Free collapse Hawking approximation and its proof obligations

Use the mathematical assumptions of the supplied chiral channel, including Countable Choice: every sequence of nonempty sets admits a choice function. Adopt SI units. Let $c>0$ have units $\mathrm{m}{\mathrm{s}}^{-1}$, $G>0$ have units ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, $\mathrm{\hslash }>0$ have units $\mathrm{J}\mathrm{s}$, and $k_B>0$ have units $\mathrm{J}{\mathrm{K}}^{-1}$. The mass $M>0$ is a fixed model parameter in kilograms. Use a Hausdorff, second-countable, oriented, time-oriented, globally hyperbolic smooth Lorentzian four-manifold $(\mathcal{M},g)$ without boundary, of signature $(-+++)$. Global hyperbolicity means strong causality and compact causal diamonds $J^{+}(p)\cap J^{-}(q)$, where $J^{\pm }$ are defined by consistently directed causal curves; strong causality means every point has arbitrarily small neighborhoods that each causal curve meets in a connected parameter interval. A Hadamard bisolution is a distribution $W_2$ on $\mathcal{M}\times \mathcal{M}$ satisfying $P_x{W}_2=P_y{W}_2=0$ and $\mathrm{WF}(W_2)={\mathcal{C}}_g^{+}$, the positive-frequency null relation whose pairs $(x,\xi ;y,-\eta )$ have nonzero future-null covectors $\xi ,\eta$ equal to a common positive multiple of the metric-dual tangent of one affinely parametrized future null geodesic at the two endpoints, with covector parallel transport and either endpoint parameter order; coincident endpoints have $\eta =\xi$. The manifold is the smooth domain on which this free-field model is posed; no singular interior is included.

Assume that after a smooth Cauchy surface ${\mathrm{\Sigma }}_0$ (a spacelike hypersurface met exactly once by each inextendible causal curve of the modeled region), the modeled future exterior is exactly the Schwarzschild exterior of fixed mass $M$, with areal radius $r>r_s$, $r_s=2GM\mathrm{/}{c}^2$, and metric $d{s}^2=-(1-\frac{r_s}{r})c^{2}d{t}^2+{(1-\frac{r_s}{r})}^{-1}d{r}^2+r^2(d{\vartheta }^2+{\sin }^2\vartheta d{\phi }^2)\mathrm{.}$ Here $t$ is the stationary Schwarzschild time in seconds, $r$ is in metres, and $\vartheta ,\phi$ are the usual dimensionless angular coordinates. The metric is a fixed background: the field does not change it. The retarded coordinate is $u=t-r_{\ast }\mathrm{/}c$, where $r_{\ast }=r+r_s\log (r\mathrm{/}{r}_s-1)$. The associated geometric surface-gravity parameter in inverse metres and its time-frequency version are, by this model's normalization, ${\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}:=\frac{1}{2r_s}=\frac{c^2}{4GM},k:=c{\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}=\frac{c^3}{4GM}>0.$

Adopt one real, linear, massless, minimally coupled scalar field on $(\mathcal{M},g)$. Its smooth classical solutions are $\varphi \in C^{\mathrm{\infty }}(\mathcal{M};\mathbb{R})$ satisfying $P\varphi =0$ for $P:=-g^{ab}{\mathrm{\nabla }}_a{\mathrm{\nabla }}_b$, where $\mathrm{\nabla }$ is the unique torsion-free connection with $\mathrm{\nabla }g=0$; the canonical scalar field has length degree ${\mathrm{m}}^{-1}$.

