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

TL;DR

This paper derives a general energy expression for 4D stationary axisymmetric spacetimes in teleparallel gravity, applicable to various asymptotic geometries, and demonstrates its use with specific spacetime examples.

Contribution

It provides a new, metric-component-based energy formula in teleparallel gravity for stationary spacetimes without restrictions on parameters.

Findings

Energy depends only on specific metric components.

The formula applies to asymptotically flat, de Sitter, and Anti-de Sitter spacetimes.

Energy calculations for Kerr-Newman and related spacetimes confirm the formula's validity.

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 $E=\frac{1}{8\pi}\int_S d\theta d\phi(sin\theta \sqrt{g_{\theta\theta}}+\sqrt{g_{\phi\phi}}-(1/\sqrt{g_{rr}})(\partial{\sqrt{g_ {\theta\theta} g_{\phi\phi}}}/\partial r))$. 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.