A nested sequence of inequivalent Rindler vacua : universal relic thermality of Planckian origin

Abstract

The Bogoliubov transformation connecting the standard inertial frame mode functions to the standard mode functions defined in the Rindler frame R0, leads to the result that the inertial vacuum appears as a thermal state with temperature where a0 is the acceleration parameter of R0. We construct an infinite family of nested Rindler-like coordinate systems within the right Rindler wedge, with time coordinates and acceleration parameters by shifting the origin along the inertial x-axis by amounts . We show that, apart from the inertial vacuum, the Rindler vacuum of the frame Rn also appears to be a thermal state in the frame with the temperature . In fact, the Rindler frame attributes to all the Rindler vacuum states of , as well as to the inertial vacuum state, the same temperature . We further show that our result is discontinuous in an essential way in the coordinate shift parameters. For a Rindler frame Ri, this thermality turns on with smallest non-zero allowed in the semiclassical framework and remains insensitive to thereafter, indicating its universal Planckian origin. Similar structures can be introduced in the right wedge of any spacetime with bifurcate Killing horizon, like, for e.g. Schwarzschild spacetime. Apart from providing unsuppressed observables capturing Planck scale effects, these results have important implications for quantum gravity (QG) when flat spacetime is treated as the ground state of QG. Furthermore, a frame with the shift and the corresponding acceleration parameter can be thought of as a Rindler frame which is instantaneously comoving with the Einstein's elevator moving with a variable acceleration. Our result suggests that the quantum temperature associated with such an Einstein's elevator is the same as that defined in the comoving Rindler frame. The implications of these results are wide ranging, from having a definitive signature of Planck shifts in the horizon to the existence of a new set of observers in black hole exterior having thermodynamic description of the horizon they perceive.