Spectral Gaps and Strong-Cosmic-Censorship Interfaces

Exterior decay and Cauchy-horizon blueshift compete in a de Sitter black hole. The useful dimensionless diagnostic is the spectral gap divided by the Cauchy-horizon surface gravity, but that ratio does not by itself settle strong cosmic censorship. Its implication depends on the perturbing field, the genericity and regularity of initial data, the extension class demanded at the inner horizon, and whether one studies a linear test field or the coupled nonlinear Einstein–matter system.

Evidence cutoff: 2026-08-12. The status and scope of the results below are assessed through this date.

Helpful background. Black-Hole Geometry and Mechanics supplies Killing and Cauchy horizons; Partial Differential Equations and Green Functions supplies resonance expansions and Sobolev regularity; Formulations of General Relativity supplies constrained perturbations; Gravitational Actions and Boundary Terms supplies flux forms; Causality and Global Structure supplies maximal developments and extendibility; Black-Hole Perturbation Theory supplies quasinormal boundary conditions.

From the exterior spectral gap to the inner horizon

Let $\omega_j$ denote the nonzero quasinormal frequencies of a fixed linear field and operator domain in a subextremal de Sitter black-hole exterior. Define the spectral gap

$$\alpha=\inf_j[-\operatorname{Im}\omega_j]>0,$$

after excluding stationary parameter or gauge modes appropriate to that problem. A pole gap does not by itself prove time decay. One also needs a resonance expansion with a controlled high-frequency remainder, data in its stated regularity class, and nonzero excitation of the relevant modes. If the infimum is not attained, a theorem may establish rates $\alpha'\lt\alpha$ without an exact leading term $e^{-\alpha t}$. Repeated poles can add powers of time. These are the same distinctions used in the retarded-response construction.

At the Cauchy horizon $r=r_-$, choose the advanced time in the same stationary normalization as $\omega_j$. A regular transverse coordinate behaves schematically as $V_-\sim-e^{-\kappa_-v}$, where $\kappa_-\gt0$ is the magnitude of the inner-horizon surface gravity. The competition is measured by

$$\beta=\frac{\alpha}{\kappa_-} =\inf_j\frac{-\operatorname{Im}\omega_j}{\kappa_-}.$$

Larger $\alpha$ means faster decay; larger $\kappa_-$ means stronger blueshift. A transmitted mode’s nonsmooth branch has a complex power $x^{p_j}$, where $x=|V_-|$ and $\operatorname{Re}p_j=\beta_j=-\operatorname{Im}\omega_j/\kappa_-$. Its envelope is $x^{\beta_j}$ and its derivative has magnitude proportional to $x^{\beta_j-1}$. The inner-horizon transmission coefficient must be nonzero for this branch to contribute. Thus an exterior pole, its excitation, and its interior transmission are three distinct ingredients.

The ratio must be computed from the dominant allowed mode family, not a visually prominent photon-sphere mode chosen in advance. The three scalar families—photon-sphere, de Sitter, and near-extremal—can exchange dominance Cardoso et al. 2018, Fig. 3 and discussion, p. 4, Open PDF. Angular sector, field charge and mass, constraints, and exceptional zero modes are part of the minimization.

Regularity thresholds are the conjecture

For a nonzero pure-power branch $\phi\sim Cx^p$ in regular coordinates with a smooth nondegenerate local measure, local $H^1$ regularity requires

$$\int_0^\epsilon |\partial_{V_-}\phi|^2dV_- \sim |Cp|^2\int_0^\epsilon x^{2\beta-2}dx\lt\infty,$$

which holds precisely when $\beta=\operatorname{Re}p\gt1/2$. The borderline pure power diverges logarithmically; additional logarithmic factors require their own borderline test. This scalar calculation is given in Dias, Eperon, Reall, and Santos 2018, §IV.A, p. 11, Eqs. (4.19)–(4.20), Open PDF. It is not a universal criterion for every variable: the gravitational argument uses a regular tetrad and a distributional curvature test Dias et al. 2018, §V.A, pp. 21–22, Open PDF. Continuous extendibility, scalar $H^1$, square-integrable connection, $C^2$ metric extendibility, and curvature blow-up ask different questions.

