Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetime

We first consider the behavior of the stress-energy component T_VV near ${\mathcal{CH}}^{R}$ for a classical field (24). This can be determined immediately from the facts that (i) the stress-energy tensor is conserved, ∇^ν T_μν = 0, and (ii) the classical stress-energy tensor is traceless, ${T}_{\;\mu }^{\mu }=0$. Thus, in the null coordinates u, v of (14), we have T_{u v} = 0. The μ = v component of the continuity equation ∇^ν T_μν = 0 then yields

$$\begin{equation}{\partial }_{u}{T}_{vv}=-f{\partial }_{v}\left({f}^{-1}{T}_{uv}\right)=0.\end{equation} \tag{ 26 }$$

Thus, T_{v v} is constant along any 'left moving' null geodesic, such as the blue line in figure 5, i.e., we have T_{v v} (U, v) = T_{v v} (U₀, v). We now take the limit as v → ∞ with U₀ < 0 and U > 0, so that (U, v) approaches ${\mathcal{CH}}^{R}$ and (U₀, v) approaches ${\mathcal{H}}_{c}^{L}$ (see figure 5). We transform to regular coordinates V = V₋ and V = V_c near ${\mathcal{CH}}^{R}$ and ${\mathcal{H}}_{c}^{L}$, respectively, (see equations (17) and (18)). We obtain

$$\begin{equation}{T}_{{V}_{-}{V}_{-}}=\frac{{\kappa }_{c}^{2}}{{\kappa }_{-}^{2}}{\left(-{V}_{-}\right)}^{2\frac{{\kappa }_{c}}{{\kappa }_{-}}-2}\;{T}_{{V}_{c}{V}_{c}}.\end{equation} \tag{ 27 }$$

If the classical solution is smooth on ${\mathcal{H}}_{c}^{L}$, then (27) shows that if ${T}_{{V}_{c}{V}_{c}}\ne 0$ on ${\mathcal{H}}_{c}^{L}$ and if κ_c < κ₋, then ${T}_{{V}_{-}{V}_{-}}$ blows up on ${\mathcal{CH}}^{R}$. However, in all cases, if ${T}_{{V}_{c}{V}_{c}}$ has a finite limit on ${\mathcal{H}}_{c}^{L}$, then ${V}_{-}^{2}{T}_{{V}_{-}{V}_{-}}\to 0$ on ${\mathcal{CH}}^{R}$.

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We now consider the corresponding behavior of the quantum stress-energy tensor near ${\mathcal{CH}}^{R}$. The only difference in our analysis from the classical case is that, as is well-known (see e.g. [38] for a thorough discussion) in any conformally invariant field theory in 2 dimensions, we have a trace anomaly of the form

$$\begin{equation}{g}^{\mu \nu }{T}_{\mu \nu }=aR,\end{equation} \tag{ 28 }$$

where a = 1/24π for the case of a massless scalar field. For the metric (25), we have R = −f'', where the prime denotes derivative w.r.t. r, so

$$\begin{equation}{T}_{uv}=\frac{a}{4}f{f}^{{\prime\prime}}.\end{equation} \tag{ 29 }$$

Therefore, the μ = v component of the continuity equation now reads

$$\begin{equation}{\partial }_{u}{T}_{vv}=-f{\partial }_{v}\left({f}^{-1}{T}_{uv}\right)=-\frac{a}{8}{f}^{2}{f}^{{\prime\prime\prime}}.\end{equation} \tag{ 30 }$$

This can be integrated to yield

$$\begin{equation}{T}_{vv}\left(U,v\right)=\frac{a}{8}{\left[2f{f}^{{\prime\prime}}-{\left({f}^{\prime }\right)}^{2}\right]}_{r\left({U}_{0},v\right)}^{r\left(U,v\right)}+{T}_{vv}\left({U}_{0},v\right).\end{equation} \tag{ 31 }$$

We choose U₀ < 0 and U > 0 and take the limit as v → ∞. Since

$$\begin{equation}\underset{_{v\to \infty}}{\mathrm{lim}}\;r\left(U,v\right)=\begin{cases}{r}_{c}\quad \hfill & U{< }0,\hfill \\ {r}_{-}\quad \hfill & U{ >}0.\hfill \end{cases}\end{equation} \tag{ 32 }$$

we obtain

$$\begin{equation}\underset{_{v\to \infty}}{\mathrm{lim}}\;{T}_{vv}\left(U,v\right)=\underset{_{v\to \infty}}{\mathrm{lim}}\;{T}_{vv}\left({U}_{0},v\right)-\frac{a}{2}\left({\kappa }_{-}^{2}-{\kappa }_{c}^{2}\right).\end{equation} \tag{ 33 }$$

This differs from the classical case by the step of $-a\left({\kappa }_{-}^{2}-{\kappa }_{c}^{2}\right)/2$ that occurs at U = 0.

When we transform (33) to regular coordinates V = V₋ and V = V_c near ${\mathcal{CH}}^{R}$ and ${\mathcal{H}}_{c}^{L}$, respectively, the first term will give rise to the behavior (27) found in the classical case. However, the second term will give rise to singular behavior of the form

$$\begin{equation}{T}_{{V}_{-}{V}_{-}}\sim \frac{a}{2}\frac{{\kappa }_{c}^{2}-{\kappa }_{-}^{2}}{{\kappa }_{-}^{2}}\frac{1}{{V}_{-}^{2}}.\end{equation} \tag{ 34 }$$

Unless κ_c = κ₋, this behavior is more singular than the classical behavior. Furthermore, in the quantum case, the ${V}_{-}^{-2}$ divergence of ${T}_{{V}_{-}{V}_{-}}$ at the Cauchy horizon ${\mathcal{CH}}^{R}$ is present whenever κ_c ≠ κ₋ even if ${T}_{{V}_{c}{V}_{c}}$ vanishes on the cosmological horizon ${\mathcal{H}}_{c}^{L}$.

Similar arguments for the RN case were previously given in [39]. Apart from not assuming stationarity, the main difference of our argument is that we impose the 'initial condition' of nonsingularity at ${\mathcal{H}}_{c}^{L}$ instead of the event horizon, as in [39], which seems physically much better motivated.

The main shortcoming of the analysis of the previous subsection is that it does not generalize to higher dimensions, as the trace anomaly does not give sufficient information to determine T_{v v} . To make an argument, one has to invoke the—very plausible—absence of numerical coincidence of a cancellation between a geometric (from the trace anomaly) and a state-dependent (from the tangential pressure) contribution to the integral of ∂_u T_{v v} [39]. ¹¹

In order to make contact with the analysis that will be given in the following sections for the four-dimensional case, it is useful to analyze the properties of the Unruh state, defined in the usual way (see e.g. [46] or (53) below) by the positive frequency mode solutions $\square {\psi }_{k}^{\mathrm{U},\mathrm{i}\mathrm{n}/\mathrm{u}\mathrm{p}}=0$ given by

$$\begin{equation}{\psi }_{k}^{\mathrm{U},\mathrm{i}\mathrm{n}}=\frac{1}{\sqrt{2\pi }}\frac{1}{\sqrt{2k}}{\mathrm{e}}^{-\text{i}k{V}_{c}},\quad {\psi }_{k}^{\mathrm{U},\mathrm{u}\mathrm{p}}=\frac{1}{\sqrt{2\pi }}\frac{1}{\sqrt{2k}}{\mathrm{e}}^{-\text{i}kU},\end{equation} \tag{ 35 }$$

We write the metric in the form

$$\begin{equation}g=G\mathrm{d}U\mathrm{d}{V}_{c}\end{equation} \tag{ 36 }$$

The expectation value of the stress tensor in the state defined by these modes is given by [47]

$$\begin{equation}{\langle {T}_{{V}_{c}{V}_{c}}\rangle }_{\mathrm{U}}=-\frac{1}{12\pi }{G}^{\frac{1}{2}}{\partial }_{{V}_{c}}^{2}{G}^{-\frac{1}{2}},\quad {\langle {T}_{UU}\rangle }_{\mathrm{U}}=-\frac{1}{12\pi }{G}^{\frac{1}{2}}{\partial }_{U}^{2}{G}^{-\frac{1}{2}}.\end{equation} \tag{ 37 }$$

One easily verifies that ${\langle {T}_{{V}_{c}{V}_{c}}\rangle }_{\mathrm{U}}$ is finite at the cosmological horizon ${\mathcal{H}}_{c}^{L}$ and that near the Cauchy horizon ${\mathcal{CH}}^{R}$

$$\begin{equation}{\langle {T}_{{V}_{-}{V}_{-}}\rangle }_{\mathrm{U}}=-\frac{1}{48\pi }\left(1-\frac{{\kappa }_{c}^{2}}{{\kappa }_{-}^{2}}\right){V}_{-}^{-2},\end{equation} \tag{ 38 }$$

consistent with (34) and the conformal anomaly coefficient $a=\frac{1}{24\pi }$ of the scalar field.

We remark that in the stationary state defined by the modes (35), also ${\langle {T}_{UU}\rangle }_{\mathrm{U}}$ diverges as U⁻² at ${\mathcal{CH}}^{L}$ in adapted coordinates U = U₋. However, following [40], it is not difficult to see that it is possible to choose a nonstationary state that is regular on ${\mathcal{CH}}^{L}$. To do so, choose a point (u₀, v₀) in the exterior region of RNdS and define Ũ(u) and (v), respectively, as the affine parameter of ingoing and outgoing null geodesics starting at (u₀, v₀), cf figure 6. Defining the state by positive frequency mode solutions ${\left(4\pi k\right)}^{-\frac{1}{2}}{\mathrm{e}}^{-\text{i}k\tilde{U}}$, ${\left(4\pi k\right)}^{-\frac{1}{2}}{\mathrm{e}}^{-\text{i}k\tilde {V}}$, one finds that ⟨T_UU ⟩ is finite on ${\mathcal{CH}}^{L}$, but $\langle {T}_{{V}_{-}{V}_{-}}\rangle$ is still singular at ${\mathcal{CH}}^{R}$. The state is Hadamard also in an extension beyond ${\mathcal{CH}}^{L}$. See figure 6 for a sketch of the setup and the domain in which the state is Hadamard. However, since regularity at ${\mathcal{CH}}^{L}$ is inessential for us, but stationarity is quite convenient for computational purposes, we shall work with the stationary Unruh state defined by the modes (35).

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Note that a single ingoing Unruh-mode ${\psi }_{k}^{\mathrm{U},\mathrm{i}\mathrm{n}}$ has a stress tensor at the Cauchy horizon ${\mathcal{CH}}^{R}$

$$\begin{equation}{T}_{{V}_{-}{V}_{-}}\left[{\psi }_{k}^{\mathrm{U},\mathrm{i}\mathrm{n}}\right]={\partial }_{{V}_{-}}{\psi }_{k}^{\mathrm{U},\mathrm{i}\mathrm{n}}{\partial }_{{V}_{-}}\overline{{\psi }_{k}^{\mathrm{U},\mathrm{i}\mathrm{n}}}=\frac{1}{4\pi }k\frac{{\kappa }_{c}^{2}}{{\kappa }_{-}^{2}}{V}_{-}^{2\frac{{\kappa }_{c}}{{\kappa }_{-}}-2},\end{equation} \tag{ 39 }$$

with a divergence which, of course, is the same as the one found in (27) for the classical stress tensor and is weaker than that found for the full quantum stress tensor. The divergence (38) of the quantum stress tensor is thus a UV effect, i.e. it stems from integration over modes of arbitrarily high frequency. This can be seen more explicitly by—instead of renormalizing the stress tensor via Hadamard point-splitting, i.e. using (37)—computing the difference of the expectation value of ${T}_{{V}_{-}{V}_{-}}$ evaluated in the state defined by the modes (35) and a 'comparison' state which is Hadamard across the Cauchy horizon ${\mathcal{CH}}^{R}$ (so that the latter has ${T}_{{V}_{-}{V}_{-}}$ finite at ${\mathcal{CH}}^{R}$). Such a state can be defined by using, instead of the modes ${\psi }_{k}^{\mathrm{U},\mathrm{i}\mathrm{n}}$, the modes $\square {\psi }_{k}^{\mathrm{C},\mathrm{o}\mathrm{u}\mathrm{t}}=0$, with

$$\begin{equation}{\psi }_{k}^{\mathrm{C},\mathrm{o}\mathrm{u}\mathrm{t}}=\frac{1}{\sqrt{2\pi }}\frac{1}{\sqrt{2k}}{\mathrm{e}}^{-\text{i}kV}\end{equation} \tag{ 40 }$$

where now V = V₋.

This difference in expectation values can be easily computed directly, with a point-split procedure. Having in mind our later discussion of the four-dimensional case, we perform this calculation differently. Restricting to the in-modes, we consider the 'Boulware' mode solutions (just as a computational device, not changing the definition of the state!)

$$\begin{equation}{\psi }_{\omega }^{\mathrm{i}\mathrm{n},\mathrm{I}}\left({V}_{c}\right)=\begin{cases}{\left(2\pi \right)}^{-\frac{1}{2}}{\left(2\omega \right)}^{-\frac{1}{2}}{\mathrm{e}}^{-\text{i}\omega v}\quad \hfill & {V}_{c}{< }0\hfill \\ 0\quad \hfill & {V}_{c}{ >}0\hfill \end{cases},\end{equation} \tag{ 41a }$$

$$\begin{equation}{\psi }_{\omega }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}\mathrm{I}}\left({V}_{c}\right)=\begin{cases}0\quad \hfill & {V}_{c}{< }0\hfill \\ {\left(2\pi \right)}^{-\frac{1}{2}}{\left(2\omega \right)}^{-\frac{1}{2}}{\mathrm{e}}^{-\text{i}\omega v}\quad \hfill & {V}_{c}{ >}0\hfill \end{cases},\end{equation} \tag{ 41b }$$

where for V_c > 0, we defined

$$\begin{equation}v={\kappa }_{c}^{-1}\;\mathrm{log}\;{V}_{c}.\end{equation} \tag{ 42 }$$

Here I stands for the original exterior region and III for the region beyond the cosmological horizon ${\mathcal{H}}_{c}^{L}$, where V_c > 0, cf figure 1. As usual [38], the formal mode integral over the Unruh modes ${\psi }_{k}^{\mathrm{U},\mathrm{i}\mathrm{n}}$ can also be expressed as

$$\begin{equation}{\langle {T}_{vv}\rangle }_{\mathrm{U}}=\int \mathrm{d}\omega \;\mathrm{coth}\left(\pi \omega {\kappa }_{c}^{-1}\right){T}_{vv}\left[{\psi }_{\omega }^{\mathrm{i}\mathrm{n},\mathrm{I}}\right]\end{equation} \tag{ 43 }$$

with

$$\begin{equation}{T}_{vv}\left[{\psi }_{\omega }^{\mathrm{i}\mathrm{n},\mathrm{I}}\right]={\partial }_{v}{\psi }_{\omega }^{\mathrm{i}\mathrm{n},\mathrm{I}}{\partial }_{v}\overline{{\psi }_{\omega }^{\mathrm{i}\mathrm{n},\mathrm{I}}}=\frac{1}{4\pi }\omega \end{equation} \tag{ 44 }$$

the evaluation of T_{v v} in the solution ${\psi }_{\omega }^{\mathrm{i}\mathrm{n},\mathrm{I}}$. For the difference between the expectation value of T_{v v} evaluated in the state defined by the modes ${\psi }_{k}^{\mathrm{U},\mathrm{i}\mathrm{n}}$ and the modes ${\psi }_{k}^{\mathrm{C},\mathrm{o}\mathrm{u}\mathrm{t}}$ defining the comparison state ⟨ . ⟩_C (the latter being Hadamard across ${\mathcal{CH}}^{R}$), one thus obtains

$$\begin{equation}{\langle {T}_{vv}\rangle }_{\mathrm{U}}-{\langle {T}_{vv}\rangle }_{\mathrm{C}}=\frac{1}{4\pi }{\int }_{0}^{\infty }\mathrm{d}\omega \;\omega \left(\mathrm{coth}\frac{\pi \omega }{{\kappa }_{c}}-\mathrm{coth}\frac{\pi \omega }{{\kappa }_{-}}\right)=\frac{1}{48\pi }\left({\kappa }_{c}^{2}-{\kappa }_{-}^{2}\right),\end{equation} \tag{ 45 }$$

which coincides with the result (33), for the anomaly coefficient a = (24π)⁻¹ of the scalar field. We note that in terms of the modes ${\psi }_{\omega }^{\mathrm{i}\mathrm{n},\mathrm{I}}$, the stress tensor for a single mode does give the correct prediction for the degree of singularity of the stress tensor at the Cauchy horizon, cf (44). There is no contradiction to the previous discussion in terms of the ${\psi }_{k}^{\mathrm{U},\mathrm{i}\mathrm{n}}$ modes, as a single ${\psi }_{\omega }^{\mathrm{i}\mathrm{n},\mathrm{I}}$ mode contains arbitrarily high frequencies w.r.t. V_c . The point is that for a single ${\psi }_{\omega }^{\mathrm{i}\mathrm{n},\mathrm{I}}$ mode the stress tensor is divergent both at the cosmological and the Cauchy horizon, but at the cosmological horizon the Boltzmann weights of these modes are exactly such that their contributions cancel upon renormalization. At the Cauchy horizon, on the other hand, the Boltzmann weights are then incommensurate, unless κ_c = κ₋, so that a finite value of T_{v v} (and, thus, a singular stress-energy tensor) at the Cauchy horizon is obtained.

Usually, one thinks of observables, such as the (smeared) stress tensor, as operators on a Hilbert space. However, properly speaking, it is problematical to fix from the outset a fixed Hilbert space representation in which the states live as vectors or statistical operators (density matrices). This is because (a) frequently states must be considered that are not expressible as density matrices on the fixed Hilbert space, and (b) the usual Hilbert space quantization is tied closely to the existence of a Cauchy surface (to define canonical fields), which is not available in our situation.

Therefore, it is formally much more convenient—and essentially requires no work (!)—to set up the quantum theory in an algebraic manner for the purpose of the discussion. Recall that a spacetime $\left(\mathcal{N},g\right)$ is called 'globally hyperbolic' if it has a Cauchy surface Σ, i.e. a spacelike or null codimension-one surface such that every inextendible causal curve hits Σ precisely once, see [44] for details. In such a situation, the Klein–Gordon wave operator □ − μ² possesses a unique retarded and advanced propagator, called E^±. We can think about the propagators as integral operators ${E}^{{\pm}}:{C}_{0}^{\infty }\left(\mathcal{N}\right)\to {C}^{\infty }\left(\mathcal{N}\right)$, which are uniquely determined by the properties

$$\begin{equation}\left(\square -{\mu }^{2}\right){E}^{{\pm}}=1,\quad \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left({E}^{{\pm}}f\right)={J}^{{\pm}}\left(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}f\right),\end{equation} \tag{ 46 }$$

where 'supp' is the support of a function f (the closure of the set where it is not zero), and where ${J}^{{\pm}}\left(\mathcal{O}\right)$ are the causal future/past of a set $\mathcal{O}$, see e.g. [44] for the detailed definition of these notions. The existence and uniqueness property of E^± express the fact that the initial value problem for (□ − μ²)Φ = 0 has a unique solution, i.e. they express determinism. The 'commutator function' is defined as E = E⁺ − E⁻, and it is often identified with a distributional kernel E(x₁, x₂) on $\mathcal{N}{\times}\mathcal{N}$ (this identification implicitly depends on the choice dη of the integration element, which we take to be that given by the metric g). Associated with $\left(\mathcal{N},g\right)$, we then define an algebra, $\mathfrak{A}\left(\mathcal{N},g\right)$, whose generators are 'smeared' field observables Φ(f), their formal adjoints ¹² Φ(f)* and an identity 1, where f is a complex-valued C^∞-function on $\mathcal{N}$ with compact support, and whose relations are:

• (A1)  
  For any f₁, f₂ and any complex numbers c₁, c₂, we have Φ(c₁ f₁ + c₂ f₂) = c₁Φ(f₁) + c₂Φ(f₂).
• (A2)  
  For any f we have Φ[(□ − μ²)f] = 0.
• (A3)  
  For any f we have ${\Phi}{\left(f\right)}^{{\ast}}={\Phi}\left(\overline{f}\right)$.
• (A4)  
  For any f₁, f₂, we have Φ(f₁)Φ(f₂) − Φ(f₂)Φ(f₁) = iE(f₁, f₂)1.

