Can reflective black-hole horizons have a causal Boltzmann response at a matching plane?

Why the exact Boltzmann modulus fails globally

Echo models of black holes replace the purely ingoing horizon condition with a near-horizon reflection amplitude, and a widely used choice is the Boltzmann flux reflectivity, whose modulus falls as an exponential of frequency times the inverse temperature. The paper asks a narrower question than whether that law is physically reasonable: can it be the modulus of an actual causal response between local scattering ports at a fixed matching plane, at every positive frequency?

The answer is no. The paper treats the reflection amplitude as a bounded analytic function in the upper half of the complex frequency plane, the Hardy space H-infinity, which is the mathematical form of a stationary, passive, causal convolution acting on wave packets. For any nonzero such function, a standard logarithmic integrability condition must hold. An exact Boltzmann modulus makes the logarithm grow linearly with frequency, so the weighted integral diverges, and no bounded causal amplitude can reproduce it on the whole positive axis.

The scope matters. The result constrains a causal transfer amplitude at a fixed plane, not a thermal state or a ratio of transition rates, and it does not invalidate a low-frequency thermal prescription. It also does not diagnose which step fails in earlier horizon-continuation derivations of a Boltzmann modulus; it only rules out identifying that modulus with a bounded causal response everywhere.

Finite bands can keep the Boltzmann modulus exactly

The obstruction is global, so the paper next shows how far a thermal law can be kept. Using outer spectral factors, it constructs causal amplitudes whose squared modulus equals the Boltzmann form exactly on any prescribed finite band. One construction keeps the exponential inside a band edge and switches to a matched power-law tail outside; a second uses a smooth crossover of the form (1 + |omega|/omega_c) raised to a negative power set by the temperature and the crossover scale.

Both constructions are passive and bounded, and in the passive thermal channel described later they also preserve the thermal equality inside the band. The price is that the ultraviolet completion still fixes part of the phase. The paper is explicit that this is an added assumption: it selects the outer factor and sets any inner factor aside, so causality alone does not determine the full reflection phase.

A shared logarithm, a completion-dependent linear term

The second main result is a low-frequency expansion of the outer phase for a class of completions that share the same Boltzmann slope but differ in their ultraviolet profile. Under stated regularity and tail conditions, the logarithmic coefficient is common to the whole class, while the linear-in-frequency constant is an explicit functional of the chosen completion. The exact-band and smooth-crossover constructions are both members of this class.

This separation is the paper's organizing distinction. Two responses that look thermally identical at low frequency can still differ in a linear phase term, and that term is where the completion enters. The remainder is controlled rather than assumed away, which is what allows the later comparison between different cutoffs to be interpreted.

Redshifting the cutoff cancels the leading matching dependence

The matching plane sits at a fixed tortoise coordinate far outside the horizon, and the response is taken to be specified in the local proper frame there. A fixed proper-frame scale Omega_UV then appears in Killing frequency as omega_c equal to the square root of f(r0) times Omega_UV, and the local temperature is the Tolman redshift of the Hawking temperature, with beta_H equal to 8 pi M.

With these two inputs, the leading matching-distance term in the combined reflection and propagation phase takes the form (beta/pi - 8M) times omega times the logarithm of d0 over 4M, where d0 is the proper distance from the horizon to the matching surface. At the Hawking value beta equals 8 pi M, that coefficient vanishes. The geometric remainder is quadratic in the proper distance, and a separate dispersive remainder depends on the completion.

The paper is careful about what this is. It is a low-energy comparison of specified response families, not an exact invariance under moving the reference plane, and not a phase fixed by causality alone. The cancellation requires an overlap regime with 4M omega much smaller than d0 Omega_UV and d0 much smaller than M, and it need not include the ringdown band.

A fixed-plane test reduces a 21% gap to 0.11%

The numerical example uses axial Schwarzschild scattering at a fixed matching plane with the Hawking temperature. Two cutoffs are compared, M omega_c equal to 20 and 40, and the quantity compared is the cavity factor, the squared magnitude of the response after propagation through the barrier and cavity.

On the sampled frequency band, the maximum relative cavity-factor difference between the two cutoffs is about 21%. Adding the predicted positive linear delay of 8M log 2 to the lower-cutoff response reduces that maximum relative difference to 0.11%. The paper notes that this reduction includes the modulus changes as well as the residual phase, so it is not a phase-only comparison, and that no delay or phase was optimized to minimize the differences.

A second diagnostic separates unaligned and delay-aligned shifts of constructive phase crossings. The unaligned shifts are of order 10 to 17 in units of 10^-3 in M omega, while the delay-aligned shifts are only 1.90 to 6.47 in units of 10^-6. The paper reads this as evidence that a substantial unaligned shift need not strongly discriminate between ultraviolet scales when an inner delay is free, and it states that establishing detectability would additionally require a source model, detector noise, and an inference calculation.

A passive completion ties reflection to absorption noise

The paper closes the quantum side with a passive completion in which the local exterior input is coupled unitarily to an environment input, and tracing over the environment output gives the reduced exterior channel. This supplies the associated absorption-noise relation and checks the consistency of the asymptotic output.

The thermal equality under finite-band Boltzmann matching is conditional on both passivity and the input temperature. The paper states that neither this construction nor the perfect-absorption limit specifies the microscopic horizon state, and that linear response and two-point noise data do not provide an injective reconstruction of horizon microstates.

An echo delay alone cannot fix the matching distance

The analysis assumes stationary, linear Schwarzschild scattering. Kerr superradiance would require the Teukolsky flux normalization and an active-channel treatment, correlated or squeezed inputs would require their full covariance, and nonstationary formation or evaporation would require two-time kernels. The strict event-horizon limit at fixed Killing frequency also lies outside the regime studied.

Within the stated families, the practical consequence is a degeneracy rather than a measurement recipe. Because an inner factor can add a delay not determined by the modulus, and because the completion enters the linear phase term, an echo delay cannot determine the matching distance without assumptions about the proper-frame cutoff and the inner phase. The paper's own priority is therefore the separation it proves: a global causal restriction on the Boltzmann modulus, explicit finite-band completions that evade it, and a phase expansion that isolates the common logarithm from completion-dependent constants and remainders. The 21% to 0.11% reduction is best read as a diagnostic of how much of a fixed-plane cutoff difference is carried by the predicted linear delay, not as evidence that the matching scale has been measured.

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