Fluxes, Memory, and Consistency Checks

Radiative modes become scientifically constraining when their integrated fluxes agree with independently measured changes in the source. Energy, momentum, angular momentum, and gravitational memory therefore form a connected set of consistency checks. Each is sensitive to waveform frame, missing modes, early junk radiation, late-time truncation, and integration constants, so closure must be demonstrated within a decomposed numerical error budget.

Helpful background. Formulations of General Relativity supplies asymptotic charges and hypersurface data; Differential Equations for Relativity supplies numerical integration and error control; Black-Hole Geometry and Mechanics supplies remnant horizon charges; Gravitational Radiation and Waveforms defines fluxes and memory; and Computational Relativity supplies the common verification framework.

News, strain, and flux conventions

In an asymptotically inertial Bondi frame, let $r_{\mathrm A}$ be areal radius and define the finite radiative strain coefficient and news by

$$H(u,\Omega)=\lim_{r_{\mathrm A}\to\infty} r_{\mathrm A}\left(h_+-ih_\times\right), \qquad N=\partial_u H,$$

with retarded time $u$ and spin weight $-2$. Thus $h_+-ih_\times=H/r_{\mathrm A}+O(r_{\mathrm A}^{-2})$ for a distant detector. With the curvature convention used in the preceding extraction page, the leading asymptotic curvature coefficient is

$$\Psi_4^0\equiv\lim_{r_{\mathrm A}\to\infty}r_{\mathrm A}\Psi_4 =\partial_u N=\partial_u^2H.$$

This sign is inherited from the site’s Newman–Penrose radiation dictionary, including its Riemann tensor, null tetrad, complex dyad, and strain convention. An opposite sign in a source or data product signals a different convention package, not a change in the physical energy flux.

The energy flux is

$$\frac{dE_{\mathrm{rad}}}{du} =\frac{1}{16\pi G}\int_{S^2}|N|^2\,d\Omega,$$

where the luminosity-distance scaling has already been removed by the definition of $H$. The linear-momentum flux inserts the asymptotic radial unit vector into the same angular integral. Angular-momentum flux involves the strain, news, and rotational derivatives, making its integration constants and frame dependence more conspicuous. Ruiz and collaborators collect compatible energy, momentum, and angular-momentum expressions in Ruiz et al. 2008, §II, Eqs. (2.7)–(2.25), and §III, Eqs. (3.3)–(3.24).

Mode sums must declare the spin-weighted-harmonic normalization and maximum $\ell$. Energy is usually dominated by low modes, while recoil and angular momentum can be more sensitive to interference with subdominant modes. Increase $\ell_{\max}$ until each target flux stabilizes; a converged dominant strain mode does not guarantee a converged kick.

From modes to a luminosity

If the spin-weighted harmonics are orthonormal, $N=\sum_{\ell m}N_{\ell m}\,{}_{-2}Y_{\ell m}$ and orthogonality gives

$$\frac{dE_{\mathrm{rad}}}{du} =\frac{1}{16\pi G}\sum_{\ell m}|N_{\ell m}|^2.$$

This supplies a direct check against angular quadrature. For the single mode $H_{22}=A e^{-i\omega u}$ with constant complex amplitude $A$ and real frequency $\omega$, its contribution is $\omega^2|A|^2/(16\pi G)$. A complete physical waveform may require additional modes; this is the contribution of the stated mode alone. With $c=1$, $H$ and $A$ have dimensions of length and $N$ is dimensionless, so the expression has dimensions of energy per time.

For a damped mode, choose the time origin at the start of the modeled interval, so $A=H_{22}(0)$, and allow $\omega$ to be complex. The same ansatz gives $N_{22}=-i\omega H_{22}$ and hence

$$\left(\frac{dE_{\mathrm{rad}}}{du}\right)_{22} =\frac{|\omega|^2|A|^2}{16\pi G} e^{2\operatorname{Im}(\omega)u}.$$

With the chosen $e^{-i\omega u}$ convention, damping requires $\operatorname{Im}(\omega)\lt0$. This expression applies over the interval where a single exponentially damped mode is an adequate model; its extrapolation to the entire inspiral or arbitrarily early times is not justified.

Momentum instead weights the angular flux by a unit direction $\widehat r_i$. Its mode integral contains products of different harmonics. That is why dropping a small mode can have a larger fractional effect on recoil than on total radiated energy.

Closing source and radiation budgets

For an asymptotically flat vacuum binary, form residuals such as

$$\delta E=E_{\mathrm{ADM}}-E_{\mathrm{rad}}-E_f, \qquad E_f=\sqrt{M_f^2+|\mathbf P_f|^2}.$$ $$\delta J_i=J_i^{\mathrm{ADM}}-J_i^{\mathrm{rad}}-J_i^f, \qquad \delta P_i=P_i^{\mathrm{ADM}}-P_i^{\mathrm{rad}}-P_i^f.$$

Here $M_f$ is the remnant’s rest mass measured from its late horizon, while $E_f$ is its total energy in the frame of the balance. They agree only for a remnant at rest. In an initial center-of-momentum frame, $E_{\mathrm{ADM}}$ equals the initial ADM rest mass, but recoil generally makes $E_f\ne M_f$. At small kick speed $v_f$ the omitted kinetic energy would be $M_fv_f^2/2+O(v_f^4)$. Likewise, $J_i^f$ is the total remnant angular momentum about the chosen origin; horizon spin alone omits an orbital contribution if the remnant is displaced and moving.