Item (A1) essentially says that Φ(f) is an operator-valued distribution. By analogy with the conventions in distribution theory, we write

$$\begin{equation}{\Phi}\left(f\right)={\int }_{\mathcal{N}}{\Phi}\left(x\right)f\left(x\right)\mathrm{d}\eta \left(x\right)\end{equation} \tag{ 47 }$$

as if Φ(x) was a function, and work from now with the formal point-like object Φ(x). Item (A2) says that (□ − μ²)Φ(x) = 0 in the sense of distributions and item (A3) that Φ(x) = Φ(x)* in the sense of distributions, i.e. we have a hermitian field. Item (A4) gives the usual commutation relation in covariant form [Φ(x₁), Φ(x₂)] = iE(x₁, x₂)1, with E(x₁, x₂) the distributional kernel of E.

In RNdS, see figure 1, the regions I, II, III separately are globally hyperbolic, as is their union. However, region IV is not globally hyperbolic. Since we somehow want to talk about the field in this region, we must find a way to generalize this construction. This can be done in the following cheap way. Let $\mathcal{M}$ be a non globally hyperbolic spacetime such as the the union I ∪ II ∪ III ∪ IV depicted in figure 1. We consider all globally hyperbolic subregions $\mathcal{N}\subset \mathcal{M}$ whose causal structure coincides with that of $\mathcal{M}$, i.e. ${J}_{\mathcal{N}}^{{\pm}}\left(p\right)={J}_{\mathcal{M}}^{{\pm}}\left(p\right)\cap \mathcal{N}$ for all $p\in \mathcal{N}$, where ${J}_{\mathcal{N}}^{{\pm}}\left(p\right)$ denotes the causal future/past of p in $\mathcal{N}$. ¹³ For two such globally hyperbolic subregions ${\mathcal{N}}_{1}$ and ${\mathcal{N}}_{2}$ such that ${\mathcal{N}}_{1}\subset {\mathcal{N}}_{2}$, one then has $\mathfrak{A}\left({\mathcal{N}}_{1}\right)\subset \mathfrak{A}\left({\mathcal{N}}_{2}\right)$. In a first step, let us define a (too large) algebra ${\bigvee }_{\mathcal{N}\subset \mathcal{M}}\mathfrak{A}\left(\mathcal{N},g\right)$ freely generated by the algebras $\mathfrak{A}\left(\mathcal{N},g\right)$ of such globally hyperbolic subregions of $\mathcal{M}$ (i.e. no relations imposed between generators of different $\mathfrak{A}\left({\mathcal{N}}_{i},g\right)$). Then we divide out the relation ${{\Phi}}_{{\mathcal{N}}_{1}}\left(f\right)={{\Phi}}_{{\mathcal{N}}_{2}}\left(f\right)$ for all f, ${\mathcal{N}}_{1}$, ${\mathcal{N}}_{2}$ such that $\text{supp}\;f\subset {\mathcal{N}}_{1},{\mathcal{N}}_{2}$, This construction gives a satisfactory way of defining the observables in a spacetime like RNdS which is not globally hyperbolic, by imposing all relations that can be obtained in globally hyperbolic subregions ¹⁴ .

Of course, to do physics, we not only need an algebra but also states. A state in this framework is simply a positive, normalized, linear functional on $\mathfrak{A}\left(\mathcal{M},g\right)$, where 'positive' means a functional $\mathfrak{A}\left(\mathcal{M},g\right)\ni A{\mapsto}{\langle A\rangle }_{{\Psi}}\in \mathbb{C}$ such that ${\langle {A}^{{\ast}}A\rangle }_{{\Psi}}{\geqslant}0$ for all A, and where 'normalized' means ⟨1⟩_Ψ = 1. Since $\mathfrak{A}\left(\mathcal{M},g\right)$ is presented in terms of generators and relations, a state is given once we know its correlation functions ${\langle {\Phi}\left({x}_{1}\right)\cdots {\Phi}\left({x}_{n}\right)\rangle }_{{\Psi}}$. Among all states, we will focus on 'Gaussian states', which are determined uniquely in terms of their 2-point correlation ('Wightman'-) function ${\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{\Psi}}$, see e.g. [35] for details. From the definitions, we must have

• (S1)  
  (Commutator) ${\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{\Psi}}-{\langle {\Phi}\left({x}_{2}\right){\Phi}\left({x}_{1}\right)\rangle }_{{\Psi}}=iE\left({x}_{1},{x}_{2}\right)$ whenever x₁, x₂ are contained in some globally hyperbolic portion $\mathcal{N}\subset \mathcal{M}$.
• (S2)  
  (Wave equation) ${\left(\square -{\mu }^{2}\right)}_{{x}_{1}}{\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{\Psi}}={\left(\square -{\mu }^{2}\right)}_{{x}_{2}}{\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{\Psi}}=0$.
• (S3)  
  (Positive type) $\int {\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{\Psi}}f\left({x}_{1}\right)\overline{f}\left({x}_{2}\right)\mathrm{d}\eta \left({x}_{1}\right)\mathrm{d}\eta \left({x}_{2}\right){\geqslant}0$ for any smooth, compactly supported function f on $\mathcal{M}$.

Conversely, any distribution ${\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{\Psi}}$ with these properties defines a Gaussian state. One may always represent the field algebra $\mathfrak{A}\left(\mathcal{M},g\right)$ on some Hilbert space such that the expectation functional ⟨.⟩_Ψ corresponds to some 'vacuum' vector in that Hilbert space, although for a generic state the terminology 'vacuum' has no physical meaning. At any rate, it will be fully sufficient for this paper to work with the expectation functionals.

While the above three conditions characterize states in general, to obtain physically reasonable states (in a sense explained below), one should impose more stringent conditions on the short-distance behavior of the 2-point function ${\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{\Psi}}$. One such condition is the 'Hadamard condition'. To state this condition, one introduces the notion of a convex normal neighborhood, $\mathcal{O}$, in the total spacetime $\mathcal{M}$, which is a globally hyperbolic sub-spacetime $\mathcal{O}\subset \mathcal{M}$ such that any pair of points x₁, x₂ from $\mathcal{O}$ can be connected by a unique geodesic. In such a neighborhood, we can define uniquely the signed squared geodesic distance σ(x₁, x₂) for x₁, x₂ from $\mathcal{O}$, and we can, non-uniquely, define a time function T(x). E.g., in RNdS, T(x) = t in region I, or T(x) = r in region II.

Definition 4.1. (Hadamard condition, see, e.g. [49]). For $\left({x}_{1},{x}_{2}\right)\in \mathcal{O}{\times}\mathcal{O}$, where $\mathcal{O}$ is any convex normal globally hyperbolic neighborhood, the 2-point function has the general form

$$\begin{equation}{\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{\Psi}}=\frac{1}{4{\pi }^{2}}\left(\frac{U\left({x}_{1},{x}_{2}\right)}{\sigma +i0T}+V\left({x}_{1},{x}_{2}\right)\mathrm{log}\left(\sigma +i0T\right)+{W}_{{\Psi}}\left({x}_{1},{x}_{2}\right)\right).\end{equation} \tag{ 48 }$$

Here U = Δ^1/2 is the square root of the VanVleck determinant [50], T = T(x₁) − T(x₂), V is determined by the Hadamard–deWitt transport equations [51] and is thereby, as U, σ, entirely determined by the local geometry within $\mathcal{O}$. The state dependence is contained in W_Ψ, which is required to be a C^∞ function on $\mathcal{O}{\times}\mathcal{O}$.

Roughly speaking, the Hadamard condition states that the singular part of the 2-point function ${\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{\Psi}}$ is entirely determined by the local geometry. In any Hadamard state, one can define the n-point correlation functions of 'renormalized composite fields'. Such fields are classically given by polynomials of the field Φ and its covariant derivatives ${\nabla }_{{\mu }_{1}}\dots {\nabla }_{{\mu }_{k}}{\Phi}$ such as the stress tensor ¹⁵

$$\begin{equation}{T}_{\mu \nu }={\nabla }_{\mu }{\Phi}{\nabla }_{\nu }{\Phi}-\frac{1}{2}{g}_{\mu \nu }\left({\nabla }^{\sigma }{\Phi}{\nabla }_{\sigma }{\Phi}+{\mu }^{2}{{\Phi}}^{2}\right).\end{equation} \tag{ 49 }$$

At the quantum level, these are defined (non-uniquely), as operator valued distributions in some larger algebra containing $\mathfrak{A}\left(\mathcal{M},g\right)$ [35, 52]. For us, it is only important how their 1-point function is defined. This is easiest to explain if there are no derivatives and if the polynomial is quadratic, i.e. for Φ². Then one defines

$$\begin{equation}{\langle {{\Phi}}^{2}\left(x\right)\rangle }_{{\Psi}}=\underset{_{{x}_{1},{x}_{2}\to x}}{\mathrm{lim}}\;{W}_{{\Psi}}\left({x}_{1},{x}_{2}\right).\end{equation} \tag{ 50 }$$

As the singular part of (48)—which is effectively subtracted—is covariant, this 'point-split renormalization prescription' ¹⁶ is covariant—in a sense it is the 'same' on all spacetimes, see [35, 54] for the precise meaning of this statement. The 1-point functions ${\langle {\nabla }_{{\mu }_{1}}\dots {\nabla }_{{\mu }_{k}}{\Phi}{\nabla }_{{\nu }_{1}}\dots {\nabla }_{{\nu }_{l}}{\Phi}\left(x\right)\rangle }_{{\Psi}}$ would be defined in the same way, except that we take the derivatives before the coincidence limit. There are certain ambiguities in the definition of the composite quantum fields ${\nabla }_{{\mu }_{1}}\dots {\nabla }_{{\mu }_{k}}{\Phi}{\nabla }_{{\nu }_{1}}\dots {\nabla }_{{\nu }_{l}}{\Phi}\left(x\right)$, in the sense that the above point-split scheme is not the only one which is 'the same on all spacetimes'. These ambiguities are fully characterized in a mathematically precise manner in [35]. They would correspond in the case of Φ² to adding a constant multiple of the scalar curvature and the mass, Φ² → Φ² + c₁ R + c₂ μ², and in the case of more general quadratic composite fields, to adding linear combinations of higher curvature invariants of the correct 'dimension'. In the case of T_μν this freedom must be used to ensure that ∇^μ T_μν = 0. These ambiguities, while important in general, are however of no consequence for our investigation here, since they only give corrections to the expectation values that are smooth functions on spacetime, whereas we are interested in their (potential) divergent part as we approach a Cauchy horizon.

For us, the following properties of Hadamard states will be relevant:

• (H1)  
  For any Hadamard state, ${\langle {\nabla }_{{\mu }_{1}}\dots {\nabla }_{{\mu }_{k}}{\Phi}{\nabla }_{{\nu }_{1}}\dots {\nabla }_{{\nu }_{l}}{\Phi}\left(x\right)\rangle }_{{\Psi}}$ is a C^∞ function of x on $\mathcal{M}$.
• (H2)  
  For any pair Ψ, Ψ' of Hadamard states, ${\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{\Psi}}-{\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{{\Psi}}^{\prime }}$ is a C^∞ function on $\mathcal{N}{\times}\mathcal{N}$ on any globally hyperbolic portion $\mathcal{N}$ of $\mathcal{M}$.
• (H3)  
  For any pair Ψ, Ψ' of Hadamard states and $x\in \mathcal{M}$, we have

  $$\begin{equation}{\langle {{\Phi}}^{2}\left(x\right)\rangle }_{{\Psi}}-{\langle {{\Phi}}^{2}\left(x\right)\rangle }_{{{\Psi}}^{\prime }}=\underset{_{{x}_{1},{x}_{2}\to x}}{\mathrm{lim}}\;\left({\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{\Psi}}-{\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{{\Psi}}^{\prime }}\right),\end{equation} \tag{ 51 }$$

  with a similar formula for general quadratic composite fields ${\nabla }_{{\mu }_{1}}\dots {\nabla }_{{\mu }_{k}}{\Phi}{\nabla }_{{\nu }_{1}}\dots {\nabla }_{{\nu }_{l}}{\Phi}\left(x\right)$.

Properties (H1) and (H3) are immediate consequences of the definition since W is locally smooth. Property (H2) is remarkable, because the Hadamard form only refers to an arbitrarily small neighborhood $\mathcal{O}$, while (H2) makes a statement about arbitrarily large globally hyperbolic subsets of $\mathcal{M}$, such as the union of regions I, II, III in RNdS. Its proof combines nontrivially the commutator, field equation, and positive type property of Hadamard states [55]. It can only be given using 'microlocal' techniques, which we will briefly describe in the next subsection. States which are not Hadamard typically have infinite fluctuations, e.g. ${\langle {{\Phi}}^{2}\left({f}_{1}\right){{\Phi}}^{2}\left({f}_{2}\right)\rangle }_{{\Psi}}=\infty$, of smeared composite fields, even if their expectation value should be finite ${\langle {{\Phi}}^{2}\left(f\right)\rangle }_{{\Psi}}{< }\infty$ (which need not be the case either, see e.g. [56]). Furthermore, interacting quantum field theories (e.g. adding a Φ⁴-interaction) require Hadamard states [52, 57]. Reasonable states in flat spacetime such as the vacuum, thermal, finite particle number, steady states are Hadamard, and if a state is Hadamard near any Cauchy surface Σ in a globally hyperbolic spacetime, i.e. 'initially', then it remains Hadamard [58].

Together, these properties strongly suggest that Hadamard states are the reasonable class to consider as 'regular states' (analogous classically to C^∞-functions) on globally hyperbolic spacetimes. In this paper, we will study in a sense whether a state which is initially Hadamard (in regions I, II, III) remains Hadamard across the Cauchy horizon ${\mathcal{CH}}^{R}$ of RNdS.

Having defined Hadamard states, we must ask whether (a) such states exist, at least on globally hyperbolic spacetimes, and (b) how to construct/characterize concretely Hadamard states on given spacetime representing given physical setups. (a) has been established by several rigorous methods, see [59–63]. In particular, given any one Hadamard state Ψ, we may go to a representation of the field algebra by operators on a Hilbert space, in which the state is represented by a vector. Then, applying any product Φ(f₁)⋯Φ(f_N ) to this vector, we get a new vector giving a new expectation functional. It can be shown that this is again a Hadamard state. Thus, in globally hyperbolic spacetimes, there is an abundance of Hadamard states. (b) In the present context, it is particularly natural to define particular Hadamard states with a concrete interpretation by 'prescribing positive frequency modes' on suitable null surfaces. This idea goes back already to the beginnings of the subject [64]. The Hadamard property of such a state was first established by [62] in the specific context of null-cones in curved space, and later independently by [65] in the case of the 'Unruh state' in Schwarzschild.

4.2.1. 'Local vacuum states' defined on null-cones.

To motivate the constructions in RNdS, we first consider the case of a massless scalar field on Minkowski spacetime. There, we have of course the global vacuum state with the usual Wightman 2-point function ${\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{0}=\frac{1}{4\pi }\;{\left[{\left({x}_{1}-{x}_{2}\right)}^{2}+i0\left({x}_{1}^{0}-{x}_{2}^{0}\right)\right]}^{-1}$. If we restrict attention to the causal future J⁺(0) of the origin 0, we can alternatively write this state in terms of modes with characteristic initial data on the null boundary, ${\dot {J}}^{+}\left(0\right)$, of this cone. For this, we let ψ_ωℓm be the mode solutions □ψ_ωℓm = 0 such that

$$\begin{equation}{\psi }_{\omega \ell m}\left(x\right)={\left(2\pi \right)}^{-\frac{3}{2}}{\left(2\omega \right)}^{-\frac{1}{2}}{Y}_{\ell m}\left(\theta ,\phi \right){r}^{-1}{\mathrm{e}}^{-\text{i}\omega r}\quad \text{on} {\dot {J}}^{+}\left(0\right).\end{equation} \tag{ 52 }$$

Here, r is viewed as an affine parameter along the null generators of ${\dot {J}}^{+}\left(0\right)$. As shown in [62], the vacuum 2-point function can be written inside the future cone as

$$\begin{equation}{\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{0}={\int }_{0}^{\infty }\sum _{\ell ,m}\overline{{\psi }_{\omega \ell m}\left({x}_{1}\right)}{\psi }_{\omega \ell m}\left({x}_{2}\right)\mathrm{d}\omega .\end{equation} \tag{ 53 }$$

This formula can be rewritten as follows. Let Φ(f) = ∫Φ(x)f(x)dη(x) be the smeared field, and let f₁, f₂ be C^∞-functions supported inside the future light cone J⁺(0). Let E = E⁺ − E⁻ be the commutator function (retarded minus advanced fundamental solution) and let ${F}_{1}={E}^{-}{f}_{1}{\vert }_{{\dot {J}}^{+}\left(0\right)}$, with a similar formula for F₂. Then we can write alternatively, with Ω = (θ, ϕ) and d²Ω = dθ² + sin² θdϕ²,

$$\begin{equation}{\langle {\Phi}\left({f}_{1}\right){\Phi}\left({f}_{2}\right)\rangle }_{0}=-\frac{1}{\pi }{\int }_{0}^{\infty }\;\;{\int }_{0}^{\infty }\;\;{\int }_{{S}^{2}}\frac{{r}_{1}{r}_{2}{F}_{1}\left({r}_{1},{\Omega}\right){F}_{2}\left({r}_{2},{\Omega}\right)}{{\left({r}_{1}-{r}_{2}-i0\right)}^{2}}{\mathrm{d}}^{2}{\Omega}\mathrm{d}{r}_{1}\mathrm{d}{r}_{2}.\end{equation} \tag{ 54 }$$

As argued in [62], and later independently in [66], one can generalize these constructions to locally define states associated with lightcones in curved spacetime $\left(\mathcal{M},g\right)$ for a conformally coupled field $\left(\square -\frac{1}{6}R\right){\Phi}=0$, in the following manner.

First, we pick a reference point $p\in \mathcal{M}$, and we let J⁺(p) be its causal future. The boundary ${\dot {J}}^{+}\left(p\right)$ will in general not have the structure of an embedded hypersurface far away from p due to caustics, but defining ${\dot {J}}^{+}\left(p\right)\cap \mathcal{O}$, with $\mathcal{O}$ a convex normal neighborhood of p with smooth boundary, ${\dot {J}}^{+}\left(p\right)\cap \mathcal{O}$ is diffeomorphic to a future lightcone in Minkowski space cut off by a spacelike plane, cf Figure 7. We can introduce coordinates (r, Ω) on ${\dot {J}}^{+}\left(p\right)\cap \mathcal{O}$ analogous to the above as follows: choose a smooth assignment ${S}^{2}\ni {\Omega}{\mapsto}l\left({\Omega}\right)\in {T}_{p}\mathcal{M}$ such that each null ray in ${T}_{p}\mathcal{M}$ is intersected exactly once my this mapping. Then (r, Ω) ↦ exp_p (rl(Ω)) is a diffeomorphism between a neighborhood of the origin in ${\mathbb{R}}_{+}{\times}{S}^{2}$ and ${\dot {J}}^{+}\left(p\right)\cap \mathcal{O}$. Then, we can define ¹⁷ a local vacuum state by

$$\begin{equation}{\langle {\Phi}\left({f}_{1}\right){\Phi}\left({f}_{2}\right)\rangle }_{p}=-\frac{1}{\pi }{\int }_{0}^{\infty }\;\;{\int }_{0}^{\infty }\;\;{\int }_{{S}^{2}}\frac{{r}_{1}{r}_{2}\left({{\Delta}}^{-\frac{1}{2}}{F}_{1}\right)\left({r}_{1},{\Omega}\right)\left({{\Delta}}^{-\frac{1}{2}}{F}_{2}\right)\left({r}_{2},{\Omega}\right)}{{\left({r}_{1}-{r}_{2}-i0\right)}^{2}}{\mathrm{d}}^{2}{\Omega}\mathrm{d}{r}_{1}\mathrm{d}{r}_{2},\end{equation} \tag{ 55 }$$

for testfunctions f₁, f₂ supported in ${J}^{+}\left(p\right)\cap \mathcal{O}$, where now ${F}_{1}=E{f}_{1}{\vert }_{{\dot {J}}^{+}\left(p\right)\cap \mathcal{O}}$, and similarly for F₂. The presence of the VanVleck determinant Δ(p, .), which is equal to 1 in Minkowski spacetime, ensures that the definition is independent of the arbitrary choice of coordinates (r, Ω) [62].

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By construction, the 2-point function (55) defines a state within ${J}^{+}\left(p\right)\cap \mathcal{O}$ because it can be seen to satisfy the commutator, field equation, and positivity requirements [62]. Furthermore, it can be viewed as a 'local vacuum state', in the sense [67] that its construction only depends on the local geometry within ${J}^{+}\left(p\right)\cap \mathcal{O}$, but not on arbitrary choices of coordinates. But is it also Hadamard inside ${J}^{+}\left(p\right)\cap \mathcal{O}$? The answer is yes [62], but the proof cannot be obtained simply by checking the definition. Instead, one must use the methods of microlocal analysis. Since we will use a similar argument below for RNdS spacetime, we here present the derivation. For details on distribution theory and functional analysis, we refer to [68].

