The Thermal Substructure of General Relativity

Abstract

In a first step we will provide arguments for the understanding of quantum space-time (QST), that means, the microscopic substructure which is assumed to underly ordinary smooth classical space-time, as a thermal system at each (macroscopic) point $x$ of the classical space-time manifold (ST). In this context we exploit among other things some recent findings in the foundations of quantum statistical mechanics. We argue that the classical metric tensor field $g_{ij}(x)$ plays the role of an order parameter field, signalling the existence of a primordial phase transition in which both space-time and (quantum) matter emerged, accompanied by the spontaneous symmetry breaking (SSB) of diffeomorphism invariance which is induced by the existence of quantum excitations in QST. We then analyze the role the Einstein equations (EEQ) are playing in this context and relate them to the local thermodynamical heat influx into QST at point $x$, thus defining the Einstein tensor or Ricci tensor as local heat tensor. We also define local gravitational temperature, entropy and internal energy. Finally, we give a thermodynamic interpretation of the conserved Noether current which derives from diffeomorphism invariance, separating it into heat contributions and work terms. In this connection we discuss the role and nature of tensor and non-tensor quantities and relate our investigation to the socalled energy vector of Hilbert`s classical analysis.