For the commutator normalization, supply continuous real-linear retarded and advanced Green maps $G_{\mathrm{r}\mathrm{e}\mathrm{t}},G_{\mathrm{a}\mathrm{d}\mathrm{v}}:C_c^{\mathrm{\infty }}(\mathcal{M};\mathbb{R})\to C^{\mathrm{\infty }}(\mathcal{M};\mathbb{R})$ for this same $P$. The domain has the test-function topology specified below; the target has uniform convergence of every coordinate derivative on compact subsets. For every real test function $f$, require $P{G}_{\mathrm{r}\mathrm{e}\mathrm{t}}f=G_{\mathrm{r}\mathrm{e}\mathrm{t}}Pf=f,P{G}_{\mathrm{a}\mathrm{d}\mathrm{v}}f=G_{\mathrm{a}\mathrm{d}\mathrm{v}}Pf=f,$ $\mathrm{supp}(G_{\mathrm{r}\mathrm{e}\mathrm{t}}f)\subseteq J^{+}(\mathrm{supp}f),\mathrm{supp}(G_{\mathrm{a}\mathrm{d}\mathrm{v}}f)\subseteq J^{-}(\mathrm{supp}f),$ where $J^{\pm }(S):={\bigcup }_{p\in S}{J}^{\pm }(p)$. With the positive metric density $d{\mathrm{v}\mathrm{o}\mathrm{l}}_g=\mathrm{\mid }\det (g_{ab}){\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\mid }d{x}^{0}d{x}^{1}d{x}^{2}d{x}^3\mathrm{\mid }$, fix the advanced-minus-retarded convention $\mathrm{\Delta }:=G_{\mathrm{a}\mathrm{d}\mathrm{v}}-G_{\mathrm{r}\mathrm{e}\mathrm{t}},E_P(f,g):={\int }_{\mathcal{M}}f\mathrm{\Delta }gd{\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$ These Green maps and their compatibility with the quantization below are supplied model data, not an existence theorem. This sign convention is for $P=-{\mathrm{\square }}_g$ and the conjugate-linear-first Hilbert pairing. In canonical normalization, tests of units ${\mathrm{m}}^{-3}$ make $\mathrm{\Phi }(f)$ and $E_P(f,g)$ dimensionless; the unsmeared commutator kernel has length degree ${\mathrm{m}}^{-2}$.

