Ray Tracing and the Hawking Bogoliubov Map

The ray-tracing function $U=p(u)$ contains the kinematic content of a collapse Hawking calculation. Its leading exponential fixes the thermal ratio; its subleading terms determine transients and finite-time corrections. Wavepackets are essential because continuum frequency eigenmodes have infinite duration and hide when the late-time approximation becomes valid.

Required background. Hawking radiation from collapse supplies the physical state and geometry, while in/out modes and number operators fixes Bogoliubov normalization.

Helpful background. WKB and turning-point matching supports the ray approximation, and moving mirrors gives an exactly tractable two-dimensional analogue.

Fourier transform of the exponential map

Take

$$p(u)=U_H-Ae^{-\kappa u}, \qquad A,\kappa>0.$$

With Klein–Gordon-normalized right-moving modes, one common phase convention gives a kernel proportional to

$$\beta_{\omega\omega'} \propto\sqrt{\frac{\omega}{\omega'}} \int_{-\infty}^{\infty}du\, e^{-i\omega u-i\omega'p(u)}.$$

Set $x=A\omega'e^{-\kappa u}$. Apart from an irrelevant phase and the normalization fixed by the full Cauchy surface,

$$\beta_{\omega\omega'} =-\frac{1}{2\pi\kappa} \sqrt{\frac{\omega}{\omega'}} e^{-i\omega'U_H} (A\omega')^{-i\omega/\kappa} e^{-\pi\omega/(2\kappa)} \Gamma\!\left(\frac{i\omega}{\kappa}\right).$$

The corresponding $\alpha$ coefficient has $e^{+\pi\omega/(2\kappa)}$ in place of the negative exponential, so

$$|\beta_{\omega\omega'}|^2 =e^{-2\pi\omega/\kappa} |\alpha_{\omega\omega'}|^2.$$

Overall phases and a possible exchange of $\omega/\omega'$ factors depend on whether the transform is written on null infinity or a complete Cauchy surface; the ratio and packet normalization are the invariant checks. Hawking’s calculation uses precisely this branch structure (Hawking 1975, § 2).

Define packets of bandwidth $\epsilon$ by

$$f_{jn}=\frac{1}{\sqrt\epsilon} \int_{j\epsilon}^{(j+1)\epsilon}d\omega\, e^{2\pi i\omega n/\epsilon}f_\omega.$$

They are centered near retarded time $u_n=2\pi n/\epsilon$ and frequency $\omega_j=(j+1/2)\epsilon$. A useful thermal statement requires $\epsilon\ll\kappa$ to resolve the spectrum while the packet duration $2\pi/\epsilon$ remains short compared with background evolution. These inequalities compete in an evaporating geometry.

The structure map places this explicit transform in the near-horizon box. It determines a mode relation; the greybody barrier and renormalized energy flux are not contained in the Gamma function.

Ray tracing inside the controlled Hawking construction. The diagram is schematic and not to scale; wavepacket Bogoliubov coefficients establish thermality only at their stated late-time resolution.

The failure map focuses attention on changing $p(u)$. A subleading perturbation can alter transients without invalidating the asymptotic temperature, whereas a change of the leading peeling law can alter the spectrum itself.

Validity boundary for the Bogoliubov map. This schematic, not-to-scale map requires the leading exponential, packet resolution, and normalization to be tested before exact thermality is claimed.

Application: packets and finite-time thermality

For the pure exponential map, packet integration removes the continuum $\delta(0)$ and gives

$$\langle N_{jn}\rangle =\frac{1}{\epsilon} \int_{j\epsilon}^{(j+1)\epsilon} \frac{d\omega}{e^{2\pi\omega/\kappa}-1}$$

after the late-time packet lies within the exponential regime. For narrow bins this equals the Planck factor at $\omega_j$ plus $O(\epsilon^2/\kappa^2)$ binning corrections. Early packets sample the nonexponential part of the collapse map and are transient, not thermal.

Now perturb

$$p(u)=U_H-Ae^{-\kappa u} \left[1+b e^{-\gamma u}+O(e^{-2\gamma u})\right], \qquad \gamma>0.$$

Expanding the Fourier phase shows relative corrections $O(be^{-\gamma u_n})$ for a packet centered at $u_n$. The leading Planck ratio survives at late time. However, the integrated transient energy can change, and a two-dimensional stress flux depends on derivatives of the full map through its Schwarzian derivative. Exact thermality at all times therefore does not follow from asymptotic thermality.

If instead $p'(u)$ has a time-dependent peeling rate

$$\kappa(u)=-\frac{p''(u)}{p'(u)},$$

then a local thermal approximation requires $|\dot\kappa|/\kappa^2\ll1$ over the packet. If $p(u)$ becomes power law, oscillatory, or terminates, the Gamma-function derivation and its Planck ratio need not survive.

Domain and failure conditions

The chapter domain and failure-conditions table supplies the surrounding distinctions. This page assumes normalized in/out modes, an affine $U$, a monotone ray map, a leading exponential over the packet support, and compatible frequency/time resolution. These inputs license the displayed Bogoliubov ratio and its late packet occupation. Subleading decaying terms license the same temperature with transient errors; a changed leading map or failed packet hierarchy requires a nonthermal spectrum. No result here licenses unit greybody transmission or a four-dimensional luminosity.

Exercise

Use $|\Gamma(iy)|^2=\pi/[y\sinh(\pi y)]$ to expose the Planck denominator in $|\beta_{\omega\omega'}|^2$.

Solution

The modulus of the displayed coefficient contains

$$e^{-\pi y}|\Gamma(iy)|^2 =\frac{\pi e^{-\pi y}}{y\sinh(\pi y)} =\frac{2\pi}{y(e^{2\pi y}-1)},$$

with $y=\omega/\kappa$. The remaining normalization and $1/\omega'$ factor encode continuum mode density; packetization makes the number finite.

Handoff

The number spectrum is one observable. A renormalized stress tensor provides an independent energy-flux calculation and exposes exactly what a two-dimensional reduction can and cannot predict in four dimensions.

References

• Hawking, Stephen W. “Particle Creation by Black Holes.” Communications in Mathematical Physics 43 (1975): 199–220. doi:10.1007/BF02345020.
• Unruh, William G. “Notes on Black-Hole Evaporation.” Physical Review D 14 (1976): 870–892. doi:10.1103/PhysRevD.14.870.