If φ is a smooth function on ${\mathbb{R}}^{n}$, then its Fourier transform $\hat{\varphi }\left(k\right)$ decays faster than any inverse power |k|⁻ⁿ as |k| → ∞. If φ is distributional, we say that a non-zero $k\in {\mathbb{R}}^{n}$ is a regular direction at x if there exists a smooth cutoff function χ of compact support such that χ(x) ≠ 0 and an open cone Γ containing k such that for all N > 0 there is a constant C_N for which

$$\begin{equation}\vert \hat{\chi \varphi }\left(p\right)\vert {\leqslant}{C}_{N}{\left(1+\vert p\vert \right)}^{-N}\quad \forall p\in {\Gamma}.\end{equation} \tag{ 56 }$$

The 'wave-front set', WF_x (φ) of φ at point x is, loosely speaking, the set of directions k such that χφ(x) fails to have a rapidly decreasing Fourier transform. More precisely, it is the complement of the regular directions at x, with k = 0 by definition never in WF_x (φ). It can be seen that under a diffeomorphism ψ of a neighborhood of x, WF_x (φ ◦ ψ) = ψ*WF_ψ(x)(φ). Thus, on an n-dimensional manifold $\mathcal{X}$, WF_x (φ) should be viewed as a dilation invariant subset of ${T}_{x}^{{\ast}}\mathcal{X}{\backslash}o$, where o denotes the zero section. One also sets $\text{WF}\left(\varphi \right)={\bigcup }_{x\in \mathcal{X}}{\text{WF}}_{x}\left(\varphi \right)\subset {T}^{{\ast}}\mathcal{X}{\backslash}o$. In the present context, we take $\mathcal{X}=\mathcal{N}{\times}\mathcal{N}$, where $\mathcal{N}$ is some globally hyperbolic subset of a spacetime $\mathcal{M}$.

By an important result of Radzikowski [55], a state is Hadamard if and only if its 2-point correlation function ${G}_{{\Psi}}\left({x}_{1},{x}_{2}\right)={\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{\Psi}}$ satisfies

$$\begin{equation}{\text{WF}}^{\prime }\left({G}_{{\Psi}}\right)={\mathcal{C}}^{+},\end{equation} \tag{ 57 }$$

where

$$\begin{equation}{\mathcal{C}}^{{\pm}}{:=}\left\{\left({x}_{1},{\xi }_{1},{x}_{2},{\xi }_{2}\right)\in {T}^{{\ast}}\left(\mathcal{N}{\times}\mathcal{N}\right){\backslash}o\vert \left({x}_{1},{\xi }_{1}\right)\sim \left({x}_{2},{\xi }_{2}\right),\;{\pm}{\xi }_{1}{\unicode{9657}}0\right\}.\end{equation} \tag{ 58 }$$

Here, the notation means ${\xi }_{1}\in {T}_{{x}_{1}}^{{\ast}}\mathcal{N}$ etc and (x₁, ξ₁) ∼ (x₂, ξ₂) means that x₁ and x₂ can be joined by a null geodesic γ such that ξ₁, ξ₂ (viewed as vectors using the metric g) are tangent to γ. ξ ⊳ 0 means that that the corresponding vector is future-pointing. If K(x₁, x₂) is a distributional kernel, the primed wave front set is defined as WF'(K) = {(x₁, ξ₁, x₂, ξ₂)|(x₁, ξ₁, x₂, −ξ₂) ∈ WF(K)}. With this characterization one can show:

Theorem 4.2. The local vacuum state (55) is Hadamard in ${J}^{+}\left(p\right)\cap \mathcal{O}$.

Proof. The full proof is given in [62]. Apart from a minor subtlety arising from tip of the lightcone at p(r = 0), it uses only straightforward techniques from microlocal analysis. The ingredients are as follows.

• The wave front set of the Schwartz kernel K_E of the commutator function satisfies ${\text{WF}}^{\prime }\left({K}_{E}\right)={\mathcal{C}}^{+}\cup {\mathcal{C}}^{-}$; this is a restatement of the 'propagation of singularities theorem' [69].
• The wave front set of the distribution kernel ${K}_{A}\left({r}_{1},{{\Omega}}_{1},{r}_{2},{{\Omega}}_{2}\right)=\delta \left({{\Omega}}_{1},{{\Omega}}_{2}\right)/{\left({r}_{1}-{r}_{2}-i0\right)}^{2}$ satisfies WF'(K_A ) = {(Ω, r, ξ_Ω, ξ_r ; Ω, r, ξ_Ω, ξ_r )|ξ_r > 0} away from the tip r = 0 by direct computation. So, microlocally, the corresponding operator A 'removes the negative frequencies'.
• The kernel K_R of the restriction operator $RF=F{\vert }_{{\dot {J}}^{+}\left(p\right)\cap \mathcal{O}}$ from C^∞-functions on $\mathcal{O}$ to C^∞-functions on ${\dot {J}}^{+}\left(p\right)\cap \mathcal{O}$, parameterized by (r, Ω) via a diffeomorphism $\psi :\left(0,{r}_{0}\right){\times}{S}^{2}\to {\dot {J}}^{+}\left(p\right)\cap \mathcal{O}{\backslash}\left\{p\right\}$ away from the tip satisfies:
  $$\begin{equation}{\text{WF}}^{\prime }\left({K}_{R}\right)\subset \left\{\left(r,{\Omega},{\xi }_{r},{\xi }_{{\Omega}};y,\eta \right)\vert y=\psi \left(r,{\Omega}\right),{\left(\mathrm{d}\psi \right)}^{t}\left(r,{\Omega}\right)\eta =\left({\xi }_{r},{\xi }_{{\Omega}}\right)\quad \;\text{if}\;r{>}0\right\}\end{equation} \tag{ 59 }$$
  by the restriction theorem (theorem 8.2.4 of [68]).

Looking at the definition of the local vacuum state, we see that the 2-point function is a composition of the kernels of E, A, R. These results are then combined with a standard result about the composition of distributional kernels. Let $B:\mathcal{D}\left({\mathcal{X}}_{1}\right)\to {\mathcal{D}}^{\prime }\left({\mathcal{X}}_{2}\right)$ and $A:\mathcal{D}\left({\mathcal{X}}_{2}\right)\to {\mathcal{D}}^{\prime }\left({\mathcal{X}}_{3}\right)$ be linear continuous maps. By the Schwartz Kernel theorem these correspond to distribution kernels ${K}_{B}\in {\mathcal{D}}^{\prime }\left({\mathcal{X}}_{2}{\times}{\mathcal{X}}_{1}\right)$ and ${K}_{A}\in {\mathcal{D}}^{\prime }\left({\mathcal{X}}_{3}{\times}{\mathcal{X}}_{2}\right)$. If $K\in {\mathcal{D}}^{\prime }\left({\mathcal{X}}_{1}{\times}{\mathcal{X}}_{2}\right)$, one defines

$$\begin{equation}\text{WF}{\left(K\right)}_{{\mathcal{X}}_{2}}{:=}\left\{\left({x}_{2},{\xi }_{2}\right)\vert \left({x}_{1},0;{x}_{2},{\xi }_{2}\right)\in \text{WF}\left(K\right)\right\}.\end{equation} \tag{ 60 }$$

Then one has (theorem 8.2.14 of [68]): if K_B has proper support, and

$$\begin{equation}{\text{WF}}^{\prime }{\left({K}_{A}\right)}_{{\mathcal{X}}_{2}}\cap {\text{WF}}^{\prime }{\left({K}_{B}\right)}_{{\mathcal{X}}_{2}}=\varnothing \end{equation} \tag{ 61 }$$

then the composition A ◦ B is well defined and

$$\begin{align}\hfill {\text{WF}}^{\prime }\left({K}_{A{\circ}B}\right)\,\subset \,& {\text{WF}}^{\prime }\left({K}_{A}\right){\circ}{\text{WF}}^{\prime }\left({K}_{B}\right)\hfill \\ \hfill & \cup \left(\left({\mathcal{X}}_{1}{\times}\left\{0\right\}\right){\times}\text{WF}{\left({K}_{B}\right)}_{{\mathcal{X}}_{3}}\right)\cup \left(\text{WF}{\left({K}_{A}\right)}_{{\mathcal{X}}_{1}}{\times}\left({\mathcal{X}}_{3}{\times}\left\{0\right\}\right)\right),\hfill \end{align} \tag{ 62 }$$

where

$$\begin{align}\hfill {\text{WF}}^{\prime }\left({K}_{A}\right){\circ}{\text{WF}}^{\prime }\left({K}_{B}\right){:=}& \left\{\left({x}_{3},{\xi }_{3},{x}_{1},{\xi }_{1}\right)\vert \text{there}\;\text{exist}\;{\xi }_{2}\ne 0\quad \;\text{and}\;\quad {x}_{2}\;\text{s.t.}\right.\hfill \\ \hfill & \left.{\times}\;\left({x}_{2},{\xi }_{2},{x}_{1},{\xi }_{1}\right)\in {\text{WF}}^{\prime }\left({K}_{B}\right)\quad \;\text{and}\;\quad \left({x}_{3},{\xi }_{3},{x}_{2},{\xi }_{2}\right)\in {\text{WF}}^{\prime }\left({K}_{A}\right)\right\}.\hfill \end{align} \tag{ 63 }$$

Combining this information, one then finds that WF'(G_p ) is contained in the set ${\mathcal{C}}^{+}$. The tip of the lightcone in effect plays no role for this argument, as backward null geodesics starting in the interior of ${J}^{+}\left(p\right)\cap \mathcal{O}$ never reach it ¹⁸ . The theorem is proven if we can show that both sets are in fact equal. To see this, introduce ${G}_{p}^{t}\left({f}_{1},{f}_{2}\right)={G}_{p}\left({f}_{2},{f}_{1}\right)$. Then, by the definition of the wave front set,

$$\begin{equation*}\text{WF}\left({G}_{p}^{t}\right)=\text{WF}{\left({G}_{p}\right)}^{t}{:=}\left\{\left({x}_{1},{\xi }_{1},{x}_{2},{\xi }_{2}\right)\vert \left({x}_{2},{\xi }_{2},{x}_{1},{\xi }_{1}\right)\in \text{WF}\left({G}_{p}\right)\right\},\end{equation*}$$

from which it follows that ${\text{WF}}^{\prime }\left({G}_{p}^{t}\right)\subset {\mathcal{C}}^{-}$. Therefore, by the commutator property of the 2-point function,

$$\begin{equation*}{\mathcal{C}}^{+}\cup {\mathcal{C}}^{-}={\text{WF}}^{\prime }\left(E\right)={\text{WF}}^{\prime }\left({G}_{p}-{G}_{p}^{t}\right)\subset {\text{WF}}^{\prime }\left({G}_{p}\right)\cup {\text{WF}}^{\prime }\left({G}_{p}^{t}\right)\subset {\mathcal{C}}^{+}\cup {\mathcal{C}}^{-},\end{equation*}$$

and the above inclusions must in fact be equalities. Since ${\mathcal{C}}^{+}\cap {\mathcal{C}}^{-}=\varnothing$, this means that in fact ${\text{WF}}^{\prime }\left({G}_{p}\right)={\mathcal{C}}^{+}$, which is the statement of the theorem. □

4.2.2. The 'Unruh' and a 'comparison' state in RNdS

We now define two states by a similar construction in RNdS which play a role for the subsequent argument.

Unruh state: the first is an analogue of the 'Unruh state' [70] in Schwarzschild. We first define (see figure 1)

$$\begin{equation}{\mathcal{H}}_{c}={\mathcal{H}}_{c}^{-}\cup {\mathcal{H}}_{c}^{R},\quad \mathcal{H}={\mathcal{H}}^{-}\cup {\mathcal{H}}^{L},\end{equation} \tag{ 64 }$$

so ${\mathcal{H}}_{c}\cup \mathcal{H}$ is formally, i.e. apart from the 'point' i⁻ at infinity, a characteristic surface for the union $\mathcal{N}$ of the regions I, II, III, where the Unruh state will be defined. Heuristically, the Unruh state is specified as in (53) by the 'positive frequency modes (k ⩾ 0)' ${\psi }_{k\ell m}^{\mathrm{U},\mathrm{i}\mathrm{n}/\mathrm{u}\mathrm{p}}$, which are solutions $\left(\square -{\mu }^{2}\right){\psi }_{k\ell m}^{\mathrm{U},\mathrm{i}\mathrm{n}/\mathrm{u}\mathrm{p}}=0$ defined on $\mathcal{N}$ in terms of their initial data on ${\mathcal{H}}_{c}\cup \mathcal{H}$: ¹⁹

$$\begin{equation}{\psi }_{k\ell m}^{\mathrm{U},\mathrm{i}\mathrm{n}}\left(x\right)=\begin{cases}{\left(2\pi \right)}^{-\frac{3}{2}}{\left(2k\right)}^{-\frac{1}{2}}{Y}_{\ell m}\left(\theta ,\phi \right){r}_{c}^{-1}{\mathrm{e}}^{-\text{i}kV}\quad \hfill & \;\text{on}\;{\mathcal{H}}_{c},\hfill \\ 0\quad \hfill & \;\text{on}\;\mathcal{H},\hfill \end{cases}\end{equation} \tag{ 65a }$$

$$\begin{equation}{\psi }_{k\ell m}^{\mathrm{U},\mathrm{u}\mathrm{p}}\left(x\right)=\begin{cases}0\quad \hfill & \;\text{on}\;{\mathcal{H}}_{c},\hfill \\ {\left(2\pi \right)}^{-\frac{3}{2}}{\left(2k\right)}^{-\frac{1}{2}}{Y}_{\ell m}\left(\theta ,\phi \right){r}_{+}^{-1}{\mathrm{e}}^{-\text{i}kU}\quad \hfill & \;\text{on}\;\mathcal{H},\hfill \end{cases}\end{equation} \tag{ 65b }$$

where V = V_c is the Kruskal type (affine) coordinate adapted to the cosmological horizon, and where U is the Kruskal type (affine) coordinate adapted to the event horizon; see section 2.

In order to see that the Unruh state is well-defined, and to understand some of its basic properties, we formally rewrite the definition (53) (with the modes ${\psi }_{k\ell m}^{\mathrm{U},\mathrm{i}\mathrm{n}/\mathrm{u}\mathrm{p}}$) in a similar way as in (55), noting that the VanVleck determinant vanishes on a Killing horizon. Let ${f}_{1},{f}_{2}\in {C}_{0}^{\infty }\left(\mathcal{N}\right)$, and define for X ∈ {+, c} the restrictions of the corresponding solutions, ${F}_{1}^{X}={E}^{-}{f}_{1}{\vert }_{{\mathcal{H}}_{X}},{F}_{2}^{X}={E}^{-}{f}_{2}{\vert }_{{\mathcal{H}}_{X}}$. Then the precise definition of the Unruh state is:

Definition 4.3. The Unruh state is defined by the 2-point function on $\mathcal{N}{\times}\mathcal{N}$ given by:

$$\begin{equation}\begin{aligned}\hfill {\langle {\Phi}\left({f}_{1}\right){\Phi}\left({f}_{2}\right)\rangle }_{\mathrm{U}}=& -\frac{1}{\pi }{\int }_{\mathbb{R}}{\int }_{\mathbb{R}}{\int }_{{S}^{2}}\frac{{F}_{1}^{+}\left({U}_{1},{\Omega}\right){F}_{2}^{+}\left({U}_{2},{\Omega}\right)}{{\left({U}_{1}-{U}_{2}-i0\right)}^{2}}{\mathrm{d}}^{2}{\Omega}\mathrm{d}{U}_{1}\mathrm{d}{U}_{2}\hfill \\ \hfill & -\frac{1}{\pi }{\int }_{\mathbb{R}}{\int }_{\mathbb{R}}{\int }_{{S}^{2}}\frac{{F}_{1}^{c}\left({V}_{1},{\Omega}\right){F}_{2}^{c}\left({V}_{2},{\Omega}\right)}{{\left({V}_{1}-{V}_{2}-i0\right)}^{2}}{\mathrm{d}}^{2}{\Omega}\mathrm{d}{V}_{1}\mathrm{d}{V}_{2}\hfill \end{aligned}\end{equation} \tag{ 66 }$$

where null-generators of the event and cosmological horizon are parameterized by U resp. V, and d²Ω is the induced integration element on the spheres of constant V resp. U.

This definition is actually still formal as it stands, because the convergence of the integrals over U and V has not been ensured. Also, one needs to show the commutator, field equation, and positivity properties (S1)–(S3), and one would also like to show that the Unruh state is Hadamard in the union $\mathcal{N}$ of the regions I, II, III where it is defined. These properties have been proven in the case of Schwarzschild spacetime (Λ = Q = 0) by [65]. A similar proof can be given in the present context, taking into account the features of the dynamical evolution special to RNdS.

First, the positivity property (S3) is obvious–assuming (66) is well defined–because −(x − i0)⁻² is a kernel of positive type (its Fourier transform is non-negative). The field equation (2)) is also obvious, because the commutator function E is a bi-solution. For a Cauchy surface Σ as in figure 1, E can be written in the form [71]:

$$\begin{equation}E\left({f}_{1},{f}_{2}\right)={\int }_{{\Sigma}}{n}^{\mu }\left({F}_{1}{\nabla }_{\mu }{F}_{2}-{F}_{2}{\nabla }_{\mu }{F}_{1}\right)\mathrm{d}{\eta }_{{\Sigma}},\end{equation} \tag{ 67 }$$

with F₁ = Ef₁, F₂ = Ef₂. Since (□ − μ²)F₁ = (□ − μ²)F₂ = 0, Gauss' theorem shows that the expression is independent of the chosen Cauchy surface for $\mathcal{N}$. Now, if we could formally deform Σ to the pair of null surfaces ${\mathcal{H}}_{c}\cup \mathcal{H}$, then we would have, formally

$$\begin{equation}E\left({f}_{1},{f}_{2}\right)={\int }_{\mathbb{R}}{\int }_{{S}^{2}}\left({F}_{1}^{+}{\partial }_{U}{F}_{2}^{+}-{F}_{2}^{+}{\partial }_{U}{F}_{1}^{+}\right){\mathrm{d}}^{2}{\Omega}\mathrm{d}U+{\int }_{\mathbb{R}}{\int }_{{S}^{2}}\left({F}_{1}^{c}{\partial }_{V}{F}_{2}^{c}-{F}_{2}^{c}{\partial }_{V}{F}_{1}^{c}\right){\mathrm{d}}^{2}{\Omega}\mathrm{d}V.\end{equation} \tag{ 68 }$$

Although this argument is not rigorous because it ignores a potential contribution from i⁻, let us proceed for the moment and assume that the integrals in (66) converge in a suitable sense. Then using the formula Im(x − i0)⁻² = −πδ'(x), on both terms, one formally gets

$$\begin{align}\hfill {\langle {\Phi}\left({f}_{1}\right){\Phi}\left({f}_{2}\right)\rangle }_{\mathrm{U}}-{\langle {\Phi}\left({f}_{2}\right){\Phi}\left({f}_{1}\right)\rangle }_{\mathrm{U}}& =2i\mathrm{Im}{\langle {\Phi}\left({f}_{1}\right){\Phi}\left({f}_{2}\right)\rangle }_{\mathrm{U}}\hfill \\ \hfill & =i{\int }_{\mathbb{R}}{\int }_{{S}^{2}}\left({F}_{1}^{+}{\partial }_{U}{F}_{2}^{+}-{F}_{2}^{+}{\partial }_{U}{F}_{1}^{+}\right){\mathrm{d}}^{2}{\Omega}\mathrm{d}U\hfill \\ \hfill & \quad +i{\int }_{\mathbb{R}}{\int }_{{S}^{2}}\left({F}_{1}^{c}{\partial }_{V}{F}_{2}^{c}-{F}_{2}^{c}{\partial }_{V}{F}_{1}^{c}\right){\mathrm{d}}^{2}{\Omega}\mathrm{d}V.\hfill \end{align} \tag{ 69 }$$

Combining with the previous equation, this would show the commutator property (S1).

