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 $R_0$, leads to the result that the inertial vacuum appears as a thermal state with temperature $T_0=a_0/2\pi$ where $a_0$ is the acceleration parameter of $R_0$. We construct an infinite family of nested Rindler-like coordinate systems $R_1, R_2, ...$ within the right Rindler wedge, with time coordinates $\tau_1, \tau_2, ...,$ and acceleration parameters $a_1, a_2, ...$ by shifting the origin along the inertial $x$-axis by amounts $\ell_1, \ell_2,\cdots$. We show that, apart from the inertial vacuum, the \textit{Rindler vacuum} of the frame $R_n$ also appears to be a thermal state in the frame $R_{n+1}$ with the temperature $a_{n+1}/2\pi$. \textit{In fact, the Rindler frame $R_{n+1}$ attributes to all the Rindler vacuum states of $R_1, R_2, ... R_n$, as well as to the inertial vacuum state, the same temperature $a_{n+1}/2\pi$.} We further show that our result is discontinuous in an essential way in the coordinate shift parameters. For a Rindler frame $R_i$, this thermality {\it turns on} with smallest non-zero $\ell_i$ allowed in the semiclassical framework and remains insensitive to $(\ell_i,a_{i-1})$ 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 when flat spacetime is treated as the ground state of quantum gravity. (Truncated Abstract)