For its quantization, choose a complex Hilbert one-particle space ${\mathcal{H}}_{\mathrm{i}\mathrm{n}}$ with inner product conjugate-linear in the first variable and linear in the second; its dense incoming packet subspace is specified below. For $n\ge 1$, let ${\mathcal{H}}_{\mathrm{i}\mathrm{n},\mathrm{s}}^{(n)}$ be the range of the symmetrizing projection $P_{s,n}:=\frac{1}{n!}{\sum }_{\pi \in S_n}{U}_{\pi }$ on the completed $n$-fold Hilbert tensor product, where $U_{\pi }$ permutes tensor factors, and put ${\mathcal{H}}_{\mathrm{i}\mathrm{n},\mathrm{s}}^{(0)}:=\mathbb{C}$. The bosonic Fock space is the Hilbert direct sum ${\mathcal{F}}_s({\mathcal{H}}_{\mathrm{i}\mathrm{n}}):=\{({\mathrm{\Psi }}_n{)}_{n\ge 0}:{\mathrm{\Psi }}_n\in {\mathcal{H}}_{\mathrm{i}\mathrm{n},\mathrm{s}}^{(n)},{\sum }_{n\ge 0}\mathrm{\parallel }{\mathrm{\Psi }}_n{\mathrm{\parallel }}^2<\mathrm{\infty }\},$ with vacuum $\mathrm{\Omega }=(1,0,0,\dots )$ and finite-particle domain ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ consisting of sequences with only finitely many nonzero components. On this domain, define creation and annihilation by $a_n^{†}(h){\mathrm{\Psi }}_n:=\sqrt{n+1}{P}_{s,n+1}(h\otimes {\mathrm{\Psi }}_n)$ and $a_n(h){\mathrm{\Psi }}_n:=\sqrt{n}(C_h\otimes I^{\otimes (n-1)}){\mathrm{\Psi }}_n$ for $n\ge 1$, where $C_h(v):=⟨h,v⟩$; set $a(h)\mathrm{\Omega }=0$. The model supplies a continuous real-linear map $K:C_c^{\mathrm{\infty }}(\mathcal{M};\mathbb{R})\to {\mathcal{H}}_{\mathrm{i}\mathrm{n}}$, for the standard test-function topology, the locally convex inductive-limit topology over compact supports of the fixed-support smooth-function spaces, whose seminorms are suprema of each finite order of coordinate derivatives on compact chart subsets, with $K(Pf)=0$ and, for all real tests $f,g$, the required symplectic normalization $2\mathrm{Im}⟨Kf,Kg⟩=E_P(f,g)$, and defines the smeared field $\mathrm{\Phi }(f):=a(Kf)+a^{†}(Kf)$ on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$. For real test functions the Fock commutation relations give $[\mathrm{\Phi }(f),\mathrm{\Phi }(g)]=2i\mathrm{Im}⟨Kf,Kg⟩I$, where $I$ is the identity; the imposed normalization therefore gives $[\mathrm{\Phi }(f),\mathrm{\Phi }(g)]=i{E}_P(f,g)I$ on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$. This normalization is an additional adoption, not a consequence of $K(Pf)=0$ or the Hadamard wavefront condition. Let ${\mathfrak{A}}_{\mathrm{i}\mathrm{n}}$ be the unital complex $\ast$-algebra of these incoming creation, annihilation and smeared-field operators on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$, with the restricted Hilbert adjoint as involution. Supply a common unital complex $\ast$-algebra $\mathfrak{A}$ and an injective unital $\ast$-homomorphism from ${\mathfrak{A}}_{\mathrm{i}\mathrm{n}}$ into it, whose images are denoted by the same symbols. The outgoing packet generators specified below belong to $\mathfrak{A}$. A compatible positive normalized state on this whole algebra, including its incoming-ground and Hadamard conditions, is supplied as model data; extension of an arbitrary previously chosen field state is not asserted. Algebra products and the involution define all expectations below without requiring outgoing unbounded operators to preserve the incoming Fock domain. If an operator realization is selected, a common invariant dense domain for all generators and their adjoints is part of that realization's supplied data; no assertion about their maximal closed domains follows. A state $\varpi$ is a positive normalized linear functional on that algebra, and its two-point distribution is $W_2(f,g)=\varpi (\mathrm{\Phi }(f)\mathrm{\Phi }(g))$. Require $W_2$ to be a Hadamard bisolution for $P$; its antisymmetric part is then $W_2(f,g)-W_2(g,f)=i{E}_P(f,g)$ for real tests by the commutator and $\varpi (I)=1$. The pairing uses $d{\mathrm{v}\mathrm{o}\mathrm{l}}_g=\mathrm{\mid }\det (g_{ab}){\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\mid }d{x}^{0}d{x}^{1}d{x}^{2}d{x}^3\mathrm{\mid }$. The one-particle space, map $K$, and state are adopted model data; their construction is not derived from collapse matter. Here positive normalized means complex-linear with $\varpi (T^{\ast }T)\ge 0$ for every $T\in \mathfrak{A}$ and $\varpi (I)=1$.

Assume a smooth conformal completion $\bar{\mathcal{M}}$ with boundary defining function $\rho$ such that $\rho >0$ on $\mathcal{M}$, $\rho =0$ and $d\rho \mathrm{\ne }0$ on the selected boundary components, and $\tilde{g}={\rho }^{2}g$ extends smoothly as a nondegenerate Lorentzian metric with those components null hypersurfaces. Denote the past and future null components by ${\mathcal{I}}^{-}$ and ${\mathcal{I}}^{+}$; they are the asymptotic endpoints of incoming and outgoing null rays in this model. Fix a smooth affine parameter $v$ in seconds on the generators of ${\mathcal{I}}^{-}$, relative to $\tilde{g}$ (so ${\tilde{\mathrm{\nabla }}}_{{\mathrm{\partial }}_v}{\mathrm{\partial }}_v=0$). The dense incoming positive-frequency one-particle packet subspace consists of the model's incoming mode packets with Fourier dependence $e^{-i\omega v}$, where $\omega >0$ is angular frequency in radians per second and radians are dimensionless. For every packet $h$ in this subspace, write $a_{\mathrm{i}\mathrm{n}}(h):=a(h)$; the incoming ground-state condition means $\varpi (a_{\mathrm{i}\mathrm{n}}(h{)}^{\ast }{a}_{\mathrm{i}\mathrm{n}}(h))=0$. This free-field equation, quantization, state, and asymptotic structure are adopted, not derived assumptions.

