The Landau–Lifshitz formulation maps a binary-black-hole spacetime onto an auxiliary flat spacetime and defines a conserved linear momentum split between the bodies and the gravitational field. Keppel, Nichols, Chen, and Thorne (2009) computed this momentum flow at 1.5PN order during the inspiral in harmonic coordinates — including the reversible body–field momentum exchange associated with spin-induced bobbing in the superkick configuration — and posed as open questions its extension through plunge, merger, and ringdown using numerical relativity, quantitative comparison of the near-zone field momentum with post-Newtonian predictions, and the gauge dependence of the momentum localization. The head-on, equal-mass, antiparallel-spin case was subsequently computed in numerical relativity (Lovelace et al. 2010), which found remarkably weak gauge dependence between two gauge families but explicitly left orbiting mergers, superkick configurations, quantitative comparisons, and a wider gauge study open; bobbing was separately reinterpreted as a kinematical effect. The surviving open problem is to establish the momentum-flow picture for inspiraling mergers: numerical-relativity computation of the near-zone and horizon momentum partitioning through merger and ringdown, quantitative comparison with the post-Newtonian momentum densities, and a systematic characterization of gauge dependence, including whether harmonic coordinates with fully fixed initial data can define a reproducible momentum-localization convention.

Background

Numerical-relativity simulations of merging binary black holes (BBHs) predict the gravitational-wave signals now observed by detectors, but the emitted waveform and the final black hole's recoil ("kick") velocity are gauge-invariant end-products: they say little about how linear momentum is distributed and exchanged inside the strong-field region while the merger unfolds. The obstacle is fundamental: momentum conservation requires translation invariance, which fails inside curved spacetime except locally and asymptotically, so there is no generally covariant local momentum density. The Landau–Lifshitz formalism circumvents this by mapping the physical spacetime onto an auxiliary flat spacetime: one chooses "preferred" coordinates, declares them globally Lorentz in the auxiliary flat spacetime, and rewrites Einstein's equations as a nonlinear field theory in flat spacetime with a total stress-energy tensor ταβ=(−g)(Tαβ+tLLαβ)\tau^{\alpha\beta}=(-g)(T^{\alpha\beta}+t^{\alpha\beta}_{\mathrm{LL}}), where gg is the determinant of the physical metric, TαβT^{\alpha\beta} is the matter stress-energy tensor, and tLLαβt^{\alpha\beta}_{\mathrm{LL}} is the Landau–Lifshitz pseudotensor, a coordinate-dependent object built from the metric and its first derivatives. In the chosen coordinates ταβ\tau^{\alpha\beta} has vanishing divergence, so its components can be read as the density and flux of a conserved linear momentum, split between the bodies (integrals over their horizons or interiors) and the gravitational field (a volume integral over the near zone, the strong-field region well inside the wave zone).

Keppel, Nichols, Chen, and Thorne used this formalism in harmonic coordinates (coordinates that each satisfy the covariant wave equation, so that metric perturbations propagate causally) at 1.5PN order — the post-Newtonian expansion in powers of v/cv/c, truncated just past the leading spin-dependent corrections — during the inspiral, the slow orbital shrinkage driven by gravitational-wave emission. They computed the momentum flow between the bodies and the near-zone field, with special attention to "bobbing": in binaries whose spins lie in the orbital plane (the "extreme-kick" or superkick configuration, where two comparable-mass holes have equal and opposite in-plane spins), the spin–velocity interaction makes the holes and the orbital plane bob up and down in phase with the orbit, and the merger then produces a recoil whose magnitude and direction depend on the orbital phase at merger. They showed that in harmonic coordinates the bobbing corresponds to a reversible exchange of momentum between the bodies and the near-zone field, with total momentum conserved, and they collected in an appendix the additional velocity-dependent and higher-order field-momentum pieces needed for later quantitative comparisons with simulations. They flagged three linked issues as open: the gauge dependence of the momentum distributions (the localization is tied to the chosen mapping to the auxiliary flat spacetime, and it remained to be shown that different coordinate choices and simulation codes give qualitatively and semiquantitatively similar results, with harmonic coordinates — whose residual freedom in the initial data must still be fixed — as a candidate community convention); the extension beyond the inspiral to the late plunge, nonlinear merger, and ringdown (the exponentially damped quasinormal ringing of the remnant), where post-Newtonian methods fail and numerical relativity is required; and detailed quantitative comparisons of the post-Newtonian near-zone momentum densities with numerical-relativity measurements.

The announced follow-up by Lovelace, Chen, Cohen, Kaplan, Keppel, Matthews, Nichols, Scheel, and Sperhake computed the Landau–Lifshitz momentum flow in full numerical relativity for the head-on plunge, merger, and ringdown of an equal-mass binary with transverse antiparallel spins: defining the momentum enclosed by each apparent horizon (the locus where outgoing null rays momentarily fail to diverge) and an effective horizon velocity, they found that the holes' transverse motion reflects momentum exchange with the surrounding field and that the final effective velocity agrees with the recoil implied by the gravitational-wave flux. Comparing evolutions from two methodologically independent numerical-relativity approaches — a pseudospectral code (expanding the fields in global spectral basis functions) in a generalized-harmonic gauge and a moving-puncture finite-difference code — they observed remarkably weak gauge dependence in that one configuration — while stating explicitly that this does not constitute a proof of gauge independence — and found qualitative (not quantitative) agreement of the near-zone momentum density with the post-Newtonian prediction before the late plunge. Their conclusion lists inspiraling (non-head-on) mergers, superkick configurations, larger spins, and a wider class of coordinate conditions as open. In parallel, Gralla, Harte, and Wald analyzed flat-spacetime and electromagnetic analog binaries and argued that the bobbing itself is a purely kinematical spin–centroid effect (analogous to Thomas precession) that appears regardless of the force holding the bodies in orbit, that momentum exchanged with the field is stored as hidden mechanical momentum, and that the kick should not be viewed as a post-merger continuation of bobbing; they also stressed that any allocation of momentum between bodies and gravitational field depends on both the pseudotensor and the gauge choice, to an extent that remained unquantified. The Caltech/Cornell program then turned to the tendex–vortex visualization of spacetime curvature, and the momentum-flow series stopped at the head-on case. Later, Woodford, Boyle, and Pfeiffer showed that large center-of-mass motion in state-of-the-art BBH waveforms is largely a gauge effect requiring empirical corrections — evidence that localization conventions for momentum-like quantities in numerical relativity remain unsettled. Most recently, Boyeneni, Wu, and Most visualized the inspiral–merger–ringdown of BBHs through an exact Maxwell-like recasting of Einstein's equations, an adjacent continuation of the field-theory-intuition program that does not address momentum localization or its gauge dependence.