Nor does a regular test scalar construct an extension of a solution to the coupled Einstein equations. Electromagnetic and gravitational constraints alter the physical mode content, quadratic stress can accumulate, and nonlinear mass inflation can reduce regularity even when a first-order field is mild. Conversely, showing curvature blow-up for a special mode is not yet a generic-data theorem. A precise SCC statement names the matter model, topology and data space, maximal globally hyperbolic development, and the regularity in which no extension should exist.

A reported mode and its integrability

For a specific numerical input, take the neutral massless scalar on Reissner–Nordström–de Sitter with geometric mass $\mu=GM$, $\Lambda\mu^2=0.06$ and $e/e_{\max}=0.996$, where $e$ is the geometric charge parameter and $e_{\max}$ is the extremal value at the same $\mu,\Lambda$. The nonstationary $\ell=0$ near-extremal mode in Cardoso et al. 2018, Table I, p. 8, Open PDF has

$$\frac{\omega_{\mathrm{NE}}}{\kappa_-} \simeq-0.8539013779\,i,\qquad \beta_{\mathrm{NE}}=0.8539013779.$$

For this purely imaginary mode, the model singular branch $Cx^{\beta_{\rm NE}}$ therefore gives

$$\int_0^\epsilon|\partial_x\phi|^2dx =|C|^2\beta_{\mathrm{NE}}^2 \frac{\epsilon^{0.7078027558}}{0.7078027558},$$

which is finite. This is arithmetic on a published frequency, not a new eigenvalue calculation or an error estimate. Crucially, the same table reports a photon-sphere mode with $\ell=100$ and $\beta=0.7643094446$, already smaller than the near-extremal value. Neither single entry certifies the infimum over all allowed modes. The example demonstrates a branch’s integrability, not a complete gap computation or a nonlinear extension.

A nonlinear comparison with specified data

To see what backreaction adds, consider a different, spherically symmetric Einstein–Maxwell–real-massless-scalar characteristic problem with $\Lambda\gt0$. Let $\varphi=\sqrt{4\pi G}\,\phi_{\rm physical}$, write $ds^2=-\Omega^2du\,dv+r^2d\Omega_2^2$, and define the geometric mass $\varpi$ and charge $e$ by

$$\begin{aligned} F&=g^{ab}\partial_a r\,\partial_b r =1-\frac{2\varpi}{r}+\frac{e^2}{r^2}-\frac{\Lambda r^2}{3},\\ \nu_r&=\partial_u r,\qquad \lambda=\partial_v r,\qquad \mathfrak k=-\frac{\Omega^2}{4\nu_r},\\ \lambda&=\mathfrak k F,\qquad \theta=r\partial_v\varphi,\qquad \partial_v\varpi=\frac{\theta^2}{2\mathfrak k}. \end{aligned}$$

Here $\Omega$ is a double-null metric coefficient, not an angular velocity. The last identity is a nonlinear mass-flux equation; fixing the geometry would remove precisely the effect being tested. These variables and the characteristic constraints are specified in Costa, Girão, Natário, and Silva 2018, §§2–3, pp. 9–18, Open PDF.

An explicit reference geometry uses a length $r_0\gt0$:

$$\begin{aligned} \varpi_0&=\frac65r_0,\qquad e^2=\frac{36}{25}r_0^2,\qquad \Lambda=\frac{3}{25r_0^2},\\ f(r)&=-\frac{(r-r_0)(r-2r_0)(r-3r_0)(r+6r_0)} {25r_0^2r^2}. \end{aligned}$$

Direct differentiation gives $r_-=r_0$, $r_+=2r_0$, $r_c=3r_0$ and

$$\kappa_-=\frac{7}{25r_0},\qquad \kappa_+=\kappa_c=\frac{1}{25r_0},\qquad \rho=\frac{\kappa_-}{\kappa_+}=7.$$

The static exterior has $f\gt0$ between $2r_0$ and $3r_0$. Now impose $\mathfrak k(0,v)=1$ and, on the outgoing initial ray,

$$\widehat\varpi(r)=\varpi_0+A(r-2r_0),\qquad A=\frac4{25}.$$

Let $F_H$ be $F$ evaluated at this prescribed mass. Evolve $r_v=F_H(r)$ from $r(0,0)=39r_0/20$, and choose $\theta=+\sqrt{2AF_H}$. This satisfies the mass constraint because $\widehat\varpi'(r)r_v=AF_H=\theta^2/2$. The elementary checks are $F_H(2r_0)=0$, $F_H'(2r_0)=-2\kappa_+$ and $F_H\gt0$ just below $2r_0$. In fact, with $y=(2r_0-r)/r_0$,