All terms must use compatible origins, asymptotic frames, signs, and units. Matter escaping the domain, electromagnetic radiation, or incoming radiation adds terms rather than becoming unexplained nonclosure. The final rest mass and intrinsic spin inherit the quasi-local definitions and errors of the remnant horizon; the radiated quantities inherit extraction and time-integration errors. If early junk radiation is discarded from the waveform, its energy and momentum must still be included when balancing against initial ADM charges. Otherwise start with Bondi charges on the same initial cut as the retained flux.

Evaluate budgets over a resolution sequence and several extraction treatments. Keep Cauchy truncation, horizon measurement, extraction-radius or characteristic error, mode truncation, and time-window error visible. A small $\delta E$ produced by cancellation of two large biased terms is not strong evidence. Check intermediate cumulative curves: they can reveal a burst of outer-boundary contamination or an incorrect sign that the final number happens to conceal.

The momentum budget also tests the center-of-mass frame. A residual boost or translation changes mode couplings and can masquerade as recoil. Fix or transform the Bondi–Metzner–Sachs frame before comparing waveform and remnant momentum. The same frame discipline is essential for angular momentum, whose value depends on the choice of origin.

Memory as a balance observable

Displacement memory is the persistent strain change

$$\Delta H=H(u_2)-H(u_1)=\int_{u_1}^{u_2}N\,du.$$

Directly integrating $\Psi_4$ twice can suppress or invent low-frequency content through drift removal. Characteristic extraction of strain can retain memory, but its finite start time and initial null data still matter. An independent route uses asymptotic charge balance laws and fluxes to reconstruct memory. Mitman and collaborators formulate that comparison in Mitman et al. 2021, §II, Eqs. (9)–(19), and §III, Figs. 1–9.

Displacement and spin memory occupy different parity and time-integral structures. A finite waveform begins after part of the inspiral, so the raw endpoint difference is generally incomplete. State how the missing early contribution is supplied, how late ringdown is extended, and how the nonradiative integration constants are fixed. The CCE strain and the BMS-balance reconstruction are meaningfully independent only if their shared Cauchy data and frame uncertainties are acknowledged. A numerical study of both displacement and spin memory, including charge-based reconstruction, appears in Mitman et al. 2020, §§II–III.

The two integration constants act differently:

$$H\mapsto H+C_0+C_1u,\qquad N\mapsto N+C_1,\qquad \Delta H\mapsto\Delta H+C_1(u_2-u_1).$$

A constant strain offset $C_0$ changes neither energy flux nor the endpoint memory difference. A constant news offset $C_1$ changes both. Consequently energy balance cannot determine an absolute strain offset, and fitting a news offset to force energy closure makes that particular check circular. Angular-momentum charges require additional frame and strain-offset care.

Documenting results and diagnosing failures

The environment record should pin the waveform and horizon datasets with checksums, extraction and integration software revisions, precision, harmonic library, frame-fixing method, quadrature, and filtering. Inputs include retarded-time interval, discarded junk interval, $\ell_{\max}$, strain integration constants, remnant charge definition, ADM convention, and any matter or boundary fluxes. Outputs should contain modes, cumulative fluxes, every balance term, residuals, and their resolution and extraction variations.

The primary benchmark is a versioned binary-black-hole evolution for which initial ADM charges, radiative modes, and final horizon charges are all available. A clean future rerun would evaluate it at three Cauchy resolutions, multiple extraction radii plus CCE when available, and increasing $\ell_{\max}$. It would close the ADM–radiation–remnant budgets and compare direct CCE memory with a BMS balance reconstruction. This is a specified workflow, not a claim that an absent program has executed it.

Reject the result if budget residuals do not decrease or stabilize with refinement, if changing $\ell_{\max}$ shifts recoil materially, if frame transformations are undocumented, if early or late time-window choices dominate memory, or if flux signs differ between formulas and source evolution. A particularly dangerous failure mode is tuning integration constants until a desired balance closes; constants must be fixed by an independent physical prescription.

Check your understanding

An energy budget closes to $10^{-4}M$, but the calculation chose the news integration constant (the linear-in-$u$ strain drift) by minimizing that same budget residual. A second integration prescription changes the memory by $30\%$. Is the closure independent evidence for the waveform?

Solution

No. The check was used to calibrate the quantity it was supposed to test, so the small residual is circular. Fix the news offset and strain conventions through an independent asymptotic or early-time prescription, recompute the energy balance, and compare memory with a charge-balance reconstruction or characteristic strain. The $30\%$ spread must remain in the memory uncertainty until the low-frequency treatment is independently constrained.

Continue with the chapter overview.

References

• Mitman, Keefe, et al. 2020. “Computation of Displacement and Spin Gravitational Memory in Numerical Relativity.” Physical Review D 102, 104007. arXiv.
• Mitman, Keefe, et al. 2021. “Adding Gravitational Memory to Waveform Catalogs Using BMS Balance Laws.” Physical Review D 103, 024031. arXiv.
• Ruiz, Milton, Ryoji Takahashi, Miguel Alcubierre, and Darío Núñez. 2008. “Multipole Expansions for Energy and Momenta Carried by Gravitational Waves.” General Relativity and Gravitation 40, 1705–1729. arXiv.