Black-Hole and Cosmological-Horizon Conditions in de Sitter

In a de Sitter black-hole exterior, the retarded problem has two future horizons and no spatial infinity. Its continued frequency-domain solutions select future event-horizon absorption and future cosmological-horizon escape. At generic frequencies these are precisely the branches smooth in the respective regular null coordinates. At exceptional Frobenius frequencies, continuation of the retarded solution, not smoothness alone, specifies the branch. Labels such as “ingoing” and “outgoing” are meaningful only with an endpoint and its flux orientation.

Helpful background. Black-Hole Geometry and Mechanics supplies Killing horizons and surface gravities; Partial Differential Equations and Green Functions supplies retarded domains and characteristic data; Formulations of General Relativity supplies perturbative variables and gauge; Gravitational Actions and Boundary Terms supplies boundary variations and flux terms; Causality and Global Structure supplies the de Sitter conformal diagram; Black-Hole Perturbation Theory supplies the master equations and Fourier convention used here.

The static patch between two horizons

For a four-dimensional Schwarzschild–de Sitter example,

$$ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega_2^2, \qquad f(r)=1-\frac{2GM}{r}-\frac{\Lambda r^2}{3},$$

with $\Lambda>0$. In the subextremal static patch, $f$ has simple positive roots $r_+<r_c$: the black-hole event horizon and cosmological horizon. The tortoise coordinate is fixed by

$$\frac{dr_*}{dr}=\frac1{f(r)}, \qquad r_*\to-\infty\ \text{at }r_+, \qquad r_*\to+\infty\ \text{at }r_c.$$

Near a simple root $r_i$, $f\simeq f'(r_i)(r-r_i)$. It is useful to define positive surface gravities

$$\kappa_+=\frac12 f'(r_+), \qquad \kappa_c=-\frac12 f'(r_c),$$

so the opposite signs of $f'$ at the two endpoints do not get hidden. The analysis below assumes both are nonzero. In the Nariai limit $r_+\to r_c$, the two simple-root expansions cease to be uniform and a separate scaled problem is required.

For a concrete reduction, take a neutral, minimally coupled, massless test scalar and separate the physical field as

$$\Phi=e^{-i\omega t}\frac{X(r)}rY_{\ell m}(\theta,\phi), \qquad V=f\left[\frac{\ell(\ell+1)}{r^2}+\frac{f'}r\right].$$

Substitution in $\Box\Phi=0$, using $\partial_{r_*}=f\partial_r$, cancels the first derivative of $X$ and gives

$$\frac{d^2X}{dr_*^2}+[\omega^2-V(r)]X=0,$$

Here $V$ is real and tends to zero exponentially at both simple horizons. Other master fields can have the same endpoint operator without sharing this potential or the scalar’s $1/r$ field reconstruction. Coupled gravitational and electromagnetic perturbations of a charged background also differ from a charged test field. We use the neutral scalar to fix the endpoint signs; changing the field requires deriving its own reduced equation.

Regularity at the future event horizon

Define $v=t+r_*$ and $u=t-r_*$. At $r=r_+$, an ingoing local solution has

$$X\sim e^{-i\omega r_*}, \qquad e^{-i\omega t}X\sim e^{-i\omega v}.$$

The advanced coordinate $v$ is regular across the future event horizon, so the selected branch has a regular reduced radial factor there. The other branch behaves as $e^{-i\omega u}$ and is adapted to the past horizon. It is excluded from the retarded exterior problem by continuation from the convergent Laplace half-plane, even at an exceptional frequency where the other reduced branch may also be smooth. Integer differences of Frobenius exponents can produce logarithms or a degeneracy of the usual normalization; one cannot then select the retarded solution by a smoothness test alone. See Berti, Cardoso, and Starinets 2009, §3.1, p. 17, Eq. (29), Open PDF.

Equivalently, the local Frobenius powers are

$$X\sim(r-r_+)^{\mp i\omega/(2\kappa_+)},$$

with the minus exponent selecting future-horizon regularity for the site’s $e^{-i\omega t}$ convention. For a rotating or charged horizon, replace $\omega$ by the generator frequency $k_+=\omega-m\Omega_+-q\Phi_+$. The geometric statement—regularity along the future generator—survives this replacement; the sign of Killing-energy flux may not.

Escape through the future cosmological horizon

At $r=r_c$, a ray leaving the static observer’s patch moves toward increasing $r_*$ and constant $u$. The retarded cosmological-horizon branch is therefore

$$X\sim e^{+i\omega r_*}, \qquad e^{-i\omega t}X\sim e^{-i\omega u}.$$

At generic frequencies this is the branch smooth across the future cosmological horizon; the same continuation prescription handles exceptional frequencies. The branch $e^{-i\omega r_*}$ depends on $v$ and represents radiation entering the static patch from the past cosmological horizon. Calling the first branch “outgoing” means “out of the static patch.”

With $x=r_c-r>0$, one has $r_*\simeq-(2\kappa_c)^{-1}\log x$. Hence

$$e^{+i\omega r_*}\sim x^{-i\omega/(2\kappa_c)}.$$

The apparently repeated minus exponent at the two horizons is not a repeated propagation direction: $r-r_+$ and $r_c-r$ have opposite radial orientations. Regular null coordinates, rather than a memorized Frobenius sign, decide the physical branch.

