Abelian-Higgs hair on a stationary axisymmetric black hole in Einstein-Maxwell-axion-dilaton gravity

We study, both analytically and numerically, an Abelian Higgs vortex in the spacetime of a stationary axisymmetric black hole being the solution of the low-energy limit of the heterotic string theory, acting as hair on the black hole background. Taking into account the gravitational backreaction of the vortex, we show that its influence is far more subtle than causing only a conical defect. As a consequence of its existence, we find that the ergosphere is shifted, and the event horizon angular velocity is affected by its presence. Numerical simulations reveal the strong dependence of the vortex behavior on the black hole charge and mass of the Higgs boson. For large masses of the Higgs boson and large values of the charge, the black hole is always pierced by an Abelian Higgs vortex, while small values of the charge and Higgs boson mass lead to the expulsion of the Higgs field.

Figure 1

The difference between locations of (a) the ergosphere and the black hole event horizon for the Kerr-Sen black hole vortex system and (b) the ergosphere for the pure Kerr-Sen black hole and for the Kerr-Sen black hole vortex system. We set $\theta =\frac{\pi }{2}$ and $ϵ={10}^{-6}$.

Figure 2

The difference between locations of (a) the ergosphere for the pure Kerr-Sen black hole and for the Kerr-Sen black hole vortex system as a function of angular-momentum parameter $a$ for fixed black hole charge $Q=0.5Q_{\max }$; (b) the ergosphere for the pure Kerr-Sen black hole and for the Kerr-Sen black hole vortex system as a function of $Q$ for $a=a_{\max }$ (extremal black hole). One considers the equatorial plane for which $\theta =\frac{\pi }{2}$.

Figure 3

The difference between locations of (a) the ergosphere and the black hole horizon for the Kerr-Sen vortex system as a function of $a$ for $Q=0.5Q_{\max }$; (b) the ergosphere and the black hole horizon for the Kerr-Sen vortex system as a function of black hole charge $Q$ for $a=a_{\max }$ (extremal black hole). The equatorial plane $\theta =\frac{\pi }{2}$ was taken into account.

Figure 4

Behavior of the Higgs and gauge fields in the background of the nonextremal Kerr-Sen black hole. One puts the boson Higgs mass equal to $m_X=0.9$.

Figure 5

Behavior of the Higgs and gauge fields in the spacetime of the nonextremal Kerr-Sen black hole. We fix the boson Higgs mass equal to $m_X=4$.

Figure 6

Behavior of the Higgs and gauge fields in the extremal Kerr-Sen black hole background with charge $Q=0.1Q_{\max }$ for the boson Higgs mass $m_X=1.2$.

Figure 7

Behavior of the Higgs and gauge fields in the spacetime of the extremal Kerr-Sen black hole with charge $Q=0.8Q_{\max }$ for the boson Higgs mass $m_X=4.0$.

Figure 8

Behavior of the Abelian Higgs vortex fields in the background of the extremal Kerr-Sen black hole. We put the black hole charge as $Q=0.1Q_{\max }$, the boson Higgs mass as $m_X=0.6$ and the Bogomol’nyi parameter as $\tilde{\beta }=1$, respectively.

Figure 9

Profiles of the Higgs field $X$ and $P^{\phi }$ gauge field component on (a) the extremal Kerr-Sen black hole event horizon. We set the black hole charge $Q=0.1Q_{\max }$ and the boson Higgs mass $m_X=0.6$. (b) The nonextremal Kerr-Sen black hole horizon with the same charge, $a=0.9a_{\max }$ and $m_X=0.9$.

Figure 10

Behavior of the Higgs and gauge fields in the spacetime of the extremal Kerr-Sen black hole for two different charges. We set $Q=0.1Q_{\max }$, $0.5Q_{\max }$ and establish the boson Higgs mass as $m_X=0.9$.