Narrow Results

The Einstein field equations can be derived in n dimensions (n>2) by the variations of the Palatini action. The Killing reduction of five-dimensional Palatini action is studied on the assumption that pentads and Lorentz connections are preserved by the Killing vector field. A Palatini formalism of four-dimensional action for gravity coupled to a vector field and a scalar field is obtained, which gives exactly the same field equations in Kaluza–Klein theory.

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.

The Galilean (and more generally Milne) invariance of Newtonian theory allows for Killing vector fields of a general kind, whereby the Lie derivative of a field is not required to vanish but only to be cancellable by some infinitesimal Galilean (respectively Milne) gauge transformation. In this paper, it is shown that both the Killing–Milne vector fields, which preserve the background Newtonian spacetime structure and the Killing–Cartan vector fields, which in addition preserve the gravitational field, form a Lie subalgebra.

Generalized Robertson–Walker spacetimes extend the notion of Robertson–Walker spacetimes, by allowing for spatial non-homogeneity. A survey is presented, with main focus on Chen's characterization in terms of a timelike concircular vector. Together with their most important properties, some new results are presented.