For each outgoing retarded-time coordinate $u$, measured in seconds, adopt the increasing null-ray label $U(u)=U_H-A{e}^{-ku},U_H\in \mathbb{R}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{s},A>0\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{s}\mathrm{.}$ The two-point current channel for this label is exactly the tested exponential channel defined by the ray map above. It is the time-coordinate version of the reduced exponential-current branch, whose length flow parameter is $u_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}=cu$, inverse-length parameter is ${\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}=k\mathrm{/}c$, and detailed-balance convention is the same. The identification of the four-dimensional near-horizon packet channel with this one-dimensional tested current is an adopted model premise, not a restriction theorem for a four-dimensional field. The separate accelerated-detector and two-state-rate branch is not part of this collapse model.

Let $\mathrm{\ell }\in {\mathbb{N}}_0$ label the degree of a spherical-harmonic angular channel; no sum over its magnetic degeneracy is taken. For angular frequency $\omega >0$, measured in radians per second (radians are dimensionless), let ${\mathrm{\Gamma }}_{\mathrm{\ell }}(\omega )\in [0,1]$ be the dimensionless transmission probability supplied by the stationary radial-scattering model. Assume ${\mathrm{\Gamma }}_{\mathrm{\ell }}$ is continuous on every compact frequency interval $K=[{\omega }_{-},{\omega }_{+}]\subset (0,\mathrm{\infty })$ with ${\omega }_{-}<{\omega }_{+}$. Put ${\mathcal{D}}_{+}:={\bigcup }_{0<a<b<\mathrm{\infty }}{L}^2([a,b],d\omega )$, with zero extension to $(0,\mathrm{\infty })$; thus packet labels have compact positive-frequency support and need not be smooth. In a specified comparison unital $\ast$-algebra, supply conjugate-linear signal and environment maps $S_{\mathrm{\ell }},E_{\mathrm{\ell }}:{\mathcal{D}}_{+}\to {\mathfrak{A}}_{\mathrm{c}\mathrm{m}\mathrm{p}}$ with bosonic CCR $[S_{\mathrm{\ell }}(f),S_{\mathrm{\ell }}(g{)}^{\ast }]=[E_{\mathrm{\ell }}(f),E_{\mathrm{\ell }}(g{)}^{\ast }]=⟨f,g⟩I$ and zero annihilator-annihilator and signal-environment commutators. Supply a positive normalized comparison state $\nu$ with the number form $q_{\mathrm{\ell },K}$ specified below and vacuum environment. Define the output and complementary packet operators by the exact equalities $b_{\mathrm{\ell }}(f)=S_{\mathrm{\ell }}(\sqrt{{\mathrm{\Gamma }}_{\mathrm{\ell }}}f)+E_{\mathrm{\ell }}(\sqrt{1-{\mathrm{\Gamma }}_{\mathrm{\ell }}}f),d_{\mathrm{\ell }}(f)=-S_{\mathrm{\ell }}(\sqrt{1-{\mathrm{\Gamma }}_{\mathrm{\ell }}}f)+E_{\mathrm{\ell }}(\sqrt{{\mathrm{\Gamma }}_{\mathrm{\ell }}}f)\mathrm{.}$ These labels belong to ${\mathcal{D}}_{+}$ since the multipliers are bounded on each band. The displayed frequency-fiber notation for this equality is $\left(\begin{array}{c}{b}_{\mathrm{\ell },\omega }\\ d_{\mathrm{\ell },\omega }\end{array}\right)=\left(\begin{array}{cc}\sqrt{{\mathrm{\Gamma }}_{\mathrm{\ell }}(\omega )}& \sqrt{1-{\mathrm{\Gamma }}_{\mathrm{\ell }}(\omega )}\\ -\sqrt{1-{\mathrm{\Gamma }}_{\mathrm{\ell }}(\omega )}& \sqrt{{\mathrm{\Gamma }}_{\mathrm{\ell }}(\omega )}\end{array}\right)\left(\begin{array}{c}{a}_{\mathrm{\ell },\omega }\\ e_{\mathrm{\ell },\omega }\end{array}\right)\mathrm{.}$ It means the smeared equalities above: $S_{\mathrm{\ell }}(f)=\int \overline{f(\omega )}{a}_{\mathrm{\ell },\omega }d\omega$, and likewise for the other symbols. The continuum symbols are operator-valued distribution notation, not operators evaluated at a point. Their delta normalization $[a_{\mathrm{\ell },\omega },a_{\mathrm{\ell },{\omega }^{\mathrm{\prime }}}^{\ast }]=\delta (\omega -{\omega }^{\mathrm{\prime }})I$ means precisely the smeared CCR; no delta label is a unit vector in $L^2$. With $[f]={\mathrm{s}}^{1\mathrm{/}2}$, $[d\omega ]={\mathrm{s}}^{-1}$ and $[a_{\mathrm{\ell },\omega }]={\mathrm{s}}^{1\mathrm{/}2}$, normalized packet operators and their number expectations are dimensionless. This comparison channel defines the limiting number form; it is not a construction of the collapse state or an assertion that its full state is quasifree. The squared coefficient ${\mathrm{\Gamma }}_{\mathrm{\ell }}(\omega )$ is the transmitted fraction; no transmission coefficient is calculated here.

