The maximum entropy principle of self-gravitating fluid system and field equations

We investigate the maximum entropy principle for general field theory, including a metric tensor $g_{ \mu \nu }$, a vector field $A_{ \mu }$, and a scalar field $\varphi$ as the fundamental fields, and find (i) imposing an ordinary constraint relation on $\varepsilon$, the field equations, which is constructed of the Euler-Lagrange derivative of an arbitrary Lagrangian density, the stress tensor of a perfect fluid, and the electric current vector, in ordinary manner, are compatible with the maximum entropy principle, and (ii) varying also the constraint relation on $\varepsilon$, the maximum entropy principle requires an extra scalar field $\eta$, which is introduced as the difference from the ordinary constraint relation on $\varepsilon$. The field $\eta$ is also interpreted as the difference between two geometries, i.e., one is the geometry, defined by $g_{ \mu \nu }$, in which the thermodynamical relations are written in the ordinary and simplest form, and the other is the geometry, defined by $$\tilde g_{ \mu \nu } = e^{ 2 \eta } g_{ \mu \nu },$$ in which a force-free fluid flows along a geodesic orbit, instead the thermodynamical relations, the first law of thermodynamics and the Gibbs-Duhem relation, are modified. We also calculate the variation of the entropy $\delta S$ in the Lovelock theory, in which $\delta S$ is expressed as a linear combination of the variations of the Kodama energy and the size of the system. Finally, with the field $\eta$ introduced, we propose a set of field equations in the thermodynamical and kinematical geometries, which possesses appropriate scaling properties, and point out that in vacuum spacetime they resemble those of the dilaton gravity in the string and Einstein frame, respectively.