Center-of-Mass Memory

Center-of-mass memory is a subleading electric-parity observable built from a time moment of the Bondi shear. It is tied to changes in super-center-of-mass charges and to center-of-mass angular-momentum flux, not simply to the final strain offset. Its definition subtracts the part that would grow linearly once ordinary displacement memory has settled, leaving a finite stationary-to-stationary quantity under stated falloffs in a specified Bondi frame and retarded-time origin. This page fixes that definition and its frame dependence; it does not make a detector or event claim.

Helpful background. Einstein Equations supplies the mass and angular-momentum-aspect constraints. Waves, Green Functions, and Geometric Optics supplies retarded-time moments and endpoint limits. Gravitational Radiation and Waveforms supplies Bondi shear, displacement memory, and flux. Causality and Global Structure supplies BMS cuts, translations, and the asymptotic charge setting.

The electric-parity time moment

Decompose the shear as

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

Displacement memory is the endpoint difference $\Delta\Phi$. Spin memory uses the time integral of the magnetic potential $\Psi$. Center-of-mass memory instead uses a particular time integral of the electric potential.

Separate the null displacement-memory potential $\Phi^{(n)}$—the part sourced by energy flux through null infinity—from the full $\Phi$. Define

$$X(u,x^A)=\Phi(u,x^A)-\Phi^{(n)}(u,x^A).$$

More precisely, set $\mathcal D=D^2(D^2+2)$ and remove the $\ell=0,1$ harmonics with $P_{\ell\ge2}$. In the volume’s news normalization,

$$\mathcal D\,\partial_u\Phi^{(n)} =P_{\ell\ge2}\left(N_{AB}N^{AB}+32\pi G\mathcal T_{uu}\right), \qquad \mathcal T_{uu}=\lim_{r\to\infty}r^2T_{uu}.$$

An initial integration constant fixes $\Phi^{(n)}$, while the low-harmonic kernel does not affect the shear. The subtraction is defined by energy flux, not by discarding all electric-parity radiation. In particular, magnetic-parity waves also carry energy and can source an electric-parity null-memory term at nonlinear order.

For stationary or suitably nonradiative endpoint cuts $u_1$ and $u_2$, the center-of-mass memory potential is

$$\Delta\mathcal K =\int_{u_1}^{u_2}X(u,x^A)\,du -\left.uX(u,x^A)\right|_{u_1}^{u_2}.$$

The subtraction removes the part of the time integral that would grow linearly after ordinary memory has made $X$ approach a new constant. The result is a potential for the portion of the time-integrated electric-parity shear that becomes constant in a stationary-to-stationary transition.

Integration by parts gives the equivalent and often more useful form

$$\Delta\mathcal K=-\int_{u_1}^{u_2}u\,\partial_uX\,du.$$

It has dimensions of length squared, since $\Phi$ has dimensions of length and $c=1$. Nichols introduces this observable and its stationary-transition properties in Nichols 2018, pp. 12–14, eqs. (3.2)–(3.12), PDF.

Why this is not displacement memory

As a resolved test of the time operation, choose a single $\ell\ge2$ harmonic $Y_{\ell m}$ and a localized pulse in the subtracted potential,

$$X(u,x^A)=B\,e^{-(u-u_0)^2/\tau^2}Y_{\ell m}(x^A),$$

with its real part understood for a real scalar potential. The endpoints vanish as $u\to\pm\infty$, so $\Delta X=0$ and

$$\Delta\mathcal K=B\tau\sqrt\pi\,Y_{\ell m}.$$

The observable can therefore be nonzero even when the subtracted potential returns to its initial value. Moving the pulse in time changes $u_0$ but leaves this integral unchanged. This example checks the definition; the mass aspect and charge/flux data of any physical realization must also satisfy the asymptotic constraints.

More generally, a constant added to $X$ contributes equally to the integral and the endpoint term and cancels. Extending a quiet endpoint interval also cancels in this way. A finite numerical waveform must nevertheless state its start, end, and missing prehistory. Taking endpoints to infinite retarded time requires sufficient decay of $u\partial_uX$, not just the assertion that news eventually becomes small.

