Geometric Optics Lies: Reproducing a 1976 Black-Hole Calculation from Scratch

Geometric Optics Lies: Reproducing a 1976 Black-Hole Calculation from Scratch<br>The standard shortcut overstates photon emission by a factor of 4.16. Six validation gates, two analytic limits the code was never tuned to, and one number I had to retract.<br>My last few articles were the same problem in different clothes: a metric that looks healthy while the thing underneath is broken. Word Error Rate passing mangled speech. A loss curve falling 62% on pure noise. An objective with a hole in it, and two training curves you cannot tell apart.<br>This one is that problem in its hardest form — when there is no experiment to check against at all.<br>If you write physics code, nobody hands you a validation set. The black hole is not going to email you. So I picked a calculation with a known answer and tried to reach it without looking: the photon emission spectrum of a Schwarzschild black hole, first computed by Don Page in 1976. The question was not really whether I could get his number. It was what it actually takes to believe your own output.<br>The shortcut, and where it breaks<br>Hawking radiation is not a blackbody spectrum. The curvature around the hole acts as a potential barrier that reflects most of what is emitted; the fraction escaping is the greybody factor , and getting it right means solving a wave equation mode by mode.<br>Almost everyone starts with the shortcut — the geometric-optics limit, which treats the hole as swallowing anything aimed inside the photon capture cross-section. It is clean, analytic, and valid only when the emitted wavelength is small compared with the hole.<br>That condition has an exact form. With u = E/(k_B·T_H):<br>E · r_s / (hbar·c) = 2GME/(hbar·c^3) = u / 4piThe thermal spectrum peaks at u = 1.594. Substitute, and you get 0.127 .<br>The typical emitted photon has a wavelength about eight times the Schwarzschild radius . The approximation is not marginally strained here — it fails across the entire range where the hole actually radiates, and the exponential cutoff kills the spectrum long before you reach the regime where it would become valid. Every textbook says the limit requires short wavelengths. Nobody makes you check.<br>Two details that decided the answer<br>A massless spin-1 field on a Schwarzschild background separates into a one-dimensional scattering problem against a barrier peaking at the photon sphere:<br>(d^2/dr*^2 + w^2 - V_l) Psi = 0, r* = r + 2 ln(r/2 - 1)

V_(S,l) = f(r) · [ l(l+1)/r^2 + (1 - S^2)·2M/r^3 ], f = 1 - 2M/rFor S = 1 the spin term vanishes identically. The greybody factor is the transmission probability through that barrier, summed over modes and over the photon's two polarization states — and a detail worth flagging: for Maxwell on Schwarzschild both field parities obey this same radial equation, and those parities are the two polarizations. That is what pins the degeneracy factor at exactly 2, with no double counting.<br>Two implementation choices determined whether the answer came out right at all.<br>Don't shoot from the horizon — expand around it. The horizon is a singular point, so integrating outward from beside it is numerically hostile. Substituting Psi = exp(-i·w·r*)·u(r) makes the equation merely regular-singular, and in x = r - 2 it becomes polynomial with a clean recursion. The nearest other singularity sits at x = -2, so the series converges comfortably and you step over the stiff region instead of fighting it.<br>The long-range tail, which nearly ruined it. The potential falls off as l(l+1)/r^2 — long-range. Matching the numerical solution to plain plane waves at large radius leaves an error of order L/(w·r), which at low frequency is ruinous.<br>This was the dominant error in my first working version, and how I found it is the point: not by looking at the answer. The spectrum looked entirely reasonable. It was caught by a flux-conservation monitor that had no business agreeing unless everything else was right. Expanding the asymptotic solution to second order fixed it and dropped the truncation to O((L/2wr)^3), which lets you match at modest radius instead of integrating to absurd distances.<br>The correction, across the spectrum

Geometric optics (dashed) overstates low-energy photon emission by up to six orders of magnitude. The curves agree only above ~7 TeV, where the wavelength finally becomes small compared with the hole.<br>The ripple in the lower panel, as the ratio approaches one, is not numerical noise — it is the absorption cross-section ringing around its geometric value, caused by surface waves trapped near the photon sphere. I did not put that in. It arrived on its own, which is the kind of thing that makes you trust a solver.<br>Six gates, two of which the code could not have anticipated<br>Agreement with one published number proves very little. Anyone can tune until a single value matches. The checks that mean something are the ones with an independently known answer the implementation had...