Gravitational energies and generalized geometric entropy

A geometrical covariant definition of the variation of conserved quantities is introduced for Lagrangian field theories, suitable for both metric and affine gravitational theories. When this formalism is applied to the Hilbert Lagrangian we obtain a covariant definition of the Hamiltonian (and consequently a definition of the variation of energy) for a gravitational system. The definition of the variation of energy depends on the boundary conditions one imposes. Different boundary conditions are introduced to define different energies: the gravitational heat (corresponding to Neumann boundary conditions) and the Brown-York quasilocal energy (corresponding to Dirichlet boundary conditions) for a gravitational system. An analogy between the behavior of a gravitational system and a macroscopical thermodynamical system naturally arises and relates control modes for the thermodynamical system with boundary conditions for the gravitational system. This geometrical and covariant framework enables one to define the entropy of gravitational systems, which turns out to be a geometric quantity with well-defined cohomological properties arising from the obstruction to foliate spacetimes into spacelike hypersurfaces. This definition of gravitational entropy is found to be very general: it can be generalized to causal horizons and multiple-horizon spacetimes and applied to define entropy for more exotic singular solutions of the Einstein field equations. The same definition is also well-suited in higher dimensions and in the case of alternative gravitational theories (e.g., Chern-Simons theories, Lovelock gravity