The Bondi--Sachs gauge, BMS frames, and memory in black hole perturbation theory

Abstract: As LISA and other next-generation detectors demand increasingly accurate waveform models, there is a growing need for these models to precisely control gauge freedoms that had previously been inconsequential. One such intrinsic freedom is the choice of the asymptotic Bondi--Metzner--Sachs (BMS) frame. The need to control the BMS frame is particularly pronounced in black hole perturbation theory, where there has been little work to this end -- most glaringly in gravitational self-force calculations, which are in an unknown frame and encounter infrared, far-zone gauge singularities at second perturbative order. Here we present a framework for iteratively transforming to the Bondi--Sachs gauge and fixing the BMS frame on a Kerr background. This includes an extension of the Bondi--Sachs formalism to the multiscale expansions that underpin most self-force-based waveforms, introducing soft hair and a concept of forgetful gauges'' in the process. Our framework evades infrared divergences and naturally incorporates memory effects that had previously only ever been addedafter the fact'' in self-force waveforms, including the recently discovered ``memory distortion''. Our formalism could also be used for ringdown analysis, and we expect it to be vital for comparisons with numerical relativity and post-Newtonian theory.

• The paper presents a framework for constructing the Bondi–Sachs gauge to fix BMS frames and avoid infrared divergences in second-order perturbation theory.
• It provides explicit procedures for transforming Kerr spacetime perturbations and isolating key BMS charges using vector harmonic decompositions and multiscale expansions.
• The study integrates gravitational memory and soft hair effects into waveform modeling, enhancing analyses for future gravitational wave detectors.

The Bondi–Sachs Gauge, BMS Frames, and Memory in Black Hole Perturbation Theory

Motivation and Overview

This paper addresses advanced issues in black hole perturbation theory (BHPT) relevant to gravitational wave astronomy, particularly as waveform modeling reaches the precision required by next-generation detectors like LISA, Cosmic Explorer, and the Einstein Telescope. The main focus is the formal control of gauge freedoms—specifically the asymptotic Bondi–Metzner–Sachs (BMS) frame—in practical waveform modeling, with a sharp emphasis on second-order perturbative calculations and gravitational memory effects.

Black hole perturbation approaches—including self-force calculations essential for extreme-mass-ratio inspirals—have traditionally disregarded the explicit fixing of the BMS frame, resulting in ambiguity and gauge singularities at second order. This paper presents a framework for constructing the Bondi–Sachs (BS) gauge systematically on a Kerr background, detailing how to fix the BMS frame iteratively, and analyzes the incorporation of gravitational-wave memory and soft hair effects into self-force theory using multiscale expansions.

Asymptotic Gauge Degrees of Freedom and the BMS Group

The asymptotic structure of spacetime, especially at future null infinity ($\mathcal{I}^+$), retains residual gauge freedom after imposing Bondi–Sachs conditions. This residual freedom constitutes the infinite-dimensional BMS group, including the Poincaré transformations (translations, rotations, boosts) and an infinite set of supertranslations. The relationship between waveform observables and physical system parameters fundamentally depends on this asymptotic frame.

The paper establishes clear operational definitions for asymptotically flat gauges and demonstrates that the BS formalism, using coordinates adapted to outgoing null cones, naturally isolates the BMS symmetry group as the only remaining gauge freedom. Explicit formalisms are provided for linearized BMS transformations and the extraction of BMS charges—mass, momentum, angular momentum, boosts, and supermomentum—through surface integrals at $\mathcal{I}^+$.

Second-Order Perturbation Theory: Gauge Invariance and Infrared Divergences

The extension of BHPT beyond linear order increases gauge complexity: at each order, new gauge degrees of freedom emerge, making previously invariant quantities, such as the perturbations to Weyl scalars $\psi_0$ and $\psi_4$, gauge-dependent at second order. The paper analyzes the reduced second-order Teukolsky equation, providing explicit expressions for source terms and examining their asymptotic behavior.

A strong result is the evasion of infrared divergences in second-order calculations when the first-order metric perturbation is transformed to a BS gauge. In generic gauges, source terms exhibit slow falloff, leading to divergent retarded integrals; in BS gauge, the falloff is sufficiently rapid to yield well-behaved solutions. This demonstrates the necessity of gauge fixing for robust second-order calculations, and provides explicit falloff conditions on tetrad contractions ensuring BS gauge regularity.