Flux and the quasinormal domain

For positive real $\omega$ and a real Schrödinger-form potential, the radial current

$$\mathcal J=\frac{1}{2i}\left(X^*\partial_{r_*}X-X\partial_{r_*}X^*\right)$$

gives $\mathcal J=-\omega|A_+|^2$ for $X=A_+e^{-i\omega r_*}$ and $\mathcal J=+\omega|A_c|^2$ for $X=A_ce^{+i\omega r_*}$. The oriented endpoint data are therefore

Endpoint	Regular coordinate	Selected radial factor	Direction out of the exterior
Future event horizon	$v=t+r_*$	$e^{-i\omega r_*}$	Decreasing $r_*$
Future cosmological horizon	$u=t-r_*$	$e^{+i\omega r_*}$	Increasing $r_*$

The Killing-energy current is a positive convention-dependent multiple of $\omega\mathcal J$. For negative real $\omega$, the Klein–Gordon current reverses sign, but the energy directions in the table do not. These are local branch fluxes, not a nonzero real-frequency source-free solution that satisfies both radiating conditions: constancy of $\mathcal J$ would forbid the opposite endpoint signs for such a solution. The QNM domain is the analytic continuation of the retarded resolvent. Decaying modes have $\operatorname{Im}\omega\lt0$.

These endpoint data differ from asymptotically flat and AdS problems. There is no asymptotically flat outgoing condition at null infinity and no timelike conformal boundary on which to choose a self-adjoint extension. The de Sitter resonance spectrum still depends on the field, angular sector and background. See Berti, Cardoso, and Starinets 2009, §7.1, pp. 49–50, Open PDF.

For compactly supported initial data on a static-patch Cauchy slice, the causal solution has no freely prescribed signal arriving from either past horizon after the initial-value problem is fixed. Its Laplace transform is initially defined where the time integral converges. Analytic continuation then exposes resonances satisfying the two future-horizon conditions above. This construction explains why a QNM profile can grow exponentially in $|r_*|$ when its frequency is damped: the profile belongs to a continued resolvent, not to the original finite-energy Hilbert space.

The same distinction controls sources. Write $F$ for the selected left solution and $G$ for the selected right solution, and differentiate with respect to $r_*$. For a unit radial delta source,

$$W=FG'-F'G,\qquad \widetilde G(r_*,r_*';\omega) =\frac{F(r_{*<},\omega)G(r_{*>},\omega)}{W(\omega)}.$$

The derivative jump is one. On a patch with analytic, nonzero endpoint normalizations, a simple zero of $W$ yields a pole unless the relevant numerator or source projection cancels it. Changing basis by $F\mapsto a(\omega)F$, $G\mapsto b(\omega)G$, with analytic nonvanishing $a,b$, multiplies both numerator and $W$ by $ab$. It changes neither the Green function nor its residue. In particular,

$$\operatorname*{Res}_{\omega=\omega_n}\widetilde G =\frac{F(r_{*<},\omega_n)G(r_{*>},\omega_n)} {\partial_\omega W(\omega_n)}$$

is invariant: the terms involving $\partial_\omega(ab)W$ vanish at the zero. A raw $1/\partial_\omega W$ without the endpoint factors is not a physical excitation amplitude. Singular or vanishing basis normalizations require separate treatment, not automatic identification of every Wronskian zero with a resonance. Initial data affect excitation, while changing a horizon condition to an artificial finite-radius condition changes the spectral problem itself.

Common pitfalls

Using the same verbal direction at both endpoints. “Ingoing at the horizon” is incomplete when the exterior has two horizons. State future event-horizon regularity and future cosmological-horizon escape, then verify both in $v$ or $u$.

Taking the extremal limit inside a simple-root ansatz. When a surface gravity vanishes, $r_*\sim\log|r-r_i|$ is no longer the correct uniform approximation. Rescale the coalescing region before comparing frequencies or boundary powers.

Exercises

Recover both retarded branches

For the subextremal metric above, show from the signs of $f'(r_+)$ and $f'(r_c)$ that $r_*\to-\infty$ at $r_+$ and $r_*\to+\infty$ at $r_c$. Then identify the branch smooth at each future horizon and compute its current for real positive $\omega$.

Solution

Near $r_+$, $f\simeq2\kappa_+(r-r_+)$, so $r_*\simeq(2\kappa_+)^{-1}\log(r-r_+)\to-\infty$. The future-regular dependence is $e^{-i\omega(t+r_*)}=e^{-i\omega v}$, hence $X\sim e^{-i\omega r_*}$ and $\mathcal J=-\omega|A_+|^2$.

Near $r_c$, $f\simeq2\kappa_c(r_c-r)$ and integration gives $r_*\simeq-(2\kappa_c)^{-1}\log(r_c-r)\to+\infty$. Escape across the future cosmological horizon has $e^{-i\omega(t-r_*)}=e^{-i\omega u}$, hence $X\sim e^{+i\omega r_*}$ and $\mathcal J=+\omega|A_c|^2$. Both signs point out of the interval $(-\infty,+\infty)$.

References

• Berti, E., Cardoso, V., and Starinets, A. O. (2009). “Quasinormal modes of black holes and black branes.” Classical and Quantum Gravity 26, 163001. DOI.