$D$-dimensional aether charged black hole and aether waves in M-subclass of Einstein-aether theory

Abstract

We obtain an exact $D$-dimensional aether charged black hole solution and gravitational wave polarizations in the M-subclass of the Einstein-aether theory with Lorentz invariance violated by an unit norm vector field---the aether field $u^{\mu }$. This aether field can be timelike or spacelike and, the aether charge $Q_{\ae }$ has a nonzero minimum value, which is different from the electric charge. The aether electric-like potential $u_t$ is regular, and the aether magnetic-like potential $u_r$ is singular at the horizons. Though the Lorentz symmetry is broken, the Smarr formula and the first law of black hole thermodynamics can be exactly constructed via the extended method of Killing potential. However, we find this conception of the aether charge doesn't exist in the $c_i$ subclass of the Einstein-aether theory. For the linearized M-subclass Einstein-aether theory with the timelike aether field, we find the speed of spin-2 modes (the usual gravitational wave) is still equal to 1. But there is only one polarization ${\gamma }_{12}$ that can propagate, while the other one, ${\gamma }_{11}$ cannot propagate for $D=4$ due to the effect of Lorentz symmetry breaking. The speed of spin-1 modes (the transverse aether wave) is also equal to 1. The third kind mode is the longitudinal aether-metric mode, which is linearly time dependent and not the spin-0 mode reported in the $c_i$ subclass Einstein-aether theory.

18 pages, 2 figure, accepted by Comm. Theor. Phys