Spin Memory

Spin memory is the magnetic-parity part of the time-integrated Bondi shear. It is sourced by angular-momentum flux per unit angle and changes in the associated asymptotic charges. An idealized Sagnac experiment—counter-propagating light rays around a large closed contour—isolates this parity sector through an accumulated relative arrival-time shift. Despite its name, the effect does not require intrinsically spinning bodies; an orbit carrying angular momentum can generate it. This page defines the asymptotic observable, not a current detection claim.

Helpful background. Einstein Equations supplies the angular asymptotic constraint. Waves, Green Functions, and Geometric Optics supplies null propagation around the contour and retarded integration. Gravitational Radiation and Waveforms supplies Bondi shear, news, and angular-momentum flux. Causality and Global Structure supplies future null infinity and its asymptotic symmetries and charges.

Electric and magnetic shear potentials

Every smooth symmetric trace-free tensor on the unit sphere can be decomposed, up to low-harmonic kernels, as

$$C_{AB} =\left(D_AD_B-\frac12q_{AB}D^2\right)\Phi +\epsilon_{C(A}D_{B)}D^C\Psi.$$

Here $\epsilon_{\theta\phi}=+\sin\theta$, $D_A$ is the unit-sphere covariant derivative, and parentheses mean symmetrization with weight $1/2$. The scalar $\Phi$ is electric parity and $\Psi$ is magnetic parity. Under the sphere parity map, the two sectors transform with opposite parity for the same $(\ell,m)$. The displacement memories treated on the preceding pages are changes in the electric potential. Spin memory is instead

$$\Delta\Sigma(x^A) =\int_{u_i}^{u_f}\Psi(u,x^A)\,du.$$

The quantity has one extra power of time compared with displacement strain. It is not generally a permanent magnetic-parity strain offset. Its retarded-time derivative, or “spin-memory mode,” can appear as a nonoscillatory magnetic-parity contribution to the instantaneous waveform while $\Delta\Sigma$ accumulates.

Nichols defines this observable and derives its charge/flux balance in Nichols 2017, pp. 7–8, eqs. (4.1)–(4.4), PDF.

Why a closed light path isolates magnetic parity

Take the sphere divergence of each parity sector:

$$D^BC^E_{AB} =\frac12D_A(D^2+2)\Phi,$$ $$D^BC^B_{AB} =-\frac12\epsilon_A{}^C D_C(D^2+2)\Psi.$$

The minus sign in the second identity follows from the ordering $\epsilon_{C(A}$ in the shear definition. Replacing it by $\epsilon_{(A|C|}$ would reverse the definition of the magnetic potential. The electric contribution is a gradient. Its line integral around a smooth closed contour $\mathcal C$ vanishes:

$$\oint_{\mathcal C}D^BC^E_{AB}\,dx^A=0.$$

The magnetic contribution is a rotated gradient and can have nonzero circulation. This motivates the contour functional

$$\mathscr S_{\mathcal C} =\int_{u_i}^{u_f}du \oint_{\mathcal C}D^BC_{AB}\,dx^A.$$

Up to the physical radius, contour length, and light-propagation normalization of a specified Sagnac setup, $\mathscr S_{\mathcal C}$ is the accumulated counter-propagating time-delay difference. By Stokes’ theorem it measures the magnetic-parity potential inside the loop. A single local detector cannot infer it without the nonlocal angular information encoded by the contour or an equivalent network of observers.

The ideal experiment holds the contour at large, constant Bondi radius and compares the two propagation orientations with the same clock and Bondi frame. At finite radius, acceleration of the apparatus and subleading metric terms must be included; the asymptotic formula is not an exact laboratory transfer function.

Angular-momentum balance

The curl of the Bondi angular-momentum-aspect constraint supplies an elliptic equation on the sphere. Define

$$\mathcal D=D^2(D^2+2), \qquad \mathcal L_B=D^2\mathcal D.$$

On a scalar spherical harmonic, $\mathcal L_BY_{\ell m}=-[\ell(\ell+1)]^2[\ell(\ell+1)-2]Y_{\ell m}$. It is invertible after its $\ell=0,1$ kernel is removed. Its source separates into:

• an ordinary spin memory from a change in the asymptotic charge;
• a matter null spin memory from angular momentum carried to null infinity by matter;
• a nonlinear null spin memory from angular momentum carried by gravitational waves.