Charge and flux balance

Let $Y^A=D^AY$ be a smooth electric-parity vector field on the sphere. The corresponding super-center-of-mass charge is the electric-parity part of the extended angular-momentum charge. The precise shear observable paired with this generator is

$$\Delta\mathcal C_Y =\frac12\int_{S^2}(D_AY^A)\,\mathcal D\Delta\mathcal K\,d\Omega.$$

This is Nichols 2018, p. 14, eq. (3.12), PDF. On $Y_{\ell m}$, the operator has eigenvalue $(\ell-1)\ell(\ell+1)(\ell+2)$, which makes both its low-harmonic kernel and its invertibility for $\ell\ge2$ explicit.

The angular Einstein constraint relates $\Delta\mathcal C_Y$ to a super-center-of-mass charge change and integrated gravitational and matter fluxes. Its derivation integrates the angular constraint once in retarded time, uses the mass constraint to remove the null displacement contribution, and integrates the remaining news term by parts. The gravitational flux is bilinear in $C_{AB}$ and $N_{AB}$; material terms also involve angular stress-energy components. A full charge calculation requires the charge and angular-momentum-aspect definitions of Nichols 2018, pp. 9 and 12–14, eqs. (2.24), (3.1), and (3.13), PDF, translated from $G=1$ as a complete set. The observable and its sphere operator are fixed above without assigning an arbitrary normalization to an undefined flux density.

This balance separates:

• an ordinary part from the change in the super-center-of-mass charge;
• a null part from center-of-mass angular momentum carried by gravitational waves or matter.

The low harmonics require special care. The $\ell=1$ sector is ordinary center-of-mass angular momentum; higher smooth electric harmonics are its “super” extension. Sphere operators used to reconstruct the tensor observable have $\ell=0,1$ kernels, so frame and charge data must be fixed before inversion.

Retarded-time origin and Bondi-frame changes

The subtraction makes the observable independent of how far the quiet endpoint intervals are extended. It does not make its numerical value independent of the retarded-time origin. To see the distinction, make the passive relabeling $u'=u+a$, keeping the angular coordinates and $Y^A$ fixed and transforming the endpoint cuts consistently. Then $X'(u')=X(u'-a)$, and

$$\begin{aligned} \Delta\mathcal K' &=-\int_{u_1}^{u_2}(u+a)\,\partial_uX\,du\\ &=\Delta\mathcal K-a\Delta X, \qquad \Delta X=X(u_2)-X(u_1). \end{aligned}$$

Consequently the paired quantity defined above transforms as

$$\Delta\mathcal C'_Y-\Delta\mathcal C_Y =-\frac a2\int_{S^2}(D_AY^A)\,\mathcal D\Delta X\,d\Omega.$$

The mass constraint and the definition of $\Phi^{(n)}$ give $\mathcal D\Delta X=8P_{\ell\ge2}\Delta m$, where $m$ is the geometrical Bondi mass aspect. Thus a change in its higher harmonics can make this origin shift nonzero. A stationary-to-stationary transition does not by itself eliminate it: the final stationary remnant may be boosted in the initial Bondi frame. If $\Delta X=0$, as in the localized-pulse example, the time moment is invariant under this constant relabeling.

Comparing BMS charges additionally requires specifying the spacetime generator. With $\zeta_Y=\tfrac12u(D_AY^A)\partial_u+Y^A\partial_A$, the generator adapted to $u'=u+a$ satisfies

$$\zeta'_Y=\zeta_Y+\frac a2(D_AY^A)\partial_u.$$

The extra term is a supertranslation generator; changing it changes which charge is being compared. Nichols 2018, pp. 7–8 and 13, eqs. (2.17)–(2.18) and (3.5)–(3.8), PDF discusses such generator-dependent comparisons, including an infinitesimal-supertranslation invariance statement. That statement is not used here as an origin-invariance theorem for the displayed $\Delta\mathcal K$ or fixed-$Y^A$ pairing. An angle-dependent supertranslation requires the complete transformation of the fields, angular derivatives, cuts, and charge generator; no general cancellation is assumed here.

A reproducible result therefore states the common Bondi frame, retarded-time origin, endpoint cuts, and generator, and distinguishes the combined shear observable from its ordinary and null charge/flux pieces.

