---
title: Christodoulou Memory Effect
url: https://www.emergentmind.com/topics/christodoulou-memory-effect
type: topic
---

# Christodoulou Memory Effect

Searching arXiv for relevant papers on Christodoulou memory and closely related memory-effect literature.
The Christodoulou Memory Effect is the nonlinear gravitational-wave memory effect: a slowly growing, non-oscillatory contribution to the gravitational-wave amplitude that leaves a permanent displacement of freely falling test masses after the wave train has passed. In operational form, a source has memory when the early-time and late-time values of a polarization differ, so that
\[
\Delta h_{+,\times}^{\rm mem} = \lim_{t\rightarrow +\infty} h_{+,\times}(t) - \lim_{t\rightarrow -\infty} h_{+,\times}(t).
\]
In detector language this is a DC-like offset or step in strain rather than an oscillatory signal; in asymptotic language it is a change in the radiative geometry at null infinity; and in compact-binary waveforms it is a hereditary contribution sourced by the entire past history of gravitational-wave emission [1003.3486] [1505.05213].

## 1. Definition and observable content

The defining observational statement is that, after a burst of gravitational radiation passes, freely falling test masses do not return exactly to their original relative positions. In an ideal interferometer this appears as a permanent displacement; in a pulsar timing array it appears as a lasting change in pulse arrival times or pulse frequency; and in waveform language it is a nonzero late-time minus early-time strain [1505.05213]. The standard detector interpretation is therefore not an instantaneous tidal deformation but the residual configuration that remains after the wave has passed.

For weak-field discussions, the displacement interpretation is often expressed through geodesic deviation. The radiative curvature drives relative acceleration,
\[
\frac{d^2 x^k}{dt^2} = - R^{k}{}_{TlT}\, x^l ,
\]
and, in the linearized setting, integrating twice in time yields a net change in separation after the burst. The permanent displacement is the memory effect in this detector-based sense [1505.05213].

A central subtlety is that memory is observable through its buildup. Very late times are equivalent to flat spacetime in unusual coordinates, so the measurable content is the accumulation of the non-oscillatory component during the event rather than a static late-time strain viewed in isolation [1003.3486]. For this reason the memory signal is intrinsically low-frequency and is experimentally more difficult to extract than the dominant oscillatory inspiral-merger-ringdown waveform.

## 2. Nonlinear origin and distinction from linear memory

Christodoulou memory is the nonlinear form of gravitational-wave memory sourced by the energy carried away by gravitational waves themselves. In relaxed Einstein equations, a piece of the gravitational-wave stress-energy behaves like
\[
T^{\rm gw}_{jk} = \frac{1}{R^2}\frac{dE^{\rm gw}}{dt\,d\Omega}\, n_j n_k,
\]
and the corresponding correction to the transverse-traceless metric is
\[
\delta h^{\rm TT}_{jk} = \frac{4}{R}\int_{-\infty}^{T_R} dt'\, \left[ \int \frac{dE^{\rm gw}}{dt'\, d\Omega'}\, \frac{n'_j n'_k}{(1-\mathbf{n}'\cdot\mathbf{N})}\, d\Omega' \right]^{\rm TT}.
\]
This exhibits the hereditary character of the effect: the memory at retarded time \(T_R\) depends on the entire past history of the emitted gravitational-wave energy flux [1003.3486].

The standard distinction is between linear memory and nonlinear memory. Linear memory is associated with systems whose matter content or radiation has a net one-way change in multipole moments, with examples including unbound scattering binaries, supernova ejecta, neutrino emission, or gamma-ray burst jets [1003.3486]. By contrast, nonlinear memory—the Christodoulou effect—is sourced by the radiated gravitational field itself. The literature surveyed in “Gravitational Waves and Their Memory in General Relativity” further emphasizes that the two effects are now understood as different phenomena: the former is due to fields that do not reach null infinity and the latter is due to fields that do reach null infinity [1505.05213].

