Thermodynamics of Gravity in Local Frames

We probe the thermodynamic structure of gravity at local scales. In any general curved spacetime, it is possible to transform to a local inertial frame at any point such that the metric is flat up to quadratic order where the curvature at that point comes in when the metric is written in Riemann normal coordinates. We consider local Rindler observers in that patch and hence the local Rindler horizon. In doing so, we find that the local horizons are also hot provided the $aL>>1$ which can always be satisfied.