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

Abstract

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 $E=\frac{1}{8\pi }{\int }_{S}d\theta d\varphi (sin\theta \sqrt{g_{\theta \theta }}+\sqrt{g_{\varphi \varphi }}-(1/\sqrt{g_{rr}})(\partial \sqrt{g_{\theta \theta }{g}_{\varphi \varphi }}/\partial r))$. It is surprised to learn that the energy expression is relevant to the metric components $g_{rr}$, $g_{\theta \theta }$ and $g_{\varphi \varphi }$ 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 Cvetič-Youm spacetimes.

12 pages

Cited by in corpus (3)

• Charged Axially Symmetric Solution and Energy in Teleparallel Theory Equivalent to General Relativity
• The Geometry of Regular Shear-Free Null Geodesic Congruences, CR functions and their Application to the Flat-Space Maxwell Equations
• Mass and angular momentum for the Kerr black hole in TEGR and STEGR