$$F_H=\frac{y(8-7y+8y^2-y^3)}{25(2-y)^2}\gt0 \quad(0\lt y\leq1/10).$$

Consequently $2r_0-r$ is comparable to $e^{-2\kappa_+v}$ and $\partial_v\varphi$ to $e^{-\kappa_+v}$ at late $v$. On the intersecting initial ray take $r(u,0)=39r_0/20-u$, $\nu_r(u,0)=-1$ and $r\partial_u\varphi(u,0)=0$; the remaining variables obey the constraints. The domain $0\leq u\leq U$ is chosen sufficiently small. These are local characteristic data near an existing horizon, not generic complete collapse data.

The time normalization matters. Changing from $(u,v)$ to $(r,v)$ gives

$$ds_{(2)}^2=-4\mathfrak k^2F\,dv^2+4\mathfrak k\,dv\,dr.$$

At $\mathfrak k=1$ the reference advanced Eddington–Finkelstein time is $v_{\rm EF}=2v$. Thus a tail $e^{-s\kappa_+v}$ has $\beta_{\rm tail}=s/(2\rho)$ in that stationary normalization; the present data have $s=1$ and $\beta_{\rm tail}=1/14$. This is an imposed tail exponent, not a computed quasinormal gap.

For these data $s\lt\min(\rho,2)$. The nonlinear result gives a positive-radius inner null boundary and continuous extensions of metric and scalar, but $\varpi(u,v)\to+\infty$ for each fixed $0\lt u\leq U$, precluding a $C^2$ metric extension Costa et al. 2018, Theorem 1.1(1)–(2), pp. 6–7, and Theorem 10.1 with proof, p. 44, Open PDF. The proof combines characteristic estimates with the positive mass-flux identity; a sufficiently slow lower tail defeats the assumed finite-mass bound. It is a nonlinear PDE theorem, not a consequence of our algebra alone.

Its geometric meaning is transparent in the ambient angular sectional curvature,

$$R_{\hat\theta\hat\phi\hat\theta\hat\phi} =\frac{1-F}{r^2} =\frac{2\varpi}{r^3}-\frac{e^2}{r^4}+\frac{\Lambda}{3}.$$

At positive limiting radius, mass inflation makes this component diverge. It is not the intrinsic curvature of a symmetry sphere. Continuous extendibility can therefore coexist with loss of higher metric regularity. This example has different parameters and data from the preceding linear mode example: it does not prove that nonlinear effects restore every claimed linear violation, nor does a no-$C^2$ conclusion exclude every weak extension.

Spectral and regularity results through 2026-08-12

Background and perturbation	Spectral evidence through the cutoff	Supported conclusion	Missing step
Reissner–Nordström–de Sitter, neutral massless scalar	All three mode families were resolved; near extremality the dominant ratio can exceed $1/2$	Linear scalar $H^1$ extendibility is possible in part of parameter space, motivating a violation claim for that formulation	Coupled nonlinear Einstein–Maxwell–scalar genericity and stronger metric regularity
Kerr–de Sitter, scalar and linearized gravity	Slowly decaying modes were found for every nonextremal parameter examined	The Christodoulou threshold is respected in these linear sectors Dias et al. 2018, §§2–5	A standalone nonlinear Kerr–de Sitter SCC theorem in every stated data class
Kerr–Newman–de Sitter, neutral massless scalar	The published 2024 numerical scan finds a candidate violation region near the small, nearly extremal RNdS corner	Rotation and charge cannot be summarized by a single monotone trend Davey, Dias, and Sola Gil 2024, §§5–6, pp. 40–41, Open PDF	Not a uniform parameter-space theorem; the sparsely sampled near-extremal region is qualified in footnote 17, p. 40; no coupled gravito-electromagnetic or nonlinear conclusion
Rough versus smooth data	Extra high-frequency or low-regularity content can change which inner-horizon singularity is generic	A smooth-data spectral statement need not answer a rough-data formulation	One must specify and prove density/genericity in the selected topology
Spherical nonlinear Einstein–Maxwell–scalar characteristic data	The explicit slow-tail example above satisfies a mass-inflation theorem	Continuous metric extension but no $C^2$ extension	Different data and parameters from the linear example; not a general collapse or weak-inextendibility theorem

Cardoso et al. concluded from neutral scalar QNMs that the Christodoulou formulation could fail near extremal RNdS Cardoso et al. 2018, pp. 1–6. Dias, Eperon, Reall, and Santos found the opposite linear threshold behavior for nonextremal Kerr–de Sitter, for scalar and gravitational perturbations Dias et al. 2018, §§2–5. The conclusions are not contradictory: the background family and perturbation system differ.

