Newman–Penrose Radiation Scalars

The Newman–Penrose formalism compresses Weyl curvature into five complex tetrad components. For an outgoing wave at future null infinity, the leading component is $\Psi_4$: it is curvature, falls as $1/r$, and equals two retarded-time derivatives of the rescaled complex strain once the null tetrad and Weyl sign are fixed. None of those qualifiers is optional. A different Riemann sign, a different definition of $\Psi_4$, or a spin-boosted tetrad changes the displayed relation. This page derives the convention used throughout the volume and states the conditional peeling behavior; rigorous peeling proofs and detector-specific strain reconstruction remain outside its scope.

Helpful background. Einstein Equations supplies vacuum Weyl curvature; Waves, Green Functions, and Geometric Optics supplies outgoing retarded waves; Causality and Global Structure supplies future null infinity; Gravitational Radiation and Waveforms supplies strain and polarization; and Formulations of General Relativity supplies tetrad and characteristic gauge choices.

The outgoing Newman–Penrose tetrad

Start with a future-directed orthonormal frame $(e_0,e_r,e_\theta,e_\phi)$ near infinity, with $e_r$ pointing outward. Define

$$\ell^a=\frac{e_0^a+e_r^a}{\sqrt2}, \qquad n^a=\frac{e_0^a-e_r^a}{\sqrt2}, \qquad m^a=\frac{e_\theta^a+i e_\phi^a}{\sqrt2}.$$

For the inherited $(-,+,+,+)$ signature,

$$\ell\!\cdot n=-1, \qquad m\!\cdot\bar m=1,$$

and all other contractions vanish. The vector $\ell^a$ is tangent to outgoing radial rays in flat space; the opposite null leg $n^a$ appears twice in the curvature scalar that detects an outgoing wave.

With the site’s Riemann tensor, define

$$\Psi_4:=-C_{abcd}n^a\bar m^b n^c\bar m^d.$$

The minus sign is part of the definition. Bishop and Rezzolla use exactly this tetrad normalization and Weyl-scalar sign in their extraction review Bishop and Rezzolla 2016, §3.3, eqs. (95)–(103), pp. 22–23, PDF.

The label “scalar” means invariant under coordinate changes after the tetrad vectors are transformed geometrically; it does not mean independent of tetrad choice. The extreme component $\Psi_0$, obtained with two outgoing legs $\ell$ and two $m$ legs, describes the opposite radiative orientation in this asymptotic tetrad. The middle component $\Psi_2$ contains leading Coulombic curvature such as the mass monopole. Their distinct radial orders prevent a stationary $1/r^3$ Coulomb field from being mistaken for the $1/r$ outgoing signal.

Plane-wave sign and normalization

Consider a TT wave moving in the $+z$ direction, with $u=t-z$ and

$$h^{\rm TT}_{ij} =h_+\left(e_{x i}e_{x j}-e_{y i}e_{y j}\right) +h_\times\left(e_{x i}e_{y j}+e_{y i}e_{x j}\right).$$

The linearized curvature convention gives

$$R_{0i0j}=-\frac12\partial_u^2h^{\rm TT}_{ij}.$$

Because the field depends only on $u$, the mixed time–$z$ components combine with $n=(e_0-e_z)/\sqrt2$ to give

$$C_{n i n j}=-\partial_u^2h^{\rm TT}_{ij}.$$

Contracting twice with $\bar m=(e_x-i e_y)/\sqrt2$ yields

$$C_{n\bar m n\bar m} =-\partial_u^2(h_+-ih_\times).$$

The defining minus sign therefore produces

$$\Psi_4=\partial_u^2(h_+-ih_\times)$$

locally in the wave zone. This calculation checks the sign rather than importing it from a source with unknown conventions.

The same calculation checks the complex polarization convention. Contracting the plus tensor with $\bar m^i\bar m^j$ gives $+1$, while contracting the cross tensor gives $-i$. Therefore the result is $h_+-ih_\times$, not its complex conjugate. Rotating the screen dyad multiplies both the strain and $\Psi_4$ by the same spin-weight $-2$ phase, so the derivative relation is covariant under a time-independent polarization rotation.

