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

TL;DR

The paper addresses black hole solutions and gravitational-wave polarizations in a D-dimensional Einstein-aether theory within the M-subset characterized by a unit-norm aether field that breaks Lorentz invariance. It derives that a timelike aether precludes static black holes, while a spacelike aether yields a RN-like black hole with a bounded aether charge and reproduces the Smarr formula and first law via a Killing-potential construction; it also performs a linearized analysis to identify unit-speed spin-2 and spin-1 modes and a dimension-dependent longitudinal mode. The results demonstrate how Lorentz-violating dynamics modify black-hole thermodynamics and wave content while preserving familiar thermodynamic relations. An Appendix generalizes the aether-charge concept to the c_i subset, indicating broader applicability of these thermodynamic and wave-structure insights.

Abstract

We study the black hole solution and gravitational wave polarizations in a M-subset of the Einstein-aether theory with Lorentz invariance violated by an unit norm vector field -- the aether field $u^a$. This M-subset of Einstein-aether theory has a form of Einstein-Maxwell theory with a term of Lagrange multiplier potential $λ(u_au^a\mp1)$. We find that if the aether field is timelike, there is no static black hole solution existing, which is different from the result reported in Phys. Rev. D {\bf64} 024028. When the aether field is spacelike, there exists a static solution -- $D$-dimensional Reissner-Nordstrom-like black hole solution, in which the aether charge have a minimum and maximum values. The conception of the aether charge also exists in the $c_i$ subset of the Einstein-aether theory. The Smarr formula and the first law can be still constructed via the extended method of Killing potential. For the linearized M-subset Einstein-aether theory with the timelike aether field, we find the speed of spin-2 modes is unit, which aren't dependent on spacetime dimensions $D$ and aether constant $b_1$, but some parts of polarizations are disappeared. The speed of spin-1 modes is unit also. The third kind mode is the longitudinal aether-metric mode, which is linearly time dependent and not the spin-0 mode.

Figures (1)

• Figure 1: Variation of the aether electric potential $u_t$ and magnetic potential $u_r$ of the aether charged black hole with different aether charge $Q_b$ when $D=4,b_1=2/\kappa=1/4\pi$. Here the aether charge $Q_b$ satisfies the condition that $M/\sqrt{2}\leq Q_b\leq M$. One can see that $u_t$ is singular at the origin, and $u_r$ is singular at the horizons $r_h/M=1\pm\sqrt{1-Q_b^2/M^2}$.