In the adopted frequency-diagonal Fock interpretation, the positive-frequency Fourier coefficient of the ordered current two-point function is $\omega \mathrm{/}2$ times the mode occupation. Thus the occupation associated with spectral coefficient $S_k(\omega )$ is, by this model's normalization, $2S_k(\omega )\mathrm{/}\omega$ for $\omega >0$. This identification is part of the adopted channel model; it is not a detector response or an observed particle count.

For each compact frequency interval $K=[{\omega }_{-},{\omega }_{+}]\subset (0,\mathrm{\infty })$ with ${\omega }_{-}<{\omega }_{+}$, the packet Hilbert space is ${\mathcal{H}}_K=L^2(K,d\omega )$ with $⟨f,g{⟩}_K:={\int }_K\overline{f(\omega )}g(\omega )d\omega$. Write $C_c^{\mathrm{\infty }}(K;\mathbb{C})$ for zero extensions to $K$ of smooth functions compactly supported in $({\omega }_{-},{\omega }_{+})$. A normalized packet is an $f\in C_c^{\mathrm{\infty }}(K;\mathbb{C})$ with ${\int }_K\mathrm{\mid }f(\omega ){\mathrm{\mid }}^{2}d\omega =1$; $f$ has units ${\mathrm{s}}^{1\mathrm{/}2}$. For each $s\in \mathbb{R}$, the model supplies a bounded sesquilinear form $B_{\mathrm{\ell },s}:{\mathcal{H}}_K\times {\mathcal{H}}_K\to \mathbb{C}$, conjugate-linear in its first entry and linear in its second. Supply a single conjugate-linear outgoing map $A_{\mathrm{\ell },\mathrm{o}\mathrm{u}\mathrm{t}}:{\mathcal{D}}_{+}\to \mathfrak{A}$ with $[A_{\mathrm{\ell },\mathrm{o}\mathrm{u}\mathrm{t}}(f),A_{\mathrm{\ell },\mathrm{o}\mathrm{u}\mathrm{t}}(g{)}^{\ast }]=⟨f,g⟩I$ and $[A_{\mathrm{\ell },\mathrm{o}\mathrm{u}\mathrm{t}}(f),A_{\mathrm{\ell },\mathrm{o}\mathrm{u}\mathrm{t}}(g)]=0$. Its restrictions to overlapping bands agree under zero extension. Define $({\tau }_{s}f)(\omega ):=e^{i\omega s}f(\omega ),a_{\mathrm{\ell },\mathrm{o}\mathrm{u}\mathrm{t},s}(f):=A_{\mathrm{\ell },\mathrm{o}\mathrm{u}\mathrm{t}}({\tau }_{s}f)\mathrm{.}$ For a smooth packet the positive-frequency profile is $p_f(u)=(2\pi {)}^{-1\mathrm{/}2}\int f(\omega )e^{-i\omega u}d\omega$, and $p_{{\tau }_{s}f}(u)=p_f(u-s)$; this fixes the retarded-time translation convention. Define on every band $B_{\mathrm{\ell },s}(f,g):=\varpi \text{\hspace{0.17em}⁣}(a_{\mathrm{\ell },\mathrm{o}\mathrm{u}\mathrm{t},s}(g{)}^{\ast }{a}_{\mathrm{\ell },\mathrm{o}\mathrm{u}\mathrm{t},s}(f))\mathrm{.}$ The reversed labels retain conjugate-linearity in $f$ and linearity in $g$, and the diagonal is the number expectation. Boundedness of this cross form on each ${\mathcal{H}}_K$ is a supplied moment condition, not boundedness of its algebra generators or their possible operator realizations.

