Exton-Newman-Penrose selection rules in the coupled Einstein-Maxwell theory: supertranslation invariance and well-definedness of the modified conserved quantities, and attainability of stationary-to-stationary electrovac transitions

In 1965 Newman and Penrose discovered ten absolutely conserved gravitational quantities at future null infinity of asymptotically flat vacuum spacetimes, whose conservation implies selection rules on transitions between stationary states. In 1969 Exton, Newman and Penrose extended the construction to the coupled Einstein-Maxwell theory: ten electromagnetic-modified gravitational constants remain conserved, as do six Maxwell-type constants (the real and imaginary parts of three complex quantities), so that sixteen absolutely conserved quantities exist in all, with the total charge conserved in addition; the implied selection rules for transitions between stationary Einstein-Maxwell states are, per the paper's abstract, 'the same as those in the pure gravitational case'. For the pure vacuum case the modern literature has resolved the central interpretive questions: the ten constants were proven invariant under BMS supertranslations (the asymptotic-symmetry ambiguity that threatens their physical meaning) and identified as subleading BMS charges; their stationary-state values were expressed through multipole moments; and the selection-rule paradox was defused heuristically by backscattered radiation tails. No analogous results exist for the coupled theory: no proof or refutation of the supertranslation invariance of the electromagnetic-modified constants is known, the inverse-edth construction on which they are built has not been rigorously treated, and no theorem constructs or excludes radiative transitions between distinct exactly stationary electrovac states. This problem asks that these three linked questions be settled for the full nonlinear Einstein-Maxwell theory, thereby determining whether the Exton-Newman-Penrose selection rules are invariant statements of physics and whether they genuinely constrain coupled gravitational-electromagnetic radiation.

Background

Setting. The problem lies in the asymptotic structure of radiating spacetimes, at the interface of general relativity, classical electromagnetism, and partial differential equations. The theory is the Einstein-Maxwell (or electrovac) theory, whose unknowns are a Lorentzian metric $g$ and a source-free Maxwell field $F$ jointly satisfying the Einstein equations with electromagnetic stress-energy. An asymptotically flat spacetime is one whose fields fall off at large distances in a controlled way; outgoing radiation escapes to future null infinity $\mathscr{I}^+$, the ideal boundary traced by outgoing light rays, whose points are labelled by a retarded time $u$, a radius $r$, and angles $(\theta,\phi)$ on the sphere.

The NP formalism and asymptotic expansions. In the Newman-Penrose (NP) formalism one introduces a spinor dyad aligned with outgoing and ingoing null directions and decomposes the gravitational field (the Weyl spinor) into five complex scalars $\psi_0,\ldots,\psi_4$ and the Maxwell field into three complex scalars $\phi_0,\phi_1,\phi_2$, ordered by how many outgoing-derivative spinor indices each carries. The peeling behavior of radiative fields makes each scalar fall off in powers of $1/r$: $\psi_n = \psi_n^0/r^{5-n} + \psi_n^1/r^{6-n} + \cdots$ and $\phi_n = \phi_n^0/r^{3-n} + \phi_n^1/r^{4-n} + \cdots$. The coefficient $\sigma^0(u,\theta,\phi)$ of the $1/r$ part of a spin coefficient is the asymptotic shear; its $u$-derivative $\dot\sigma^0$ is the news function, the standard measure of outgoing gravitational radiation, whose squared integral over the sphere gives the Bondi mass-loss rate (the rate of decrease of the total mass of the isolated system as measured at null infinity).

The BMS group and supertranslations. The asymptotic symmetry group of $\mathscr{I}^+$ is the Bondi-Metzner-Sachs (BMS) group. Besides Lorentz transformations it contains supertranslations, shifts of retarded time $u \mapsto u + \alpha(\theta,\phi)$ by an arbitrary angle-dependent function, which move one family of cuts $u=\mathrm{const}$ of $\mathscr{I}^+$ to another. A quantity constructed on a particular choice of cuts has unambiguous physical meaning only if its value is unchanged under supertranslations; otherwise a 'conserved' quantity is an artifact of an arbitrary cut choice. Establishing this invariance is the crux of much of the modern literature on asymptotic charges.

