Natural and Gauge Natural Formalism for Classical Field Theorie

Geometric Entropy of Self-Gravitating Systems

“…In all this Section the main reference for the formalism will be [22]. We shall omit the (sometimes long) details, which can be found there.…”

“…purely metric, Palatini metric-affine, purely affine, purely tetrad, tetrad-affine, Ashtekar's variables, Chern-Simons formulations, just to mention the most commonly used; see [22]). The configuration bundle is assumed to be natural which means that spacetime diffeomorphisms act on configuration fields.…”

“…However, due to the low order of the Lagrangian (k = 1 or k = 2), in all cases analysed below S will be uniquely defined (see [25]). The reason why we use the variational morphisms formulation is that being an algebrization of the classical Variational Calculus, it captures the geometrical essence of it and provide the tools sufficient to deal with global properties of field equations, dependence on boundary conditions and conservation laws (see [16], [22], [26], [27], [28] for more details on this).…”

“…As a consequence of conservation laws (11) one has B = 0 even off-shell (result known as Bianchi identities; see [16], [22], [24], [29]),Ẽ ≈ 0 (i.e.,Ẽ = 0 on-shell), while U will enter into the definition of conserved quantities (later appearing in the definition of the entropy) in a Gauss-like fashion.…”

“…This Lagrangian presents an advantage with respect to the Hilbert's one (see [16], [22], [30], [39], [41]): the superpotential ensuing from L gḡ provides a conserved quantity which depends on both g and g and satisfactorily represents the relative energy-momentum between the two field configurations. In this wayḡ can be fixed to be any kind of solution (e.g.…”