Black holes with many horizons in the theories of nonlinear electrodynamics

In this article, we construct exact black hole solutions with many horizons (more than two) in the Einstein-nonlinear electrodynamic theories. In particular, we acquire the explicit expression of a nonlinear electrodynamic Lagrangian for the 3-horizon black holes. Then we make the investigations of three-horizon black holes on the horizons, the null and timelike geodesics, the Love numbers, and the thermodynamics.

Figure 1

The locations of three horizons for $M=1$, $Q=0.7$, ${\alpha }_2=-0.001$.

Figure 2

The locations of four horizons for $M=1$, $Q=0.74$, ${\alpha }_2=-0.0009$, ${\alpha }_3=3.14\times 0^{-6}$.

Figure 3

Five critical curves for $V$ with $Q^2>\frac{25}{24}{M}^2$, $Q^2=\frac{25}{24}{M}^2$, $M^2<Q^2<\frac{25}{24}{M}^2$, $Q^2=M^2$, and $Q^2<M^2$, from up to down, respectively.

Figure 4

The disposition of the roots of the eight-degree equation $f(u)=0$ for ${\alpha }_2<0$.

Figure 5

The disposition of the five roots of the eight-degree equation $f(u)=0$ for general timelike geodesics when ${\alpha }_2<0$.