Kerr stability with BMS charges and nonlinear memory

Summary

Package asymptotic charges, BMS gauge choices, and nonlinear memory into a stability statement compatible with rigorous Kerr stability theory—a synthesis rather than a single classical one-line conjecture.

Why this matters

A black-hole stability theorem should align with the asymptotic observables used in waveform and memory analysis, not only with interior decay rates.

Exact scope

Background / setting

Asymptotically flat vacuum Einstein equations; future null infinity and BMS framework.

Equation type

Full nonlinear vacuum Einstein (stability-scale regularity).

Linearity

Nonlinear memory and BMS transformations are inherently nonlinear in the metric.

Regularity

Must match the function spaces of the chosen stability theorem (e.g. weighted Sobolev / polyhomogeneous classes at null infinity).

Parameter regime

Compatible with near-subextremal Kerr stability regimes as they are proved (currently including small-$|a|$ windows).

Asymptotics

Asymptotically flat.

Gauge / formulation

BMS gauge fixing and memory definitions must be tied to the gauge used in the nonlinear stability construction.

Status explanation

This entry bundles infrastructure problems around a future stability statement; it is not a standard single-sentence “famous conjecture” but a quantitative/synthesis frontier built on stability theory.

Problem statement

Build a formulation of asymptotically flat vacuum stability compatible with Bondi–Metzner–Sachs (BMS) charge bookkeeping and nonlinear gravitational memory: define charges and memory observables in the regularity class used by the stability proof, prove their finiteness and flux laws, and show consistency with convergence to a final Kerr state modulo gauge and parameter modulation.

What is already known

• Rigorous Bondi–Sachs frameworks define BMS charges, flux laws, and gravitational memory in asymptotically flat settings—including nonlinear memory contributions accessible in PDE formulations (e.g. curve-deviation memory).
  Regime: Asymptotically flat vacuum; null-infinity bookkeeping.
  Supplies honest definitions for the observables that a Kerr stability theorem should eventually align with.
• Klainerman–Szeftel’s Kerr stability program proves nonlinear stability in restricted subextremal windows with explicit Cauchy/topology choices; BMS/memory compatibility is not yet a standard packaged corollary.
  Regime: Nonlinear vacuum near Kerr (small $|a|/M$ regime).
  Identifies where charge/memory theorems would need to be embedded in the stability bootstrap.
• Linearized BMS/memory on fixed backgrounds is more developed than the fully nonlinear Kerr-stability-compatible synthesis targeted here.
  Regime: Linearized or Minkowski-centric memory literature.
  Clarifies that the open issue is **compatibility** with the nonlinear Kerr proof’s gauge and regularity.

Progress summary: Nonlinear Kerr stability exists in restricted regimes; BMS/memory theorems on fixed or Minkowski-like backgrounds are separate. The gap is unifying these threads in one stability-compatible theorem.

What remains open

Decompose the target into: (i) define BMS charges and memory in the gauge of the near-Kerr stability framework; (ii) control nonlinear memory contributions under the same bootstrap; (iii) connect the charge/memory picture to the final Kerr identification and modulation of parameters. All three must be executed at theorem level.

Mathematical prerequisites

Bondi–Sachs expansions; asymptotic symmetries; flux laws; gauge fixing at null infinity; nonlinear radiation estimates.

Completion criteria

A complete answer must define charges and memory in the chosen regularity class, prove finiteness and flux laws, and show consistency with final Kerr parameters.

Implications if solved

Connects rigorous PDE stability with the mathematical structure behind gravitational-wave observables.

Formal verification suitability

FV: medium

Some subquestions may formalize before the full statement.

See Formal verification for how this database uses these labels.

References

• primary Persistent gravitational wave observables: curve deviation in asymptotically flat spacetimes — Grant, Nichols (2021)
  Rigorous Bondi–Sachs framework for displacement/spin/center-of-mass memory and nonlinear flux contributions—mathematical template for memory observables.
• survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022)
  States the current nonlinear Kerr stability landscape with which BMS/memory bookkeeping must be compatible.

Related problems

Related by shared tags

Heuristic matches on family, cluster, equation level, asymptotics, and relevance.

• K-004 — Peeling and polyhomogeneous expansions at null infinity for nonlinear near-Kerr evolutions
• K-014 — Nonlinear decay versus resonance expansions
• K-602 — Prove nonlinear stability of Kerr for the full subextremal range |a|<M.
• K-604 — Establish a sharp peeling/polyhomogeneity theorem at future null infinity ($\mathcal{I}^+$) for nonlinear near-Kerr evolutions.