In order to make this argument rigorous, we should first check (68), and for that, we need the asymptotic behavior of F₁, F₂ near i⁻. This asymptotic behavior has been analyzed by Hintz and Vasy [1] and is best expressed resolving i⁻ by a blow up procedure sketched in section 2. The results of [1] express the regularity of the forward solution F = E⁺ f near i⁺ in terms of certain 'variable order' Sobolev spaces. By time reflection symmetry, these analogously hold for F = E⁻ f near i⁻. The variable orders of (fractional) differentiability express the different regularity properties of the solution near the various horizons, i.e. the different locations of r, see appendix A of [1] for the precise definitions. The variable forward order functions s(r) considered by these authors are such that s(r) is constant for r < r₋ + and r > r₋ + 2, with

$$\begin{equation}s\left(r\right)\begin{cases}{< }1/2+\alpha /{\kappa }_{-}\quad \hfill & \text{for} r{< }{r}_{-}+{\epsilon}\hfill \\ {\geqslant}1/2+\mathrm{max}\left\{\alpha /{\kappa }_{c},\alpha /{\kappa }_{-}\right\}\quad \hfill & \text{for} r{ >}{r}_{-}+2{\epsilon}\hfill \end{cases}\end{equation} \tag{ 70 }$$

and s'(r) ⩾ 0. The parameter α of main interest is the spectral gap, but forward order functions can in principle be considered for any $\alpha \in \mathbb{R}$. Based on a forward order function, a certain Sobolev space ${H}_{\mathrm{b},+}^{s}\left(\mathcal{R}\right)$ of functions supported toward the future of the Cauchy surface Σ, is then defined, see appendix B of [1] for details, and figure 4 for the definition of the domain $\mathcal{R}$. Further, [1] define the spaces ${H}_{\mathrm{b},+}^{s,\alpha }\left(\mathcal{R}\right)={\tau }^{\alpha }{H}_{\mathrm{b},+}^{s}\left(\mathcal{R}\right)$ of functions with an exponential decay toward t_* → ∞, i.e. toward i⁺, see section 2 for the definition of t_* = −log τ. Their main result can be phrased as follows:

Theorem 4.4. If ²⁰ μ² > 0 and $f\in {C}_{0}^{\infty }\left(\mathcal{N}\right)$, then $F={E}^{+}f\in {H}_{\mathrm{b},+}^{s,\alpha }\left(\mathcal{R}\right)$ for any forward order function, and furthermore ${\Vert}F{{\Vert}}_{{H}_{\mathrm{b},+}^{s,\alpha }\left(\mathcal{R}\right)}\overset{<}{\sim} {\Vert}f{{\Vert}}_{{C}^{m}\left(\mathcal{N}\right)}$ for some m depending on s.

Proof. The proof of this theorem is essentially contained in that of proposition 2.17 of [1]. Here we only somewhat pedantically go through the norm estimates provided by [1]. [1] first modify the wave operator to P = □ − μ² − iQ, where Q is a suitable formally self-adjoint pseudo-differential operator having its support in the region r < r₋ inside the black hole behind the Cauchy horizon ${\mathcal{CH}}^{L}$. The construction of Q is roughly made in such a way that singularities propagating into the interior of the black hole get 'absorbed' by Q. Then they show (theorem 2.13) that with a suitable choice of $-{\epsilon}{< }\tilde {\alpha }{< }0$ and of Q, the map P is a Fredholm operator between their weighted 'forward' Sobolev spaces ${H}_{\mathrm{b},+}^{\tilde {s},\tilde {\alpha }}\left(\mathcal{R}\right)\to {H}_{\mathrm{b},+}^{\tilde {s}-1,\tilde {\alpha }}\left(\mathcal{R}\right)$, with $\tilde {s}$ a forward order function for $\tilde {\alpha }$, and with $\mathcal{R}$ a domain of the shape indicated in figure 4. More precisely, $\mathcal{R}$ should be extended to the left beyond the Cauchy horizon r < r₋ by effectively modifying the metric function f(r) such that the singularity at r = 0 gets replaced with the origin of polar coordinates or another horizon. By the standard theory of Fredholm operators, the inverse P⁻¹ is defined as a bounded operator on a subspace of ${H}_{\mathrm{b},+}^{\tilde {s}-1,\tilde {\alpha }}\left(\mathcal{R}\right)$ of finite co-dimension. In fact, it is shown in [1], lemma 2.15 that the projector onto this subspace may be chosen as $Rf=f-{\sum }_{i=1}^{N}{\phi }_{i}{v}_{i}\left(f\right)$, with ϕ_i smooth and supported in r < r₋ and v_i distributions. Thus, if $f\in {C}_{0}^{\infty }\left(\mathcal{M}\right)$, it follows that the forward solution of PF = f has

$$\begin{equation}{\Vert}F{{\Vert}}_{{H}_{\mathrm{b},+}^{\tilde {s},\tilde {\alpha }}\left(\mathcal{R}\right)}\overset{<}{\sim} {\Vert}f{{\Vert}}_{{C}^{m}\left(\mathcal{M}\right)}\end{equation} \tag{ 71 }$$

for a sufficiently large m depending on the choice of $\tilde {s}$. Then, by the propagation of singularities theorem, proposition 2.9 of [1], one has

$$\begin{equation}{\Vert}F{{\Vert}}_{{H}_{\mathrm{b},+}^{s,\tilde {\alpha }}\left(\mathcal{R}\right)}\overset{<}{\sim} {\Vert}F{{\Vert}}_{{H}_{\mathrm{b},+}^{\tilde {s},\tilde {\alpha }}\left(\mathcal{R}\right)}+{\Vert}PF{{\Vert}}_{{H}_{\mathrm{b},+}^{\tilde {s},\tilde {\alpha }}\left(\mathcal{R}\right)}\overset{<}{\sim} {\Vert}f{{\Vert}}_{{C}^{m}\left(\mathcal{M}\right)},\end{equation} \tag{ 72 }$$

with $s{\leqslant}\tilde {s}+1$ a forward order function for any weight α > 0 such that α/κ₋ < 1. By [1], section 2 and [26], lemma 3.1 and remark 3.4, (essentially a Fourier–Laplace-transform argument in the variable t_* = −log τ), F can be developed in an asymptotic expansion

$$\begin{equation}F\left(x,{t}_{{\ast}}\right)=\sum _{j}{t}_{{\ast}}^{{m}_{j}}{\mathrm{e}}^{{\sigma }_{j}{t}_{{\ast}}}{a}_{j}\left(x\right)+{F}^{\prime }\left(x,{t}_{{\ast}}\right)\end{equation} \tag{ 73 }$$

with $-\alpha {< }{\sigma }_{j}{< }-\tilde {\alpha }$ the resonances of the Fourier–Laplace-transformed operator $\hat{P}\left(\sigma \right)$, m_j their multiplicity, and with a norm bound

$$\begin{equation}{{\Vert}{F}^{\prime }{\Vert}}_{{H}_{\mathrm{b},+}^{s,\alpha }\left(\mathcal{R}\right)}\overset{<}{\sim} {\Vert}f{{\Vert}}_{{C}^{m}\left(\mathcal{M}\right)}.\end{equation} \tag{ 74 }$$

By [1], lemma 2.14, F' = F in the region r > r₋ if α is chosen so that there are no quasi-normal frequencies for the scattering problem in region I with 0 ⩽ Im(ω_nℓm ) > α—in fact they argue that any resonance of $\hat{P}\left(\sigma \right)$ not equal to such a quasi-normal mode ²¹ has a_j (x) = 0 for r > r₋.

In the case at hand, we have μ² > 0. It follows that there can be no such quasi-normal frequencies on the real line (unlike for the massless wave operator considered by [1], who have to consider separately the constant mode), as one can see by a simple argument involving the Wronskian of the radial equation, cf (104) below, for real ω. Thus, we may put α to be the value (3). □

These results will now be used in order to investigate the properties of the Unruh state. As already mentioned, theorem 4.4 analogously holds for the behavior of the backward solution F = E⁻ f near i⁻. We are interested in its behavior in region I. But, as no lower bound is imposed on the forward order function s(r) in the region r₊ ⩽ r ⩽ r_c , using Sobolev embedding, there follows a bound

$$\begin{equation}\vert {\partial }^{N}F\left({t}_{{\ast}},r,{\Omega}\right)\vert \overset{<}{\sim} {\mathrm{e}}^{\alpha {t}_{{\ast}}}{\Vert}f{{\Vert}}_{{C}^{m}\left(\mathcal{N}\right)}\quad \text{for} {r}_{+}{\leqslant}r{\leqslant}{r}_{c},\end{equation} \tag{ 75 }$$

where ∂^N is a derivative in (${t}_{{\ast}}$, r, Ω) of order N > 0 and m depends only on N. By choosing in (67) a sequence of Cauchy surfaces as depicted in figure 8 moving downwards toward the horizontal boundary, it immediately follows by writing (67) out in the coordinates (${t}_{{\ast}}$, r, Ω) that in the limit only the contributions from the two vertical boundaries ${\mathcal{H}}^{-},{\mathcal{H}}_{c}^{-}$ contribute near i⁻, but not the interpolating surface depicted in green. Hence, in this limit, (68) holds true.

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We may also see immediately that the integrals in (66) converge when f₁, f₂ are functions in ${C}_{0}^{\infty }\left(\mathcal{N}\right)$, making the Unruh state actually well defined. For this, we note that ${\mathrm{e}}^{\alpha {t}_{{\ast}}}\sim \vert U{\vert }^{-\alpha /{\kappa }_{+}}$ for U → −∞ on $\mathcal{H}$ and ${\mathrm{e}}^{\alpha {t}_{{\ast}}}\sim \vert V{\vert }^{-\alpha /{\kappa }_{c}}$ for V → −∞ on ${\mathcal{H}}_{c}$. In combination with (75), this easily implies convergence of the integrals (69). We therefore conclude:

Proposition 4.5. The Unruh state is well-defined on the algebra $\mathcal{A}\left(\mathcal{N}\right),\mathcal{N}=\mathrm{I}\cup \mathrm{I}\mathrm{I}\cup \mathrm{I}\mathrm{I}\mathrm{I}$ generated by Φ(f) with $f\in {C}_{0}^{\infty }\left(\mathcal{N}\right)$.

We now verify that the Unruh 2-point function ${G}_{\mathrm{U}}\left({x}_{1},{x}_{2}\right)={\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{\mathrm{U}}$ is Hadamard in the union of regions I, II, III (but not necessarily across ${\mathcal{CH}}^{R}$!). Let us begin by recalling basic results from the analysis of partial differential equations. For any distribution φ on $\mathcal{X}$ and (pseudo-) differential operator A, it is known that

$$\begin{equation}\text{WF}\left(\varphi \right)\subset \text{WF}\left(A\varphi \right)\cup \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\left(A\right),\end{equation} \tag{ 76 }$$

where $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\left(A\right)=\left\{\left(x,\xi \right)\in {T}^{{\ast}}\mathcal{X}{\backslash}o\vert a\left(x,\xi \right)=0\right\}$, and where a(x, ξ) is the principal symbol of A, see theorem. 8.3.1 of [68]. Furthermore, the propagation of singularities theorem [69] states that if the differential operator A has real principal symbol a and if Aφ is smooth then WF(φ) is contained in the intersection of char(A) and the Hamiltonian orbits of the symbol a(x, ξ) on 'phase space' ${T}^{{\ast}}\mathcal{X}{\backslash}o$. In the case of A = □ − μ², the principal symbol is g^μν (x)ξ_μ ξ_ν , and the Hamiltonian orbits in the characteristic set char(□ − μ²) are precisely the null geodesics with initial condition (x, ξ). We denote one such an orbit, called bicharacteristic, as $\mathcal{B}\left(x,\xi \right)$. Based on this definition, we distinguish three cases:

(1) Neither the bicharacteristic $\mathcal{B}\left({x}_{1},{\xi }_{1}\right)$ nor $\mathcal{B}\left({x}_{2},{\xi }_{2}\right)$ enters region I: in this case, the claim follows by exactly the same argument as given in the previous section for lightcones, because every backward null geodesic emanating from these regions will eventually hit $\mathcal{H}\cup {\mathcal{H}}_{c}$.

(2) The bicharacteristics $\mathcal{B}\left({x}_{1},{\xi }_{1}\right)$ and $\mathcal{B}\left({x}_{2},{\xi }_{2}\right)$ enter region I: by propagation of singularities, it suffices to consider x₁, x₂ ∈ I. In region I, past-directed null geodesics trapped on the photon ring (see e.g. [44]), will not come from $\mathcal{H}\cup {\mathcal{H}}_{c}$ but begin their lives in the 'point' i⁻. Thus, the type of argument made in the case of lightcones—where this cannot occur—does not work, and one needs another argument worked out in detail in the case of Schwarzschild by [65]. The argument is based on the fact that the Unruh state is static, i.e. invariant under translations in the coordinate t. A simplified version of the argument which also works in the present case runs as follows.

First, for an arbitrary but fixed open region $\mathcal{O}$ whose closure is contained in region I (i.e. not touching the horizons $\mathcal{H}\cup {\mathcal{H}}_{c}$), we introduce the maps, X ∈ {c, +},

$$\begin{equation}{K}^{X}:{C}_{0}^{\infty }\left(\mathcal{O}\right)\to {L}^{2}\left({\mathbb{R}}_{\omega }{\times}{S}_{{\Omega}}^{2}\right),\quad {K}^{X}f\left(\omega ,{\Omega}\right)={r}_{X}{\left(\frac{\omega {\mathrm{e}}^{2\pi {r}_{X}\omega }}{\mathrm{sinh}\left(2\pi {r}_{X}\omega \right)}\right)}^{\frac{1}{2}}{\int }_{\mathbb{R}}{F}^{X}\left(s,{\Omega}\right){\mathrm{e}}^{\text{i}\omega s}\mathrm{d}s\end{equation} \tag{ 77 }$$

where ${F}^{+}\left(u,{\Omega}\right)=Ef{\vert }_{{\mathcal{H}}^{-}}\left(u,{\Omega}\right)$ and ${F}^{c}\left(v,{\Omega}\right)=Ef{\vert }_{{\mathcal{H}}_{c}^{-}}\left(v,{\Omega}\right)$. Next, we let Kf := K⁺ f ⊕ K^c f ∈ L² ⊕ L², and then we can the write the Unruh 2-point function as [49]

$$\begin{equation}{\langle {\Phi}\left({f}_{1}\right){\Phi}\left({f}_{2}\right)\rangle }_{\mathrm{U}}={\langle K{f}_{1},K{f}_{2}\rangle }_{{L}^{2}\oplus {L}^{2}},\end{equation} \tag{ 78 }$$

which follows from (66) by Fourier transformation and the relations $U=-{\mathrm{e}}^{-{\kappa }_{+}u},V=-{\mathrm{e}}^{-{\kappa }_{c}v}$. The decay and regularity results, theorem 4.4, together with the Sobolev embedding theorem and the relation between ${t}_{{\ast}}$ and u resp. v on ${\mathcal{H}}^{-}$ resp. ${\mathcal{H}}_{c}^{-}$ imply that $\vert {\partial }_{s}^{N}{F}^{X}\left(s,{\Omega}\right)\vert {\leqslant}{C}_{N}{\mathrm{e}}^{-\alpha \vert s\vert }$, where the constant C_N is controlled by some C^m norm of f by (75). As a consequence, K = K⁺ ⊕ K^c is shown to be a distribution in $\mathcal{O}$ with values in the Hilbert space L² ⊕ L²: since $\vert {\partial }_{s}^{N}{F}^{X}\left(s,{\Omega}\right)\vert < sim {\Vert}f{{\Vert}}_{{C}^{m}\left(\mathcal{O}\right)}{\mathrm{e}}^{-\alpha \vert s\vert }$ for all $f\in {C}_{0}^{\infty }\left(\mathcal{O}\right)$, we find ${\Vert}Kf{{\Vert}}_{{L}^{2}\oplus {L}^{2}}< sim {\Vert}f{{\Vert}}_{{C}^{m}\left(\mathcal{O}\right)}$, which is the continuity required from a distribution.

Next, invariance of E under translations of t in the coordinates (t, ${r}_{{\ast}}$, Ω) and the exponentially decreasing prefactor of order $O\left({\mathrm{e}}^{2\pi {r}_{X}\omega }\right)$ in (77) for ω → −∞ imply that the distribution K has a holomorphic extension to the strip {t + is|s ∈ (0, 2πr₊)} in the t-coordinate. Moreover it is the boundary value of this holomorphic extension, in the sense of distributions, as s → 0⁺, for all test-functions from ${C}_{0}^{\infty }\left(\mathcal{O}\right)$. By a very slight modification of the proof of theorem 3.1.14 of [68] we then get, for some m, the inequality

$$\begin{equation}{\Vert}K\left(t+is,r,{\Omega}\right){{\Vert}}_{{L}^{2}\oplus {L}^{2}}\overset{<}{\sim} {s}^{-m}\quad \;\text{for}\;\left(t,r,{\Omega}\right)\in \mathcal{O}\;\text{and}\;s\in \left(0,\pi {r}_{+}\right).\end{equation} \tag{ 79 }$$

In view of theorem 8.4.8 of [68], we then further get

$$\begin{equation}\text{WF}\left(K\right){\vert }_{\mathcal{O}}\subset \left\{\left(x,\xi \right)\in {T}^{{\ast}}\mathcal{O}{\backslash}o\vert \langle \xi ,{\partial }_{t}\rangle { >}0\right\}.\end{equation} \tag{ 80 }$$

Combining (78) and the rules for the wave front set under the composition of distributions, we finally get

$$\begin{equation}{\text{WF}}^{\prime }\left({G}_{\mathrm{U}}\right){\vert }_{\mathcal{O}{\times}\mathcal{O}}\subset \left\{\left({x}_{1},{\xi }_{1},{x}_{2},{\xi }_{2}\right)\in {T}^{{\ast}}\left(\mathcal{O}{\times}\mathcal{O}\right){\backslash}o\vert \langle {\xi }_{i},{\partial }_{t}\rangle { >}0\right\}.\end{equation} \tag{ 81 }$$

Next, using the relationship between t and u, v, it follows immediately from (77), (78) that the 2-point function G_U is invariant under translation of t in both arguments. In infinitesimal form, $\left({\partial }_{{t}_{1}}+{\partial }_{{t}_{2}}\right){G}_{\mathrm{U}}=0$. From (76), it follows that in fact

$$\begin{equation}{\text{WF}}^{\prime }\left({G}_{\mathrm{U}}\right){\vert }_{\mathcal{O}{\times}\mathcal{O}}\subset \left\{\left({x}_{1},{\xi }_{1},{x}_{2},{\xi }_{2}\right)\in {T}^{{\ast}}\left(\mathcal{O}{\times}\mathcal{O}\right){\backslash}o\vert \langle {\xi }_{1},{\partial }_{t}\rangle =\langle {\xi }_{2},{\partial }_{t}\rangle { >}0\right\}.\end{equation} \tag{ 82 }$$

If we assume that $\mathcal{O}$ is the entire (open) region I, viewed as a globally hyperbolic spacetime in its own right with Cauchy-surface ${{\Sigma}}_{\mathcal{O}}$, then it follows from the propagation of singularities theorem that

$$\begin{equation}{\text{WF}}^{\prime }\left({G}_{\mathrm{U}}\right){\vert }_{\mathcal{O}{\times}\mathcal{O}}\subset \left\{\mathcal{B}\left({x}_{1},{\xi }_{1}\right){\times}\mathcal{B}\left({x}_{2},{\xi }_{2}\right){\backslash}o\vert {x}_{i}\in {{\Sigma}}_{\mathcal{O}},g\left({\xi }_{i},{\xi }_{i}\right)=0\right\}.\end{equation} \tag{ 83 }$$

We now distinguish the cases (a) $\mathcal{B}\left({x}_{1},{\xi }_{1}\right)=\mathcal{B}\left({x}_{2},{\xi }_{2}\right)$ and (b) $\mathcal{B}\left({x}_{1},{\xi }_{1}\right)\ne \mathcal{B}\left({x}_{2},{\xi }_{2}\right)$. In case (a), we obtain, from (82) and (83), that (x₁, ξ₁, x₂, ξ₂) ∉ WF'(G_U), unless ξ₁ ∼ ξ₂ and ξ₁ ⊳ 0. In case (b), we may, without loss of generality, assume that ${x}_{1},{x}_{2}\in {{\Sigma}}_{\mathcal{O}}$ are spacelike separated. We may then choose real-valued cutoff functions χ₁, χ₂ supported in a coordinate chart near x₁, x₂ such that χ_i (x_i ) ≠ 0, with supp(χ₁) remaining spacelike to supp(χ₂). Since G_U is a distribution of positive type, it follows in view of the Cauchy–Schwarz inequality that

$$\begin{equation}\vert {G}_{\mathrm{U}}\left({\chi }_{1}{e}_{-{k}_{1}},{\chi }_{2}{e}_{{k}_{2}}\right)\vert {\leqslant}\vert {G}_{\mathrm{U}}\left({\chi }_{1}{e}_{-{k}_{1}},{\chi }_{1}{e}_{{k}_{1}}\right){\vert }^{\frac{1}{2}}\vert {G}_{\mathrm{U}}\left({\chi }_{2}{e}_{-{k}_{2}},{\chi }_{2}{e}_{{k}_{2}}\right){\vert }^{\frac{1}{2}},\end{equation} \tag{ 84 }$$

where e_k (x) := e^−ikx . Now let

$$\begin{equation}{\mathcal{V}}_{i}^{{\pm}}=\left\{\left(x,\xi \right)\in {T}^{{\ast}}\mathcal{O}{\backslash}o\vert x\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}{\chi }_{i},g\left(\xi ,\xi \right)=0,\langle \xi ,{\partial }_{t}\rangle \gtrless 0\right\}.\end{equation} \tag{ 85 }$$

