Tensorial Mode Stability of Kerr–de Sitter Spacetimes

Tensorial Mode Stability of Kerr–de Sitter Spacetimes

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.

A tensorial Einstein mode is a separated solution of the linearized Einstein equations, rather than a scalar-wave solution. Ungauged mode stability concerns geometric perturbations modulo infinitesimal diffeomorphisms and family-parameter changes. The full subextremal Kerr--de Sitter range excludes degeneracy of its distinct horizon roots.

Catalogue wording retained for traceability.

Prove ungauged tensorial Einstein mode stability throughout the full subextremal Kerr–de Sitter parameter range.

Scope and source qualification.

Scalar mode stability or a gauge-fixed result in a small-rotation or small-cosmological-parameter range does not cover the entire requested tensorial range.

Short English statement

Prove geometric tensorial mode stability for every subextremal Kerr--de Sitter parameter choice.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Derived pure-gauge linearized tensor solutions, including separated apparent growth, to test the correct geometric quotient in mode stability.

Sources and literature check.

Read the full supplied definition, catalogue wording, source qualifications and listed references for 30006049. No fresh status or novelty search was performed. The argument below proves its auxiliary statements directly and does not import an unverified external theorem.

Why raw tensor modes contain false instabilities.

Write the geometric Einstein operator as $\mathcal E(g)=\operatorname{Ric}(g)+\Lambda g$ in the convention of the source. For any smooth diffeomorphism $\Phi$, tensor covariance gives $\mathcal E(\Phi^*g)=\Phi^*\mathcal E(g)$. If $\mathcal E(g)=0$ and $\Phi_s$ is the local flow of a vector field $X$, differentiating at $s=0$ gives \[ D\mathcal E_g(\mathcal L_Xg)=\mathcal L_X\mathcal E(g)=0. \] Thus every infinitesimal coordinate change supplies a linearized tensor solution.

Separated growing gauge fields.

On a stationary coordinate patch choose $X=e^{\sigma t}Y(x)$, $\sigma>0$, with smooth spatially supported $Y$. Since the background is time independent, $\mathcal L_Xg$ is also proportional to $e^{\sigma t}$ and generically nonzero. Taking the spatial support away from horizon and asymptotic boundaries avoids artificial local boundary singularities. This produces apparent growing solutions of the ungauged coordinate equation, but they represent zero geometric perturbation after quotienting by infinitesimal diffeomorphisms.

First physical-mode gap.

The calculation explains why componentwise scalar estimates or absence of raw growing tensors is not the requested statement. One must prove that every admissible nondecaying tensor mode in the full subextremal range is gauge or a permitted family-parameter variation, with the actual horizon conditions and regularity. No classification of the quotient kernel is proved here, and no estimate uniform near excluded degenerate horizons is asserted. The first unresolved step is precisely that geometric kernel analysis beyond the small-parameter regimes.

Verification and adversarial audit.

Checked covariance differentiation, stationary separation, local support and the distinction between gauge growth and a physical instability; no scalar-to-tensor inference. Model self-audit only; no independent human mathematical verification. Applied contribution prompt A before drafting and B afterwards. Checked covariance differentiation, stationary separation, local support and the distinction between gauge growth and a physical instability; no scalar-to-tensor inference. First remaining gap: Classify admissible tensor modes modulo gauge and family variations throughout the full source subextremal parameter range. The inherited source status is preserved; original-target completion remains zero. Additional GPT-6 Astra reviewer (ultra reasoning): Differentiated diffeomorphism covariance of Ric(g)+Lambda g at an Einstein background to obtain D E_g$L\_X g$=0. For a stationary coordinate patch and X=exp(sigma t)Y(x), checked that all Lie-derivative terms have the same exp(sigma t) factor, including derivatives in time; generic Y gives a nonzero pure-gauge tensor. Checked that compact spatial support away from coordinate boundaries is sufficient for the local countercheck, while the draft correctly leaves actual admissible horizon conditions and quotient-kernel classification unproved. Confirmed no scalar or small-rotation result is substituted for full subextremal ungauged tensorial stability.

Tools, process and contributions.

Python 3 and local file tools used to read definitions and serialize manuscripts. Calculations are symbolic reasoning; no numerical experiment or theorem prover. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.

Final status and first remaining gap.

UNRESOLVED. Classify admissible tensor modes modulo gauge and family variations throughout the full source subextremal parameter range.

Completion Estimate

COMPLETION: 0% This is an elementary auxiliary calculation or an explicit test of a restricted strategy. No new portion of the full source target is established.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath, record 30006049 (OWR-14298591-013). Catalogue curation under CC BY 4.0; definitions and English exposition added locally. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Mathematical Aspects of General Relativity $2024$, Oberwolfach Reports, DOI 10.4171/owr/2024/36, pp. 2033--2112. Original report and mathematical context. Incorporated report material remains under the source CC BY-SA 4.0 terms unless otherwise noted. https://doi.org/10.4171/owr/2024/36.