Stability ranges of magnetic black holes and mirror (topological) stars in 5D gravity

Abstract

We discuss static, spherically symmetric solutions to the 5D Einstein-Maxwell equations (belonging to wide classes of multidimensional solutions known at least from the 1990s) and select among them those which must observationally look like local objects whose surface reflects back particles or signals getting there, the so-called mirror stars (also called ``topological stars'' by some authors). Their significant parameters are the Schwarzschild mass $m$ and the magnetic charge $q$, such that $q^2 > 3m^2$, while the radius of their mirror surface is $r_b = 2q^2/(3m) > 2m$. We also discuss their black hole counterparts for which $q^2 \leq 3m^2$. For both these objects, we study spherically symmetric time-dependent perturbations and determine the stability regions in their parameter spaces. Thus, mirror stars turn out to be stable only at $r_b < r_b^{\rm crit} \approx 4.004\,m$, while the black holes prove to be stable in the whole range of their parameters. We calculate the fundamental frequencies and decay rates of black hole perturbations using the WKB and time domain methods. Our stability results disagree with some of those previously announced in the literature.

Figures (8)

• Figure 1: The Kruskal diagram for the metric \ref{['TScw']} with a temporal extra dimension.
• Figure 2: A regular $r{-}v$ surface in a T-Schwarz-schild space-time with a spatial extra dimension.
• Figure 3: (Left): The effective potential $V_{\rm eff}$ for mirror star perturbations as a function of $y = r-r_b$ for $m=1$ and $r_b -2m= 0.02, 0.1, 0.3, 0.7, 1.3$ (upside-down at larger $y$). (Right): 3D plot of $V_{\rm eff}(y)$ for $m=1$ and $r_b - 2m \in (0,4)$. The transparent level $V_{\rm eff} =0$ visualizes the region where $V_{\rm eff} >0$. Note that the charge $q$ is given by $q^2 = (3/2) m r_b$.
• Figure 4: The eigenvalue $\omega^2$ as a function of $r_b$. The red dashed line corresponds to $r_b = r_b^{\rm crit}\simeq 4.003996$, which shows the right boundary of the stability region.
• Figure 5: (Left): numerical curves $X_{\rm num}(y)$ for various $\omega^2$ and $r_b$. (Right): vanishing of the right-end value $X_{\rm num}(y_1)\bigr|_{\omega^2=0}$ as a function of $r_b$ allows us to find the critical value $r_b ^{\rm crit}\simeq 4.003996$.