Physical interpretation and fundamental significance of the absolutely conserved Newman-Penrose constants for zero rest-mass fields in asymptotically flat space-time

The Newman-Penrose (NP) constants - ten real quantities for the gravitational field (five complex $G_m$), with analogues $F_m$ for the Maxwell field and $N_m$ for the neutrino field - are defined by conformally invariant Kirchhoff-d'Adhemar-type light-cone integrals on cuts of future null infinity $\mathscr{I}^+$ of an asymptotically flat spacetime and are 'absolutely conserved': they take the same value on every cut even while gravitational radiation (Bondi news) flows through null infinity, and they are invariant under Bondi-Metzner-Sachs (BMS) supertranslations. In flat spacetime an infinite hierarchy of such constants exists and simply encodes the incoming radiation; in the full non-linear Einstein theory only the first tier survives, and its physical meaning has remained unclear since Newman and Penrose posed the question in 1968. The source-faithful open problem is the fundamental interpretation of the constants in generic non-stationary radiative spacetimes - are they bona fide charges of the sources, of incoming radiation, or of initial data at spatial infinity? - and the intervening literature additionally traces closely related questions: whether the constants can be derived from a Noether/variational principle, whether further absolutely conserved quantities beyond the first tier can be constructed in the non-linear theory, and what dynamical constraints absolute conservation imposes on gravitational collapse. The stationary case is settled (the constants are translation-invariant quadratic combinations of the Geroch-Hansen multipole moments, the invariant multipoles of a stationary spacetime defined at spatial infinity), the infinite linear tower is identified with subleading (dual) BMS charges, a partial Hamiltonian derivation of the ten non-linear charges exists, the back-scattered tail decay question has been largely resolved by Price-law-type results, and the constants have recently been computed for the first time in a non-stationary non-linear setting. The surviving open core is the complete physical interpretation of the absolutely conserved non-linear constants in generic radiative spacetimes; the problem remains open.

Background

An isolated gravitating system in general relativity is modeled by an asymptotically flat spacetime: in Bondi-Sachs coordinates $(u,r,\theta,\phi)$, where $u$ is retarded time labeling outgoing null hypersurfaces and $r$ is a radial parameter, the metric approaches the Minkowski form as $r$ grows without bound. Future null infinity, $\mathscr{I}^+$, is the idealized boundary through which outgoing radiation escapes; it is constructed rigorously by Penrose's conformal compactification, rescaling the metric by a conformal factor that vanishes at the boundary, which zero-rest-mass fields tolerate because their field equations are conformally invariant. Bondi, van der Burg and Metzner, and Sachs, showed that the mass aspect on $\mathscr{I}^+$ decreases at a rate equal to the squared norm of the news function (the outgoing radiation flux), and that the asymptotic symmetry group is not the Poincaré group but the Bondi-Metzner-Sachs (BMS) group: Lorentz transformations together with supertranslations, an infinite-dimensional family of angle-dependent shifts of the cuts of $\mathscr{I}^+$. In the Newman-Penrose (NP) spinor formalism the gravitational field is represented by five complex Weyl-spinor components $\Psi_0,\ldots,\Psi_4$ relative to a null tetrad (a basis of four lightlike vector fields); for a peeling field (Penrose's asymptotic property whereby successive components fall off with successive powers of $1/r$), the leading component satisfies $\Psi_0=O(r^{-5})$.