The gravitational null contribution can be stated explicitly in this convention. Set $N_{AB}=\partial_uC_{AB}$ and define

$$\begin{aligned} \mathcal B_A={}&C^{BC}D_BN_{AC}-N^{BC}D_BC_{AC}\\ &+3N_{AB}D_CC^{BC}-3C_{AB}D_CN^{BC}. \end{aligned}$$

Then the nonlinear null part obeys

$$\mathcal L_B\Delta\Sigma_{\mathrm{GW}} =P_{\ell\ge2}\int_{u_i}^{u_f} \epsilon^{AD}D_D\mathcal B_A\,du.$$

This is Nichols 2017, p. 8, eq. (4.4), PDF, with the same shear normalization and sphere orientation. No factor of $G$ remains when the gravitational source is written in terms of shear and news: $G$ in the response cancels the inverse $G$ in the flux. Both sides have dimensions of length squared. The full ordinary and matter terms in eq. (4.3) additionally require the source’s improved angular-momentum aspect and the appropriate factor of $G$ multiplying material stress-energy. They must be included when those sources are present; the displayed equation isolates only the gravitational null contribution.

The source vanishes when the relevant magnetic-parity charge change and angular-momentum-flux moments vanish. This is the independent zero-control. Conversely, nonspinning compact bodies in an orbit can have nonzero gravitational angular-momentum flux because orbital angular momentum is radiated.

Relation to waveform multipoles

Expand the shear in electric- and magnetic-parity tensor harmonics. The magnetic coefficients are related to current-type radiative multipoles. Products of oscillatory mass and current modes enter the nonlinear flux, with angular selection rules set by three-harmonic integrals. Nichols finds that, for the stated nonspinning quasicircular approximation and a frame aligned with orbital angular momentum, the leading accumulated signal is the magnetic-parity $\ell=3,m=0$ harmonic Nichols 2017, p. 9, eqs. (4.6)–(4.10), PDF.

That result is model- and frame-dependent, not the definition of spin memory. The definition remains the magnetic-parity time integral at null infinity. A finite waveform must state its starting cut: extending an idealized inspiral to infinite separation can make the formal time integral ill behaved, while a real model covers only a finite history.

Distinctions that must remain visible

Spin memory is not:

• ordinary displacement memory, which is an electric-parity strain difference;
• Christodoulou memory, which is an electric-parity response to energy flux;
• merely the cross polarization, since plus/cross rotate into each other within one screen while electric/magnetic sphere parity is global;
• the intrinsic spin of a compact object;
• a detector-level conclusion.

Instrument response, finite angular sampling, low-frequency noise, and evidence for a particular event belong to Gravitational-Wave Astronomy and dated Research records.

Common pitfalls

Dropping the time integral. The defining observable is $\Delta\Sigma=\int\Psi\,du$. Its derivative is a waveform component but is not the accumulated spin memory itself.

Using a local plus/cross rotation as a parity test. Electric and magnetic parity are properties of a field over the sphere. They cannot be decided from one line of sight alone.

Ignoring the low-harmonic kernel. The magnetic elliptic operator must be inverted only after its kernel is projected out and the Bondi frame and charge convention are fixed.

Exercises

1. Electric-parity null test. Prove that the contour functional $\mathscr S_{\mathcal C}$ vanishes for a purely electric shear on any contractible closed curve.

Solution

For electric shear,

$$D^BC^E_{AB}=\frac12D_A(D^2+2)\Phi.$$

This is the gradient of the single-valued scalar $(D^2+2)\Phi/2$. Its line integral around a closed contractible curve is therefore zero by the fundamental theorem for line integrals. Time integration preserves zero. A nonzero result requires magnetic parity, a singular/global obstruction, or a failure of the assumed contour and Bondi-frame construction.

2. Zero-source control. Suppose the superspin charge change and both matter and gravitational fluxes have zero $\ell\ge2$ magnetic-parity source. What spin memory remains?

Solution

The projected right-hand side of the balance law vanishes, so $\mathcal L_B\Delta\Sigma=0$. After the low-harmonic kernel is removed by the stated frame and charge convention, the only solution is $\Delta\Sigma=0$. This does not forbid ordinary displacement or nonlinear energy-flux memory, which live in the electric-parity sector.

References

• Nichols, David A. “Spin Memory Effect for Compact Binaries in the Post-Newtonian Approximation.” Physical Review D 95 (2017): 084048. DOI. Open PDF.