Narrow Results

In a space–time M with a Killing vector field ξ^a which is either everywhere timelike or everywhere spacelike, the collection of all trajectories of ξ^a gives a three-dimensional space S. Besides the symmetry-reduced action from that of Einstein–Hilbert, an alternative action of the fields on S is also proposed which gives the same fields equations as those reduced from the vacuum Einstein equation on M.

A general paradigm for describing classical (and semiclassical) gravity is presented. This approach brings to the center-stage a holographic relationship between the bulk and surface terms in a general class of action functionals and provides a deeper insight into several aspects of classical gravity which have no explanation in the conventional approach. After highlighting a series of unresolved issues in the conventional approach to gravity, we show that (i) principle of equivalence, (ii) general covariance and (iii) a reasonable condition on the variation of the action functional, suggest a generic Lagrangian for semiclassical gravity of the form L = Q_a^bcdR^a_bcd with ∇_b Q_a^bcd = 0. The expansion of Q_a^bcd in terms of the derivatives of the metric tensor determines the structure of the theory uniquely. The zeroth order term gives the Einstein–Hilbert action and the first order correction is given by the Gauss–Bonnet action. Any such Lagrangian can be decomposed into surface and bulk terms which are related holographically. The equations of motion can be obtained purely from a surface term in the gravity sector. Hence the field equations are invariant under the transformation T_ab → T_ab + λg_ab and gravity does not respond to the changes in the bulk vacuum energy density. The cosmological constant arises as an integration constant in this approach. The implications are discussed.

We considered derivation of Vlasov–Einstein–Maxwell system of equations from the first principles, i.e. using classical Maxwell–Einstein–Hilbert action principle. We know many papers in which the theories indicated as Einstein–Vlasov, Vlasov–Maxwell–Einstein, Einstein–Maxwell–Boltzmann are discussed, and we discuss difficulties of usually used equations. We use another way of derivation and obtain an alternative version based on the generalized Fock–Weinberg form of equation of motion.

Lagrangian and Hamiltonian descriptions of the dynamics of a self-gravitating matter shell in General Relativity are discussed in general. The case of a spherical shell composed of an elastic fluid is then considered, its Lagrangian function is derived from first principles and the Hamiltonian is calculated. Known results for dust shells are recovered as particular cases.