This distinction is conceptually important because the Christodoulou effect is not merely a reformulation of source kinematics. It is a manifestation of gravitational self-interaction: gravitational waves carry energy and momentum, and that energy contributes to the geometry that later observers measure. Thorne’s interpretation, summarized in the review literature, is that the nonlinear memory can be viewed as a kind of linear memory sourced by the “unbound particles” of the radiated gravitons [1003.3486].

The physical interpretation is not entirely exhausted by the phrase “null flux to infinity.” “Retarded Fields of Null Particles and the Memory Effect” shows that an eternal null source yields an Aichelburg–Sexl-type pure tidal field with no memory, whereas finite-time creation of a null source produces a radiative burst that does generate memory. That analysis strongly suggests that memory, including nonlinear memory, should not be interpreted as arising simply from the passage of null stress energy to null infinity but rather from a burst of radiation associated with the creation or evolution of the source [1401.5831].

## 3. Null infinity, asymptotic shear, and the geometric memory formula

In the nonlinear asymptotic treatment, Christodoulou memory is encoded in the change of asymptotic shear at future null infinity. The outgoing null hypersurfaces have a shear whose asymptotic limits define radiative data, and the permanent displacement is governed by the difference
\[
\Sigma^+ - \Sigma^- .
\]
The asymptotic quantities satisfy transport relations of the form
\[
\frac{\partial \Sigma}{\partial u} = -\frac{1}{2}\,\Xi, \qquad \frac{\partial \Xi}{\partial u} = -\frac{1}{4}\,A_W ,
\]
so the curvature drives the null-infinity radiation field and its integrated change [1505.05213].

The Bondi mass-loss law gives the flux interpretation. In vacuum,
\[
\frac{\partial M}{\partial u} = -\frac{1}{8\pi}\int_{S^2} |\Xi|^2\, d\omega ,
\]
while in the presence of matter fields reaching null infinity the more general form is
\[
\frac{\partial M}{\partial u} = -\frac{1}{8\pi}\int_{S^2}\left(|\Xi|^2 + C\,T_{uu}\right)\, d\omega .
\]
The total energy radiated per solid angle is then
\[
E = \int_{-\infty}^{\infty}\left(|\Xi|^2 + C\,T_{uu}\right)\,du ,
\]
and the memory equation is stated as
\[
\Delta h = E - \overline{E}, \qquad \operatorname{div}\big(\Sigma^+ - \Sigma^-\big) = \nabla h .
\]
Accordingly, the change in shear across the burst is determined by the angular distribution of energy radiated to null infinity [1505.05213].

The detector displacement follows directly from this shear change:
\[
\Delta x = -\frac{d_0}{r}\,(\Sigma^+ - \Sigma^-).
\]
This is the geometric core of the Christodoulou effect: a permanent offset in separation proportional to a change in asymptotic shear, itself sourced by the integrated radiative flux through null infinity [1505.05213].

The same asymptotic framework makes clear why additional null radiation can enlarge the memory. In the Einstein–Maxwell case the flux acquires an extra leading-order term,
\[
F(\cdot)=\int_{-\infty}^{\infty} \left(|\Xi(u,\cdot)|^2+\frac12|A_F(u,\cdot)|^2\right)\,du,
\]
and in the null-fluid model for neutrinos the total radiated energy per unit solid angle becomes
\[
F(\omega) = \int_{-\infty}^{+\infty} \Big( |\Xi(u,\omega)|^2 + 4\pi\,T_{\underline L \underline L}^{\infty}(u,\omega) \Big)\,du.
\]
In both cases the permanent displacement is still governed by the same shear-difference equation, but the source term is enlarged by the extra flux [1110.0410] [1308.3100].

## 4. Compact-binary waveforms, post-Newtonian structure, and numerical modeling

For quasicircular compact binaries, the Christodoulou memory contributes a growing, nonoscillatory change to the plus polarization and produces a permanent displacement after the wave has passed. A striking result of the post-Newtonian analyses is that, although the interaction that generates memory first appears at \(2.5\)PN order in the multipolar-post-Minkowskian expansion, the memory enters the observable waveform amplitude at leading Newtonian-quadrupole order [0812.0069] [1003.3486].