It follows from the wave front conditions (82), (83) applied to the right side of (84) that the left side of (84) is decaying rapidly in (k₁, k₂) (i.e. is bounded by ${\leqslant}{C}_{N}{\left(1+\vert {k}_{1}\vert +\vert {k}_{2}\vert \right)}^{-N}$ for any N) in any conic neighborhood not intersecting ${\mathcal{V}}_{1}^{+}{\times}{\mathcal{V}}_{2}^{+}$. However, since the arguments are spacelike, the commutator property of 2-point functions also gives

$$\begin{equation}{G}_{\mathrm{U}}\left({\chi }_{1}{e}_{-{k}_{1}},{\chi }_{2}{e}_{{k}_{2}}\right)={G}_{\mathrm{U}}\left({\chi }_{2}{e}_{{k}_{2}},{\chi }_{1}{e}_{-{k}_{1}}\right).\end{equation} \tag{ 86 }$$

Applying the same reasoning to ${G}_{\mathrm{U}}\left({\chi }_{2}{e}_{{k}_{2}},{\chi }_{1}{e}_{-{k}_{1}}\right)$, it follows that the left side of (84) is decaying as fast in (k₁, k₂) in any conic neighborhood not intersecting ${\mathcal{V}}_{1}^{-}{\times}{\mathcal{V}}_{2}^{-}$. For supp χ_1/2 small enough, ${\mathcal{V}}_{1}^{+}{\times}{\mathcal{V}}_{2}^{+}\cap {\mathcal{V}}_{1}^{-}{\times}{\mathcal{V}}_{2}^{-}=\varnothing$, so that (x₁, ξ₁, x₂, ξ₂) ∉ WF'(G_U) in case (b). Combining this information, we have

$$\begin{equation}\begin{aligned}\hfill {\text{WF}}^{\prime }\left({G}_{\mathrm{U}}\right){\vert }_{\mathcal{O}{\times}\mathcal{O}}\subset & \left\{\left({x}_{1},{\xi }_{1},{x}_{2},{\xi }_{2}\right)\in {T}^{{\ast}}\left(\mathcal{O}{\times}\mathcal{O}\right){\backslash}o\vert {x}_{1},{x}_{2}\;\text{connected}\;\text{by}\;\text{null}\;\text{geodesic}\;\gamma ,\right.\hfill \\ \hfill & \left.\quad {\xi }_{1}={c}_{1}g\left(\;.\;,{\dot {\gamma }}_{1}\right),{\xi }_{2}={c}_{2}g\left(\;.\;,{\dot {\gamma }}_{2}\right),\langle {\xi }_{1},{\partial }_{t}\rangle =\langle {\xi }_{2},{\partial }_{t}\rangle { >}0\right\}.\hfill \end{aligned}\end{equation} \tag{ 87 }$$

The set on the right hand side is easily seen to be equal to ${\mathcal{C}}^{+}$, thus we learn ${\text{WF}}^{\prime }\left({G}_{\mathrm{U}}\right){\vert }_{\mathcal{O}{\times}\mathcal{O}}\subset {\mathcal{C}}^{+}$, and we actually get equality by the commutator argument used at the end of the proof of theorem 4.2.

(3) One bicharacteristic, say $\mathcal{B}\left({x}_{1},{\xi }_{1}\right)$, will enter I, but $\mathcal{B}\left({x}_{2},{\xi }_{2}\right)$ not: we can argue as in subcase (b) above, i.e. using that the right hand side of the inequality (84) is decaying rapidly in (k₁, k₂), due to (1), (2).

Comparison state: for the sake of the discussion in the next section, it is also convenient to have another state that is stationary and that is manifestly Hadamard in a two-sided open neighborhood of the Cauchy horizon ${\mathcal{CH}}^{R}$, see figure 1. This state is not of particular physical interest and introduced purely in order to make our arguments with greater ease. In order to define it, we modify the metric function f(r) beyond the Cauchy horizon for r < r₋ − δ with a small δ > 0 in such a way that it smoothly interpolates between f(r) and the constant function 1 in a neighborhood of r = 0. We call the new function f_*(r). Evidently, it defines the RNdS metric in regions I, II, III and an open neighborhood of ${\mathcal{CH}}^{R}$, but replaces the singularity in region IV with a smooth origin of polar coordinates r = 0, see region IV' in figure 9.

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In this—made-up unphysical—spacetime, we now define a vacuum state called ⟨.⟩_C for regions II, IV' by the 'positive frequency modes (k ⩾ 0)' ${\psi }_{k\ell m}^{\mathrm{C},\mathrm{o}\mathrm{u}\mathrm{t}}$, which are solutions $\left(\square -{\mu }^{2}\right){\psi }_{k\ell m}^{\mathrm{C},\mathrm{o}\mathrm{u}\mathrm{t}}=0$ defined on the union of II, IV' in terms of their final data on $\mathcal{CH}={\mathcal{CH}}^{+}\cup {\mathcal{CH}}^{L}$:

$$\begin{equation}{\psi }_{k\ell m}^{\mathrm{C},\mathrm{o}\mathrm{u}\mathrm{t}}\left(x\right)={\left(2\pi \right)}^{-\frac{3}{2}}{\left(2k\right)}^{-\frac{1}{2}}{Y}_{\ell m}\left(\theta ,\phi \right){r}_{c}^{-1}{\mathrm{e}}^{-\text{i}kV}\quad \;\text{on} \mathcal{CH},\end{equation} \tag{ 88 }$$

where V = V₋ is the Kruskal type (affine) coordinate adapted to this Cauchy horizon, see figure 9. Again, it can be shown by exactly the same methods as in the lightcone case that these modes define via (53) a Hadamard 2-point function ${\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{\mathrm{C}}$ in regions II, IV', which we call the 'comparison state'.

We have shown:

Theorem 4.6. The Unruh state has a 2-point function of Hadamard form in the union of region I, II, III. The comparison state is Hadamard in the union of region II and an open (two-sided) neighborhood of the Cauchy horizon ${\mathcal{CH}}^{R}$.

In this subsection, we show that ${\langle {T}_{\mu \nu }\rangle }_{{\Psi}}-{\langle {T}_{\mu \nu }\rangle }_{\mathrm{U}}$ has the same behavior near ${\mathcal{CH}}^{R}$ as the stress-energy tensor of a classical field. Let ⟨.⟩_Ψ be an arbitrary state that is Hadamard near the Cauchy surface Σ as drawn in figure 1. By the propagation of singularities [58], it remains Hadamard throughout regions I, II, III, but it is not defined (at least not uniquely) in region IV, just as if we were to specify initial data for the classical wave equation on Σ. Since the state is Hadamard in I, II, III, the expectation value ${\langle {T}_{\mu \nu }\rangle }_{{\Psi}}$ is finite and smooth in I, II, III, but may diverge ²² as we move toward the Cauchy horizon, ${\mathcal{CH}}^{R}$. We would like to see how it diverges, if at all. The first step is to relate the behavior of ${\langle {T}_{\mu \nu }\rangle }_{{\Psi}}$ to behavior of the corresponding quantity ${\langle {T}_{\mu \nu }\rangle }_{\mathrm{U}}$ for the Unruh state, which we have a better way of calculating.

Since we are interested in the local behavior, we pick a point p on the Cauchy horizon ${\mathcal{CH}}^{R}$ contained in a coordinate chart, (x^μ ) = (V, yⁱ ), where V = V₋ is the Kruskal-type coordinate and yⁱ , i = 1, 2, 3 are local coordinates parameterizing ${\mathcal{CH}}^{R}$. So, V = 0 locally defines ${\mathcal{CH}}^{R}$, and the intersection of this chart with region II locally corresponds to V < 0. Let $\mathcal{U}$ be a small open neighborhood contained in this coordinate chart. Then the following proposition basically follows from theorem 4.4.

Proposition 5.1. Let Ψ, Ψ' be Hadamard states near the Cauchy surface Σ as drawn in figure 1, and assume β > 1/2, see (2). Then:

• each component of ${t}_{\mu \nu }{:=}{\langle {T}_{\mu \nu }\rangle }_{{\Psi}}-{\langle {T}_{\mu \nu }\rangle }_{{{\Psi}}^{\prime }},$ is smooth in y, and viewed as a function of V < 0, is locally in ${L}^{p}\left({\mathbb{R}}_{-}\right)$, where 1/p = 2 − 2β + 2 for arbitrarily small > 0 (thus p > 1), with norm uniformly bounded in y within $\mathcal{U}$.
• viewed as a function of V, t_μν has s = β − 1/2 − derivatives locally in ${L}^{p}\left({\mathbb{R}}_{-}\right)$ for 1/p = 3/2 − β + and arbitrarily small .

Remarks: (1) For a classical field with smooth initial data on Σ, the stress tensor is a function known to have the same regularity as t_μν , so the result is that the difference between the quantum stress tensors in two states that are initially Hadamard behaves basically like the classical stress tensor near the Cauchy horizon. (2) The restriction to β > 1/2 is of technical nature. For β ⩽ 1/2, already a classical solution ϕ would generically fail to be of Sobolev class H¹ near ${\mathcal{CH}}^{R}$, thus making it impossible to extend the solution through ${\mathcal{CH}}^{R}$ already at the classical level, see [13] for a discussion of such matters. Thus, β > 1/2 is the interesting case for the discussion of sCC.

Proof. Let $\mathcal{N}=\mathrm{I}\cup \mathrm{I}\mathrm{I}\cup \mathrm{I}\mathrm{I}\mathrm{I}$. Since $\mathcal{N}$ is globally hyperbolic and since both states are Hadamard near Σ, it follows by propagation of singularities that the difference between the corresponding two point functions $W\left({x}_{1},{x}_{2}\right)={\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{\Psi}}-{\langle {\Phi}\left({x}_{1}\right){\Phi}\left({x}_{2}\right)\rangle }_{{{\Psi}}^{\prime }}$ is in ${C}^{\infty }\left(\mathcal{N}{\times}\mathcal{N}\right)$. It has been shown in [72], lemma 3.7 that given a causal normal neighborhood $\tilde {\mathcal{N}}$ of Σ (see [49] for the definition), and any open subset $\mathcal{O}$ with compact closure contained in $\tilde {\mathcal{N}}$, there exists a smooth function B with support in $\mathcal{O}{\times}\mathcal{O}$ such that

$$\begin{equation}W\left({f}_{1},{f}_{2}\right)={\int }_{\mathcal{O}{\times}\mathcal{O}}\left(E{f}_{1}\right)\left({x}_{1}\right)\left(E{f}_{2}\right)\left({x}_{2}\right)B\left({x}_{1},{x}_{2}\right)\mathrm{d}\eta \left({x}_{1}\right)\mathrm{d}\eta \left({x}_{2}\right),\end{equation} \tag{ 90 }$$

for any testfunctions f₁, f₂ with support in the domain of dependence $D\left(\mathcal{O}\right)$. By making $\mathcal{O}$ sufficiently large, we may assume that the neighborhood $\mathcal{U}$ of interest near the Cauchy horizon is contained in $D\left(\mathcal{O}\right)$. Furthermore, by the algebraic relations (A3), (A4) and the positivity property (S3) of states, we have $B\left({x}_{1},{x}_{2}\right)=\overline{B\left({x}_{1},{x}_{2}\right)}=B\left({x}_{2},{x}_{1}\right)$.

Let m be a fixed natural number. Using the result [72], appendix B, and the symmetry properties of B, one can see that there exist ${C}_{0}^{m}$ functions {b_j } on $\mathcal{O}$ such that

$$\begin{equation}B\left({x}_{1},{x}_{2}\right)=\sum _{j}{c}_{j}\overline{{b}_{j}\left({x}_{1}\right)}{b}_{j}\left({x}_{2}\right),\quad \sum _{j}{{\Vert}{b}_{j}{\Vert}}_{{C}^{m}\left(\mathcal{O}\right)}^{2}{< }\infty ,\quad {c}_{j}\in \left\{{\pm}1\right\}.\end{equation} \tag{ 91 }$$

Now let ψ_j = E⁺ b_j , with E⁺ the retarded fundamental solution. Then (□ − μ²)ψ_j = b_j , and furthermore,

$$\begin{equation}W\left({x}_{1},{x}_{2}\right)=\sum _{j}{c}_{j}\overline{{\psi }_{j}\left({x}_{1}\right)}{\psi }_{j}\left({x}_{2}\right)\quad \;\text{for} {x}_{1},{x}_{2}\in \mathcal{U}.\end{equation} \tag{ 92 }$$

To estimate ψ_j , we can apply the results of [1] already mentioned in theorem 4.4. First, we define new local coordinates (z, yⁱ ) in the following way. First we locally parameterize the orbit ${S}_{p}^{2}$ of $p\in {\mathcal{CH}}^{R}$ by angles Ω. Then we transport these coordinates away from ${\mathcal{CH}}^{R}$ along ∂/∂V and let z be the parameter along this curve. Finally, we transport the coordinates (z, Ω) along the Killing vector field ∂/∂t, and then (yⁱ ) = (t, Ω).

If we apply an arbitrary product ${\prod }_{j=1}^{N}{X}^{\left({a}_{j}\right)}$ of the vector fields X^(a), a = 1, 2, 3, 4 generating the symmetry group $\mathbb{R}{\times}SO\left(3\right)$ of RNdS to the defining equation (□ − μ²)ψ_j = b_j (with retarded initial conditions) and commute these through □ − μ², the function ${\prod }_{a=1}^{N}{X}^{\left(a\right)}{\psi }_{j}$ is seen to satisfy an equation of the same form and initial conditions with a new source. By theorem 4.4 and the construction of (z, yⁱ ), this gives

$$\begin{equation}{{\Vert}{\partial }_{y}^{N}{\psi }_{j}{\Vert}}_{{H}^{1/2+\beta -{\epsilon}}\left(\mathcal{U}\right)}^{2}\overset{<}{\sim} {{\Vert}{b}_{j}{\Vert}}_{{C}^{m}\left(\mathcal{O}\right)}^{2}\end{equation} \tag{ 93 }$$

for an implicit constant independent of j and a sufficiently large m depending on N. We can also use our expression for the difference of the expected stress tensor in two Hadamard states, see (51). This shows in combination with (92) that

$$\begin{equation}{t}_{\mu \nu }\left(x\right)=\sum _{j}{c}_{j}\left\{{\partial }_{\mu }\overline{{\psi }_{j}\left(x\right)}{\partial }_{\nu }{\psi }_{j}\left(x\right)-\frac{1}{2}{g}_{\mu \nu }\left({\partial }_{\gamma }\overline{{\psi }_{j}\left(x\right)}{\partial }^{\gamma }{\psi }_{j}\left(x\right)+{\mu }^{2}\overline{{\psi }_{j}\left(x\right)}{\psi }_{j}\left(x\right)\right)\right\}.\end{equation} \tag{ 94 }$$

A standard argument based on the Fourier transform in y, Cauchy's inequality and Parseval's theorem shows that for φ supported in the coordinate patch covered by (z, yⁱ ),

$$\begin{equation}\begin{aligned}\hfill \vert \varphi \left(z,y\right)\vert & =\int {\mathrm{d}}^{3}\xi \tilde {\varphi }\left(z,\xi \right){\mathrm{e}}^{\text{i}y\xi }\hfill \\ \hfill & {\leqslant}{\left(\int {\mathrm{d}}^{3}\xi {\left(1+{\xi }^{2}\right)}^{-N/2}\right)}^{1/2}{\left(\int {\mathrm{d}}^{3}\xi {\left(1+{\xi }^{2}\right)}^{N/2}\vert \tilde {\varphi }\left(z,\xi \right){\vert }^{2}\right)}^{1/2}\hfill \\ \hfill & \overset{<}{\sim} {{\Vert}{\left(-{\partial }_{y}^{2}+1\right)}^{N/2}\varphi \left(z,-\right){\Vert}}_{{L}^{2}\left({\mathbb{R}}_{y}^{3}\right)},\hfill \end{aligned}\end{equation} \tag{ 95 }$$

taking N > 3. In combination with the Sobolev embedding theorem, p as in the statement of the proposition,

$$\begin{equation}{\Vert}\varphi \left(-,y\right){{\Vert}}_{{L}^{2p}\left({\mathbb{R}}_{-}\right)}\overset{<}{\sim} {\Vert}{\left(-{\partial }_{z}^{2}+1\right)}^{\left(\beta -1/2-{\epsilon}\right)/2}\varphi \left(-,y\right){{\Vert}}_{{L}^{2}\left({\mathbb{R}}_{-}\right)}\overset{<}{\sim} {\Vert}{\left(-{\partial }_{y}^{2}+1\right)}^{N/2}\varphi {{\Vert}}_{{H}^{\beta -1/2-{\epsilon}}\left(\mathcal{U}\right)},\end{equation} \tag{ 96 }$$

uniformly in y within $\mathcal{U}$. We now take φ = ∂_μ ψ_j or φ = ψ_j and combine this with (93), thereby obtaining

$$\begin{equation}{\Vert}{\partial }_{\mu }{\psi }_{j}\left(-,y\right){{\Vert}}_{{L}^{2p}\left({\mathbb{R}}_{-}\right)}\overset{<}{\sim} {{\Vert}{b}_{j}{\Vert}}_{{C}^{m}\left(\mathcal{O}\right)}.\end{equation} \tag{ 97 }$$

Therefore, using the Cauchy–Schwarz inequality,

$$\begin{align}\hfill {\Vert}{t}_{\mu \nu }\left(-,y\right){{\Vert}}_{{L}^{p}\left({\mathbb{R}}_{-}\right)}& \overset{<}{\sim} \sum _{j}{\left(\sum _{\gamma =1}^{4}{\Vert}{\partial }_{\gamma }{\psi }_{j}\left(-,y\right){{\Vert}}_{{L}^{2p}\left({\mathbb{R}}_{-}\right)}+{\Vert}{\psi }_{j}\left(-,y\right){{\Vert}}_{{L}^{2p}\left({\mathbb{R}}_{-}\right)}\right)}^{2}\hfill \\ \hfill & \overset{<}{\sim} \sum _{j}{{\Vert}{b}_{j}{\Vert}}_{{C}^{m}\left(\mathcal{O}\right)}^{2}{< }\infty .\hfill \end{align} \tag{ 98 }$$

By considering $\varphi ={\partial }_{\mu }{\partial }_{y}^{M}{\psi }_{j}$, we can similarly obtain a corresponding bound for ${\Vert}{\partial }_{y}^{M}{t}_{\mu \nu }\left(-,y\right){{\Vert}}_{{L}^{p}\left({\mathbb{R}}_{-}\right)}$. Transforming from the coordinates (z, yⁱ ) to (V, yⁱ ) gives the first statement.