Put $\theta :=2\pi \mathrm{/}k$ seconds and $n_{\theta }(\omega ):=(e^{\theta \omega }-1{)}^{-1}$. Define the incoming number form on ${\mathcal{H}}_K\oplus {\mathcal{H}}_K$ by $q_{\mathrm{\ell },K}((f_1,f_2),(g_1,g_2)):={\int }_K\overline{f_1(\omega )}{g}_1(\omega )n_{\theta }(\omega )d\omega ,$ where, for $T_{\mathrm{\ell }}(f_1,f_2):=S_{\mathrm{\ell }}(f_1)+E_{\mathrm{\ell }}(f_2)$, the comparison state is required to satisfy $\nu (T_{\mathrm{\ell }}(g_1,g_2{)}^{\ast }{T}_{\mathrm{\ell }}(f_1,f_2))=q_{\mathrm{\ell },K}((f_1,f_2),(g_1,g_2))$. The first component is the near-horizon signal and the second the independent incoming vacuum channel. This specifies comparison moments, not the full collapse state. Define the packet map induced by the first row of the passive mode transformation by $J_{\mathrm{\ell },K}f:=(\sqrt{{\mathrm{\Gamma }}_{\mathrm{\ell }}}f,\sqrt{1-{\mathrm{\Gamma }}_{\mathrm{\ell }}}f)\mathrm{.}$ The exact limiting form is the adopted direct-integral model input $B_{\mathrm{\ell },\mathrm{\infty }}(f,g):=q_{\mathrm{\ell },K}(J_{\mathrm{\ell },K}f,J_{\mathrm{\ell },K}g)={\int }_K\overline{f(\omega )}g(\omega ){\mathrm{\Gamma }}_{\mathrm{\ell }}(\omega )n_{\theta }(\omega )d\omega \mathrm{.}$ Thus the signal has the exponential-channel occupation $n_{\theta }$, the environment and signal-environment cross terms vanish by the incoming vacuum and independence premises, and the passive map supplies the transmission factor. Assume that for each fixed $K$ the bounded forms converge in operator norm as $s\to +\mathrm{\infty }$ to this $B_{\mathrm{\ell },\mathrm{\infty }}$, meaning ${\sup }_{\mathrm{\parallel }f{\mathrm{\parallel }}_2,\mathrm{\parallel }g{\mathrm{\parallel }}_2\le 1}\mathrm{\mid }{B}_{\mathrm{\ell },s}(f,g)-B_{\mathrm{\ell },\mathrm{\infty }}(f,g)\mathrm{\mid }⟶0.$ No uniform error rate over bands, angular indices, or packet families is assumed. This particular fixed-band operator-norm premise is stronger than pointwise late-packet convergence: with the exact translation just defined it forces $B_{\mathrm{\ell },s}=B_{\mathrm{\ell },\mathrm{\infty }}$ for all $s$. Indeed $B_{\mathrm{\ell },s}(f,g)=B_{\mathrm{\ell },0}({\tau }_{s}f,{\tau }_{s}g)$; ${\tau }_s$ is unitary and the multiplier ${\mathrm{\Gamma }}_{\mathrm{\ell }}{n}_{\theta }$ commutes with it, so $\mathrm{\parallel }{B}_{\mathrm{\ell },s}-B_{\mathrm{\ell },\mathrm{\infty }}\mathrm{\parallel }=\mathrm{\parallel }{B}_{\mathrm{\ell },0}-B_{\mathrm{\ell },\mathrm{\infty }}\mathrm{\parallel }$. A constant nonnegative norm tending to zero is zero. This exact number-form stationarity is a qualification of the adopted strong premise, not a four-dimensional scattering estimate, a late-time rate or stationarity of all state correlations.

