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The Hamiltonian formulation of the theory of a gravitational field interacting with a perfect fluid is considered. There is a natural gauge related to the mechanical and thermodynamical properties of the fluid, which enables us to describe 2 degrees of freedom of the gravitational field and 4 degrees of freedom of the fluid (together with 6 conjugate momenta) by nonconstrained data (g, P) where g is a 3-dimensional metric and P is the corresponding Arnowitt-Deser-Misner momentum. The Hamiltonian of the theory, numerically equal to the entropy of the fluid, generates uniquely the evolution of the data. The Hamiltonian vanishes on the data satisfying the vacuum constraint equations and tends to infinity elsewhere as the amount of the matter tends to zero. In this way the vacuum theory with constraints is obtained as a limiting case of a "deep potential well" theory.
Kijowski et al. (Thu,) studied this question.