Strain and Curvature Mode Decompositions

Strain, Bondi news, and the outgoing Weyl scalar describe successive retarded-time derivatives of the same asymptotic radiation field. They are not interchangeable data products: strain contains offsets and memory, news controls energy flux, and curvature emphasizes rapid variation while losing integration constants. Once the null tetrad, complex polarization, spin-weighted basis, and Fourier sign are fixed, the relation holds mode by mode. This page derives that dictionary at future null infinity; finite-radius extrapolation and detector reconstruction remain outside its scope.

Required background. Spacetime Symmetries and Killing Fields supplies the rotational mode labels and fixed angular basis.

Helpful background. Einstein Equations supplies Weyl curvature; Waves, Green Functions, and Geometric Optics supplies outgoing retarded fields and Fourier differentiation; and Gravitational Radiation and Waveforms supplies strain, news, and flux conventions.

The asymptotic derivative chain

Use retarded Bondi time $u$, an areal radius $r$, and a unit-sphere dyad with $m^A\bar m_A=1$. Define

$$H=C_{AB}\bar m^A\bar m^B =\lim_{r\to\infty}r(h_+-ih_\times), \qquad \mathcal N=\partial_uH.$$

Order the null tetrad as $(\ell,n,m,\bar m)$ with $\ell\!\cdot n=-1$ and $m\!\cdot\bar m=1$. Fix its remaining boost normalization asymptotically by $\ell=(e_t+e_r)/\sqrt2$ and $n=(e_t-e_r)/\sqrt2$, where $e_t$ is the unit time vector of the chosen Bondi frame and $e_r$ points outward. Take

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

For an outgoing wave in the curvature convention inherited by the site,

$$\Psi_4^0:=\lim_{r\to\infty}r\Psi_4 =\partial_u\mathcal N =\partial_u^2H.$$

All three complex fields have spin weight $-2$ in this tetrad and dyad convention. A source using $h_++ih_\times$, the opposite Riemann tensor, the opposite sign in the definition of $\Psi_4$, or an exchanged tetrad leg can display a minus sign or a complex conjugate instead. Even $\ell\mapsto b\ell$, $n\mapsto n/b$ preserves $\ell\cdot n=-1$ while changing $\Psi_4\mapsto b^{-2}\Psi_4$; the symmetric normalization above is therefore essential. Bishop and Rezzolla give this tetrad and strain relation in Bishop and Rezzolla 2016, pp. 22–24, eqs. (95)–(105), PDF.

Projection onto one common basis

In a time-independent Bondi angular frame, expand every field in the same unit-normalized harmonics:

$$H=\sum_{\ell,m}H_{\ell m}\,{}_{-2}Y_{\ell m}, \qquad \mathcal N=\sum_{\ell,m}N_{\ell m}\,{}_{-2}Y_{\ell m},$$ $$\Psi_4^0=\sum_{\ell,m}\psi_{4,\ell m}\,{}_{-2}Y_{\ell m}.$$

Because the basis does not depend on $u$, projection commutes with differentiation:

$$N_{\ell m}=\dot H_{\ell m}, \qquad \psi_{4,\ell m}=\dot N_{\ell m}=\ddot H_{\ell m}.$$

This simple result fails if one differentiates coefficients in a rotating basis while pretending the basis is fixed. If ${}_{-2}Y_{\ell m}$ is carried by a time-dependent frame, its derivative contributes Wigner-matrix terms; the inertial news must include them.

The mode dictionary independently reconstructs the positive outward luminosity:

$$\mathcal F_E^\infty =\frac{1}{16\pi G}\int_{S^2}\lvert\mathcal N\rvert^2d\Omega =\frac{1}{16\pi G}\sum_{\ell,m}\lvert\dot H_{\ell m}\rvert^2.$$

Agreement between the sphere integral and the mode sum checks the harmonic normalization, omitted modes, and the factor relating tensor to scalar news.

