Energy of general 4-dimensional stationary axisymmetric spacetime in the teleparallel geometry

The field equation with the cosmological constant term is derived and the energy of the general 4-dimensional stationary axisymmetric spacetime is studied in the context of the hamiltonian formulation of the teleparallel equivalent of general relativity (TEGR). We find that, by means of the integral form of the constraints equations of the formalism naturally without any restriction on the metric parameters, the energy for the asymptotically flat/de Sitter/Anti-de Sitter stationary spacetimes in the Boyer-Lindquist coordinate can be expressed as $E=\frac{1}{8\pi}\int_S d\theta d\phi(sin\theta \sqrt{g_{\theta\theta}}+\sqrt{g_{\phi\phi}}-(1/\sqrt{g_{rr}})(\partial{\sqrt{g_ {\theta\theta} g_{\phi\phi}}}/\partial r))$. It is surprised to learn that the energy expression is relevant to the metric components $g_{rr}$, $g_{\theta\theta}$ and $g_{\phi\phi}$ only. As examples, by using this formula we calculate the energies of the Kerr-Newman (KN), Kerr-Newman Anti-de Sitter (KN-AdS), Kaluza-Klein, and Cveti\v{c}-Youm spacetimes.Comment: 12 page

Geodesics in a quasispherical spacetime: A case of gravitational repulsion

Geodesics are studied in one of the Weyl metrics, referred to as the M--Q solution. First, arguments are provided, supporting our belief that this space--time is the more suitable (among the known solutions of the Weyl family) for discussing the properties of strong quasi--spherical gravitational fields. Then, the behaviour of geodesics is compared with the spherically symmetric situation, bringing out the sensitivity of the trajectories to deviations from spherical symmetry. Particular attention deserves the change of sign in proper radial acceleration of test particles moving radially along symmetry axis, close to the $r=2M$ surface, and related to the quadrupole moment of the source.Comment: 30 pages late