Competing claims and what would decide between them

One defensible position is that a region with $\beta>1/2$ constitutes a genuine counterexample to Christodoulou SCC for the stated smooth linear model. Another is that SCC is fundamentally a nonlinear, generic-data statement and remains undecided until coupled backreaction and the chosen metric regularity are controlled. The first isolates a sharp failure of a linear mechanism; the second refuses to identify that mechanism with the whole conjecture. Neither supports the unqualified sentence “SCC is violated in de Sitter.”

Counterevidence to a proposed violation can come from a previously missed, more slowly decaying physical mode; a different admissible data class; or a nonlinear obstruction in the same proposed extension class. Conversely, discovering a mode with a larger $\beta$ cannot remove an existing slower generic contribution. For mode sets $\mathcal S,\mathcal T$,

$$\inf_{\mathcal S\cup\mathcal T}\beta_j =\min\left(\inf_{\mathcal S}\beta_j,\inf_{\mathcal T}\beta_j\right).$$

Adding $\beta=4/5$ to a family containing $\beta=2/5$ leaves the infimum at most $2/5$. To challenge a proposed restoration, one must show that the presumed slow contribution is not physical or generically excited in the stated problem, and control the remaining families, or construct the nonlinear weak extension at issue. A cancellation for specially tuned data does not establish a generic cancellation.

A nonlinear inextendibility or stability theorem, a newly dominant gravito-electromagnetic mode, a corrected near-extremal spectrum, or a different data or regularity class can change the conclusion. Each comparison must therefore recompute $\alpha$, $\kappa_-$, mode genericity, and the local integrability exponent under the new hypotheses.

Common pitfalls

Treating one mode as the gap. The least damped physical mode can change family across parameter space. Track modes continuously and minimize only after imposing the full domain and constraint conditions.

Equating $\beta>1/2$ with a smooth extension. The threshold concerns a particular local $H^1$ test-field criterion. It does not promise bounded curvature, uniqueness, or a nonlinear Einstein extension.

Exercises

Derive the scalar H¹ threshold

Assume $\phi(V_-)=|V_-|^\beta$ near $V_-=0$ with $\beta>0$. Determine when $\phi$ is continuous and when its first weak derivative is locally square integrable. Explain why the answer alone does not settle nonlinear SCC.

Solution

For every $\beta\gt0$, $\phi\to0$, so this model field is continuous. Its derivative has magnitude $\beta|V_-|^{\beta-1}$. For $\beta\gt1/2$,

$$\int_0^\epsilon |\partial_{V_-}\phi|^2dV_- =\beta^2\frac{\epsilon^{2\beta-1}}{2\beta-1}$$

is finite. For $0\lt\beta\lt1/2$ the improper integral diverges, while at $\beta=1/2$ it diverges logarithmically. This calculation fixes only the local scalar $H^1$ threshold. It neither enforces the Einstein constraints nor controls connection, curvature, nonlinear stress, generic excitation, or uniqueness of an extended maximal development, so those hypotheses require separate analysis.

References

• Cardoso, V., Costa, J. L., Destounis, K., Hintz, P., and Jansen, A. (2018). “Quasinormal modes and strong cosmic censorship.” Physical Review Letters 120, 031103. DOI.
• Costa, J. L., Girão, P. M., Natário, J., and Silva, J. D. (2018). “On the occurrence of mass inflation for the Einstein–Maxwell-scalar field system with a cosmological constant and an exponential Price law.” Communications in Mathematical Physics 361, 289–341. DOI. Open PDF.
• Davey, A., Dias, O. J. C., and Sola Gil, D. (2024). “Strong cosmic censorship in Kerr–Newman–de Sitter.” Journal of High Energy Physics 07, 113. DOI. Open PDF.
• Dias, O. J. C., Eperon, F. C., Reall, H. S., and Santos, J. E. (2018). “Strong cosmic censorship in de Sitter space.” Physical Review D 97, 104060. DOI.