Only after this late-time convergence is fixed may a stationary-frequency limit be taken. For each ${\omega }_0$ in the interior of such an interval $K$, use a specified family $f_{ϵ}\in {\mathcal{H}}_K$ with common compact support $K$, $\mathrm{\parallel }{f}_{ϵ}{\mathrm{\parallel }}_2=1$, and spectral probability measures $\mathrm{\mid }{f}_{ϵ}(\omega ){\mathrm{\mid }}^{2}d\omega$ converging weakly to the point mass ${\delta }_{{\omega }_0}$. This is concentration of normalized packets, not $L^2$-norm convergence to a non-normalizable monochromatic mode.

This is an adopted fixed-background scattering model. The Hadamard state, incoming ground condition, exact late Schwarzschild exterior, passive channel, transmissivity, packet-pairing convergence, and packet-family limits are premises; no Einstein evolution, four-dimensional scattering estimate, quantitative finite-time error bound, trans-Planckian robustness, detector construction, observed temperature, backreaction, or evaporation history is derived or asserted.

Wavefront and positive-frequency Hadamard data

Specified chiral collapse channel

Reduced exponential current and defined spectral detector adoption

Schwartz/distribution spaces and Hilbert Fock conventions

Bosonic and fermionic Fock operators with exact domains

Unital star algebra and positive normalized state

The packet comparison data have a concrete consistency witness without requiring the actual supplied collapse state to be quasifree. Fix a band $K$, write $H={\mathcal{H}}_K$, let $Cf=\bar{f}$, and use bosonic Fock space over $H\oplus H\oplus H$ with vacuum $\mathrm{\Omega }$ and its finite-particle domain. With $n=n_{\theta }$ restricted to $K$, define $S(f)=a(\sqrt{1+n}f\oplus 0\oplus 0)+a^{†}(0\oplus \sqrt{n}Cf\oplus 0),E(f)=a(0\oplus 0\oplus f)\mathrm{.}$ Both are conjugate-linear in $f$. These formulas are well defined on the invariant finite-particle domain because $n$ is bounded on $K$; the sector estimates $\mathrm{\parallel }a(h){\mathrm{\Psi }}_m\mathrm{\parallel }\le \sqrt{m}\mathrm{\parallel }h\mathrm{\parallel }\mathrm{\parallel }{\mathrm{\Psi }}_m\mathrm{\parallel }$ and $\mathrm{\parallel }{a}^{†}(h){\mathrm{\Psi }}_m\mathrm{\parallel }\le \sqrt{m+1}\mathrm{\parallel }h\mathrm{\parallel }\mathrm{\parallel }{\mathrm{\Psi }}_m\mathrm{\parallel }$ follow directly from contraction and the norm-one symmetrizing projection. Hence each generator and its restricted adjoint maps a finite number of sectors to a finite number of sectors. Their polynomial algebra has the positive normalized vacuum functional $\nu (X)=⟨\mathrm{\Omega },X\mathrm{\Omega }⟩$, since $\nu (X^{\ast }X)=\mathrm{\parallel }X\mathrm{\Omega }{\mathrm{\parallel }}^2$. The CCR give $[S(f),S(g{)}^{\ast }]=⟨f,(1+n)g⟩-⟨Cg,nCf⟩=⟨f,g⟩I,$ and $[S(f),S(g)]=0$ because the annihilation and creation labels occupy orthogonal summands. Also $[E(f),E(g{)}^{\ast }]=⟨f,g⟩I$, and signal/environment commutators vanish. Finally $S(f)\mathrm{\Omega }=a^{†}(0\oplus \sqrt{n}Cf\oplus 0)\mathrm{\Omega }$, so $\nu (S(g{)}^{\ast }S(f))=⟨\sqrt{n}Cg,\sqrt{n}Cf⟩={\int }_K\bar{f}gnd\omega ;$ the vacuum environment and cross number moments vanish. Thus the exact smeared definitions of $b,d$ give the displayed $q$ and $\mathrm{\Gamma }n$ output form. Equivalently, $b(f)=a(\sqrt{1+n}\sqrt{\mathrm{\Gamma }}f\oplus 0\oplus \sqrt{1-\mathrm{\Gamma }}f)+a^{†}(0\oplus \sqrt{n}\sqrt{\mathrm{\Gamma }}Cf\oplus 0)$ is an explicit Bogoliubov formula, not merely a definition of its expected number. This bandwise witness concerns the outgoing comparison algebra only. It constructs neither the collapse spacetime, its Hadamard field state nor a compatible extension of an arbitrary field state, and supplies no global unitary implementer or maximal closed-domain assertion. The actual model retains all its separately supplied state and scattering premises.

