Near-Kerr rigidity with computable geometric constants
Summary
Upgrade qualitative near-Kerr uniqueness/rigidity into explicit geometric inequalities with computable constants; the qualitative rigidity technology exists, the effective bounds do not.
Why this matters
Numerical relativity and inverse problems need constants, not only existence of a small neighborhood where rigidity holds.
Exact scope
- Background / setting
- Stationary vacuum, asymptotically flat, near-Kerr in rigidity-theoretic sense.
- Equation type
- Stationary Einstein vacuum reduction.
- Linearity
- Stationary nonlinear elliptic-type constraints; linearization may appear as a tool.
- Regularity
- Smooth or high Sobolev as in near-Kerr rigidity papers.
- Parameter regime
- Smallness measured by explicit curvature or Mars–Simon-type tensors, not an abstract implicit topology.
- Asymptotics
- Asymptotically flat.
- Gauge / formulation
- Invariants (Mars–Simon, Weyl alignment) should be stated in a gauge-fixed form amenable to constants.
Status explanation
Open as a quantitative / effective problem; not a separate classical conjecture from the existence of qualitative rigidity.
Problem statement
Upgrade perturbative or near-Kerr uniqueness theorems for stationary vacuum asymptotically flat black-hole exteriors so that hypotheses are checkable geometric inequalities (tetrad or curvature norms) with explicit constants controlling closeness to an exact Kerr metric—not only asymptotic “smallness in implicit norms.”
What is already known
- Qualitative rigidity and local uniqueness statements for near-Kerr stationary vacuum solutions under hypotheses used in the Mars/Simon tensor framework.Regime: Stationary vacuum, asymptotically flat, near-Kerr.Provides the qualitative theorem that quantitative sharpening aims to make effective.
Progress summary: Qualitative near-Kerr rigidity exists; explicit constants and checkable inequalities remain open.
What remains open
Replace “sufficiently small in an implicit norm” with finite, computable bounds linking invariant sizes to Kerr distance functionals.
Mathematical prerequisites
Perturbative uniqueness; unique continuation; invariant boundary data; effective estimates.
Completion criteria
Finite, computable conditions implying exact Kerrness with quantitative parameter control.
Implications if solved
Makes rigidity usable for analysis and numerical stability certificates.
Formal verification suitability
FV: high
Stationary, algebraic, ODE/separable, or finite-dimensional substatements admit clearer formalization boundaries.
See Formal verification for how this database uses these labels.
References
- primary A spacetime characterization of the Kerr metric — Mars (1999) Classical tensor characterization underlying qualitative Kerr rigidity; baseline “known qualitative” statement.
- primary A characterization of 3+1 spacetimes via the Simon–Mars tensor — García-Parrado Gómez-Lobo, Mars, Simon (2014) Tensor toolkit used in modern rigidity and numerical comparisons—natural language for quantitative bounds.
Related problems
Related by shared tags
- K-508 — Stability or obstruction for approximate Killing–Yano tensors near Kerr
- K-617 — Prove a quantitative distance-to-Kerr estimate from a small invariant (Mars-Simon-type) with computable constants.
- K-618 — Prove global Kerr uniqueness without analyticity under minimal smoothness/decay hypotheses.
- K-633 — Prove that near-Kerr spacetimes admit approximate Carter operators with controlled commutators (usable in PDE estimates).
- K-646 — Prove global uniqueness/rigidity for Kerr with minimal assumptions on horizon regularity and connectedness.
- K-647 — Prove a minimal-data theorem: finite set of horizon multipoles determines Kerr parameters with stability bounds.