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The Bogoliubov transformation connecting the standard inertial frame modefunctions 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 withtemperature $T_0=a_0/2\pi$ 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 theorigin along the inertial $x$-axis by amounts $\ell_1, \ell_2,\cdots$. We showthat, apart from the inertial vacuum, the Rindler vacuum of the frame $R_n$also appears to be a thermal state in the frame $R_{n+1}$ with the temperature$a_{n+1}/2\pi$. In fact, the Rindler frame $R_{n+1}$ attributes to all theRindler vacuum states of $R_1, R_2, ... R_n$, as well as to the inertial vacuumstate, the same temperature $a_{n+1}/2\pi$. The frame with the shift $\ell$ andthe corresponding acceleration parameter $a(\ell)$ can be thought of as aRindler frame which is instantaneously comoving with the Einstein's elevatormoving with a variable acceleration. Our result suggests that the quantumtemperature associated with such an Einstein's elevator is the same as thatdefined in the comoving Rindler frame. The shift parameters $\ell_j$ arecrucial for the inequivalent character of these vacua and encode the fact thatRindler vacua are not invariant under spatial translation. We further show thatour result is discontinuous in an essential way in the coordinate shiftparameters. Similar structures can be introduced in the right wedge of anyspacetime with bifurcate Killing horizon, like, for e.g., Schwarzschildspacetime. This has important implications for quantum gravity when flatspacetime is treated as the ground state of quantum gravity.

A nested sequence of comoving Rindler frames, the corresponding vacuum states, and the local nature of acceleration temperature