Quantitative Mars–Simon tensor gap for near-Kerr stationary vacuum data
Summary
Bound how far stationary vacuum data can deviate from Kerr before the Mars–Simon invariant package becomes quantitatively nonzero; complements exact vanishing characterizations.
Why this matters
Turns qualitative rigidity into a stability statement for curvature invariants used in geometric inverse problems.
Exact scope
- Background / setting
- Four-dimensional stationary asymptotically flat vacuum near Kerr.
- Equation type
- Stationary vacuum Einstein.
- Linearity
- Stationary nonlinear problem; estimates may linearize near Kerr.
- Regularity
- Sobolev or $C^k$ norms on metric/invariants as stated in the theorem.
- Parameter regime
- Uniformity over compact subsets of subextremal $(M,a)$ if claiming uniform bounds.
- Asymptotics
- Asymptotically flat.
- Gauge / formulation
- Invariant tensor norms in a fixed gauge class suitable for global elliptic estimates.
Status explanation
Quantitative sharpening relative to classical Mars–Simon rigidity—not a new unrelated conjecture.
Problem statement
Quantify, in a stated Sobolev or $C^k$ topology on stationary vacuum data, how small deviations from Kerr can be before the Mars–Simon tensor (or an agreed equivalent invariant package) is forced away from zero—i.e. a quantitative gap theorem complementing exact vanishing characterizations on Kerr.
What is already known
- Exact vanishing of the Mars–Simon package characterizes Kerr (local/global statements as in cited references).Regime: Stationary vacuum, asymptotically flat.Qualitative target; this entry asks for quantitative stability of the vanishing condition.
Progress summary: Exact vanishing characterizations exist; quantitative stability of smallness of the invariant package is the open effective problem.
What remains open
Explicit bounds of the form “small invariant norm $\Rightarrow$ small Kerr distance” with computable constants.
Mathematical prerequisites
Four-dimensional Lorentz geometry; Newman–Penrose or orthonormal tensor calculus; uniqueness literature; stability of PDE constraints under perturbation.
Completion criteria
Theorem with explicit bounds relating a norm of Mars–Simon (or substitute) to a distance-to-Kerr functional.
Implications if solved
Sharpens uniqueness technology for inverse problems and numerical certification.
Formal verification suitability
FV: high
Tensor identities and stationary reductions yield finite lists of algebraic conditions suitable for proof-assistant formalization before any global dynamics.
See Formal verification for how this database uses these labels.
References
- primary A spacetime characterization of the Kerr metric — Mars (1999) Exact characterization underlying the Mars–Simon vanishing picture.
- primary A characterization of 3+1 spacetimes via the Simon–Mars tensor — García-Parrado Gómez-Lobo, Mars, Simon (2014) Concrete tensor definitions used in quantitative comparisons.
Related problems
Related by shared tags
- K-502 — Horizon-data rigidity and effective reconstruction of Kerr parameters
- K-503 — Algebraic uniqueness of quadratic symmetry operators commuting with $\Box_g$ on exact Kerr ($\mathcal{D}_{\le 2}$ class)
- K-510 — Ernst equation on stationary axisymmetric vacuum exteriors — sharp uniqueness class for asymptotically flat Kerr (boundary-value formulation)
- K-102 — Derive the interior theorem directly from exterior data