M\"obius Mirrors

An accelerating boundary (mirror) acts as a horizon and black hole analog, radiating energy with some particle spectrum. We demonstrate that a M\"obius transformation on the null coordinate advanced time mirror trajectory uniquely keeps invariant not only the energy flux but the particle spectrum. We clarify how the geometric entanglement entropy is also invariant. The transform allows generation of families of dynamically distinct trajectories, including $\mathcal{PT}$-symmetric ones, mapping from the eternally thermal mirror to the de Sitter horizon, and different boundary motions corresponding to Kerr or Schwarzschild black holes.