5D black holes and mirror stars from nonlinear electrodynamics: Existence and stability

Abstract

We consider static, spherically symmetric solutions of 5D general relativity with magnetic fields governed by nonlinear electrodynamics (NED) with the Lagrangian $L(F)$, $F = F_{AB} F^{AB}$, and show that generic solutions describe either 5D black holes (also called black strings due to a circular extra dimension) or so-called mirror stars (also called topological stars) with perfectly reflecting boundary surfaces. Two particular examples of such solutions have been obtained, admitting analytic expressions for the metric coefficients and $L(F)$, and their stability under radial (monopole) perturbations is studied. While the whole obtained family of black hole solutions turns out to be stable, mirror star solutions prove to be stable only in a certain range in the parameter space. We thus extend to the Einstein-NED system the results previously obtained for Einstein-Maxwell fields.

Figures (5)

• Figure 1: Metric coefficients in Examples 2 (mirror stars) and 3 (black holes). Left: ${\,\rm e}^{2\xi(r)}$ in Example 2, for $m=1$ and $N = 0, 0.2, 0.5, 1$ (upside down). Zeroes of ${\,\rm e}^{2\xi}$ at $N > 0$ correspond to mirror surfaces. Right: ${\,\rm e}^{2\gamma(r)}$ in Example 3, for $p=1$ and $N = 0, -0.2, -0.5, -1$ (upside down). Zeroes of ${\,\rm e}^{2\gamma}$ at $N < 0$ correspond to event horizons. The curves for $N=0$ in the two panels correspond to the same black hole solution with the metric \ref{['ds-N=0']}, where $\xi \equiv \gamma$.
• Figure 2: The effective potential $V_{\rm eff}(r)$ for mirror star solutions of Example 2. Only the surface to the right of the deep holes is relevant.
• Figure 3: The eigenvalue $\omega^2$ as a function of $\lg N$. The red dashed line corresponds to $N = N_{\rm crit}\simeq 0.6719642$, showing the left boundary of the instability region.
• Figure 4: Left: numerical curves $X_{\rm num}(y)$ for various $\omega^2$ and $N$. Right: The right-end value $X_{\rm num}(y_1)\bigr|_{\omega^2=0}$ as a function of $N$ allows one to find the critical value $N_{\rm crit}\simeq 0.6719642$.
• Figure 5: The effective potential $V_{\rm eff}(r)$ for black hole solutions of Example 3. Left: $V_{\rm eff}(r)$ for $N=0, -0.2, -0.5, -1$ (upside down). Right: a 3D picture, in which the line $V_{\rm eff} =0$ corresponds to $r=r_h$, and the transparent zero level illustrates that $V_{\rm eff} > 0$ outside the horizon.