  It is well known that Minkowski vacuum appears as a thermal bath in the
Rindler spacetime when the modes on the left wedge are traced out. We introduce
the concept of a Rindler-Rindler spacetime, obtained by a further coordinate
transformation from the Rindler spacetime, in a manner similar to the
transformation from inertial to Rindler frame. We show that the Rindler vacuum
appears as a thermal state in the Rindler-Rindler frame. Further, the spectrum
of particles seen by the Rindler-Rindler observers in the original Minkowski
vacuum state is shown to be identical to that seen by detector accelerating
through a real thermal bath. Thus the Davies-Unruh effect acts as a proxy for a
real thermal bath, for a certain class of observers in the Rindler-Rindler
spacetime. We interpret this similarity as indicating further evidence of the
indistinguishablity between thermal and quantum fluctuations along the lines of
the recent work in arXiv:1308.6289. The implications are briefly discussed.