Spin-weighted functions and the eth operator. Functions on the sphere that pick up a phase $e^{is\chi}$ under a rotation of the dyad carry spin weight $s$; they expand in spin-weighted spherical harmonics ${}_sY_{\ell,m}$ with $\ell \ge |s|$. The eth operator $\eth$ is a first-order angular derivative that raises spin weight by one. Its formal inverse $\bar\eth^{-1}$, which enters the Exton-Newman-Penrose construction, is defined by inverting mode-by-mode; this requires the low modes $\ell \le s$ to vanish and raises questions of existence, uniqueness, and convergence of the resulting harmonic series - one of the open issues about the ENP construction.

The NP constants and the selection rules. Newman and Penrose discovered that in vacuum, ten quantities (five complex) assembled from the coefficients $\psi_n^0$ and $\psi_n^1$ are absolutely conserved along $\mathscr{I}^+$: their values are identical on every cut, in any asymptotically flat vacuum spacetime, no matter how violent the radiation. For stationary vacuum spacetimes the constants reduce to fixed combinations of mass, dipole (angular momentum) and quadrupole moments, so conservation appears to forbid a radiating body from settling to a stationary state whose constants differ from the initial ones: a selection rule. The apparent paradox - a radiating quadrupole that cannot reach a final stationary state with a changed quadrupole - is the historical heart of the subject.

The 1969 Einstein-Maxwell extension. Exton, Newman and Penrose showed that in the coupled theory the ten gravitational constants remain conserved when corrected by terms built from the Maxwell coefficients $\phi_n^0$ and $\phi_n^1$ (the asymptotic Bianchi identities acquire Maxwell source terms in electrovac), and that six further constants from the Maxwell field - the real and imaginary parts of three complex quantities $F_m$, $m=-1,0,1$ - are conserved as well, so that sixteen absolutely conserved quantities exist in all (ten gravitational plus six Maxwell), with the total charge conserved in addition. In the original paper the corrected gravitational constants $G_m$ are defined through the inverse-edth construction described above; an explicit equivalent definition, following the Penrose-Rindler conventions and used in the modern electrovac literature (Zhang, Wu and Gao 2009), is $G_m=\int_{S^2}{}_2Y_{2,m}\,\Psi_0^1\,dS$ over a cut of $\mathscr{I}^+$, where $\Psi_0^1$ is the subleading coefficient in the $1/r$-expansion of the outgoing Weyl scalar $\Psi_0$ of the Einstein-Maxwell peeling expansion, and exact conservation holds only together with the Maxwell-modified asymptotic Bianchi identities. The paper's official abstract states the conservation of the ten suitably modified gravitational constants and that the implied selection rules for transitions between stationary Einstein-Maxwell states are the same as those in the pure gravitational case; the retention of the six Maxwell constants alongside them is documented by Penrose (2024), citing the 1969 paper. Three issues about this construction remain open: whether the selection rules ever actually bind (can a radiating electrovac spacetime pass from one exactly stationary state to a distinct one), whether the constructed constants are supertranslation-invariant (Glass and Goldberg discussed invariance in 1970, but only in a linearized shock-wave setting), and whether the inversion defining them is well-defined.

Tools from the modern literature. (i) Stationary multipole moments: the Geroch-Hansen moments, a sequence of mass and current multipoles $(P_n, J_n)$, characterize any stationary asymptotically flat spacetime; Bäckdahl (2009) showed the vacuum NP constants are translation-invariant combinations of these moments up to quadrupole order. (ii) Asymptotic charges: each BMS symmetry carries a conserved charge, of which the Bondi mass is the leading one; Godazgar-Godazgar-Pope (2019) showed the vacuum NP constants appear among the subleading BMS charges, at order $1/r^3$ in the standard charge expansion. (iii) Characteristic gluing: the problem of matching asymptotic data across null hypersurfaces, whose obstructions are precisely conserved quantities (Aretakis 2013 for the scalar wave equation; Aretakis-Czimek-Rodnianski for vacuum general relativity).

