Third-approximation asymptotics in asymptotically null-spherical coordinates: tetrad-gauge-invariant curvature falloff orders and the angular-momentum flux

The 1970 analysis of Persides and Chandrasekhar (Proc. R. Soc. Lond. A 320, 349) developed the vacuum Einstein equations for asymptotically flat spacetimes in coordinates $(u,r,\theta,\phi)$ where $u=\mathrm{const}$ is a null hypersurface only at infinity and $(r,\theta,\phi)$ reduce to spherical polar coordinates, using a completely arbitrary null tetrad and a metric expanded in inverse powers of $r$ through the second approximation ($r^{-2}$). It derived the peeling behaviour of the tetrad components of the Riemann tensor, a generalized news function, flux formulas for energy and linear momentum, and a criterion for non-radiating fields, and it explicitly left the third approximation undone. The surviving problem is to carry the expansion and the vacuum field equations to the third approximation (the $r^{-3}$ coefficients) in the same coordinate- and tetrad-general framework, in order to settle (a) which of the leading falloff orders of the five complex curvature scalars $R_A$ are fixed by the metric and which can be lowered by the six-parameter tetrad freedom — in particular whether the order of $R_4$ can be lowered — (b) the flux law for the angular momentum carried by the waves, and (c) whether the integer inverse-power metric ansatz remains consistent at that order or must be enlarged (for example by logarithmic terms). The underlying questions have parallel, partly answered treatments in standard gauges, but the problem in its source-faithful generality has no closure in the literature.

Background

General relativity describes an isolated gravitating body by an asymptotically flat spacetime — one whose metric approaches the Minkowski metric at large distance, so that the source's total mass, momentum and radiated energy can be read off from the asymptotic behaviour of the curvature. The classical radiation theory was built by Bondi, van der Burg and Metzner (1962) and Sachs (1962) in retarded coordinates $(u,r,\theta,\phi)$ in which every hypersurface $u=\mathrm{const}$ is exactly null (a Bondi system), $r$ parametrizing the outgoing null rays. There the asymptotic field is characterized by the news function — essentially the retarded-time derivative of the shear of the outgoing null congruence, whose non-vanishing signals outgoing radiation — and by the Bondi mass-loss formula, which equates the decrease of the source's mass to a positive quadratic functional of the news.

Newman and Penrose (1962) reformulated this analysis using a null tetrad — a basis of two real null and two complex-conjugate null vector fields — in which the Weyl curvature (the trace-free part of the Riemann tensor, equal to it in vacuum) is represented by five complex scalars $\Psi_0,\dots,\Psi_4$, and the connection is encoded in the rotation coefficients (the tetrad components of the covariant derivatives of the tetrad vectors, also called spin coefficients). Sachs' peeling theorem states that along outgoing null geodesics these scalars fall off as $\Psi_k\sim r^{k-5}$, so that asymptotically only the radiative component survives at leading order. Peeling is a central structural statement about radiating fields, but it was derived under special gauge assumptions: exact Bondi coordinates and a tetrad adapted to them. Newman and Penrose (1965) further exhibited ten absolutely conserved quantities at null infinity — the Newman–Penrose constants — whose physical meaning remained obscure.

The 1970 paper of Persides and Chandrasekhar addressed a practical gap in this theory. Metrics produced by source models (for instance post-Newtonian expansions for bounded sources) are usually given in coordinates that become spherical or Cartesian at spatial infinity, in which $u=\mathrm{const}$ is null only asymptotically, and there is no preferred null tetrad. The paper therefore re-derived the asymptotics in coordinates $(u,r,\theta,\phi)$ where $u=\mathrm{const}$ is null only at infinity and $(r,\theta,\phi)$ reduce to ordinary spherical polar coordinates, with the only restriction on the tetrad being that its vectors are null everywhere, and with the metric expanded in inverse powers of $r$,

$$g_{\mu\nu}=b_{\mu\nu}+\frac{a_{\mu\nu}}{r}+\frac{A_{\mu\nu}}{r^{2}}+O(r^{-3}),$$

where $b_{\mu\nu}$ is the flat retarded metric and the coefficients depend on $(u,\theta,\phi)$ only. Computing the rotation coefficients and the tetrad components $R_{mnpq}$ of the Riemann tensor order by order and imposing the vacuum Einstein equations, the paper obtained: the constraints on the first- and second-order coefficients $a_{\mu\nu}$, $A_{\mu\nu}$; the peeling statement in this general setting, phrased through five complex scalars

$$R_1=R_{0202},\quad R_2=-R_{0223},\quad R_3=\tfrac{1}{2}(R_{0123}-R_{0101}),\quad R_4=R_{1223},\quad R_5=R_{1212},$$

