Nonlinear Christodoulou Memory

Gravitational radiation carries energy, and that energy flux gravitates. The resulting second-order field contains a nonoscillatory displacement memory even when the material source remains bound. This nonlinear, or Christodoulou, memory is therefore not an optional add-on to the radiative solution: it is the zero-frequency response required by the asymptotic Einstein constraint once the wave’s own positive energy flux is included. This page derives the constraint and normalization; it does not make a current detection claim.

Helpful background. Einstein Equations supplies the nonlinear constraint in which radiative energy is a source. Waves, Green Functions, and Geometric Optics supplies retarded propagation and the zero-frequency limit. Gravitational Radiation and Waveforms supplies Bondi news, outward flux, and strain. Causality and Global Structure supplies future null infinity, its sphere of generators, and stationary endpoint cuts.

The news-squared source

Use

$$N_{AB}=\partial_uC_{AB}, \qquad \mathcal N=N_{AB}\bar m^A\bar m^B, \qquad N_{AB}N^{AB}=2\lvert\mathcal N\rvert^2.$$

The Bondi mass-aspect constraint in this volume’s convention is

$$\partial_um =\frac14D_AD_BN^{AB} -\frac18N_{AB}N^{AB} -4\pi G\,\mathcal T_{uu},$$

where $\mathcal T_{uu}=\lim_{r\to\infty}r^2T_{uu}$ is any material null-energy flux per unit solid angle. The gravitational part of the outward radiated energy per unit solid angle, integrated across the burst, is

$$F_{\mathrm{GW}}(x^A) =\frac{1}{32\pi G} \int_{u_i}^{u_f}N_{AB}N^{AB}\,du =\frac{1}{16\pi G} \int_{u_i}^{u_f}\lvert\mathcal N\rvert^2\,du.$$

It is nonnegative at every angle. Integrating the constraint in retarded time gives

$$D_AD_B\Delta C^{AB} =4\Delta m+16\pi G \left(F_{\mathrm{GW}}+F_{\mathrm{matter}}\right),$$

with $F_{\mathrm{matter}}=\int\mathcal T_{uu}\,du$. This equation contains both a change in Coulombic mass data and energy that actually reaches null infinity. Ordinary and null memory are different source terms in one balance law.

Here $m$ is the geometric Bondi mass aspect, with dimensions of length; the total Bondi energy is $(4\pi G)^{-1}\int m\,d\Omega$. Integrating the displayed equation over the sphere gives $\Delta E_{\mathrm B}=-\int(F_{\mathrm{GW}}+F_{\mathrm{matter}})d\Omega$, because a sphere divergence integrates to zero. This independently checks the sign and factor of the energy-flux source.

Favata derives the nonlinear memory from the gravitational-wave energy flux and translates it into waveform modes in Favata 2010, §4, pp. 5–6, PDF. The factors above use $N_{AB}=\partial_uC_{AB}$; sources defining news as half this tensor must be translated before their flux formula is used.

Electric-parity inversion on the sphere

Decompose the electric-parity shear as

$$C^{E}_{AB} =\left(D_AD_B-\frac12q_{AB}D^2\right)\Phi.$$

On the unit sphere,

$$D_AD_BC_E^{AB} =\frac12D^2(D^2+2)\Phi.$$

The operator annihilates the $\ell=0,1$ harmonics. Those modes encode total four-momentum and coordinate-origin information rather than a tensor memory pattern. After projecting the balance law onto $\ell\ge2$,

$$D^2(D^2+2)\Delta\Phi =8(\Delta m)_{\ell\ge2} +32\pi G \left(F_{\mathrm{GW}}+F_{\mathrm{matter}}\right)_{\ell\ge2}.$$

If the mass aspect has no $\ell\ge2$ component on either endpoint cut in the same Bondi frame used for the flux integral, the radiative-flux term alone determines the memory potential. Independently boosting each endpoint to its own rest frame would change the comparison and can incorrectly erase ordinary recoil memory. The positive scalar flux does not imply that every component of $\Delta C_{AB}$ has the same sign: removing low harmonics and differentiating $\Phi$ produces a trace-free angular pattern.

For a weak first-order oscillatory wave $N^{(1)}_{AB}=O(\varepsilon)$, the source $N^{(1)}_{AB}N_{(1)}^{AB}$ is $O(\varepsilon^2)$. The nonlinear memory $\Delta C^{(2)}_{AB}$ is therefore a second-order field sourced by first-order radiation. Its slow growth can accumulate across many wave cycles even though each instantaneous contribution is perturbatively small.

An axisymmetric check

Let the total null flux have an axisymmetric quadrupolar part

$$F(\theta)=F_0\left[1+aP_2(\cos\theta)\right],$$

with $F_0\gt0$ and $-1\le a\le2$ so that the flux is nonnegative, and suppose $(\Delta m)_{\ell\ge2}=0$. Only the $aP_2$ term sources tensor memory. Since

$$D^2P_2=-6P_2, \qquad D^2(D^2+2)P_2=24P_2,$$

the memory potential is

$$\Delta\Phi =\frac{4\pi G}{3}F_0a\,P_2(\cos\theta),$$

up to the irrelevant $\ell=0,1$ kernel. The result vanishes for isotropic flux ($a=0$), even though the total radiated energy can be nonzero. Memory records anisotropic energy transport, not merely the scalar total energy.

