Accelerating Bertotti-Robinson Black Holes in a Uniform Magnetic Field

Abstract

We study the Hawking temperature, geodesic motion, and observable signatures of the accelerating Bertotti-Robinson (BR) spacetime, a vacuum black-hole solution deformed by a uniform magnetic field $B$ and an acceleration parameter $α$. In the timelike sector, we derive the effective potential for massive particles, determine the specific energy and angular momentum for equatorial circular orbits, and determine how $(B,α)$ shifts the ISCO; we also illustrate representative trajectories of massive particles. We then compute the radial and latitudinal epicyclic frequencies for small perturbations about circular orbits, quantifying how the magnetic field and acceleration modify local radial and vertical stability. In the null sector, we derive the photon effective potential and obtain analytical expressions for the photon-sphere radius, critical impact parameter, and shadow radius, complemented by photon trajectories, the effective radial force, and the Lyapunov exponent controlling the instability of circular null orbits; we also provide parameter-space maps for the photon sphere and shadow. Finally, we obtain the energy emission rate emitted from the black hole, showing how the acceleration parameter and the magnetic field affect this.

Figures (14)

• Figure 1: The behavior of the refractive index as a function $r/m$ by varying $B$ and $\alpha$.
• Figure 2: The behavior of the Hawking temperature as a function of the magnetic field $B$ and acceleration $\alpha$.
• Figure 3: Effective potential $U_{\rm eff}(r)$ for massive test particles [Eq. (\ref{['bb6']})] as a function of the radial coordinate $r/m$, with $m=1$ and angular momentum $\mathcal{L}=4$. Panel (a) shows the effect of varying the magnetic field strength $B \in \{0, 0.08, 0.14, 0.20\}$ at fixed acceleration $\alpha=0.05$. Panel (b) shows the effect of varying the acceleration parameter $\alpha \in \{0, 0.03, 0.07, 0.10\}$ at fixed $B=0.12$. In panel (a), increasing $B$ systematically raises the height of the potential barrier and shifts its local maximum inward, reflecting the magnetic confinement that tightens circular orbits. In panel (b), increasing $\alpha$ lowers and broadens the barrier, indicating that acceleration reduces the orbital binding energy. The standard Schwarzschild effective potential is recovered in the $B=\alpha=0$ limit (solid dark curve in panel (a)). The vertical dashed lines at small $r$ indicate the respective event-horizon radii $r_h = 2m/(1-B^2m^2)$ for each value of $B$.
• Figure 4: Specific angular momentum $\mathcal{L}_{\rm sp}$ (panel a, Eq. (\ref{['bb8']})) and specific energy $\mathcal{E}_{\rm sp}$ (panel b, Eq. (\ref{['bb10']})) for circular orbits as functions of the radial coordinate $r/m$, shown for four representative parameter combinations: $(B,\alpha) = (0,0)$ (Schwarzschild, solid dark), $(0.10, 0)$, $(0.10, 0.05)$, and $(0.15, 0.05)$. The dotted vertical lines mark the corresponding ISCO radii obtained numerically from the marginal stability condition [Eq. (\ref{['bb11']})]; the minimum of each $\mathcal{L}_{\rm sp}$ curve coincides with the respective ISCO. The horizontal dotted line in panel (b) marks the marginal energy $\mathcal{E}=1$ separating bound ($\mathcal{E}<1$) from unbound ($\mathcal{E}>1$) circular orbits. Increasing $B$ generally raises the angular momentum curve and shifts the ISCO to larger radii, while increasing $\alpha$ at fixed $B$ lowers both $\mathcal{L}_{\rm sp}$ and $\mathcal{E}_{\rm sp}$. The divergence of $\mathcal{L}_{\rm sp}$ near the photon sphere and its minimum at the ISCO are characteristic features of every Schwarzschild-like geometry; the parameters $B$ and $\alpha$ control the location and height of these features.
• Figure 5: ISCO radius $r_{\rm ISCO}/m$ as a function of the magnetic field parameter $Bm$ for four values of the acceleration parameter: $\alpha \in \{0, 0.01, 0.03, 0.05\}$. The dotted horizontal line marks the Schwarzschild reference value $r_{\rm ISCO} = 6m$. Each curve was obtained by numerically solving the marginal stability condition $d^2U_{\rm eff}/dr^2=0$ along the family of circular orbits parameterized by $L_{\rm sp}(r)$. For fixed $\alpha$, increasing $B$ moves the ISCO to larger radii; the curves terminate at the maximum $B$ for which a stable circular orbit can exist outside the horizon, beyond which no ISCO is found in the scanned range. For fixed $B$, increasing $\alpha$ shifts the entire curve downward, indicating that acceleration counteracts the magnetic confinement and can restore stability at smaller radii. At small $B$ and $\alpha=0$, all curves start above $6m$ because the conformal factor $\Omega^2$ modifies the effective potential even at leading order in $B$; only in the strict double limit $B,\alpha\to 0$ is the Schwarzschild value recovered.