Kerr black hole in Einstein–æther gravity

While nonrotating black-hole solutions are well known in Einstein–æther gravity, no axisymmetric solutions endowed with Killing horizons have been so far found outside of the slowly rotating limit. Here we show that the Kerr spacetime is also an exact vacuum solution of Einstein–æther gravity in a phenomenologically viable corner of the parameter space; the corresponding æther flow is characterized by a vanishing expansion. Such a solution displays all the characteristic features of the Kerr metric (inner and outer horizons, ergoregion, etc.) with the remarkable exception of the causality-violating region in proximity of the ring singularity. However, due to the associated æther flow, it is endowed with a special surface, inside the Killing horizon, which exhibits many features normally related to the universal horizon of the nonrotating solutions—to which it tends in the limit of zero angular momentum. Hence, these Kerr black holes are very good mimickers of their general relativistic counterparts while sporting important differences and specific structures. As such, they appear particularly well-suited candidates for future phenomenological studies.

Figure 1

Plots of $r_{\mathrm{QUH}}(\theta )$ for several values of the spin $a$. The dotted line, which is reported for reference, corresponds to $r=2M$, i.e. to the Killing horizon’s radius of a Schwarzschild black hole of the same mass as our solution. Note that the curves look circles but, generically, they are not.

Figure 2

Integral lines of the vector $s^{\mu }$, at different angles $\theta$. The vertical lines mark $r=r_{\mathrm{QUH}}(\theta )$. The spin is $a=0.9M$.

Figure 3

Relative difference of the two definitions of surface gravity, as a function of position on the surface $r=r_{\mathrm{QUH}}(\theta )$. The spin is $a=0.9M$.