Fourier signs and zero frequency

Define the transform pair

$$\widetilde f(\omega)=\int_{-\infty}^{\infty}f(u)e^{+i\omega u}du, \qquad f(u)=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi} \widetilde f(\omega)e^{-i\omega u}.$$

Then $\partial_u\leftrightarrow-i\omega$, so

$$\widetilde N_{\ell m}=-i\omega\widetilde H_{\ell m}, \qquad \widetilde\psi_{4,\ell m}=-\omega^2\widetilde H_{\ell m}.$$

The second relation is safe to invert only away from $\omega=0$. Memory, a strain offset, and slow numerical drift all occupy the low-frequency sector, but they are physically different. Dividing by $\omega^2$ with an arbitrary cutoff can remove genuine memory together with unwanted integration drift.

What curvature integration cannot recover

Given curvature on a finite interval beginning at $u_0$, twice integrating yields

$$H_{\ell m}(u) =H_{\ell m}(u_0) +(u-u_0)N_{\ell m}(u_0) +\int_{u_0}^{u}(u-u')\psi_{4,\ell m}(u')du'.$$

The two constants are not determined by $\Psi_4$. Physical endpoint conditions, a matched strain calculation, or a justified low-frequency prescription must fix them. In an early stationary era one often imposes $N_{\ell m}=0$, but a nonzero constant $H_{\ell m}$ may remain because of the Bondi frame. If the early and late constants differ, that difference is memory and should not be forced to zero.

At finite radius, tetrad contamination, gauge terms, areal-radius errors, and subleading $1/r$ corrections can also spoil the chain. The asymptotic limit and a controlled frame must be established before a residual is called a failure of general relativity or of the waveform model.

A convention-safe comparison

When two waveform products use different radiative variables, compare them in five steps. First, place both in the same Bondi frame, retarded time, areal-radius scaling, and polarization dyad. Second, translate the Riemann and Newman–Penrose signs rather than assuming that the symbol $\Psi_4$ has one universal meaning. Third, project onto harmonics with the same Condon–Shortley and $e^{im\phi}$ phases. Fourth, differentiate the strain rather than integrate curvature whenever both are sufficiently smooth, because differentiation introduces no arbitrary constants. Finally, reconstruct the news-squared flux from each route.

The comparison should distinguish three outcomes. A constant strain difference leaves news and curvature unchanged and usually reflects a shear-frame or integration constant. A linear-in-time strain difference leaves curvature unchanged but changes news and creates an unphysical persistent flux unless supported by the boundary data. An oscillatory residual that survives two derivatives can indicate a tetrad, retarded-time, resolution, or physical modeling difference. These patterns localize the problem more effectively than a single norm of $H_1-H_2$.

A time shift deserves special care. If $H_2(u)=H_1(u-\Delta u)$, then every frequency-domain mode gains $e^{+i\omega\Delta u}$ in the transform convention above, while the derivative chain remains intact. Aligning strain and curvature with independent time shifts can therefore manufacture agreement in phase but violate their common physical retarded-time origin.

Exercise

For one normalized mode $H_{22}=Ae^{-i\omega u}$ with real $\omega>0$, compute its news, curvature, and energy flux. State what information is lost if only the curvature mode is supplied.

Solution

Direct differentiation gives

$$N_{22}=-i\omega Ae^{-i\omega u}, \qquad \psi_{4,22}=-\omega^2Ae^{-i\omega u}.$$

Orthonormality gives

$$\mathcal F_E^\infty =\frac{\omega^2\lvert A\rvert^2}{16\pi G}.$$

Curvature fixes the oscillatory strain amplitude for $\omega\ne0$ through $H_{22}=-\psi_{4,22}/\omega^2$, but it does not determine an added constant or a term linear in $u$. Those are the two integration constants of the time-domain equation; endpoint physics must fix them.

References

• Bishop, Nigel T., and Luciano Rezzolla. “Extraction of Gravitational Waves in Numerical Relativity.” Living Reviews in Relativity 19 (2016): 2. DOI. Open PDF.