Thermality of the Boxed Worldline in Large-r or Small-s Limits

Establish whether the classical radiation emitted by the finite-distance worldline t(x) = x(1/s − 1/r) + (2/κ) tanh^{-1}(κ x / 2 r) is thermal with temperature T = κ/(2π) in the limit of large box size r → ∞ or small maximum speed s ≪ 1 by deriving the spectrum and identifying a Planck factor.

Background

Using the concept of peel acceleration, the authors show that their boxed trajectory has approximately constant peel in the large-r or small-s regimes, suggesting thermality analogous to the Carlitz–Willey mirror and the beta-decay electron trajectories, both associated with temperature κ/(2π).

While energy considerations and kinematic properties suggest thermality in these limits, a direct demonstration via the spectral Planck factor in the large-box regime remains analytically intractable, prompting the authors to state the thermality as a conjecture by analogy.