For quasi-circular binaries the plus-polarization memory is written as
\[
h_{+}^{\rm (mem)} = \frac{2\eta M x}{R}\sum_{n=0}^{\infty} x^{n/2} H_+^{(n/2,{\rm mem})},
\]
with leading term
\[
H_{+}^{(0,{\rm mem})} = \frac{1}{96}s_\Theta^2(17+c_\Theta^2).
\]
The full 3PN calculation shows that the \(0.5\)PN correction vanishes and the \(1.5\)PN memory correction also vanishes,
\[
H_{+}^{(0.5,{\rm mem})}=0, \qquad H_{+}^{(1.5,{\rm mem})}=0,
\]
while the \(2.5\)PN term is proportional to \((1-4\eta)\) and therefore vanishes for equal masses [0812.0069]. Combined with the oscillatory results of Blanchet et al., these results complete the waveform polarizations to 3PN order [0812.0069] [1003.3486].

The time development is not a sharp jump. During inspiral the memory grows slowly and monotonically; it continues increasing through merger; and it saturates after ringdown to a final constant value [0902.3660]. The review literature gives a leading-order inspiral growth \(h_+^{\rm mem,insp}\propto (t_c-t)^{-1/4}\), while full-event modeling combines PN inspiral information with minimal-waveform or effective-one-body input and computes the memory through
\[
h_{+}^{\rm (mem)} \approx \frac{R}{192\pi}s_\Theta^2(17+c_\Theta^2) \int_{-\infty}^{T_R} |\dot h_{22}|^2\, dt.
\]
This makes explicit that the buildup is controlled by the squared amplitude of the dominant oscillatory \(h_{22}\) mode [1003.3486].

The main practical challenge is numerical extraction. The memory sits in the \(m=0\) modes rather than the commonly plotted \(l=m=2\) mode, and in \(\Psi_4\) it is suppressed by several PN orders relative to the oscillatory signal [1003.3486]. The 3PN study emphasizes the same point: the physical memory lives mainly in \(h_{20}\) and \(h_{40}\), two time integrations of \(\Psi_4\) introduce integration-constant ambiguities, and finite initial binary separation causes numerical underestimation of the accumulated memory [0812.0069]. These difficulties explain why memory is hard to extract directly from standard numerical-relativity outputs despite entering the waveform amplitude at leading order.

## 5. Detection channels and observational status

In an ideal interferometer such as LIGO or LISA, the Christodoulou effect is a permanent displacement of the test masses; in pulsar timing arrays such as NANOGrav, it appears as a lasting change in pulse arrival times or pulse frequency [1505.05213]. Because the signal is nonoscillatory and low-frequency, it is less amenable to matched filtering than the usual inspiral-merger waveform and is vulnerable to low-frequency detector noise [1505.05213].

Early detectability estimates concluded that initial LIGO could detect memory only from stellar-mass black-hole mergers in the Local Group, advanced LIGO could reach roughly \(\lesssim 20\) Mpc, and LISA could detect memory from supermassive black-hole mergers out to redshifts \(z\lesssim 2\) [1003.3486]. The binary-black-hole coalescence calculations that include inspiral, merger, and ringdown similarly found that the memory is very difficult to detect with ground-based interferometers but is likely to be observable in supermassive black-hole mergers with LISA out to a redshift of two [0902.3660].

A more detector-specific study using a GW150914-like source and a practical approximation tied to numerical-relativity waveforms found that a single event is not detectable in current second-generation ground-based detectors. For an optimally oriented GW150914-like event, the reported memory signal-to-noise ratios are \(\rho \approx 0.45\) for aLIGO, \(\rho \approx 0.238\) for AdV, and \(\rho \approx 0.243\) for KAGRA, whereas future detectors give \(\rho \approx 9.726\) for ET, \(\rho \approx 27.73\) for CE, \(\rho \approx 96.53\) for DECIGO, and \(\rho \approx 177.2\) for BBO [1810.09563]. The same analysis reports that at current aLIGO sensitivity a detectable GW150914-like memory event would need to be at roughly \(65\) Mpc for \(\rho=3\) or \(39\) Mpc for \(\rho=5\) [1810.09563].

