Relativistic dynamics of moving mirrors in ${\mathrm{CFT}}_2$: Quantum backreaction and black holes

There is a well-known correspondence between the physics of black hole evaporation and that of moving mirrors in quantum field theory (QFT). However, most analyses in this subject rely on prescribed mirror trajectories. Here, we study the flat-space dynamics of $1+1$-dimensional conformal field theories interacting with a relativistic boundary particle of mass $m$ acting as a perfect mirror. The trajectory of the latter is not fixed but follows its own relativistic equation of motion $F^{\mu }=m{a}^{\mu }$. For given initial conditions at past null infinity, we find the boundary particle's trajectory and the reflected energy-momentum of the quantum fields. For incoming vacuum states, the solution yields mirror orbits that correspond to extremal black holes. For the class of incoming states that produce orbits becoming null in finite proper time—corresponding to the formation of a horizon—at the classical level, the quantum backreaction avoids this endpoint rendering the mirror's velocity finite in light cone coordinates. We investigate the behavior of the averaged null energy condition, which in this setup reduces to a boundary term.