Newman and Penrose (1968) applied the generalized Kirchhoff-d'Adhemar integral - the conformally invariant formula expressing a zero-rest-mass field at a point as an integral over a cut of that point's light cone - formally at timelike infinity, obtaining quantities defined on cuts of $\mathscr{I}^+$: for a spin-$s$ zero-rest-mass field, $2s+1$ complex constants ($4s+2$ real), denoted $N_m$ for the neutrino field ($s=1/2$), $F_m$ for the Maxwell field ($s=1$) and $G_m$ for gravitation ($s=2$; five complex constants with $m=-2,-1,0,1,2$), transforming in a $D(s,0)$ irreducible representation of the BMS group (the spin-$s$ representation of its Lorentz subgroup). In flat spacetime an infinite hierarchy of such constants exists for each spin and is essentially trivial: it encodes the coefficients of a power series expansion of the incoming field. In an asymptotically flat spacetime only the first tier survives, and it is absolutely conserved: the same value is obtained on every cut of $\mathscr{I}^+$ even when the news is non-zero, and the values are invariant under supertranslations - a far stronger conservation property than the Bondi mass enjoys. For stationary fields the 1968 analysis already showed that the constants are not pure incoming-radiation data: they receive contributions from the source's multipole structure; in modern rigorous form (Bäckdahl 2009), the $G_m$ of a stationary spacetime are translation-invariant quadratic combinations of the Geroch-Hansen multipole moments - the invariant multipole moments of a stationary spacetime defined at spatial infinity - specifically combinations of monopole-times-quadrupole and dipole-squared terms.

This raised the interpretive question that has stood since: the constants are conserved even during fully radiative epochs, yet they are not obviously charges of anything, and no symmetry principle generating them was known. In the subsequent literature, whether they admit a Noether or variational derivation has remained open, and whether the hierarchy can be extended in the full non-linear theory - for instance by formal inverses of the spin-weight-raising angular operator (eth) of the Geroch-Held-Penrose formalism (a covariant reformulation of the Newman-Penrose spinor calculus), which generate the higher tiers in flat spacetime - has likewise remained open, with no further non-linearly conserved quantities constructed to date. Finally, the conservation argument implied that a stationary asymptotically flat field cannot radiate and become exactly stationary again after a finite time, except possibly if the conserved quadratic combination of multipole moments returns to its original value, which pointed to the existence of back-scattered 'tails' whose long-time behavior was not settled.

Problem Statement

Consider the full non-linear asymptotically flat Einstein (or Einstein-Maxwell) theory, with zero-rest-mass fields satisfying the standard peeling-type fall-off at future null infinity $\mathscr{I}^+$. The first-tier Newman-Penrose quantities - the five complex gravitational constants $G_m$ with $m=-2,-1,0,1,2$ (ten real; analogously $F_m$ for spin 1 and $N_m$ for spin 1/2) - are absolutely conserved: they take the same value on every cut of $\mathscr{I}^+$ even when the news (radiation flux) is non-zero, and they are invariant under BMS supertranslations. The problem is to establish their fundamental physical significance:

Give a complete physical interpretation of these constants in generic, non-stationary, radiative asymptotically flat spacetimes - what property of the gravitational field, of its sources, or of its initial data near spatial infinity they measure.

Closely related questions documented in the literature - whether the constants admit a Noether or variational derivation from a symmetry principle of the full non-linear theory, whether any further absolutely conserved quantities beyond the first tier of $4s+2$ real quantities per spin can be constructed in the full non-linear theory (Macaulay 2022 records the expectation, without proof, that none exist without restricting the metric), and what dynamical constraints absolute conservation imposes on radiative processes such as the late stages of gravitational collapse - are traced under Background. The 1968 tail-decay question (whether the back-scattered tails predicted by the conservation argument decay so that the field becomes asymptotically stationary) is largely settled by Price-law-type decay results and their modern refinements.

The fundamental significance of the constants is realized in the generic non-linear theory: flat spacetime, stationary fields, the linearized theory, and special solution classes capture only part of the phenomenon.