The finite curvature field at null infinity

Under the regular asymptotics used in this chapter, the Weyl scalars satisfy the classical hierarchy

$$\Psi_0=O(r^{-5}),\quad \Psi_1=O(r^{-4}),\quad \Psi_2=O(r^{-3}),\quad \Psi_3=O(r^{-2}),\quad \Psi_4=O(r^{-1}).$$

This peeling sequence is conditional on sufficient conformal regularity. Polyhomogeneous expansions can weaken it, so the finite leading field below is not advertised as a proof of the full hierarchy.

Define

$$\Psi_4^0(u,x^A):=\lim_{r\to\infty}r\Psi_4.$$

For $H=\lim r(h_+-ih_\times)=C_{AB}\bar m^A\bar m^B$ and $\mathcal N=\partial_uH$,

$$\Psi_4^0=\partial_u^2H=\partial_u\mathcal N.$$

Mädler and Winicour’s scalar news is half the retarded-time derivative in the tensor normalization used here, so their notation gives $\Psi_4^0=2\partial_u\overline{N}_{\rm MW}$ after translating their opposite complex-strain choice. Bishop and Rezzolla derive the corresponding strain–news–curvature chain in Bondi gauge Bishop and Rezzolla 2016, §6.2, eqs. (271)–(276), p. 55, PDF.

This limit is taken at fixed $u$ and angle in an asymptotic tetrad. Evaluating $r\Psi_4$ on one finite coordinate sphere leaves corrections from radius choice, gauge, tetrad, near-zone curvature, and backscatter. Those corrections may be estimated by extrapolation or characteristic extraction, but neither procedure is part of the definition above. Agreement of two extraction methods is stronger evidence than apparent $1/r$ scaling alone.

Tetrad freedom and Fourier sign

A residual spin-boost

$$\ell\mapsto A\ell, \qquad n\mapsto A^{-1}n, \qquad m\mapsto e^{i\chi}m$$

sends

$$\Psi_4\mapsto A^{-2}e^{-2i\chi}\Psi_4.$$

Thus a component called $\Psi_4$ is not tetrad invariant. The Bondi normalization fixes the asymptotic boost, while the dyad phase records the same spin weight $-2$ as $H$. At finite radius, null rotations can also mix Weyl components; a coordinate sphere and an arbitrary tetrad do not automatically supply Bondi curvature.

With time dependence $e^{-i\omega u}$,

$$\widetilde\Psi_4^0(\omega) =-\omega^2\widetilde H(\omega).$$

For $\omega\ne0$, $\widetilde H=-\widetilde\Psi_4^0/\omega^2$. The zero-frequency sector contains integration constants and memory, so naive division by $\omega^2$ is ill-conditioned and cannot determine the strain offset. This is a mathematical limitation, not evidence for removing low-frequency physics.

In the time domain, double integration yields $H(u)$ only up to $A+Bu$. The constant $A$ fixes the strain offset in the chosen shear frame, while $B$ fixes an undetermined constant news. A constant news over the entire real time axis would radiate infinite total energy, so stationary or finite-energy endpoint conditions exclude it. Over a finite interval it cannot be discarded solely from knowledge of $\Psi_4^0$. Numerical integration constants can also create a linear drift; a constant curvature offset instead produces a quadratic drift. Physical boundary information must distinguish memory from these integration artifacts.

Exercise

For $H(u)=H_0e^{-i\omega u}$ with real nonzero $\omega$, compute $\Psi_4^0$ and verify the phase relation between curvature and strain.

Solution

Two retarded-time derivatives give $\partial_u^2H=(-i\omega)^2H=-\omega^2H$. Hence $\Psi_4^0=-\omega^2H$. Curvature differs from strain by a real negative factor, corresponding to a phase shift of $\pi$ for this monochromatic complex representation. The result agrees with the Fourier formula and fixes both the inherited $e^{-i\omega u}$ sign and the absence of an extra complex conjugation.

References

• Nigel T. Bishop and Luciano Rezzolla, “Extraction of Gravitational Waves in Numerical Relativity,” Living Reviews in Relativity 19 (2016) 2, arXiv:1606.02532v2, DOI, Open PDF.
• Thomas Mädler and Jeffrey Winicour, “Bondi–Sachs Formalism,” Scholarpedia 11 (2016) 33528, updated arXiv version (2018), arXiv:1609.01731v3, Open PDF.