Problem Statement

The problem is to turn the Landau–Lifshitz momentum-flow picture of black-hole binaries into a quantitatively verified and gauge-characterized description of the full coalescence. Concretely, for binary black holes (or compact binaries) evolved through the late inspiral, plunge, merger, and ringdown:

  1. Momentum flow through coalescence: compute, in the Landau–Lifshitz formulation on an auxiliary flat spacetime, the near-zone field momentum density and the momenta of the individual holes, and determine how linear momentum is partitioned among the bodies, the near-zone field, and the emitted gravitational waves as functions of time — in particular for orbiting (non-head-on) binaries and the superkick (in-plane antiparallel spin) configuration, not only the head-on case already treated — and how the near-zone field momentum relates to the radiated momentum and the final kick.

  2. Quantitative post-Newtonian comparison: compare the numerical-relativity near-zone momentum densities and integrated momenta quantitatively with the post-Newtonian predictions in harmonic-compatible coordinates, including the velocity-dependent and higher-order post-Newtonian field-momentum pieces beyond the spin-induced bobbing terms (Appendix A of the source paper), stating the domain of validity (separations, times before merger) and the accuracy of agreement.

  3. Gauge dependence: characterize, across a wide class of coordinate conditions and simulation codes — at a minimum the orbiting and superkick configurations, evolved in at least one harmonic and one non-harmonic coordinate condition, extending the two gauge families already compared for the head-on case — to what extent the qualitative and semiquantitative features of the momentum distributions are gauge-independent; and determine whether harmonic coordinates — with the residual freedom in the harmonic initial data fixed — can yield a fully specified, reproducible momentum localization suitable for cross-code comparison.

The three components serve one common objective — establishing (or refuting) the momentum-flow formalism as a reliable, convention-independent window on the nonlinear dynamics of merging binaries, for the configurations and gauge families fixed as the minimum in component 3 — by determining which momentum-flow features are convention-independent across simulations and the post-Newtonian analysis and which are gauge artifacts.

References

  1. D. Keppel, D. A. Nichols, Y. Chen, K. S. Thorne, 'Momentum Flow in Black Hole Binaries: I. Post-Newtonian Analysis of the Inspiral and Spin-Induced Bobbing', Phys. Rev. D 80, 124015 (2009), DOI: 10.1103/PhysRevD.80.124015, arXiv:0902.4077, https://arxiv.org/abs/0902.4077
  2. G. Lovelace, Y. Chen, M. Cohen, J. D. Kaplan, D. Keppel, K. D. Matthews, D. A. Nichols, M. A. Scheel, U. Sperhake, 'Momentum flow in black-hole binaries. II. Numerical simulations of equal-mass, head-on mergers with antiparallel spins', Phys. Rev. D 82, 064031 (2010), DOI: 10.1103/PhysRevD.82.064031, arXiv:0907.0869, https://arxiv.org/abs/0907.0869
  3. S. E. Gralla, A. I. Harte, R. M. Wald, 'Bobbing and Kicks in Electromagnetism and Gravity', Phys. Rev. D 81, 104012 (2010), DOI: 10.1103/PhysRevD.81.104012, arXiv:1004.0679, https://arxiv.org/abs/1004.0679
  4. C. J. Woodford, M. Boyle, H. P. Pfeiffer, 'Compact Binary Waveform Center-of-Mass Corrections', Phys. Rev. D 100, 124010 (2019), DOI: 10.1103/PhysRevD.100.124010, arXiv:1904.04842, https://arxiv.org/abs/1904.04842
  5. D. A. Nichols, R. Owen, F. Zhang, A. Zimmerman, J. Brink, Y. Chen, J. D. Kaplan, G. Lovelace, K. D. Matthews, M. A. Scheel, K. S. Thorne, 'Visualizing Spacetime Curvature via Frame-Drag Vortexes and Tidal Tendexes I. General Theory and Weak-Gravity Applications', Phys. Rev. D 84, 124014 (2011), DOI: 10.1103/PhysRevD.84.124014, arXiv:1108.5486, https://arxiv.org/abs/1108.5486
  6. S. Boyeneni, J. Wu, E. R. Most, 'Unveiling the electrodynamic nature of spacetime collisions', Phys. Rev. Lett. 135, 101401 (2025), DOI: 10.1103/995s-wxl7, arXiv:2504.15978, https://arxiv.org/abs/2504.15978
  7. C.-C. Tsai, Z. Cao, C.-Y. Lin, H.-J. Yo, 'Pseudotensor applied to Numerical Relativity in Calculating Global Quantities', Chinese Journal of Physics (2015), DOI: 10.6122/CJP.20150903, arXiv:1507.08166, https://arxiv.org/abs/1507.08166