Kerr Spacetime in Bondi–Sachs Coordinates

The background Kerr metric is recast in BS coordinates using results from Bai et al. [108], yielding explicit asymptotic expansions. These coordinates provide a shear-free, regular specification at null infinity, facilitating extraction and manipulation of BMS charges and the transformation of the metric perturbation to BS gauge.

The paper clarifies that the background BS coordinates are not unique but subject to residual BMS transformations. Using stationarity and axial symmetry, it proves that time translations and axial rotations remain unfixed (Killing symmetries), but all other BMS freedoms are fully determined by the metric form.

Transformation to the BS Gauge and BMS Frame Fixing

A hierarchical procedure is given to transform an arbitrary first-order asymptotically flat metric perturbation to BS gauge. The gauge vector is constructed by solving ODEs along null rays, iteratively enforcing BS conditions to required orders in $1/r$ for extracting BMS charges. The method ensures that the mass, linear momentum, and angular momentum aspects are accessible and fixable at first order.

After transformation, residual gauge freedom corresponding to BMS transformations is isolated and fully characterized. The paper employs explicit vector harmonic decompositions to fix the BMS frame: zeroing linear momentum and boost charges (center-of-mass frame), aligning the angular momentum axis, and eliminating the even-parity shear component (supertranslation-fixing) at a reference time. The remaining time translation and axial rotation are left to waveform alignment in practical comparison.

Incorporation of Memory and Soft Hair: Multiscale Expansion in Self-Force Theory

For asymmetric mass binaries modeled with self-force theory, the multiscale expansion encodes periodic orbital dynamics and slow evolution of system parameters ($\mathcal{O}(\epsilon)$ timescales). The paper demonstrates that traditional self-force gauges (Lorenz, radiation, Regge–Wheeler–Zerilli) are "forgetful": they fail to propagate nonlinear memory effects into the near zone, requiring complex matched-expansion or post-Minkowskian treatments.

By transforming to BS gauge at each step of the iterative multiscale procedure, the need for near/far matching is eliminated. Slowly evolving BMS transformations, with supertranslations and rotations as functions of "slow time", are introduced, which install soft hair (nontrivial shear) on the perturbed spacetime. Gravitational-wave memory and memory distortion (discovered recently [69]) arise naturally and extend continuously from near to far zones. Explicit expressions are provided, showing how coupling between oscillatory and memory modes generates the distortion observable in 1PA waveform fluxes.

Comparison to Numerical Relativity BMS Frame Fixing

The paper contrasts its frame-fixing scheme with that developed by the SXS collaboration in numerical relativity [35, 36], noting that SXS approaches iteratively fix BMS charges using full metric data and angular velocity vectors extracted from waveforms, while the formalism here fixes all BMS charges independently at first perturbative order, with robust scalar invariants ensuring comparisons through second order. The distinction is emphasized in the rotational frame: this work aligns the $z$ axis with total Bondi angular momentum, while SXS uses waveform angular velocity.

Implications and Future Directions

This work provides a systematic methodology for gauge fixing in BHPT, enabling physically unambiguous extraction of waveform parameters, robust second-order calculations free from infrared gauge singularities, and natural inclusion of gravitational memory in both near and far zones. The theoretical implications include:

• Sharp parameter identification: Waveform models and the physical meaning of orbital parameters (eccentricity, spins, mass ratio) can be compared without ambiguity due to gauge or BMS frame choices.
• Interoperability across modeling approaches: Consistent BMS frame fixing enables rigorous comparisons between self-force, post-Newtonian, post-Minkowskian, ringdown, and numerical relativity models.
• Practical computational gains: The iterative BS gauge transformation eliminates the need for complicated matched expansions, allowing straightforward second-order computations and waveform generation for extreme-mass-ratio inspirals.

Anticipated future developments include further extension of frame-fixing beyond first-order perturbation theory, incorporation of memory distortion into 1PA waveform models, and refined BMS charge-based alignment procedures in waveform catalogs. Accurate BMS frame control is expected to be vital for precision GW parameter estimation and black hole spectroscopy as detector sensitivity increases.

Conclusion

The framework elaborated in this paper for constructing the Bondi–Sachs gauge and fixing the BMS frame in black hole perturbation theory addresses longstanding issues in second-order calculations, gravitational memory incorporation, and waveform comparison. Its rigorous mathematical foundation, practical iterative scheme, and multifaceted applications to self-force and numerical relativity position it as a necessary formalism for the future of gravitational wave modeling and interpretation (2606.24816).