Free collapse Hawking approximation and its proof obligations — Examples

Use $r_s=2GM\mathrm{/}{c}^2$ and the SI constants of the preceding postulate. Define the two-sided eternal Schwarzschild spacetime by ${\mathcal{S}}_{\mathrm{e}\mathrm{x}\mathrm{t}}:=\{(T,X)\in {\mathbb{R}}^2:X^2-T^2>-1\}\times S^2,$ where $T,X$ are dimensionless, $S^2$ has its unit round metric $d{\mathrm{\Omega }}^2$, and $r>0$ is the unique solution of $(r\mathrm{/}{r}_s-1)e^{r\mathrm{/}{r}_s}=X^2-T^2$. Uniqueness follows because the left side increases strictly from $-1$ to $+\mathrm{\infty }$ as $r$ increases from zero. Give this manifold the metric and time orientation $d{s}^2=\frac{4r_s^3}{r}{e}^{-r\mathrm{/}{r}_s}(-d{T}^2+d{X}^2)+r^{2}d{\mathrm{\Omega }}^2,{\mathrm{\partial }}_T\text{ future-directed}\mathrm{.}$ Its right exterior is ${\mathcal{E}}_R:=\{X>\mathrm{\mid }T\mathrm{\mid }\}\times S^2$, with $r>r_s$; there $t=(r_s\mathrm{/}c)\log ((X+T)\mathrm{/}(X-T))$. The adjacent future and past horizons are respectively $T=X>0$ and $T=-X<0$ (times $S^2$), and the black-hole interior is $T>\mathrm{\mid }X\mathrm{\mid }$ (times $S^2$). A positive Schwarzschild-frequency mode is one with time dependence $e^{-i\omega t}$ for $\omega >0$, where $t$ is the stationary time of the Schwarzschild exterior. Here Hadamard means that the two-point distribution has the wavefront-set condition $\mathrm{WF}(W_2)={\mathcal{C}}_g^{+}$ defined in the preceding postulate. The Boulware state on ${\mathcal{E}}_R$ is the ground state for these exterior modes: for every normalized exterior positive-frequency packet $h$, its number expectation is $\varpi (a(h{)}^{\ast }a(h))=0$. Its two-point function is Hadamard within ${\mathcal{E}}_R$, but it does not extend as a Hadamard state across either horizon branch.

The Unruh state is a different state on the connected open domain ${\mathcal{D}}_U:=\{(T,X)\in {\mathbb{R}}^2:X^2-T^2>-1,T+X>0\}\times S^2$ of ${\mathcal{S}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$. Define ${\mathcal{H}}_R^{+}:=\{T=X>0\}\times S^2$ and $\mathcal{B}:=\{(T,X)\in {\mathbb{R}}^2:T>\mathrm{\mid }X\mathrm{\mid },X^2-T^2>-1\}\times S^2$; then ${\mathcal{D}}_U={\mathcal{E}}_R\cup {\mathcal{H}}_R^{+}\cup \mathcal{B}$, and every point of the right future horizon has an open neighborhood in this domain. Its incoming positive-frequency modes from ${\mathcal{I}}^{-}$ are in their ground state, while the outgoing sector associated with modes emerging from the past horizon has occupation $n_H(\omega ):=(e^{\mathrm{\hslash }\omega \mathrm{/}(k_B{T}_H)}-1{)}^{-1}$ for angular frequency $\omega >0$, before exterior transmission filtering. Its two-point function satisfies the Hadamard condition across the future horizon but has no Hadamard extension across the past horizon. In contrast, the Hartle–Hawking state on ${\mathcal{S}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ is invariant under Schwarzschild time translations and satisfies the Hadamard condition there; its restriction to ${\mathcal{E}}_R$ has both incoming and outgoing mode occupations $n_H(\omega )$, with $T_H:=\mathrm{\hslash }{c}^3\mathrm{/}(8\pi GM{k}_B)$, and satisfies the Hadamard condition across both horizon branches. These are distinct mode conditions and domains.

The collapse in-state adopted on the preceding page belongs to a dynamical collapse spacetime, not to ${\mathcal{S}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$. Its incoming-ground and late-time packet assumptions do not identify it with the eternal Unruh state. No rotating black-hole state is asserted here.

Specified late-time free-field collapse scattering model