CAT in large boxes: Planckian spectrum in the large-displacement limit

Show that, for the finite-distance worldline t(x)=x(1/s−1/r)+(2/κ) tanh⁻¹(κx/(2r)) in the limit of large displacement r→∞, the emitted radiation exhibits a Planck spectrum with temperature T=κ/(2π), thereby establishing that large boxes contain a Classical Acceleration Temperature (CAT).

Background

The authors find that the total emitted energy approaches a finite value independent of the distance traveled in the large-r limit, and they demonstrate that the trajectory’s peel becomes constant in regimes suggestive of thermality.

Despite these hints, they state that a Planck distribution for the large-box spectrum cannot be confirmed due to its seemingly intractable form and consequently advance a conjecture that large boxes contain CATs, motivating a precise spectral proof in the r→∞ limit.