de Sitter and AdS Boundary Conditions

Changing the asymptotic geometry changes the perturbation operator. In de Sitter space the exterior is bounded by a black-hole horizon and a future cosmological horizon; in anti-de Sitter space the outer endpoint is a timelike conformal boundary whose domain must be prescribed. The resulting poles, decay rates, energy identities, and stability statements cannot be imported from asymptotic flatness by changing only a potential term.

Helpful background. Black-Hole Geometry and Mechanics supplies cosmological black-hole horizons; Partial Differential Equations and Green Functions supplies operator domains, resolvents, and spectral gaps; Formulations of General Relativity supplies field-specific perturbation variables; Gravitational Actions and Boundary Terms supplies symplectic flux and boundary variations.

One boundary dictionary

In four spacetime dimensions, write

$$\Lambda=\begin{cases} +3/L^2,&\text{de Sitter},\\ -3/L^2,&\text{anti-de Sitter}. \end{cases}$$

The shared Fourier convention is $e^{-i\omega t}$. A stable resonance therefore has $\operatorname{Im}\omega<0$. What differs is the endpoint data:

Geometry	Inner endpoint	Outer endpoint	Retarded or spectral condition
asymptotically de Sitter black hole	future event horizon	future cosmological horizon	continued retarded absorption on the left and escape on the right
asymptotically AdS black hole	future event horizon	timelike conformal boundary	continued retarded horizon branch; one declared outer boundary domain
pure global AdS	regular center	timelike conformal boundary	center regularity plus a declared positive self-adjoint zero-flux domain gives real normal modes

For the de Sitter static patch, the tortoise coordinate runs from $-\infty$ at the event horizon to $+\infty$ at the cosmological horizon. The retarded phases are $e^{-i\omega r_*}$ and $e^{+i\omega r_*}$, respectively. They are the future-smooth branches at generic frequencies; exceptional Frobenius frequencies require continuation of the retarded solution, not smoothness alone. For AdS, “outgoing at infinity” is not a default condition. In AdS$_{d+1}$, a scalar of real effective mass squared $m_{\rm eff}^2$ has, for noncoincident indicial powers near the inverse-radius coordinate $z=L/r=0$,

$$\phi\sim\alpha z^{\Delta_-}+\beta z^{\Delta_+}, \qquad \Delta_\pm=\frac d2\pm\nu,\qquad \nu=\sqrt{\frac{d^2}{4}+m_{\rm eff}^2L^2},$$

where the four-dimensional case has $d=3$. In the two-normalizable-power window $0\lt\nu\lt1$, the raw boundary symplectic form is proportional to $\alpha_1^*\beta_2-\beta_1^*\alpha_2$. The coincident case $\nu=0$ requires logarithms, and arbitrary slow data outside this window need a different prescription. Dirichlet, Neumann, and real Robin choices are different domains. The exact-AdS positive-extension classification is not automatically a black-hole positivity theorem Ishibashi and Wald 2004, Theorem 3.2 and proof, pp. 35–38; discussion p. 39, Open PDF.

Six pages in logical order

1. Black-Hole and Cosmological-Horizon Conditions in de Sitter derives both future-horizon phases in regular null coordinates and checks their signed currents.
2. Spectral Gaps and Strong-Cosmic-Censorship Interfaces compares exterior decay with Cauchy-horizon blueshift while keeping matter model, data class, and extension regularity explicit.
3. Reflecting, Transparent, and Mixed AdS Conditions uses the boundary symplectic form to classify conservative domains and distinguishes symmetry from positive energy.
4. AdS Normal Modes and Black-Hole Quasinormal Modes explains how selected global-AdS cavity modes should be followed into lossy black-hole resonances and separates small- from large-hole scaling.
5. Superradiant and Trapping Instabilities in AdS distinguishes upper-half-plane modes, stable-trapping decay obstructions, boundary-driven growth, and nonlinear endpoint conjectures.
6. Numerical Spectra with Nonflat Boundaries constructs a horizon-regular Schwarzschild–AdS pencil, reproduces a standard mode with convergence data, and distinguishes a pure-de Sitter formula smoke check from the two-horizon operator test still required.

The order moves from causal endpoint orientation to consequences. The SCC question requires the de Sitter domain before a spectral gap can be interpreted. AdS spectra require a boundary domain before normal modes, QNMs, or instabilities can be compared. Numerical work comes last because a discretization cannot repair an unspecified operator.

Three reader routes

Route	Pages	Decisive question
de Sitter decay and determinism	1 → 2	Which physical mode fixes $\alpha$, what is $\beta=\alpha/\kappa_-$, and which field/data/regularity formulation is being tested?
AdS domains and resonances	3 → 4 → 6	Is boundary symplectic flux zero, which falloff is fixed, and does the tracked pole recover the correct pure-AdS or black-hole benchmark?
AdS stability	3 → 5 → 6	Is there a positive energy, an exponentially growing mode, or a long-lived approximate solution obstructing uniform decay?

The first route prevents an exterior linear calculation from becoming an unqualified verdict on nonlinear strong cosmic censorship. Neutral scalar QNMs suggest a Christodoulou-threshold issue near extremal Reissner–Nordström–de Sitter Cardoso et al. 2018, pp. 1–6, while scalar and gravitational Kerr–de Sitter modes give a different result Dias et al. 2018, §§2–5. Background, field, and regularity explain the difference.