This example also checks dimensions: $GF_0$ has dimensions of length and so does $C_{AB}$ and $\Phi$ in the expansion $g_{AB}=r^2q_{AB}+rC_{AB}+\cdots$.

To connect the potential to an actual strain, use the oriented $(e_\theta,e_\phi)$ screen. Axisymmetry gives

$$\Delta H_+ =\frac12(\partial_\theta^2-\cot\theta\,\partial_\theta)\Delta\Phi =2\pi GF_0a\sin^2\theta, \qquad \Delta H_\times=0.$$

Thus $\Delta h_+=\Delta H_+/r$. The memory vanishes along the symmetry axis and is largest on the equator; changing the sign of the flux anisotropy reverses the plus component without making the energy flux negative.

Relation to waveform modes

The nonlinear memory can be evaluated from oscillatory waveform modes because $F_{\mathrm{GW}}$ is quadratic in news. Let $H=\sum H_{\ell m}\,{}_{-2}Y_{\ell m}$ use unit-normalized spin-weighted harmonics, and let $Y_{\ell m}$ denote the corresponding scalar harmonics. Since electric shear obeys $H^E=\bar\eth^2\Phi/2$, the pure gravitational null-memory contribution satisfies

$$\begin{aligned} &H^{\mathrm{mem}}_{\ell m}(u)-H^{\mathrm{mem}}_{\ell m}(u_i)\\ &\quad=\sqrt{\frac{(\ell-2)!}{(\ell+2)!}}\, \int_{u_i}^{u}du'\int_{S^2}d\Omega\, |\mathcal N(u',\Omega)|^2\bar Y_{\ell m}(\Omega), \qquad \ell\ge2. \end{aligned}$$

The $G$ in the response cancels the $1/G$ in the energy flux. Each $H_{\ell m}$ has dimensions of length, $\mathcal N=\dot H$ is dimensionless, and the time integral again has dimensions of length. The observed dimensionless mode is $h^{\mathrm{mem}}_{\ell m}=H^{\mathrm{mem}}_{\ell m}/r$. This is the normalization of Favata 2010, p. 5, eqs. (12)–(14), PDF, translated from $G=1$ and finite-distance modes to the rescaled strain.

Expanding the square gives products $\dot H_{\ell_1m_1}\dot{\bar H}_{\ell_2m_2}$ multiplied by the explicitly defined angular integral

$$\mathcal K^{\ell m}_{\ell_1m_1\ell_2m_2} =\int_{S^2} {}_{-2}Y_{\ell_1m_1}\, {}_{-2}\bar Y_{\ell_2m_2}\, \bar Y_{\ell m}\,d\Omega.$$

Its azimuthal integral requires $m=m_1-m_2$. For example, a mode multiplied by its own conjugate sources $m=0$ memory even when the input oscillates rapidly. At leading nonlinear order, insert the first-order oscillatory news; feedback of memory into its own flux belongs to higher orders. A calculation must include sufficient angular modes, a stable early-time baseline, and the same frame for every input mode.

The integration constant is physical bookkeeping, not something fixed by the flux integrand. A finite waveform must state which early cut defines zero memory and what prehistory is omitted.

Scope and observation boundary

The classical balance law predicts a nonoscillatory signal component. Instrumental high-pass behavior, calibration, low-frequency noise, finite observation windows, and current event evidence are separate questions treated in Gravitational-Wave Astronomy and dated Research records. The existence of positive news-squared source does not by itself establish that a particular data set resolves the offset.

Common pitfalls

Calling all displacement memory “Christodoulou memory.” Matter/Coulombic endpoint changes generate ordinary memory. The Christodoulou term is specifically the contribution from gravitational-wave energy flux.

Inverting the sphere operator without removing $\ell=0,1$. $D^2(D^2+2)$ has a kernel. Its low harmonics must be fixed by frame and charge data, not divided by zero.

Assuming positive flux means positive strain everywhere. The tensor pattern is trace free and obtained after angular projection and differentiation. Positivity applies to energy density, not to every polarization component.

Exercises

1. Isotropic-flux control. Show that a purely isotropic null flux produces no $\ell\ge2$ displacement-memory tensor in the stationary-endpoint case.

Solution

An isotropic flux is proportional to $Y_{00}$ and therefore has no $\ell\ge2$ projection. The electric tensor operator also annihilates an $\ell=0$ potential:

$$\left(D_AD_B-\frac12q_{AB}D^2\right)Y_{00}=0.$$

The flux changes the total Bondi mass balance, but it cannot produce a trace-free angular deformation. This separates total radiated energy from anisotropic memory.

2. Zero-news limit. Set $N_{AB}=0$ and $\mathcal T_{uu}=0$ throughout the interval. What does the integrated constraint imply?

Solution

The fluxes vanish and $\Delta C_{AB}=0$ because the news is its retarded-time derivative. The integrated constraint then gives $\Delta m=0$. These statements concern shear and mass aspect; the angular-momentum aspect is governed by another constraint and need not be time independent in an arbitrary nonradiative Bondi frame. A nonzero time-independent shear can still label the chosen vacuum, but it does not change and therefore supplies no memory across the interval.

References

• Favata, Marc. “The Gravitational-Wave Memory Effect.” Classical and Quantum Gravity 27 (2010): 084036. DOI. Open PDF.

Dated Research assessment. For open questions and recent results, follow the link below. Each assessment states the date through which it considers results and the limits of its conclusions.

• Detectability of Nonlinear Gravitational-Wave Memory