To prove the second result, we use that (95) also implies

$$\begin{equation}{\Vert}{D}_{z}^{s/2}\varphi \left(-,y\right){{\Vert}}_{{L}^{2p}\left({\mathbb{R}}_{-}\right)}\overset{<}{\sim} {\Vert}{\left(-{\partial }_{y}^{2}+1\right)}^{N/2}\varphi {{\Vert}}_{{H}^{s}\left(\mathcal{U}\right)}\end{equation} \tag{ 99 }$$

for ${\left(-{\partial }_{z}^{2}+1\right)}^{s/2}={D}_{z}^{s}$, s and p as indicated in the second statement, and φ of compact support in $\mathcal{U}$. With φ = ∂_μ ψ_j , we get

$$\begin{equation}{{\Vert}{D}_{z}^{s/2}{\partial }_{\mu }{\psi }_{j}\left(-,y\right){\Vert}}_{{L}^{2p}\left({\mathbb{R}}_{-}\right)}\overset{<}{\sim} {{\Vert}{\partial }_{y}^{N}{\psi }_{j}{\Vert}}_{{H}^{s+1}\left(\mathcal{U}\right)}^{2},\end{equation} \tag{ 100 }$$

for s < β − 1/2. We have classical commutator estimates of the type [73], theorem A.8,

$$\begin{equation}{{\Vert}{D}^{s}\left({f}_{1}{f}_{2}\right)-{D}^{s}{f}_{1}{f}_{2}-{f}_{1}{D}^{s}{f}_{2}{\Vert}}_{{L}^{p}\left({\mathbb{R}}^{n}\right)}\overset{<}{\sim} {{\Vert}{D}^{{s}^{\prime }}{f}_{1}{\Vert}}_{{L}^{{p}^{\prime }}\left({\mathbb{R}}^{n}\right)}{{\Vert}{D}^{{s}^{\prime \prime }}{f}_{1}{\Vert}}_{{L}^{{p}^{\prime \prime }}\left({\mathbb{R}}^{n}\right)}\end{equation} \tag{ 101 }$$

for 0 < s = s' + s'', s', s'' > 0, 0 < 1/p = 1/p' + 1/p'' < 1. These basically allow us to distribute the fractional derivatives ${D}_{z}^{s}$ on a product ${\partial }_{\mu }{\overline{\psi }}_{j}{\partial }_{\nu }{\psi }_{j}$ such as in t_μν up to an error term which is controlled in a lower Sobolev norm using ${{\Vert}{\partial }_{y}^{N}{\psi }_{j}{\Vert}}_{{H}^{s+1}\left(\mathcal{U}\right)}^{2}\overset{<}{\sim} {{\Vert}{b}_{j}{\Vert}}_{{C}^{m}\left(\mathcal{O}\right)}$. The second statement then follows taking s' = s'' = s/2, p' = p'' = 2p. This concludes the proof of the proposition. □

The proposition says that for the purposes of calculating ${\langle {T}_{\mu \nu }\rangle }_{{\Psi}}$ near the Cauchy horizon in a state Ψ which starts out as a Hadamard state near the initial time surface Σ, we may work with the Unruh state ${\langle {T}_{\mu \nu }\rangle }_{\mathrm{U}}$ because, as we have shown in thm. 4.6, it is also a Hadamard state on the initial time surface Σ. In doing this, we must accept an error t_μν which has the regularity described in proposition 5.1 at the Cauchy horizon, and which, as we have hinted, is the behavior of the classical stress tensor. This is acceptable since, as we will show, the behavior of ${\langle {T}_{\mu \nu }\rangle }_{\mathrm{U}}$ is more singular than this at the Cauchy horizon ${\mathcal{CH}}^{R}$.

We turn, now to the calculation of the middle term in (89), namely ${\langle {T}_{\mu \nu }\rangle }_{\mathrm{U}}-{\langle {T}_{\mu \nu }\rangle }_{\mathrm{C}}$. By stationarity of the Unruh and the comparison state, it suffices to perform the computation in a neighborhood of ${\mathcal{CH}}^{L}$. The calculation is outlined in subsection 5.2, with the details of the mode calculations relegated to subsection 5.3.

The mode calculation is a straightforward effective one-dimensional scattering construction. It proceeds in two steps:

• We solve a scattering problem in region I, with asymptotic conditions determined by our choice of initial state on the surfaces ${\mathcal{H}}^{-},{\mathcal{H}}_{c}^{-}$.
• We subsequently solve a scattering problem in region II, with asymptotic conditions determined by our choice of initial state on ${\mathcal{H}}^{L}$ and by the data from the previous step on ${\mathcal{H}}^{R}$.

Because similar constructions have been described in the literature previously for the case of RN [37], we will be brief and focus on the novel aspects. Actually, the calculation is not more difficult for the operators ${\left({\partial }_{V}^{N}{\Phi}\right)}^{2}$, which are equal to T_VV for N = 1, using the obvious generalization of (51) to the operators ${\left({\partial }_{V}^{N}{\Phi}\right)}^{2}$.

Since $\partial v/\partial V=-{\left({\kappa }_{-}V\right)}^{-1}$ with V = V₋ the outgoing Kruskal-type coordinate regular in a neighborhood of ${\mathcal{CH}}^{R}$, we can work instead with the variable v, so we are interested in ${\langle {\left({\partial }_{v}^{N}{\Phi}\right)}^{2}\rangle }_{\mathrm{U}}-{\langle {\left({\partial }_{v}^{N}{\Phi}\right)}^{2}\rangle }_{\mathrm{C}}$. As in [37], for computational purposes only (not changing the state!), it is useful to consider the 'Boulware' modes ²³ , defined by the wave equation and asymptotic data ²⁴

$$\begin{equation}{\psi }_{\omega \ell m}^{\mathrm{i}\mathrm{n},\mathrm{I}}=\begin{cases}{\left(2\pi \right)}^{-\frac{3}{2}}{\left(2\vert \omega \vert \right)}^{-\frac{1}{2}}{Y}_{\ell m}\left(\theta ,\phi \right){r}_{c}^{-1}{\Theta}\left(-V\right){\mathrm{e}}^{-\text{i}\omega v}\quad \hfill & \;\text{on} {\mathcal{H}}_{c},\hfill \\ 0\quad \hfill & \;\text{on} \mathcal{H},\hfill \end{cases}\end{equation} \tag{ 102a }$$

$$\begin{equation}{\psi }_{\omega \ell m}^{\mathrm{u}\mathrm{p},\mathrm{I}}=\begin{cases}0\quad \hfill & \;\text{on} {\mathcal{H}}_{c}\hfill \\ {\left(2\pi \right)}^{-\frac{3}{2}}{\left(2\vert \omega \vert \right)}^{-\frac{1}{2}}{Y}_{\ell m}\left(\theta ,\phi \right){r}_{+}^{-1}{\Theta}\left(-U\right){\mathrm{e}}^{-\text{i}\omega u}\quad \hfill & \;\text{on} \mathcal{H},\hfill \end{cases}\end{equation} \tag{ 102b }$$

$$\begin{equation}{\psi }_{\omega \ell m}^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}=\begin{cases}0\quad \hfill & \;\text{on} {\mathcal{H}}_{c}\hfill \\ {\left(2\pi \right)}^{-\frac{3}{2}}{\left(2\vert \omega \vert \right)}^{-\frac{1}{2}}{Y}_{\ell m}\left(\theta ,\phi \right){r}_{+}^{-1}{\Theta}\left(U\right){\mathrm{e}}^{-\text{i}\omega u}\quad \hfill & \;\text{on} \mathcal{H},\hfill \end{cases}\end{equation} \tag{ 102c }$$

$$\begin{equation}{\psi }_{\omega \ell m}^{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{I}\mathrm{I}}=\begin{cases}{\left(2\pi \right)}^{-\frac{3}{2}}{\left(2\vert \omega \vert \right)}^{-\frac{1}{2}}{Y}_{\ell m}\left(\theta ,\phi \right){r}_{-}^{-1}{\Theta}\left(-V\right){\mathrm{e}}^{-\text{i}\omega v}\quad \hfill & \;\text{on} {\mathcal{CH}}^{L}\hfill \\ 0\quad \hfill & \;\text{on} {\mathcal{CH}}^{+},\hfill \end{cases}\end{equation} \tag{ 102d }$$

with Θ the step function, and where we used the notation (64). Furthermore, V = V_c/− in the first/last equation. For later convenience, we defined these modes also for ω < 0. We omitted the modes ψ^in,III and ψ^out,IV, which are not relevant for the calculations that we are going to perform. In the next section 5.3, we shall find the exact solutions to the above characteristic initial value problems in mode form, i.e. in the general form

$$\begin{equation}{\psi }_{\omega \ell m}\left(t,r,\theta ,\phi \right)={\left(2\pi \right)}^{-\frac{3}{2}}{\left(2\vert \omega \vert \right)}^{-\frac{1}{2}}{Y}_{\ell m}\left(\theta ,\phi \right){r}^{-1}{R}_{\omega \ell }\left(r\right){\mathrm{e}}^{-\text{i}\omega t},\end{equation} \tag{ 103 }$$

where the radial function R has to solve

$$\begin{equation}-{\partial }_{{r}_{{\ast}}}{\partial }_{{r}_{{\ast}}}{R}_{\omega \ell }+\left({V}_{\ell }-{\omega }^{2}\right){R}_{\omega \ell }=0,\end{equation} \tag{ 104 }$$

with a smooth effective potential

$$\begin{equation}{V}_{\ell }=f\left(\frac{\ell \left(\ell +1\right)}{{r}^{2}}+\frac{{f}^{\prime }}{r}+{\mu }^{2}\right).\end{equation} \tag{ 105 }$$

The particular solutions corresponding to various asymptotic conditions are denoted by superscripts, such as in ${\psi }_{\omega \ell m}^{\mathrm{i}\mathrm{n},\mathrm{I}}$, with corresponding radial function ${R}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}}$, etc. For example, ${R}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}}$ should satisfy the asymptotic condition

$$\begin{equation}{R}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}}\left(r\right)=\begin{cases}{\mathrm{e}}^{-\text{i}\omega {r}_{{\ast}}}+{\mathcal{R}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}}{\mathrm{e}}^{\text{i}\omega {r}_{{\ast}}}\quad \hfill & {r}_{{\ast}}\to \infty ,\hfill \\ {\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}}{\mathrm{e}}^{-\text{i}\omega {r}_{{\ast}}}\quad \hfill & {r}_{{\ast}}\to -\infty .\hfill \end{cases}\end{equation} \tag{ 106 }$$

Since the effective potential is smooth and, when expressed in terms of ${r}_{{\ast}}$, decays, with all its derivatives, faster than any power, we must have [74], proposition 5.2.9,

$$\begin{equation}{\mathcal{R}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}}=\begin{cases}O\left({\omega }^{-\infty }\right)\quad \hfill & \omega \to \infty ,\hfill \\ 1+O\left({\ell }^{-\infty }\right)\quad \hfill & \ell \to \infty ,\hfill \end{cases}\vert {\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}}{\vert }^{2}=\begin{cases}1+O\left({\omega }^{-\infty }\right)\quad \hfill & \omega \to \infty ,\hfill \\ O\left({\ell }^{-\infty }\right)\quad \hfill & \ell \to \infty .\hfill \end{cases}\end{equation} \tag{ 107 }$$

where O(ω^−∞) indicates a decay faster than any power. This provides asymptotic data on ${\mathcal{H}}^{R}$ and ${\mathcal{H}}^{L}$ (the latter vanish), which can be transported to ${\mathcal{CH}}^{L}$ by again solving (104), now with the asymptotic condition

$$\begin{equation}{R}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}\left(r\right)={\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}}\begin{cases}{\mathrm{e}}^{-\text{i}\omega {r}_{{\ast}}}\quad \hfill & {r}_{{\ast}}\to -\infty ,\hfill \\ {\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}{\mathrm{e}}^{-\text{i}\omega {r}_{{\ast}}}+{\mathcal{R}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}{\mathrm{e}}^{\text{i}\omega {r}_{{\ast}}}\quad \hfill & {r}_{{\ast}}\to \infty .\hfill \end{cases}\end{equation} \tag{ 108 }$$

Note that the new coefficients fulfill the relation

$$\begin{equation}\vert {\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}{\vert }^{2}-\vert {\mathcal{R}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}{\vert }^{2}=1,\end{equation} \tag{ 109 }$$

which in particular means that they need not be bounded. Nevertheless, we again have

$$\begin{equation}{\mathcal{R}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}=O\left({\omega }^{-\infty }\right),\vert {\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}{\vert }^{2}=1+O\left({\omega }^{-\infty }\right).\end{equation} \tag{ 110 }$$

We thus obtain, for the restriction of the 'Boulware' in-mode to ${\mathcal{CH}}^{L}$,

$$\begin{equation}{\psi }_{\omega \ell m}^{\mathrm{i}\mathrm{n},\mathrm{I}}{\vert }_{{\mathcal{CH}}^{L}}={\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}{\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}}{\psi }_{\omega \ell m}^{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{I}\mathrm{I}}.\end{equation} \tag{ 111 }$$

For the up-modes ${\psi }_{\omega \ell m}^{\mathrm{u}\mathrm{p},\mathrm{I}}$, we proceed similarly, i.e., we first scatter them to ${\mathcal{H}}^{R}$ by solving again (104), now with the asymptotic condition

$$\begin{equation}{R}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}}\left(r\right)=\begin{cases}{\mathrm{e}}^{\text{i}\omega {r}_{{\ast}}}+{\mathcal{R}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}}{\mathrm{e}}^{-\text{i}\omega {r}_{{\ast}}}\quad \hfill & {r}_{{\ast}}\to -\infty ,\hfill \\ {\mathcal{T}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}}{\mathrm{e}}^{\text{i}\omega {r}_{{\ast}}}\quad \hfill & {r}_{{\ast}}\to \infty .\hfill \end{cases}\end{equation} \tag{ 112 }$$

One then scatters to ${\mathcal{CH}}^{L}$ as for the in-modes and obtains

$$\begin{equation}{\psi }_{\omega \ell m}^{\mathrm{u}\mathrm{p},\mathrm{I}}{\vert }_{{\mathcal{CH}}^{L}}={\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}{\mathcal{R}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}}{\psi }_{\omega \ell m}^{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{I}\mathrm{I}},\end{equation} \tag{ 113 }$$

For the up-modes ${\psi }_{\omega \ell m}^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}$, only a single scattering is necessary. The radial function ${R}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}$ is defined by the asymptotic condition

$$\begin{equation}{R}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}\left(r\right)=\begin{cases}{\mathrm{e}}^{\text{i}\omega {r}_{{\ast}}}\quad \hfill & {r}_{{\ast}}\to -\infty ,\hfill \\ {\mathcal{T}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}{\mathrm{e}}^{\text{i}\omega {r}_{{\ast}}}+{\mathcal{R}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}{\mathrm{e}}^{-\text{i}\omega {r}_{{\ast}}}\quad \hfill & {r}_{{\ast}}\to \infty .\hfill \end{cases}\end{equation} \tag{ 114 }$$

This gives

$$\begin{equation}{\psi }_{\omega \ell m}^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}{\vert }_{{\mathcal{CH}}^{L}}={\mathcal{R}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}{\psi }_{\omega \ell m}^{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{I}\mathrm{I}}.\end{equation} \tag{ 115 }$$

Comparing (114) with (108) yields

$$\begin{equation}{\mathcal{R}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}=\overline{{\mathcal{R}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}},\quad \quad {\mathcal{T}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}=\overline{{\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}}.\end{equation} \tag{ 116 }$$

The symmetrized two-point function, i.e., the Hadamard function, of the Unruh state defined by the modes (65) can be expressed in terms of the Boulware modes (102) as, see [75] for a similar calculation (but note the change of sign of u in the interior region w.r.t. that reference),

$$\begin{align}\hfill {\langle \left\{{\Phi}\left({x}_{1}\right),{\Phi}\left({x}_{2}\right)\right\}\rangle }_{\mathrm{U}}=& \sum _{\ell m}{\int }_{0}^{\infty }\mathrm{d}\omega \left[\mathrm{coth}\frac{\pi \omega }{{\kappa }_{c}}\left\{{\psi }_{\omega \ell m}^{\mathrm{i}\mathrm{n},\mathrm{I}}\left({x}_{1}\right),\overline{{\psi }_{\omega \ell m}^{\mathrm{i}\mathrm{n},\mathrm{I}}\left({x}_{2}\right)}\right\}\right.\hfill \\ \hfill & +2\text{csch}\frac{\pi \omega }{{\kappa }_{+}}\text{Re}\left\{{\psi }_{\omega \ell m}^{\mathrm{u}\mathrm{p},\mathrm{I}}\left({x}_{1}\right),\overline{{\psi }_{\omega \ell m}^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}\left({x}_{2}\right)}+\;\mathrm{coth}\frac{\pi \omega }{{\kappa }_{+}}\left(\left\{{\psi }_{\omega \ell m}^{\mathrm{u}\mathrm{p},\mathrm{I}}\left({x}_{1}\right),\overline{{\psi }_{\omega \ell m}^{\mathrm{u}\mathrm{p},\mathrm{I}}\left({x}_{2}\right)}\right\}\right.\right\}\hfill \\ \hfill & \left.\left.+\left\{{\psi }_{-\omega \ell m}^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}\left({x}_{1}\right),\overline{{\psi }_{-\omega \ell m}^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}\left({x}_{2}\right)}\right\}\right)\right],\hfill \end{align} \tag{ 117 }$$

where we omitted the contribution from ${\psi }_{\omega \ell m}^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}\mathrm{I}}$, which is absent in I ∪ II, and used the notation

$$\begin{equation}\left\{A\left({x}_{1}\right),B\left({x}_{2}\right)\right\}=\frac{1}{2}\left(A\left({x}_{1}\right)B\left({x}_{2}\right)+A\left({x}_{2}\right)B\left({x}_{1}\right)\right).\end{equation} \tag{ 118 }$$

We now turn to the comparison state. The specification of this state in terms of modes on a null hypersurface has the property that, after selecting an angular momentum component,

$$\begin{equation}{{\Phi}}_{\ell m}\left(U,V\right)=r\int \overline{{Y}_{\ell m}\left({\Omega}\right)}{\Phi}\left(U,V,{\Omega}\right){\mathrm{d}}^{2}{\Omega},\end{equation} \tag{ 119 }$$

and taking at least one derivative w.r.t. V = V₋, one can evaluate the two-point function on the null hypersurface, i.e.,

$$\begin{equation}\underset{_{U\to \infty }}{\mathrm{lim}}{\langle {\partial }_{V}^{n}{{\Phi}}_{{\ell }_{1}{m}_{1}}^{{\ast}}\left(U,{V}_{1}\right){\partial }_{{V}_{2}}^{n}{{\Phi}}_{{\ell }_{2}{m}_{2}}\left(U,{V}_{2}\right)\rangle }_{\mathrm{C}}={\delta }_{{\ell }_{1}{\ell }_{2}}{\delta }_{{m}_{1}{m}_{2}}\frac{1}{16{\pi }^{3}}{\int }_{0}^{\infty }\mathrm{d}k\;{k}^{2n-1}{\mathrm{e}}^{-\text{i}k\left({V}_{1}-{V}_{2}-i0\right)}.\end{equation} \tag{ 120 }$$

When both points are on ${\mathcal{CH}}^{L}$, one may reexpress this as, see [75] for example,

$$\begin{equation}\underset{_{U\to \infty }}{\mathrm{lim}}{\langle {\partial }_{v}^{n}{{\Phi}}_{{\ell }_{1}{m}_{1}}^{{\ast}}\left(U,{v}_{1}\right){\partial }_{v}^{n}{{\Phi}}_{{\ell }_{2}{m}_{2}}\left(U,{v}_{2}\right)\rangle }_{\mathrm{C}}={\delta }_{{\ell }_{1}{\ell }_{2}}{\delta }_{{m}_{1}{m}_{2}}\frac{1}{16{\pi }^{3}}{\int }_{0}^{\infty }\mathrm{d}\omega \;{\omega }^{2n-1}\mathrm{coth}\frac{\pi \omega }{{\kappa }_{-}}{\mathrm{e}}^{-\text{i}\omega \left({v}_{1}-{v}_{2}-i0\right)}.\end{equation} \tag{ 121 }$$

With (111), (113), (115) and (117), we thus obtain for the difference of the expectation value in the Unruh and the comparison state, in a coinciding point limit,

$$\begin{align}\hfill & \underset{_{U\to \infty ,{v}_{1}\to {v}_{2}}}{{\mathrm{lim}}}\left[{\langle {\partial }_{v}^{n}{{\Phi}}_{{\ell }_{1}{m}_{1}}^{{\ast}}\left(U,{v}_{1}\right){\partial }_{v}^{n}{{\Phi}}_{{\ell }_{2}{m}_{2}}\left(U,{v}_{2}\right)\rangle }_{\mathrm{U}}-{\langle {\partial }_{v}^{n}{{\Phi}}_{{\ell }_{1}{m}_{1}}^{{\ast}}\left(U,{v}_{1}\right){\partial }_{v}^{n}{{\Phi}}_{{\ell }_{2}{m}_{2}}\left(U,{v}_{2}\right)\rangle }_{\mathrm{C}}\right]\hfill \\ \hfill & \quad ={\delta }_{{\ell }_{1}{\ell }_{2}}{\delta }_{{m}_{1}{m}_{2}}\frac{1}{16{\pi }^{3}}{\int }_{0}^{\infty }\mathrm{d}\omega \;{\omega }^{2n-1}{n}_{{\ell }_{1}}\left(\omega \right),\hfill \end{align} \tag{ 122 }$$

where

$$\begin{align}\hfill {n}_{\ell }\left(\omega \right)=& \;\vert {\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}}{\vert }^{2}\vert {\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}{\vert }^{2}\mathrm{coth}\frac{\pi \omega }{{\kappa }_{c}}+\left(\vert {\mathcal{R}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}}{\vert }^{2}\vert {\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}{\vert }^{2}+\vert {\mathcal{R}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}{\vert }^{2}\right)\mathrm{coth}\frac{\pi \omega }{{\kappa }_{+}}\hfill \\ \hfill & +2\text{csch}\frac{\pi \omega }{{\kappa }_{+}}\text{Re}\left({\mathcal{R}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}}{\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}\overline{{\mathcal{R}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}}\right)-\mathrm{coth}\frac{\pi \omega }{{\kappa }_{-}}.\hfill \end{align} \tag{ 123 }$$

From the asymptotic behavior of the transmission and reflection coefficients, it follows that (123) falls off faster than any power in ω for |ω| → ∞, so that the integral (122) is UV finite for all n ⩾ 1. However, one may worry that there is an IR divergence for n = 1, due to the fact that generically ${\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}$ and hence, by (109), also ${\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}$ have a simple pole at ω = 0, [76], proposition 6.2, so that the second and the third term on the r.h.s. are individually IR divergent for n = 1 (in the first term, $\vert {\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}}{\vert }^{2}$ vanishes rapidly as ω → 0). ²⁵ However, the potential IR divergences due to these terms cancel: we first notice that ${\mathcal{R}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}}=-1+i{C}_{\ell }\omega +O\left({\omega }^{2}\right)$ as ω → 0, where ${C}_{\ell }\in \mathbb{R}$, see [37]. The statement then follows from (116) and the fact, derivable from inspection of the proof of [76], proposition 6.2, that, in case of a divergent ${\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}$,

$$\begin{equation}{\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}=\frac{i{C}_{\ell }^{\prime }}{\omega }+O\left({\omega }^{0}\right),\quad {\mathcal{R}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}=-\frac{i{C}_{\ell }^{\prime }}{\omega }+O\left({\omega }^{0}\right),\end{equation} \tag{ 124 }$$

where ${C}_{\ell }^{\prime }\in \mathbb{R}$. ²⁶ It is then straightforward to see that the potential IR divergences in (122) cancel.