Problem Statement

Give the Exton-Newman-Penrose (ENP) selection rules for the coupled Einstein-Maxwell theory invariant physical content, and determine whether they actually constrain the dynamics. The ENP construction yields sixteen absolutely conserved quantities at future null infinity of an asymptotically flat electrovac spacetime - ten electromagnetic-modified gravitational constants $G_m$ (five complex), given in the modern electrovac literature (Zhang, Wu and Gao 2009, following the Penrose-Rindler conventions) by the explicit cut integral $G_m=\int_{S^2}{}_2Y_{2,m}\,\Psi_0^1\,dS$, where $\Psi_0^1$ is the subleading coefficient in the $1/r$-expansion of the outgoing Weyl scalar and exact conservation holds only with the Maxwell-modified asymptotic Bianchi identities, this being equivalent to the inverse-edth construction of the original 1969 paper (contemporaneous treatments of the coupled constants are Robinson 1969 and Glass and Goldberg 1970) - and six Maxwell-type constants $F_m$, the real and imaginary parts of the three complex integrals $F_m=\int_{S^2}{}_1\bar{Y}_{1,m}\,\phi_0^1\,dS$ for $m=-1,0,1$, where $\phi_0^1$ is the subleading coefficient in the $1/r$-expansion of the outgoing Maxwell scalar - with the total charge $e=\int_{S^2}{}_0Y_{0,0}\,\phi_1^0\,dS$ (electric plus $i$ times magnetic) conserved in addition; conservation of these constants implies selection rules on transitions between stationary Einstein-Maxwell states. Three linked questions must be settled, in the full nonlinear Einstein-Maxwell theory, within the class of asymptotically flat (peeling) electrovac spacetimes where the ENP construction is defined, and without symmetry restrictions such as axisymmetry, time-symmetry, algebraic speciality (a degeneracy of the Weyl spinor, such as the Kerr-Newman family possesses), or restriction to the vacuum or flat-space sub-cases:

(1) Invariance. Prove that the ten modified gravitational constants $G_m$, and the six Maxwell constants $F_m$, are invariant under BMS supertranslations of the cuts of future null infinity - or refute this by exhibiting an electrovac solution and a finite supertranslation under which one of the constants changes.

(2) Well-definedness. Establish that the inverse-edth operations ($\bar\eth^{-1}$) used to define the modified gravitational constants are well-defined on the asymptotic data of a general asymptotically flat electrovac spacetime - existence, uniqueness up to the explicitly projected-out low spherical-harmonic modes, convergence of the harmonic series, and coordinate-independence - or else replace the inversion by an equivalent well-defined construction and prove that the replacement computes the same conserved quantities.

(3) Effectiveness. Determine whether an asymptotically flat electrovac spacetime can evolve, through a period of gravitational and electromagnetic radiation, from one exactly stationary state to a distinct exactly stationary state (an exactly stationary state means here an interval of retarded time on which the news $\dot\sigma^0$ and the asymptotic shear $\sigma^0$ vanish and the asymptotic data are time-independent, so that the region is a stationary electrovac solution characterizable, for instance, by its Geroch-Hansen multipole moments; distinct means differing in that data): either construct such a transition, or prove a no-go theorem (for example, that backscattered radiation - outgoing radiation scattered off the spacetime curvature into late-time tails - always prevents the return to exact stationarity).

The three questions are jointly posed and together determine whether the ENP selection rules are invariant statements of physics and whether they genuinely constrain coupled gravitational-electromagnetic radiation.

References

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