The second route begins with the boundary form, not a customary label. The conformal scalar example on the boundary page has two positive domains, with lowest frequencies $2/L$ and $1/L$. A third, real Robin choice admits a negative spatial eigenvalue despite zero flux. Replacing the center by a future horizon adds absorption; damping follows when the actual black-hole domain retains positive energy. Horowitz and Hubeny compute the standard scalar problem Horowitz and Hubeny 2000, §§II–IV. Changing the Robin parameter while changing horizon size conflates two operator deformations.

The third route separates amplification from confinement and growth. A globally timelike generator and compatible positive energy can exclude growth in its domain. Outside that regime, a growing mode still has to be constructed. Separately, exponentially accurate real-frequency quasimodes obstruct rapid uniform decay. The quasimode construction does not itself locate exact poles; the chapter keeps that additional spectral question separate.

A recurring scalar example

Use one neutral scalar to carry the distinctions. In Schwarzschild–de Sitter, the retarded endpoint phases are $e^{-i\omega r_*}$ at $r_+$ and $e^{+i\omega r_*}$ at $r_c$. If a charged extension has an inner Cauchy horizon, compare $\alpha$ with $\kappa_-$ only after checking all physical mode families, the resonance-expansion bounds, and excitation and transmission of the relevant branch. Adding a faster-decaying mode cannot remove an already established slower contribution.

In global AdS$_{d+1}$, use regularity at the center and fast falloff. The real normal frequencies are

$$\omega L=2n+\ell+\Delta_+.$$

For the fixed-label, massless, fast-falloff small-hole example, selected Schwarzschild–AdS branches approach these values with a small negative imaginary part. This is a source-backed matched limit, not a numerical continuation performed here. For large horizon radius $r_+\gg L$, $\omega/T$ is the useful thermal scaling for the specified sectors. With rotation, real-frequency horizon flux changes sign in $0\lt\omega\lt m\Omega_H$ in the nonrotating boundary frame. A growing mode must still solve the complex return equation, including phase and all losses; at its root the complete return factor equals one.

This one field therefore yields four distinct objects: de Sitter two-horizon resonances, an inner-horizon regularity diagnostic, global-AdS normal modes, and AdS black-hole QNMs. The differential equation alone does not identify which one has been computed.

Distinctions that prevent false conclusions

Cosmological escape versus AdS reflection. A future cosmological horizon is a characteristic sink. A conservative AdS boundary is a chosen zero-flux operator domain.

Spectral gap versus censorship theorem. A gap controls linear decay only with the appropriate resolvent and data hypotheses. The explicit nonlinear characteristic example on the SCC page permits a continuous metric while mass inflation rules out a $C^2$ extension. That example does not settle every weak-extension formulation or every exterior initial-data problem.

Zero flux versus positive energy. A real Robin condition can make the boundary form vanish while admitting a negative eigenvalue. Self-adjointness alone is not stability.

Long life versus instability. Stable trapping supplies exponentially accurate approximate solutions and a logarithmic uniform-decay obstruction. A law for exact QNM widths needs separate justification. Linear exponential growth requires $\operatorname{Im}\omega\gt0$.

Residual versus correct spectrum. A tiny matrix residual only verifies a discretized pencil. Boundary orientation, falloff, branch tracking, convergence, and an independent method verify the physical mode.

Review the chapter

A numerical code returns the same complex number for three runs labeled “dS,” “AdS Dirichlet,” and “AdS Robin.” Give one diagnostic for each run that must differ if the problems were implemented correctly, and explain why equal eigenvalues alone prove nothing.

Solution

For de Sitter, reconstruct the retarded endpoint branches in regular null coordinates, using analytic continuation at exceptional frequencies. For positive real-frequency branch tests, the currents point out of opposite ends of the tortoise interval. For AdS Dirichlet, verify that the slow coefficient $\alpha$ vanishes. In the admissible two-power Robin window, verify $\beta=\kappa\alpha$ for the stated real $\kappa$ and check zero symplectic flux; check positivity separately if stability is claimed.

The same number can occur accidentally, through a limiting symmetry, or because all three labels call one unchanged matrix. Only endpoint residuals and continuation under changes of $\Lambda$, horizon size, and Robin parameter discriminate the domains. A trustworthy comparison also increases resolution and reproduces a pure de Sitter or global-AdS analytic limit.

Limits of the linear boundary problem

This chapter treats linear perturbation spectra and causal response when cosmological or timelike boundaries replace asymptotically flat infinity. The comparisons of SCC and AdS-instability results cover the literature through 2026-08-12; their conclusions depend on boundary conditions and regularity. Causality and Global Structure develops the global causal classification. A holographic interpretation requires further assumptions about the boundary theory.

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.
• 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.
• Horowitz, G. T., and Hubeny, V. E. (2000). “Quasinormal modes of AdS black holes and the approach to thermal equilibrium.” Physical Review D 62, 024027. DOI.
• Ishibashi, A., and Wald, R. M. (2004). “Dynamics in non-globally-hyperbolic static spacetimes. III. Anti-de Sitter spacetime.” Classical and Quantum Gravity 21, 2981–3014. DOI.