An arXiv preprint dated 26 August 2026 presents an analytic model of a localized spin-1/2 detector falling into a Schwarzschild–MOG black hole. The detector’s internal two-level transition couples to a neutral, massless scalar field, while its translational motion follows the MOG-charged Dirac equation. The setup is a theoretical detector–field model rather than an empirical measurement.
One of the calculation’s first checks concerns whether the trajectory can physically reach the horizon. It places every physical crossing trajectory at positive horizon energy. The formal zero-trajectory-function threshold, written as UH = 0, lies outside the model’s timelike crossing sector, so it is not a future-directed timelike crossing trajectory.
Thermality comes with conditions
The calculation’s main warning is that thermality is not a blanket description of the detector’s full response. It defines detailed balance as the ratio between excitation and absorption in a chosen channel. In the outgoing channel, that ratio reaches exp(−2πν/κα) only when the analysis is restricted to the near-horizon region, the motion is adiabatic, the detector has a high energy gap, and the outgoing branch is isolated. The result is a conditional limit, not a generic prediction for every version of the response.
The complete mode matters
The branch-isolation requirement is central because the mode used in the calculation is global. It is normalized as a Boulware scattering mode that is outgoing at infinity, yet near the horizon it retains both radial-flux branches. A local outgoing ansatz is therefore not treated as a complete mode. The branch coefficients and their relative phase are global scattering quantities, and they are not numerically supplied in the analysis.
To preserve both pieces, the model uses a finite radial gate and calculates closed-form excitation and absorption probability densities. Those densities include the ingoing contribution and interference between the branches. The result is not an outgoing term with a greybody multiplier—an extra factor pasted on afterward. The regular ingoing branch enters detailed balance through interference at the amplitude level, tied to the phase of the global scattering solution.
Two different thermal questions
Under the controlled near-horizon approximation, the complete detailed-balance ratio separates into an outgoing-branch ratio and a distinct two-branch factor. That factor records what changes when the full mode, rather than an isolated branch, is retained. For the full response to share the outgoing thermal limit, the two-branch factor must approach unity. The analysis does not show that this condition holds generically.
Where the MOG correction enters
The paper also expands the balance in the weak-MOG regime. Its local correction splits into terms associated with surface gravity, the trajectory prefactor and the finite-gate protocol; the full response adds a global-scattering term. This decomposition keeps local geometric and detector-protocol effects separate from propagation effects that depend on the complete scalar scattering solution.
The switching correction has a scale that depends on frequency as well as gate length. Its leading form is (ν/κα)/(S± Lχ), rather than simply (S± Lχ)−1. In plain terms, the size of the correction depends on the mode frequency relative to surface gravity, not only on the gate scale.
At fixed M and ν, the normalization-reduced thermal spectrum is lower for a positive weak-MOG deformation. That is a statement about a reduced thermal factor, not the complete probability that a detector would register. The complete response still carries branch, trajectory, gate and global-scattering dependence.
A calculation with clear boundaries
These results come from a model calculation, not an empirical dataset. No datasets were generated or analyzed, and the equation used to generate the figure is provided in the text. The Dirac sector describes translational propagation, while the two-level monopole is an independent effective degree of freedom; the setup therefore does not describe a detector coupled to a quantized fermion field.
The unresolved part is the global correction. Without numerical scalar scattering coefficients and reflection phase, the full two-branch contribution cannot be reduced to a number from surface gravity alone. The balance therefore remains dependent on the global scattering solution and on the specified detector protocol.
Disclosure
The preprint reports institutional support from De La Salle University and the DLSU Theoretical Physics Group, along with research support from DOST–ASTHRDP. The authors declare no competing interests.
Paper data and sources
Original title: Two-branch detector response for Dirac infall into a Schwarzschild--MOG black hole
Authors: Nikko John Leo S. Lobos, Emmanuel T. Rodulfo
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text