The spacetime DoF surface density as canonically conjugate variables driven by acceleration

Padmanabhan found a definition for the surface density of spacetime degrees of freedom assuming a canonical ensemble for accelerating observers in stationary systems. We find that this density is constructed from a pair of canonically conjugate variables. These variables should be derived with respect to a special space like vector field, which is in the direction of the accelerating observers. Moreover, we show that this density is actually a pair of canonically conjugate variables projected along the observer's velocity vector field, during one period of the Euclidean time of the canonical ensemble. We use our derivation to estimate the number of internal states of the “atoms of spacetime”. We find that this number is independent on the gravitational theory and its dimensions.