Near horizon local instability and quantum thermality

We revisit our previous proposed conjecture -- horizon creates a local instability which acts as the source of quantum temperature of black hole. It is found that a chargesless massless particle moving along the null trajectory in Eddington-Finkelstein (EF) coordinates feels instability in the vicinity of the horizon. Such instability is observer independent for this particle motion. Moreover, an observer associated to EF coordinates finds the local Hamiltonian as $xp$ where $p$ is the canonical momentum corresponding the coordinate $x$. Finally, using this Hamiltonian we notice that at the quantum level this class of observers feel the horizon as thermal object with temperature is given by the Hawking expression. We provide this by using various techniques in quantum mechanics and thereby bolstered our earlier claim -- the automatic local instability can be a mechanism for emerging horizon as a thermal object. In this process, the present analysis provides another set of coordinates (namely EF frame), in addition to our earlier Painleve ones, in which the null trajectory of the massless particle is governed by $xp$ type Hamiltonian in near the horizon regime.