The detector response is commonly described as step-like, with a smooth rise during merger and a permanent offset afterward. Its Fourier transform is approximately \(1/f\) at low frequencies [1810.09563]. This low-frequency character is precisely what makes third-generation ground detectors, decihertz missions, and pulsar timing arrays the most promising observational settings.

## 6. Enlargements, contrasts, and later generalizations

The Christodoulou effect has been extended beyond vacuum general relativity to settings in which additional radiation reaches null infinity. In Einstein–Maxwell spacetimes the electromagnetic field contributes at the same highest asymptotic order as the gravitational term to the permanent memory, enlarging the displacement of test masses after a burst. For binary neutron star mergers the relevant flux is
\[
F(\cdot)=\int_{-\infty}^{\infty} \left(|\Xi|^2+\frac12|A_F|^2\right)\,du,
\]
and the detector displacement remains
\[
\Delta x^A_{(B)} = -\frac{d_0}{r}\left(\Sigma^+_{AB}-\Sigma^-_{AB}\right).
\]
The key distinction is that electromagnetism does not affect the leading-order instantaneous displacement in the Jacobi equation, but it does enlarge the nonlinear permanent memory [1105.6054] [1110.0410].

Neutrino radiation can enlarge the Christodoulou memory in an analogous way when modeled as a null fluid in the Einstein equations. In that setting the Bondi mass-loss law becomes
\[
\frac{dM}{du}(u) = -\frac{1}{8\pi} \int_{S^2} \Big( |\Xi(u,\omega)|^2 + 4\pi\, T_{\underline L \underline L}^{\infty}(u,\omega) \Big)\, d\omega,
\]
so the neutrino flux contributes at the same leading asymptotic order as the gravitational flux to the permanent displacement formula [1308.3100].

Not every matter-radiation memory is identical to Christodoulou memory. “Memory effect from supernova neutrino shells” studies a permanent detector response sourced by a spherically symmetric shell of relativistic neutrinos rather than by a freely propagating non-spherical gravitational wave. That effect is memory by the operational criterion of a permanent change caused by a transient event, but the paper emphasizes two differences from standard Christodoulou memory: there is a longitudinal component, and the transverse part contains a term that grows linearly with time [1605.05399]. The contrast is useful because it isolates what is specific to the Christodoulou effect: null-infinity flux, asymptotic shear change, and the standard displacement-memory observable in the transverse sector.

Later work also broadened the conceptual setting. “Gravitational memory in the bulk” replaces Christodoulou’s timelike test particles with ingoing null geodesics and defines a finite-radius bulk memory observable \(\Delta \eta^a\), reducing at null infinity to the standard shear-jump formula and making the connection to BMS supertranslations explicit in Newman–Unti gauge [1908.07505]. “The Cosmological Memory Effect” shows that in spatially flat FLRW cosmology only the tensor sector contributes to memory and that, at fixed luminosity distance, the memory is enhanced over the Minkowski case by a factor of \((1+z)\) [1606.04894]. “Angular Momentum Memory Effect” places Christodoulou’s nonlinear displacement memory alongside spin memory and a new angular-momentum memory effect, with the displacement memory written in terms of a permanent difference \(\Xi^+-\Xi^-\) determined by integrated flux data at future null infinity [2403.11133]. At the perturbative quantum level, “Finite-time memory detectors and fully constraining Faddeev-Kulish dressings in QED and gravity” argues that a carefully defined finite-time memory detector acting on uniquely fixed finite-time dressed states reproduces both first-order gravitational memory and higher-order Christodoulou contributions from the gravitational field [2605.06774].

Across these extensions, the core content remains stable. Christodoulou memory is the nonlinear displacement memory effect of gravitational radiation: a hereditary, non-oscillatory, permanent change in detector configuration determined by the net flux of radiation through null infinity and encoded geometrically by a change in asymptotic shear [1003.3486] [1505.05213].

Source: https://www.emergentmind.com/topics/christodoulou-memory-effect