Geometry as Thermodynamics: Deriving Gravitational Dynamics from Entropic Principles

We present a conceptual and mathematical framework in which spacetime geometry and Einstein's gravitational dynamics emerge from thermodynamic or entropic principles. In this view, the Einstein field equations are obtained as an extremization condition of a suitable entropy functional, rather than from a traditional action principle. We introduce an entropy functional $S[g,\xi^\mu] = \int (\nabla_\mu \xi_\nu)(\nabla^\mu \xi^\nu)\sqrt{-g}\,d^4x$ depending on the spacetime metric $g_{ab}$ and an auxiliary vector field $\xi^\mu$, and show that its extremum (for all null $\xi^\mu$) reproduces the vacuum Einstein equations (with a cosmological constant) as well as the appropriate generalization in presence of matter. In particular, the variational principle $\delta S=0$ yields the geometric field equations $R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G\,T_{\mu\nu}$, highlighting gravity as an equation of state of spacetime thermodynamics \cite{Jacobson:1995,Padmanabhan:2007}. Furthermore, by considering a complex analytic extension of the entropy functional, we relate the Laurent series expansion of a potential function $f(z) = \exp(S[g,\xi^\mu])$ to gravitational dynamics. We demonstrate that the absence of higher-order poles in this expansion is equivalent to the satisfaction of Einstein's equations, while the coefficient $a_{-1}$ of the simple pole (the entropy residue) corresponds to conserved Noether charges such as the horizon entropy.