the analogues, in the paper's own tetrad labelling and conventions — which are to be followed from the source paper; no explicit identification of the $R_A$ with particular Newman–Penrose scalars $\Psi_k$ is asserted here — of the Newman–Penrose Weyl scalars, with the second-approximation orders $R_A=O(r^{-A})$ for $A\le4$ and $R_5=O(r^{-4})$ reported from the source on the branch $a_{11}=0$ (the branch capable of carrying radiation); a generalized news function $N=a_{22}+i\,a_{23}/\sin\theta$; flux formulas for total energy and linear momentum, derived via the Landau–Lifshitz energy-momentum complex (a quadratic, coordinate-dependent but conserved expression for the total energy-momentum of the gravitational field); and a simple criterion for non-radiating fields (if the first-order coefficient $a_{11}\neq0$, the field equations force $N_{,0}=0$, so no perturbation preserving the asymptotic form can radiate). A covariant formulation of the Landau–Lifshitz complex was later given by Persides and Papadopoulos (1979).

The analysis stopped at the second approximation, and the paper records that this leaves unsettled: the third approximation ($r^{-3}$), needed to decide whether the order of $R_4$ can be lowered by tetrad choice and to count how many of the falloff orders of the $R_A$ are tetrad-gauge artifacts rather than metric-determined; the flux of angular momentum, whose integrands likewise require third-order data; the justification of the assumed integer inverse-power ansatz itself; and, as the long-range objective, the integration of the asymptotic constraints toward global solutions and the relation of the Newman–Penrose constants to explicit source models.

Problem Statement

Within the framework of Persides and Chandrasekhar (1970) — vacuum, asymptotically flat spacetimes written in coordinates $(u,r,\theta,\phi)$ in which $u=\mathrm{const}$ is a null hypersurface only at infinity and $(r,\theta,\phi)$ reduce to ordinary spherical polar coordinates, with the metric expanded as $g_{\mu\nu}=b_{\mu\nu}+a_{\mu\nu}/r+A_{\mu\nu}/r^{2}+O(r^{-3})$ (coefficients depending on $(u,\theta,\phi)$ only) and with the only restriction on the null tetrad being that its vectors are null everywhere, the tetrad index labelling, normalizations and complex-conjugation conventions being those fixed by the source paper — carry the analysis to the third approximation: compute the $O(r^{-3})$ contributions to the rotation coefficients and to the tetrad components $R_{mnpq}$ of the Riemann tensor, and solve the vacuum Einstein equations at that order. Use the result to settle, in this coordinate- and tetrad-general setting:

1. (Gauge content of the falloff orders.) For each of the five complex curvature scalars $R_1=R_{0202}$, $R_2=-R_{0223}$, $R_3=\tfrac{1}{2}(R_{0123}-R_{0101})$, $R_4=R_{1223}$, $R_5=R_{1212}$, determine whether its leading large-$r$ falloff order is fixed by the metric coefficients or can be lowered by the six-parameter null-tetrad freedom, and by how much. An order is lowered if an admissible null-tetrad gauge transformation weakens the leading falloff — for example, turning an established $O(r^{-4})$ into $O(r^{-3})$ — and "by how much" asks for the worst admissible leading order; the baseline orders are those reported above from the source's second-approximation results, in the tetrad labelling and conventions of Persides and Chandrasekhar (1970), which a solver may rederive within the framework. In particular, determine whether the order $R_4=O(r^{-4})$ established at the second approximation can be lowered, and how many of the $R_A$ are lowerable in total. The classification must account for the global existence and regularity of the tetrad gauge functions on the sphere, including at the coordinate poles.

2. (Angular-momentum flux.) Determine the balance law for the angular momentum carried by the radiation in this framework, as an explicit surface integral over the spheres of constant retarded time, expressed in terms of the asymptotic metric coefficients, and state which integrands are metric-determined and which are tetrad-gauge dependent.

3. (Consistency of the ansatz.) Establish whether the integer inverse-power expansion posited in the source remains consistent under the third-approximation field equations, or whether solutions of physical interest require enlarging the expansion class (for example, to polyhomogeneous expansions containing powers of $\log r$), thereby justifying or relaxing the assumed metric form.

These three deliverables share one objective: to determine which parts of the asymptotic vacuum field at and beyond the second approximation are genuine physical data of an isolated source and which are artifacts of coordinate or tetrad choice, and to complete the radiation balance picture by including angular momentum. The paper's stated long-range objective — integrating the hierarchy of asymptotic constraints toward global solutions and relating the asymptotic data, in particular the Newman-Penrose constants, to explicit source models — is the program context this computation feeds, not a completion requirement of this problem.

References

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