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Entropy for horizons in theories of gravitation // Tomsk State Pedagogical University Bulletin. 2004. Issue 7 (44). P. 80-87

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 Energy) for a gravitational system The definition of the variation of Energy depends on 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 thermodynamics 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 entropy of gravitational systems, wnich results 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 turns out 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 Einstein field equations. The same definition results also to be well-suited in higher dimensions and in the case of alternative gravitational theories.