References

1. E. T. Newman and Roger Penrose, 'New conservation laws for zero rest-mass fields in asymptotically flat space-time', Proceedings of the Royal Society of London A 305, 175-204 (1968). DOI: 10.1098/rspa.1968.0112, https://doi.org/10.1098/rspa.1968.0112.
2. Hermann Bondi, M. G. J. van der Burg and A. W. K. Metzner, 'Gravitational waves in general relativity. VII. Waves from axi-symmetric isolated systems', Proceedings of the Royal Society of London A 269, 21-52 (1962). DOI: 10.1098/rspa.1962.0161, https://doi.org/10.1098/rspa.1962.0161.
3. R. K. Sachs, 'Gravitational waves in general relativity. VIII. Waves in asymptotically flat space-time', Proceedings of the Royal Society of London A 270, 103-126 (1962). DOI: 10.1098/rspa.1962.0206, https://doi.org/10.1098/rspa.1962.0206.
4. Roger Penrose, 'Zero rest-mass fields including gravitation: asymptotic behaviour', Proceedings of the Royal Society of London A 284, 159-203 (1965). DOI: 10.1098/rspa.1965.0058, https://doi.org/10.1098/rspa.1965.0058.
5. R. Geroch, A. Held and R. Penrose, 'A space-time calculus based on pairs of null directions', Journal of Mathematical Physics 14, 874-881 (1973). DOI: 10.1063/1.1666410, https://doi.org/10.1063/1.1666410.
6. Albert R. Exton, Ezra T. Newman and Roger Penrose, 'Conserved Quantities in the Einstein-Maxwell Theory', Journal of Mathematical Physics 10, 1566-1570 (1969). DOI: 10.1063/1.1665006, https://doi.org/10.1063/1.1665006.
7. William H. Hallidy and Allen I. Janis, 'Gravitational Radiation: Cutting the Tail', Journal of Mathematical Physics 11, 578-589 (1970). DOI: 10.1063/1.1665172, https://doi.org/10.1063/1.1665172.
8. Richard H. Price, 'Nonspherical Perturbations of Relativistic Gravitational Collapse. I. Scalar and Gravitational Perturbations', Physical Review D 5, 2419-2438 (1972). DOI: 10.1103/PhysRevD.5.2419, https://doi.org/10.1103/PhysRevD.5.2419.
9. R. Gómez, J. Winicour and Bernd G. Schmidt, 'Newman-Penrose constants and the tails of self-gravitating waves', Physical Review D 49, 2828-2836 (1994). DOI: 10.1103/PhysRevD.49.2828, https://doi.org/10.1103/PhysRevD.49.2828.
10. Juan Antonio Valiente-Kroon, 'Logarithmic Newman-Penrose constants for arbitrary polyhomogeneous spacetimes', Classical and Quantum Gravity 16, 1653-1665 (1999). DOI: 10.1088/0264-9381-16-5-314, https://arxiv.org/abs/gr-qc/9812004.
11. Helmut Friedrich and János Kánnár, 'Bondi-type systems near spacelike infinity and the calculation of the Newman-Penrose constants', Journal of Mathematical Physics 41, 2195-2232 (2000). DOI: 10.1063/1.533235, https://doi.org/10.1063/1.533235.
12. Sergio Dain and J. A. Valiente-Kroon, 'Conserved quantities in a black hole collision', Classical and Quantum Gravity 19, 811-815 (2002). DOI: 10.1088/0264-9381-19-4-312, https://arxiv.org/abs/gr-qc/0105109.
13. Xiaoning Wu and Yu Shang, 'On Newman-Penrose constants of stationary spacetimes', Classical and Quantum Gravity 24, 679-690 (2007). DOI: 10.1088/0264-9381-24-3-011, https://arxiv.org/abs/gr-qc/0612109.
14. Thomas Bäckdahl, 'Relating the Newman-Penrose constants to the Geroch-Hansen multipole moments', Classical and Quantum Gravity 26, 175021 (2009). DOI: 10.1088/0264-9381-26-17-175021, https://arxiv.org/abs/0905.3292.