We also note that due to $\vert {\mathcal{T}}_{\omega \ell }^{\mathrm{i}\mathrm{n},\mathrm{I}}{\vert }^{2}=\vert {\mathcal{T}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}}{\vert }^{2}$, it suffices to determine ${\mathcal{R}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}}$, ${\mathcal{T}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}$, and ${\mathcal{R}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}$ in order to evaluate (123).

If the integral on the r.h.s. of (122) gives a non-zero value for any n, then the Unruh state is singular at ${\mathcal{CH}}^{R}$: by stationarity and continuity, the values on ${\mathcal{CH}}^{R}$ and ${\mathcal{CH}}^{L}$ coincide. Converting the resulting tensor to adapted coordinates V = V₋ instead of v, one obtains

$$\begin{equation}{\langle {\partial }_{V}^{n}{{\Phi}}_{\ell m}^{{\ast}}{\partial }_{V}^{n}{{\Phi}}_{\ell m}\rangle }_{\mathrm{U}}-{\langle {\partial }_{V}^{n}{{\Phi}}_{\ell m}^{{\ast}}{\partial }_{V}^{n}{{\Phi}}_{\ell m}\rangle }_{\mathrm{C}}\sim {c}_{n\ell }\vert V{\vert }^{-2n},\end{equation} \tag{ 125 }$$

where c_nℓ is ${\kappa }_{-}^{2n}$ times the value of the integral on the right hand side of (122). That the integral vanishes for all ℓ and all n ⩾ 1 seems quite improbable, and this expectation is confirmed in section 5.4. Because the comparison state C is Hadamard in an open neighborhood on ${\mathcal{CH}}^{R}$ by theorem 4.6, any singular behavior must be due to the Unruh state U and not the comparison state C. Therefore, in view of property (H1) of subsection 4.1, we conclude:

Conclusion 1: unless the quantity on the right side of (122) vanishes for all ℓ and all n ⩾ 1, the Unruh state cannot be a Hadamard state in a two sided open neighborhood of ${\mathcal{CH}}^{R}$ (of course it is Hadamard away from ${\mathcal{CH}}^{R}$ by theorem 4.6).

While for the angular momentum components Φ_ℓm the limits U → ∞ and v₁ → v₂ on the left hand side of (122) can be controlled and shown to converge to the right hand side, this is not straightforward for Φ, required for the evaluation of the difference of expectation values of local Wick powers such as the stress tensor T_{v v} on ${\mathcal{CH}}^{L}$. Noting that we can write

$$\begin{equation}{\partial }_{v}{\Phi}\left({x}_{1}\right){\partial }_{v}{\Phi}\left({x}_{2}\right)=\frac{1}{{r}_{1}{r}_{2}}\sum _{{\ell }_{1},{\ell }_{2}}\sum _{{m}_{1},{m}_{2}}{Y}_{{\ell }_{1}{m}_{1}}^{{\ast}}\left({{\Omega}}_{1}\right){Y}_{{\ell }_{2}{m}_{2}}\left({{\Omega}}_{2}\right){\partial }_{v}{{\Phi}}_{{\ell }_{1}{m}_{2}}^{{\ast}}\left({U}_{1},{V}_{1}\right){\partial }_{v}{{\Phi}}_{{\ell }_{2}{m}_{2}}\left({U}_{2},{V}_{2}\right),\end{equation} \tag{ 126 }$$

and assuming that we may interchange the limits U → ∞ and v₁ → v₂ with the summations, we obtain

$$\begin{equation}{\langle {T}_{vv}\rangle }_{\mathrm{U}}-{\langle {T}_{vv}\rangle }_{\mathrm{C}}=\frac{1}{16{\pi }^{3}{r}_{-}^{2}}\sum _{\ell }\frac{2\ell +1}{4\pi }{\int }_{0}^{\infty }\mathrm{d}\omega \;\omega {n}_{\ell }\left(\omega \right),\end{equation} \tag{ 127 }$$

with n_ℓ (ω) as in (123). As discussed above, the integral on the right hand side converges for all ℓ. However, the convergence of the sum over ℓ is not obvious. Numerical evidence, see below, indicates that it does converge, which we take as an indication that the above interchange of limits is justified. In view of the regularity of C on ${\mathcal{CH}}^{R}$, the relationship between v and V₋, and proposition 5.1, we then also conclude:

Conclusion 2: if the sum on right side of (127) converges to a nonzero value, then for any initially a Hadamard state Ψ in a neighborhood of the Cauchy surface Σ as in figure 1 and for any RNdS spacetime with β > 1/2 (see (2)), then the expected stress tensor behaves in region II near ${\mathcal{CH}}^{R}$ as

$$\begin{equation}{\langle {T}_{VV}\rangle }_{{\Psi}}\sim C\vert V{\vert }^{-2}+{t}_{VV}.\end{equation} \tag{ 128 }$$

Here C is given by ${\kappa }_{-}^{2}$ times the right side of (127) and only depends on the parameters of the black hole but not on Ψ. t_VV , which depends on Ψ, has the regularity proposition 5.1, i.e. t_VV ∈ L^p as a function of V locally near ${\mathcal{CH}}^{R}$ with 1/p = 2 − 2β + 2 and with β − 1/2 − fractional derivatives w.r.t. V in L^{1/(3/2−β+)}, where > 0 is arbitrarily small.

Based on the results of the next section 5.3, we shall numerically evaluate the right sides of (123) and (127) in the case of the conformally coupled scalar field and find them to be nonvanishing. In particular, this provides evidence that (128) holds with C ≠ 0 for generic values of the black hole parameters.

Remark: these results should be compared with the expression (45) in the two-dimensional case. In particular, in (123) one explicitly sees the effect of backscattering in 4 dimensions, as the up modes also contribute. Analogous expressions for the evaluation of T_vv in the Unruh state on RN (without the subtraction of the contribution of the comparison state) can be found in [36].

We next describe how to determine the scattering coefficients ${\mathcal{R}}^{\mathrm{u}\mathrm{p},\mathrm{I}}$, ${\mathcal{T}}^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}$, ${\mathcal{R}}^{\mathrm{i}\mathrm{n},\mathrm{I}\mathrm{I}}$ needed for the evaluation of (123). When expressed in terms of r, the radial equation (104) has regular singular points at r = r₋, r₊, r_c , r_o , ∞ (recall (10) for the definition of r_o ). The transformation

$$\begin{equation}x=\frac{\left({r}_{-}-{r}_{o}\right)\left(r-{r}_{+}\right)}{\left({r}_{-}-{r}_{+}\right)\left(r-{r}_{o}\right)},\end{equation} \tag{ 129 }$$

maps r₋, r₊, r_c , r_o , ∞ to

$$\begin{equation}{x}_{-}=1,\quad {x}_{+}=0,\quad {x}_{c}=\frac{\left({r}_{-}-{r}_{o}\right)\left({r}_{c}-{r}_{+}\right)}{\left({r}_{-}-{r}_{+}\right)\left({r}_{c}-{r}_{o}\right)},\quad {x}_{o}=\infty ,\quad {x}_{\infty }=\frac{{r}_{-}-{r}_{o}}{{r}_{-}-{r}_{+}}\end{equation} \tag{ 130 }$$

which are the singular points of the radial equation written in terms of x. In the conformally coupled case, ${\mu }^{2}=\frac{1}{6}R=\frac{2}{3}{\Lambda}$, it is possible to factor out the singularity at x_∞ by a suitable transformation [77] of the radial equation ²⁷ . On account of this simplification, we restrict consideration to the conformally coupled case from this point on.

We define

$$\begin{equation}{b}_{+}=+i\frac{\omega }{2{\kappa }_{+}},\quad {b}_{-}=-i\frac{\omega }{2{\kappa }_{-}},\quad {b}_{c}=-i\frac{\omega }{2{\kappa }_{c}},\quad {b}_{o}=+i\frac{\omega }{2{\kappa }_{o}},\end{equation} \tag{ 131 }$$

and make the ansatz

$$\begin{equation}\left({r}^{-1}R\right)\left(r\left(x\right)\right)={\left({\pm}x\right)}^{{\sigma }_{+}{b}_{+}}{\left(1-x\right)}^{{\sigma }_{-}{b}_{-}}{\left(\frac{x-{x}_{c}}{1-{x}_{c}}\right)}^{{\sigma }_{c}{b}_{c}}\frac{x-{x}_{\infty }}{1-{x}_{\infty }}H\left(x\right),\end{equation} \tag{ 132 }$$

where σ_X = ±1 and the sign in the first factor is chosen below depending on whether one considers solutions in I or II. It is then found [77] that H(x) satisfies Heun's equation [79] with the singular points x₋, x₊, x_c , ∞. The parameters of this equation can be written in terms of these and x_∞.

It is known that solutions to Heun's equation which are analytic in a domain of the complex plane containing two of the four singular points can be constructed as series of various special functions, see [78, 80, 81] for the origins of this method. For us, it will be useful to follow a variant of this method where the special functions are hypergeometric functions, see [82], chapter 15. Of particular interest for us are the domains containing {x₊, x₋} respectively {x_c , x_∞}. In the first case one can define solutions fulfilling 'in' or 'up' boundary conditions at r₊ and compute the scattering coefficients ${\mathcal{T}}^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}$, ${\mathcal{R}}^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}$. In the second case, one can define solutions fulfilling specified boundary conditions at r_c . As the domains overlap, one can compute the scattering coefficients ${\mathcal{R}}^{\mathrm{u}\mathrm{p},\mathrm{I}}$.

The solution to the radial equation (104) with 'up' boundary condition at r₊ in region II can be written, up to normalization, as

$$\begin{align}\hfill {r}^{-1}{R}_{\omega \ell ,\nu }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}\left(r\right)& ={x}^{{b}_{+}}{\left(1-x\right)}^{{b}_{-}}{\left(\frac{x-{x}_{c}}{1-{x}_{c}}\right)}^{{b}_{c}}\frac{x-{x}_{\infty }}{1-{x}_{\infty }}\hfill \\ \hfill & \quad {\times}\sum _{n=-\infty }^{\infty }{a}_{n}^{\nu }F\left(-n-\nu +{b}_{+}+{b}_{-},n+\nu +{b}_{+}+{b}_{-}+1;1+2{b}_{+};x\right),\hfill \end{align} \tag{ 133 }$$

with F = ₂ F₁ the Gauss hypergeometric function. The so-called characteristic exponent ν is, a priori, an undetermined parameter unrelated to the parameters appearing in the radial—resp. Heun equation. The ansatz can be shown [77] to yield a formal solution for any ν, provided the coefficients ${a}_{n}^{\nu }$ solve the three term recursion relation

$$\begin{equation}{\alpha }_{n}^{\nu }{a}_{n+1}^{\nu }+{\beta }_{n}^{\nu }{a}_{n}^{\nu }+{\gamma }_{n}^{\nu }{a}_{n-1}^{\nu }=0\end{equation} \tag{ 134 }$$

with initial condition

$$\begin{equation}{a}_{0}^{\nu }=1\end{equation} \tag{ 135 }$$

and coefficients

$$\begin{align}\hfill {\alpha }_{n}^{\nu }=-\frac{\left(n+\nu -{b}_{+}-{b}_{-}+1\right)\left(n+\nu -{b}_{c}+{b}_{o}+1\right)\left(n+\nu -{b}_{c}-{b}_{o}+1\right)\left(n+\nu -{b}_{+}+{b}_{-}+1\right)}{2\left(n+\nu +1\right)\left(2n+2\nu +3\right)},\end{align} \tag{ 136a }$$

$$\begin{align}\hfill {\beta }_{n}^{\nu }& =\frac{\left({b}_{+}+{b}_{-}\right)\left({b}_{+}-{b}_{-}\right)\left({b}_{c}-{b}_{o}\right)\left({b}_{c}+{b}_{o}\right)}{2\left(n+\nu \right)\left(n+\nu +1\right)}+\left(\frac{1}{2}-{x}_{c}\right)\left(n+\nu \right)\left(n+\nu +1\right)\hfill \\ \hfill & \quad +\frac{1}{2}\left(\left(2{b}_{c}+1\right)\left({b}_{+}-{b}_{-}\right)-\left(2{b}_{-}+1\right)\left({b}_{+}+{b}_{-}\right)+{\left({b}_{+}+{b}_{-}+{b}_{c}+1\right)}^{2}-{b}_{o}^{2}\right)\hfill \\ \hfill & \quad +{x}_{c}\left({\left({b}_{+}+{b}_{-}\right)}^{2}+{b}_{+}+{b}_{-}\right)+v,\hfill \end{align} \tag{ 136b }$$

$$\begin{align}\hfill {\gamma }_{n}^{\nu }=-\frac{\left(n+\nu +{b}_{+}+{b}_{-}\right)\left(n+\nu +{b}_{c}-{b}_{o}\right)\left(n+\nu +{b}_{c}+{b}_{o}\right)\left(n+\nu +{b}_{+}-{b}_{-}\right)}{2\left(n+\nu \right)\left(2n+2\nu -1\right)},\end{align} \tag{ 136c }$$

where

$$\begin{align}\hfill v& =2{\omega }^{2}\frac{{r}_{+}^{3}\left({r}_{+}{r}_{-}-2{r}_{-}{r}_{c}+{r}_{+}{r}_{c}\right){\left({r}_{o}^{2}-{r}_{+}{r}_{-}-{r}_{+}{r}_{c}-{r}_{-}{r}_{c}\right)}^{2}}{{\left({r}_{+}-{r}_{-}\right)}^{3}{\left({r}_{+}-{r}_{c}\right)}^{2}\left({r}_{+}-{r}_{o}\right)\left({r}_{c}-{r}_{o}\right)}-2\frac{{r}_{o}^{2}}{\left({r}_{+}-{r}_{-}\right)\left({r}_{c}-{r}_{o}\right)}\hfill \\ \hfill & \quad -{x}_{\infty }-\left(1+{x}_{c}\right){b}_{+}-{x}_{c}{b}_{-}-{b}_{c}-2{b}_{+}\left({x}_{c}{b}_{-}+{b}_{c}\right)-\ell \left(\ell +1\right)\frac{{r}_{o}^{2}-{r}_{+}{r}_{-}-{r}_{+}{r}_{c}-{r}_{-}{r}_{c}}{\left({r}_{+}-{r}_{-}\right)\left({r}_{c}-{r}_{o}\right)}.\hfill \end{align} \tag{ 137 }$$

However, for generic values of the characteristic exponent ν, the solution is only formal in the sense that the series in (133) fails to converge. To get a convergent (bilateral) series, ν has to be determined such that the recursion relation (134) has a solution which is 'minimal' in the sense of [83], both for the ascending n → +∞ and for the descending n → −∞ series. By the general theory of three-term relations, a solution which is minimal in both directions can be determined as follows. Using one of the algorithms for finding the minimal solution to a three-term recursion relation described in [83], one determines

$$\begin{equation}{\rho }_{n}^{\nu }=\frac{{a}_{n}^{\nu }}{{a}_{n-1}^{\nu }},\quad {\lambda }_{n}^{\nu }=\frac{{a}_{n}^{\nu }}{{a}_{n+1}^{\nu }},\end{equation} \tag{ 138 }$$

the former by recursion starting at n = +∞, the latter by recursion starting at n = −∞ (for numerical investigations, at some integer of large modulus). The critical exponent ν is then determined by solving the transcendental equation

$$\begin{equation}{\rho }_{n}^{\nu }{\lambda }_{n-1}^{\nu }=1.\end{equation} \tag{ 139 }$$

Choosing this minimal solution, it follows by theorem 2.2 of [83] and the well-known asymptotics of F = ₂ F₁ that the series for (133) converges in some ellipse in the complex plane with focal points {x₊, x₋} [77].