Parity and symmetry checks

Center-of-mass memory is electric parity, like displacement memory, but has one additional retarded-time moment. Spin memory is magnetic parity. These statements give immediate controls:

• the magnetic potential $\Psi$ does not enter the defining time moment directly, although its energy flux can enter the subtraction $\Phi^{(n)}$;
• if the relevant electric charge change and center-of-mass flux moments vanish, the projected memory vanishes;
• extending either stationary endpoint interval does not change the subtracted result;
• under a constant change of retarded-time origin, the observable obeys the transformation law above.

For source models with an exact exchange or inversion symmetry, particular odd multipoles of center-of-mass flux can vanish. Such a symmetry result must be derived in the specified frame rather than assumed from the phrase “center of mass.”

Scope and observation boundary

The observable is constructed from strain, displacement memory, time integrals, and asymptotic frame data. Turning it into an instrumental estimator requires finite-duration response, calibration, low-frequency control, and a statistical model. Those current and detector-specific questions belong to Gravitational-Wave Astronomy and dated Research records.

Common pitfalls

Using $\int\Phi\,du$ without the endpoint subtraction. A settled ordinary memory would make that integral grow with the arbitrarily chosen length of the final quiet era.

Calling $\Delta\mathcal K$ another strain offset. It has an extra time moment and probes center-of-mass charge/flux information not contained in $\Delta\Phi$ alone.

Conflating endpoint independence with origin independence. Extending quiet endpoint intervals leaves the subtraction unchanged, but shifting the clock origin changes the time moment by $-a\Delta X$. State both the common Bondi frame and time origin.

Exercises

1. Endpoint-independence check. Let $X(u)$ become constants $X_i$ before $u_a$ and $X_f$ after $u_b$. Show that moving $u_2\gt u_b$ later does not change $\Delta\mathcal K$.

Solution

Increase the endpoint by $\delta u$ within the final stationary era. The integral gains $X_f\delta u$. The boundary term $u_2X_f$ also gains $X_f\delta u$, and it is subtracted. Their changes cancel exactly. The same reasoning applies at the initial endpoint. Convergence still requires that the endpoint values and approach rates are well defined.

2. A complete-subtraction control. Suppose the electric potential satisfies $\Phi=\Phi^{(n)}$ throughout the interval. What center-of-mass memory follows? Why would setting only $\Phi=0$ be insufficient?

Solution

The assumed equality gives $X=0$ and therefore

$$\Delta\mathcal K=0.$$

By contrast, $\Phi=0$ alone does not establish $\Phi^{(n)}=0$: a magnetic-parity wave can carry nonzero $N_{AB}N^{AB}$ and hence source the null-memory subtraction. One must inspect the flux and the asymptotic constraints as well as the instantaneous shear parity.

3. Shift a smooth step. For one real $\ell\ge2$ harmonic, take $X(u)=\tfrac A2[1+\tanh((u-u_0)/\tau)]$, where $\tau\gt0$ and the common harmonic factor is suppressed. Compute $\Delta\mathcal K$ between infinite endpoint times and check the effect of $u'=u+a$.

Solution

The endpoint difference is $\Delta X=A$, and

$$\partial_uX=\frac{A}{2\tau}\operatorname{sech}^2\!\left(\frac{u-u_0}{\tau}\right).$$

Write $u=u_0+\tau v$ in $-\int u\partial_uX\,du$. The term odd in $v$ integrates to zero and $\int_{-\infty}^{\infty}\operatorname{sech}^2v\,dv=2$, giving $\Delta\mathcal K=-Au_0$. In the shifted coordinates the step center is $u'_0=u_0+a$, so $\Delta\mathcal K'=-A(u_0+a)=\Delta\mathcal K-a\Delta X$. The subtraction removes dependence on the lengths of quiet intervals while retaining the specified clock origin. As with the pulse example, a physical realization must also satisfy the Bondi constraints.

References

• Nichols, David A. “Center-of-Mass Angular Momentum and Memory Effect in Asymptotically Flat Spacetimes.” Physical Review D 98 (2018): 064032. DOI. Open PDF.