15. Xiangdong Zhang, Xiaoning Wu and Sijie Gao, 'Newman-Penrose constants of stationary electrovacuum space-times', Physical Review D 79, 104001 (2009). DOI: 10.1103/PhysRevD.79.104001, https://arxiv.org/abs/0904.2240.
16. Réka Farkas and László B. Szabados, 'On quasi-local charges and Newman-Penrose type quantities in Yang-Mills theories', Classical and Quantum Gravity 28, 145013 (2011). DOI: 10.1088/0264-9381-28-14-145013, https://arxiv.org/abs/1012.4662.
17. Edgar Gasperín and Juan Antonio Valiente Kroon, 'Zero rest-mass fields and the Newman-Penrose constants on flat space', Journal of Mathematical Physics 61, 122503 (2020). DOI: 10.1063/5.0034784, https://arxiv.org/abs/1608.05716.
18. Mirjam Cvetič and Alejandro Satz, 'General relation between Aretakis charge and Newman-Penrose charge', Physical Review D 98, 124035 (2018). DOI: 10.1103/PhysRevD.98.124035, https://arxiv.org/abs/1811.05627.
19. Hadi Godazgar, Mahdi Godazgar and C. N. Pope, 'Subleading BMS charges and fake news near null infinity', Journal of High Energy Physics 2019, 143 (2019). DOI: 10.1007/JHEP01(2019)143, https://arxiv.org/abs/1809.09076.
20. Hadi Godazgar, Mahdi Godazgar and Malcolm J. Perry, 'Asymptotic gravitational charges', Physical Review Letters 125, 101301 (2020). DOI: 10.1103/PhysRevLett.125.101301, https://arxiv.org/abs/2007.01257.
21. George Macaulay (journal version published under the name George Long), 'Linear Newman-Penrose charges as subleading BMS and dual BMS charges', Physica Scripta 97, 125209 (2022). DOI: 10.1088/1402-4896/aca0cf, https://arxiv.org/abs/2206.11097.
22. Dejan Gajic and Leonhard M. A. Kehrberger, 'On the relation between asymptotic charges, the failure of peeling and late-time tails', Classical and Quantum Gravity 39, 195006 (2022). DOI: 10.1088/1361-6382/ac8863, https://arxiv.org/abs/2202.04093.
23. Jörg Frauendiener, Alex Goodenbour and Chris Stevens, 'The non-linear perturbation of a black hole by gravitational waves. III. Newman-Penrose constants', Classical and Quantum Gravity 41, 065005 (2024). DOI: 10.1088/1361-6382/ad2288, https://arxiv.org/abs/2301.05268.
24. Marc Geiller, 'Celestial $w_{1+\infty}$ charges and the subleading structure of asymptotically-flat spacetimes', SciPost Physics 18, 023 (2025). DOI: 10.21468/SciPostPhys.18.1.023, https://arxiv.org/abs/2403.05195.
25. Nicolas Cresto and Laurent Freidel, 'Asymptotic higher spin symmetries II: Noether realization in gravity', Journal of High Energy Physics 2026, 147 (2026). DOI: 10.1007/JHEP03(2026)147, https://arxiv.org/abs/2410.15219.
26. Shreyansh Agrawal, Panagiotis Charalambous and Laura Donnay, 'Null infinity as an inverted extremal horizon: Matching an infinite set of conserved quantities for gravitational perturbations', SciPost Physics 20, 054 (2026). DOI: 10.21468/SciPostPhys.20.2.054, https://arxiv.org/abs/2506.15526.
27. Berend Schneider, 'Newman-Penrose-like exact and approximate conservation laws: a covariant and conformal formulation', Classical and Quantum Gravity 43, 065012 (2026). DOI: 10.1088/1361-6382/ae4e10, https://arxiv.org/abs/2507.02237.
28. R. Penrose and M. A. H. MacCallum, 'Twistor theory: An approach to the quantisation of fields and space-time', Physics Reports 6, 241-315 (1973). DOI: 10.1016/0370-1573(73)90008-2, https://doi.org/10.1016/0370-1573(73)90008-2.