It is obvious from the series (133) that the critical exponent is only determined up to an integer. Furthermore, one can show [77] that if ν is a solution to (139), then so is −ν − 1, with the corresponding coefficients related by

$$\begin{equation}{a}_{-n}^{-\nu -1}={a}_{n}^{\nu }.\end{equation} \tag{ 140 }$$

Although the solution of the transcendental equation (139) does not seem possible in closed form, one can in practice get solutions as power series expansions in suitably small parameters. For example, one can obtain the following (not necessarily convergent) asymptotic expansion for large r_c (near RN) and small ω,

$$\begin{equation}\nu \sim \ell \left[1+\sum _{m,n{\geqslant}0}^{\prime }{\nu }_{n,m}{\left(\frac{\omega }{\ell }\right)}^{n}{\left(\frac{1}{{r}_{c}}\right)}^{m}\right],\end{equation} \tag{ 141 }$$

where the prime indicates that (m, n) = (0, 0) is excluded. The coefficients depend on ℓ and r₋, r₊, remain bounded for ℓ → ∞, and can be found by direct substitution into the defining equation for ν. The lowest order in n + m coefficients turn out to be

$$\begin{equation}{\nu }_{2,0}=\frac{\ell \left(11{r}_{-}^{2}+14{r}_{-}{r}_{+}+11{r}_{+}^{2}\right)-{\ell }^{2}\left(\ell +1\right)\left(15{r}_{-}^{2}+18{r}_{-}{r}_{+}+15{r}_{+}^{2}\right)}{2\left(2\ell +1\right)\left(4\ell \left(\ell +1\right)-3\right)},\end{equation} \tag{ 142a }$$

$$\begin{equation}{\nu }_{0,2}=\frac{\left(15{\ell }^{4}+30{\ell }^{3}+9{\ell }^{2}-6\ell -4\right)\left({r}_{-}^{2}+{r}_{+}^{2}\right)+\left(18{\ell }^{4}+36{\ell }^{3}+10{\ell }^{2}-8\ell -4\right){r}_{-}{r}_{+}}{2\ell \left(2\ell +1\right)\left(4\ell \left(\ell +1\right)-3\right)},\end{equation} \tag{ 142b }$$

$$\begin{equation}{\nu }_{0,3}=\frac{-\left(15{\ell }^{4}+30{\ell }^{3}+9{\ell }^{2}-6\ell -4\right){\left({r}_{-}^{2}+{r}_{+}^{2}\right)}^{2}-\left(33{\ell }^{4}+66{\ell }^{3}+19{\ell }^{2}-14\ell -8\right){r}_{-}{r}_{+}\left({r}_{-}+{r}_{+}\right)}{2\ell \left(8{\ell }^{3}+12{\ell }^{2}-2\ell -3\right)}.\end{equation} \tag{ 142c }$$

To determine the normalization of the solution (133), we note that, by F(a, b; c; 0) = 1, it behaves as

$$\begin{equation}{r}^{-1}{R}_{\omega \ell ,\nu }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}\left(x\right)\sim {\mathrm{e}}^{\text{i}\omega \left({r}_{{\ast}}+{D}^{\prime }\right)}{\left(\frac{-{x}_{c}}{1-{x}_{c}}\right)}^{{b}_{c}}\frac{-{x}_{\infty }}{1-{x}_{\infty }}\sum _{n=-\infty }^{\infty }{a}_{n}^{\nu }\end{equation} \tag{ 143 }$$

near r₊, with a constant D', related to the integration constant D in (12). To determine the asymptotic behavior at the Cauchy horizon, we use [82], 15.3.6,

$$\begin{align}\hfill F\left(a,b;c;x\right)& =\frac{{\Gamma}\left(c\right){\Gamma}\left(c-a-b\right)}{{\Gamma}\left(c-a\right){\Gamma}\left(c-b\right)}F\left(a,b;a+b-c+1;1-x\right)\hfill \\ \hfill & \quad +{\left(1-x\right)}^{c-a-b}\frac{{\Gamma}\left(c\right){\Gamma}\left(a+b-c\right)}{{\Gamma}\left(a\right){\Gamma}\left(b\right)}F\left(c-a,c-b;c-a-b+1;1-x\right).\hfill \end{align} \tag{ 144 }$$

It follows that near r₋, the solution (133) behaves as

$$\begin{align}\hfill {r}^{-1}{R}_{\omega \ell ,\nu }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}& \sim {\mathrm{e}}^{\text{i}\omega \left({r}_{{\ast}}+{D}^{\prime \prime }\right)}\sum _{n=-\infty }^{\infty }{a}_{n}^{\nu }\frac{{\Gamma}\left(1+2{b}_{+}\right){\Gamma}\left(-2{b}_{-}\right)}{{\Gamma}\left({b}_{+}-{b}_{-}+n+\nu +1\right){\Gamma}\left({b}_{+}-{b}_{-}-n-\nu \right)}\hfill \\ \hfill & \quad +\;{\mathrm{e}}^{-\text{i}\omega \left({r}_{{\ast}}+{D}^{\prime \prime }\right)}\sum _{n=-\infty }^{\infty }{a}_{n}^{\nu }\frac{{\Gamma}\left(1+2{b}_{+}\right){\Gamma}\left(2{b}_{-}\right)}{{\Gamma}\left({b}_{+}+{b}_{-}-n-\nu \right){\Gamma}\left({b}_{+}+{b}_{-}+n+\nu +1\right)},\hfill \end{align} \tag{ 145 }$$

with another integration constant D''. From this, one can read off

$$\begin{align}\hfill {\mathcal{T}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}& ={\mathrm{e}}^{-\text{i}\omega \left({D}^{\prime }-{D}^{\prime \prime }\right)}\frac{{r}_{-}}{{r}_{+}}{\left(\frac{1-{x}_{c}}{-{x}_{c}}\right)}^{{b}_{c}}\frac{1-{x}_{\infty }}{-{x}_{\infty }}{\Gamma}\left(1+2{b}_{+}\right){\Gamma}\left(-2{b}_{-}\right)\hfill \\ \hfill & \quad {\times}\frac{{\sum }_{n=-\infty }^{\infty }{a}_{n}^{\nu }{\left[{\Gamma}\left({b}_{+}-{b}_{-}+n+\nu +1\right){\Gamma}\left({b}_{+}-{b}_{-}-n-\nu \right)\right]}^{-1}}{{\sum }_{n=-\infty }^{\infty }{a}_{n}^{\nu }},\hfill \end{align} \tag{ 146a }$$

$$\begin{align}\hfill {\mathcal{R}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}\mathrm{I}}& ={\mathrm{e}}^{-\text{i}\omega \left({D}^{\prime }+{D}^{\prime \prime }\right)}\frac{{r}_{-}}{{r}_{+}}{\left(\frac{1-{x}_{c}}{-{x}_{c}}\right)}^{{b}_{c}}\frac{1-{x}_{\infty }}{-{x}_{\infty }}{\Gamma}\left(1+2{b}_{+}\right){\Gamma}\left(2{b}_{-}\right)\hfill \\ \hfill & \quad {\times}\;\frac{{\sum }_{n=-\infty }^{\infty }{a}_{n}^{\nu }{\left[{\Gamma}\left({b}_{+}+{b}_{-}-n-\nu \right){\Gamma}\left({b}_{+}+{b}_{-}+n+\nu +1\right)\right]}^{-1}}{{\sum }_{n=-\infty }^{\infty }{a}_{n}^{\nu }}.\hfill \end{align} \tag{ 146b }$$

From the factors Γ(±2b₋), one easily verifies the asymptotic behavior (124). Note that in both expressions the fraction of the sums converges to

$$\begin{equation}-\frac{1}{\pi }\mathrm{sin}\pi \nu \frac{{\sum }_{n=-\infty }^{\infty }{\left(-1\right)}^{n}{a}_{n}^{\nu }}{{\sum }_{n=-\infty }^{\infty }{a}_{n}^{\nu }}\end{equation} \tag{ 147 }$$

in the limit as ω → 0, so that the divergence at ω → 0 cannot be enhanced but possibly canceled if $\nu \to \mathbb{Z}$ for ω → 0, which, as can be seen from (141), happens in the RN limit, consistent with results by [76].

In order to determine the coefficient ${\mathcal{R}}^{\mathrm{u}\mathrm{p},\mathrm{I}}$, we first define the up mode ${R}_{\omega \ell ,\nu }^{\mathrm{u}\mathrm{p},\mathrm{I}}$ in region I by replacing in (133) the first factor by ${\left(-x\right)}^{{b}_{+}}$. Then, by [82], 15.3.8, and (140), one can write

$$\begin{equation}{R}_{\omega \ell ,\nu }^{\mathrm{u}\mathrm{p},\mathrm{I}}\left(x\right)={R}_{\omega \ell ,\nu }^{+,\mathrm{I}}\left(z\right)+{R}_{\omega \ell ,-\nu -1}^{+,\mathrm{I}}\left(z\right),\end{equation} \tag{ 148 }$$

where

$$\begin{align}\hfill {r}^{-1}{R}_{\omega \ell ,\nu }^{+,\mathrm{I}}\left(z\right)& ={\left(z-1\right)}^{{b}_{+}}{z}^{{b}_{-}}{\left(1-\frac{z}{{z}_{c}}\right)}^{{b}_{c}}\left(1-\frac{z}{{z}_{\infty }}\right){\Gamma}\left(1+2{b}_{+}\right)\hfill \\ \hfill & \quad {\times}\;\sum _{n=-\infty }^{\infty }{a}_{n}^{\nu }\frac{{\Gamma}\left(2n+2\nu +1\right)}{{\Gamma}\left(n+\nu +{b}_{+}+{b}_{-}+1\right){\Gamma}\left(n+\nu +{b}_{+}-{b}_{-}+1\right)}{z}^{n+\nu -{b}_{+}-{b}_{-}}\hfill \\ \hfill & \quad {\times}\;F\left({b}_{+}+{b}_{-}-n-\nu ,{b}_{+}-{b}_{-}-n-\nu ;-2n-2\nu ;1/z\right).\hfill \end{align} \tag{ 149 }$$

Here z = 1 − x, and analogously for z_c , z_∞. By (144), the behavior near the event horizon is given by

$$\begin{align}\hfill {r}^{-1}{R}_{\omega \ell ,\nu }^{+,\mathrm{I}}& \sim {\mathrm{e}}^{\text{i}\omega \left({r}_{{\ast}}+{D}^{{\prime\prime\prime}}\right)}{\left(\frac{-{x}_{c}}{1-{x}_{c}}\right)}^{{b}_{c}}\frac{-{x}_{\infty }}{1-{x}_{\infty }}\frac{\mathrm{sin}\left[\pi \left(\nu +{b}_{+}+{b}_{-}\right)\right]\mathrm{sin}\left[\pi \left(\nu +{b}_{+}-{b}_{-}\right)\right]}{\mathrm{sin}\;2\pi {b}_{+}\;\mathrm{sin}\;2\pi \nu }\hfill \\ \hfill & \quad {\times}\sum _{n=-\infty }^{\infty }{a}_{n}^{\nu }+\;{\mathrm{e}}^{-\text{i}\omega \left({r}_{{\ast}}+{D}^{{\prime\prime\prime}}\right)}{\left(\frac{-{x}_{c}}{1-{x}_{c}}\right)}^{{b}_{c}}\frac{-{x}_{\infty }}{1-{x}_{\infty }}{\Gamma}\left(1+2{b}_{+}\right){\Gamma}\left(2{b}_{+}\right){\Gamma}\left(1+2\nu \right){\Gamma}\left(-2\nu \right)\hfill \\ \hfill & \quad {\times}\;\sum _{n=-\infty }^{\infty }{a}_{n}^{\nu }\left[{\Gamma}\left(1+n+\nu +{b}_{+}+{b}_{-}\right){\Gamma}\left(1+n+\nu +{b}_{+}-{b}_{-}\right)\right.\hfill \\ \hfill & \quad {\times}\;{\left.{\Gamma}\left(-n-\nu +{b}_{+}+{b}_{-}\right){\Gamma}\left(-n-\nu +{b}_{+}-{b}_{-}\right)\right]}^{-1},\hfill \end{align} \tag{ 150 }$$

with some integration constant D^‴, where we also used

$$\begin{equation}{\Gamma}\left(z\right){\Gamma}\left(1-z\right)=\pi \mathrm{csc}\pi z.\end{equation} \tag{ 151 }$$

An analogous solution in region I which is regular at the cosmological horizon r_c is given by

$$\begin{align}\hfill {r}^{-1}{R}_{\omega \ell ,\nu }^{c,\mathrm{I}}\left(z\right)& ={\left(z-1\right)}^{{b}_{+}}{z}^{{b}_{-}}{\left(1-\frac{z}{{z}_{c}}\right)}^{{b}_{c}}\left(1-\frac{z}{{z}_{\infty }}\right)\frac{1}{{\Gamma}\left({b}_{+}+{b}_{-}-{b}_{c}-{b}_{o}+1\right)}\hfill \\ \hfill & \quad {\times}\;\sum _{n=-\infty }^{\infty }{a}_{n}^{\nu }\frac{{\Gamma}\left(n+\nu -{b}_{c}+{b}_{o}+1\right){\Gamma}\left(n+\nu -{b}_{c}-{b}_{o}+1\right)}{{\Gamma}\left(2n+2\nu +2\right)}{\left(\frac{z}{{z}_{c}}\right)}^{n+\nu -{b}_{+}-{b}_{-}}\hfill \\ \hfill & \quad {\times}\;F\left(n+\nu +{b}_{+}-{b}_{-}+1,n+\nu +{b}_{+}+{b}_{-}+1;2n+2\nu +2;z/{z}_{c}\right).\hfill \end{align} \tag{ 152 }$$

It follows from the coefficients in our three-term relation, by theorem 2.2 of [83] and the well-known asymptotics of F = ₂ F₁ that the series for (133) converges in some ellipse in the complex z_c /z plane with focal points {0, 1}, containing in particular {x_c , x_∞} in the complex x plane. By comparison of coefficients one then finds that [77]

$$\begin{equation}{R}_{\omega \ell ,\nu }^{+,\mathrm{I}}={K}_{\omega \ell }^{\nu }{R}_{\omega \ell ,\nu }^{c,\mathrm{I}}\end{equation} \tag{ 153 }$$

with

$$\begin{align}\hfill & {K}_{\omega \ell }^{\nu }={z}_{c}^{p+\nu -{b}_{+}-{b}_{-}}\hfill \\ \hfill & {\times}\frac{{\Gamma}\left(1+2{b}_{+}\right){\Gamma}\left({b}_{+}+{b}_{-}-{b}_{c}-{b}_{o}+1\right)}{{\Gamma}\left(p+\nu -{b}_{+}-{b}_{-}+1\right){\Gamma}\left(p+\nu -{b}_{+}+{b}_{-}+1\right){\Gamma}\left(p+\nu +{b}_{c}-{b}_{o}+1\right){\Gamma}\left(p+\nu +{b}_{c}+{b}_{o}+1\right)}\hfill \\ \hfill & {\times}\sum _{n=p}^{\infty }{\left(-1\right)}^{n-p}{a}_{n}^{\nu }\frac{{\Gamma}\left(n+\nu -{b}_{+}-{b}_{-}+1\right){\Gamma}\left(n+\nu -{b}_{+}+{b}_{-}+1\right){\Gamma}\left(p+n+2\nu +1\right)}{{\Gamma}\left(n+\nu +{b}_{+}+{b}_{-}+1\right){\Gamma}\left(n+\nu +{b}_{+}-{b}_{-}+1\right)\left(n-p\right)!}\hfill \\ \hfill & {\times}{\left(\sum _{n=-\infty }^{p}{a}_{n}^{\nu }\frac{{\Gamma}\left(n+\nu -{b}_{c}+{b}_{o}+1\right){\Gamma}\left(n+\nu -{b}_{c}-{b}_{o}+1\right)}{{\Gamma}\left(n+\nu +{b}_{c}-{b}_{o}+1\right){\Gamma}\left(n+\nu +{b}_{c}+{b}_{o}+1\right){\Gamma}\left(p+n+2\nu +2\right)\left(p-n\right)!}\right)}^{-1}.\hfill \end{align} \tag{ 154 }$$

Here p is an arbitrary integer, on which the result does not depend (keeping this parameter provides a check for the subsequent formulas). The solutions ${R}_{\omega \ell ,\nu }^{c,\mathrm{I}}$ and ${R}_{\omega \ell ,-\nu -1}^{c,\mathrm{I}}$ are linearly independent. Using (140), (144), and the identity (151), one can show that the linear combination

$$\begin{align}\hfill {R}_{\omega \ell ,\nu }^{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{I}}& =\mathrm{sin}\left[\pi \left({b}_{c}+{b}_{o}-\nu \right)\right]\mathrm{sin}\left[\pi \left({b}_{c}-{b}_{o}-\nu \right)\right]{R}_{\omega \ell ,\nu }^{c,\mathrm{I}}\hfill \\ \hfill & \quad -\mathrm{sin}\left[\pi \left({b}_{c}+{b}_{o}+\nu \right)\right]\mathrm{sin}\left[\pi \left({b}_{c}-{b}_{o}+\nu \right)\right]{R}_{\omega \ell ,-\nu -1}^{c,\mathrm{I}}\hfill \end{align} \tag{ 155 }$$

is given by

$$\begin{align}\hfill {r}^{-1}{R}_{\omega \ell ,\nu }^{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{I}}\left(z\right)& ={\left(z-1\right)}^{{b}_{+}}{z}^{{b}_{-}}{\left(1-\frac{z}{{z}_{c}}\right)}^{{b}_{c}}\left(1-\frac{z}{{z}_{\infty }}\right)\frac{1}{{\Gamma}\left({b}_{+}+{b}_{-}-{b}_{c}-{b}_{o}+1\right)}\frac{\pi \;\mathrm{sin}\;2\pi \nu }{{\Gamma}\left(1+2{b}_{c}\right)}\hfill \\ \hfill & \quad {\times}\;\sum _{n=-\infty }^{\infty }{a}_{n}^{\nu }{\left(\frac{z}{{z}_{c}}\right)}^{n+\nu -{b}_{+}-{b}_{-}}\hfill \\ \hfill & \quad {\times}F\left(n+\nu +{b}_{c}-{b}_{o}+1,n+\nu +{b}_{c}+{b}_{o}+1;1+2{b}_{c};1-z/{z}_{c}\right).\hfill \end{align} \tag{ 156 }$$

Its asymptotic form near r = r_c is given by ${R}_{\omega \ell ,\nu }^{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{I}}\sim {D}_{\omega }{\mathrm{e}}^{\text{i}\omega {r}_{{\ast}}}$ with some irrelevant factor D_ω . Thus, it represents an outgoing solution near the cosmological horizon. Using (153), we may reexpress ${R}_{\omega \ell ,\nu }^{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{I}}$ as

$$\begin{equation}{R}_{\omega \ell ,\nu }^{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{I}}=\mathrm{sin}\left[\pi \left({b}_{c}+{b}_{o}-\nu \right)\right]\mathrm{sin}\left[\pi \left({b}_{c}-{b}_{o}-\nu \right)\right]{\left({K}_{\omega \ell }^{\nu }\right)}^{-1}{R}_{\omega \ell ,\nu }^{+,\mathrm{I}}-\left\{\nu {\leftrightarrow}-\nu -1\right\}.\end{equation} \tag{ 157 }$$

From (150), we can thus read off

$$\begin{align}\hfill {\mathcal{R}}_{\omega \ell }^{\mathrm{u}\mathrm{p},\mathrm{I}}=& -{\mathrm{e}}^{-2\text{i}\omega {D}^{\prime }}\pi \mathrm{sin}\left(2\pi {b}_{+}\right){\Gamma}\left(1+2{b}_{+}\right){\Gamma}\left(2{b}_{-}\right)\hfill \\ \hfill & {\times}\;\left[\mathrm{sin}\left[\pi \left({b}_{c}+{b}_{o}-\nu \right)\right]\mathrm{sin}\left[\pi \left({b}_{c}-{b}_{o}-\nu \right)\right]{\left({K}_{\omega \ell }^{\nu }\right)}^{-1}+\left\{\nu {\leftrightarrow}-\nu -1\right\}\right]\hfill \\ \hfill & {\times}\;\sum _{n=-\infty }^{\infty }{a}_{n}^{\nu }\left[{\Gamma}\left(1+n+\nu +{b}_{+}+{b}_{-}\right){\Gamma}\left(1+n+\nu +{b}_{+}-{b}_{-}\right){\Gamma}\left({b}_{+}+{b}_{-}-n-\nu \right)\right.\hfill \\ \hfill & \quad {\times}\;{\left.{\Gamma}\left({b}_{+}-{b}_{-}-n-\nu \right)\right]}^{-1}\left[\mathrm{sin}\left[\pi \left({b}_{c}+{b}_{o}-\nu \right)\right]\mathrm{sin}\left[\pi \left({b}_{c}-{b}_{o}-\nu \right)\right]\right.\hfill \\ \hfill & \quad {\times}\mathrm{sin}\left[\pi \left({b}_{+}+{b}_{-}+\nu \right)\right]\mathrm{sin}\left[\pi \left({b}_{+}-{b}_{-}+\nu \right)\right]{\left({K}_{\omega \ell }^{\nu }\right)}^{-1}\hfill \\ \hfill & {\left.\quad \,+\left\{\nu {\leftrightarrow}\nu -1\right\}\right]}^{-1}{\left[\sum _{n=-\infty }^{\infty }{a}_{n}^{\nu }\right]}^{-1}.\hfill \end{align} \tag{ 158 }$$

Regarding the integration constants D', D'', one easily checks that these cancel in the next-to-last term in (123), which is the only term sensitive to phases.

From the discussion in the previous sections, two open questions remain: (1) Is the 'density of states' n_ℓ (ω) as given in (123) non-vanishing ²⁸ , with generically non-vanishing integrals of the form (122)? (2) Does the sum (127) over angular momenta converge and give a generically non-vanishing result, corresponding to a nonzero C in (128), resp. (5)? Without performing an exhaustive parameter scan, we provide numerical evidence that the answer to both questions is affirmative for the case of a conformally coupled scalar field.

As the near extremal case seems to be the most interesting one, due to large violations of sCC in the classical case, we pick an example from that regime. Figure 10 shows ωn₀(ω) for the parameters r_c = 100, r₊ = 2, r₋ = 1.95, corresponding to κ_c = 0.0094, κ₊ = 0.0062, κ₋ = 0.0066. Shown in the figure are results obtained with the method based on Heun functions described in section 5.3, combined with results obtained by direct numerical integration of the radial equation, which is the method employed in [36] for the RN case. The consideration of the results obtained by direct integration of the radial equation not only provides a consistency check for the results obtained with the Heun function method, but is at present also necessary as the search for a solution of (139), i.e. for the critical exponent ν, becomes difficult for large frequencies ω for which the asymptotic expansion (141) does not provide a useful estimate for the starting point of a numerical search algorithm ²⁹ . On the other hand, the Heun method seems to have advantages for ω → 0 where some scattering coefficients diverge.

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At any rate, clearly ωn₀(ω) is not only non-vanishing, but also has a non-vanishing integral, answering the first of the above questions. As for the second question, one finds that ωn₁(ω) is already of order 10⁻¹⁵, i.e. suppressed by eight orders of magnitude. The contributions of higher angular momenta are already drowned in numerical noise. The same behavior, i.e. non-vanishing integrands ωn_ℓ (ω) with non-vanishing integrals and a rapid decrease in angular momentum were also found for the other parameters r_X at which we numerically evaluated ωn_ℓ (ω) (both in the near-extremal and the 'normal' regime κ₋ > κ_c ). We thus conclude that the coefficient C in (5) is indeed generically non-vanishing.