Assume the fixed-background collapse model and all its conditions stated in the preceding postulate on this page. Let $\mathrm{\ell }\in {\mathbb{N}}_0$, let $K=[{\omega }_{-},{\omega }_{+}]\subset (0,\mathrm{\infty })$ with ${\omega }_{-}<{\omega }_{+}$, and let $f\in C_c^{\mathrm{\infty }}(K;\mathbb{C})$ satisfy ${\int }_K\mathrm{\mid }f(\omega ){\mathrm{\mid }}^{2}d\omega =1$. Write ${\mathcal{N}}_{\mathrm{\ell },f}(s):=B_{\mathrm{\ell },s}(f,f),n_{\theta }(\omega ):=\frac{1}{e^{\theta \omega }-1},\theta :=\frac{2\pi }{k}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{s}\mathrm{.}$ Then the late-time packet occupation exists and is ${\lim }_{s\to +\mathrm{\infty }}{\mathcal{N}}_{\mathrm{\ell },f}(s)={\int }_K\mathrm{\mid }f(\omega ){\mathrm{\mid }}^2{\mathrm{\Gamma }}_{\mathrm{\ell }}(\omega )n_{\theta }(\omega )d\omega \mathrm{.}$ Thus a finite-width packet has a spectral integral; its value is not, in general, the integrand at a chosen central frequency.

If ${\omega }_0$ is in the interior of $K$ and $f_{ϵ}$ is a specified normalized family supported in $K$ with $\mathrm{\mid }{f}_{ϵ}(\omega ){\mathrm{\mid }}^{2}d\omega \rightharpoonup {\delta }_{{\omega }_0}$, then the ordered limit, first $s\to +\mathrm{\infty }$ and then $ϵ\downarrow 0$, is ${\lim }_{ϵ\downarrow 0}{\lim }_{s\to +\mathrm{\infty }}{\mathcal{N}}_{\mathrm{\ell },f_{ϵ}}(s)=\frac{{\mathrm{\Gamma }}_{\mathrm{\ell }}({\omega }_0)}{e^{2\pi {\omega }_0\mathrm{/}k}-1}\mathrm{.}$ Here $M$ is the fixed mass in kilograms, $G$ has units ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$, $\mathrm{\hslash }$ has units $\mathrm{J}\mathrm{s}$, and $k_B$ has units $\mathrm{J}{\mathrm{K}}^{-1}$. Matching the Bose exponent to $\mathrm{\hslash }\omega \mathrm{/}(k_B{T}_{\mathrm{\infty }})$ defines the asymptotic spectral temperature parameter $T_{\mathrm{\infty }}:=\frac{\mathrm{\hslash }k}{2\pi k_B}=\frac{\mathrm{\hslash }{c}^3}{8\pi GM{k}_B}\mathrm{k}\mathrm{e}\mathrm{l}\mathrm{v}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{.}$ This parameter labels the unfiltered Planck factor; a frequency-dependent ${\mathrm{\Gamma }}_{\mathrm{\ell }}$ need not leave the spectrum Planckian. It is not an observed temperature.

Specified late-time free-field collapse scattering model

Exponential channel thermality and greybody filtered occupations