-1 (Maxwell) and spin-2 fields' asymptotic charges at the critical sets where spatial infinity meets null infinity can be expressed in terms of initial data, using Friedrich's cylinder at spatial infinity and the F-gauge. By expanding initial data in spin-weighted harmonics, it derives regularity conditions under which the BMS-type charges at I± are well-defined and shows that charges at I+ and I− are determined by the same freely specifiable data for each harmonic, yielding a natural matching. The results provide explicit, data-driven expressions for the charges and establish a clear correspondence between future and past charges, laying groundwork for extending the analysis to full General Relativity in stationary spacetimes. This framework highlights the role of boosted initial data and regularity in defining conserved asymptotic charges for massless fields at null infinity.

Abstract

The asymptotic charges of spin-1 and spin-2 fields are studied near spatial infinity. We evaluate the charges at the critical sets where spatial infinity meets null infinity with the aim of finding the relation between the charges at future and past null infinity. To this end, we make use of Friedrich's framework of the cylinder at spatial infinity to obtain asymptotic expansions of the Maxwell and spin-2 fields near spatial infinity, which are fully determined in terms of initial data on a Cauchy hypersurface. Expanding the initial data in terms of spin-weighted spherical harmonics, it is shown that only a subset of the initial data, that satisfies certain regularity conditions, gives rise to well-defined charges at the point where future (past) infinity meets spatial infinity. Given such initial data, the charges are shown to be fully expressed in terms of the freely specifiable part of the data. Moreover, it is shown that there exists a natural correspondence between the charges defined at future and past null infinity.

Asymptotic charges for spin-1 and spin-2 fields at the critical sets of null infinity

TL;DR

The paper analyzes how massless spin-1 (Maxwell) and spin-2 fields' asymptotic charges at the critical sets where spatial infinity meets null infinity can be expressed in terms of initial data, using Friedrich's cylinder at spatial infinity and the F-gauge. By expanding initial data in spin-weighted harmonics, it derives regularity conditions under which the BMS-type charges at I± are well-defined and shows that charges at I+ and I− are determined by the same freely specifiable data for each harmonic, yielding a natural matching. The results provide explicit, data-driven expressions for the charges and establish a clear correspondence between future and past charges, laying groundwork for extending the analysis to full General Relativity in stationary spacetimes. This framework highlights the role of boosted initial data and regularity in defining conserved asymptotic charges for massless fields at null infinity.

Abstract

The asymptotic charges of spin-1 and spin-2 fields are studied near spatial infinity. We evaluate the charges at the critical sets where spatial infinity meets null infinity with the aim of finding the relation between the charges at future and past null infinity. To this end, we make use of Friedrich's framework of the cylinder at spatial infinity to obtain asymptotic expansions of the Maxwell and spin-2 fields near spatial infinity, which are fully determined in terms of initial data on a Cauchy hypersurface. Expanding the initial data in terms of spin-weighted spherical harmonics, it is shown that only a subset of the initial data, that satisfies certain regularity conditions, gives rise to well-defined charges at the point where future (past) infinity meets spatial infinity. Given such initial data, the charges are shown to be fully expressed in terms of the freely specifiable part of the data. Moreover, it is shown that there exists a natural correspondence between the charges defined at future and past null infinity.
Paper Structure (15 sections, 10 theorems, 94 equations, 1 figure)

This paper contains 15 sections, 10 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The cylinder at spatial infinity and the F-gauge
  3. The cylinder at spatial infinity
  4. F-gauge
  5. The Newman-Penrose gauge
  6. The Maxwell field and asymptotic charges in the F-gauge
  7. The spinorial Maxwell equations
  8. Initial data for the Maxwell equations
  9. The Maxwell charges
  10. The spin-2 field and the asymptotic charges in the F-gauge
  11. The spin-2 equation
  12. Initial data for the spin-2 equations
  13. The asymptotic BMS charges for the spin-2 field
  14. Conclusions
  15. Acknowledgements

Key Result

Theorem 1

For generic boosted initial data for the Maxwell and spin-2 fields, the asymptotic charges at $\mathcal{I}^{\pm}$ are well-defined if and only if the freely specifiable data on the initial hypersurface $\mathcal{S}_\star$ satisfy certain regularity conditions ---given by Lemma Maxwell-conditions; Le

Figures (1)

  • Figure 1: Schematic representation of the sets of the cylinder at spatial infinity. Note that the two-dimensional representation on the right is not a conformal diagram.

Theorems & Definitions (23)

  • Theorem
  • Remark 1
  • Proposition 1
  • Remark 2
  • Proposition 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • Remark 5
  • Remark 6
  • ...and 13 more