Optical properties of black holes in regularized Maxwell theory

Regularized Maxwell electrodynamics is a recently discovered theory of nonlinear electrodynamics, with a minimally regularized field strength of a point charge, that is very close to the Maxwell theory in many aspects. In this paper we investigate some of the optical properties of its black holes. Namely, we study geodesics, gravitational redshift, black hole shadow, as well as investigate the relationship between the behavior of (null geodesic) Lyapunov exponents and the existence of thermodynamic critical points in both canonical and grand-canonical ensembles.

Figure 1

Black hole horizons. We display the behavior of the metric function $f(r)$ for fixed values $\alpha =1$, $Q=1$, and $\mathrm{\Lambda }=0$, as it depends on the mass parameter $m$. Namely, from top to bottom, we set $m=0.5137$, $m=0.6156$, $m=0.6497$, $m=m_{\text{marginal}}=2/3$, and $m=0.7771$; pure Schwarzschild with $m=2/3$ is shown in dashed red. We observe two distinct regimes: the Schwarzschild-like regime denoted by red curves that occurs for $m>m_{\text{marginal}}$, and the Reissner-Nordström-like regime, for $m<m_{\text{marginal}}$, indicated by blue curves. The two regimes are separated by the marginal case, denoted by a solid black curve.

Figure 2

Circular orbits: Timelike geodesics. In this figure, we display the dependence of the first derivative of the effective potential on particle’s angular momentum $L$; a $V_r^{\prime }-L$ diagram. The dot-dashed line corresponds to $L=30$, the solid line to $L=18$, and the dashed one to $L=10$, with other parameters taking values $\alpha =1$, $Q=1$, $P=0.01$, $r_{+}=2$. As is clear from the plot, function $V_r^{\prime }(r)$ admits nontrivial roots only for sufficiently large $L$; in that case, the larger root corresponds to the stable circular trajectory and the smaller root to the unstable circular trajectory.

Figure 3

Circular orbits: Null geodesics. This figure depicts the dependence of the first derivative of the potential $V_r^{\prime }(r)$ on angular momentum $L$ for null effective (red) and null background (blue) geodesics; the dependence on $E$ for null effective geodesics was eliminated via the condition, (36), $V_r(r_c)=0$. Values of angular momentum are $L=5$ for solid curves, $L=10$ for dashed curves and $L=20$ for dashed-dot curves; other parameters take values $\alpha =1$, $Q=1$, $r_{+}=0.5$, $P=0.01$. We can see that $L$ only affects prominence of local minima while the root of $V_r^{\prime }(r)$ stays the same; the circular radius $r_c$ is independent of $L$ in the case of null geodesics. While in both cases we observe two roots, the smaller one is below the black hole horizon and is unphysical. We thus observe $r_c^{\mathrm{eff}}>r_c^{\mathrm{bcg}}$.

Figure 4

Phase velocities: In this figure we can clearly see the birefringence for the case of motion in the azimuthal direction; red curve stands for $v_{\mathrm{eff}}^{(\phi )}$, blue curve for $v^{(\phi )}$. In the case of motion in radial direction, phase velocities for both effective and background metric are the same and coincide with $v^{(\phi )}$. Here, we have set $m=1$, $Q=0.5$, and $\alpha =0.75$. Note that the curves emerge from $r=1.90567$ which is the horizon radius.

Figure 5

Light trajectories around black holes. In this figure we display trajectories of light rays emerging from the vicinity of the photon sphere, as they inspiral towards the black hole for Schwarzschild (solid black), Reissner-Nordström (dashed black), and RegMax black holes in the case of effective (red) and background (blue) metrics. In all figures, the RegMax black holes are characterized by $\alpha =1/\sqrt{2}$ and $Q=1/2$, the Reissner-Nordström black holes by $Q=1/2$, and the light ray is sent along the null geodesics just below the particular photon sphere, at $r_0=0.999r_c$. Moreover, if different, we display the black hole horizons with circles of the corresponding color. (a) Equal horizon radius. In this subfigure we fix $r_{+}=2$ for all black holes. Other parameters are as follows. Schwarzschild ($r_c=3$, $M=1$), Reissner-Nordström ($r_c=3.0022,M=1.0625$), and RegMax ($M=1.0448$) with background ($r_c=3.010$) and effective ($r_c=3.1492$). (b) Equal photon sphere radius. In this subfigure all cases share the same photon sphere radius $r_c=3$; other parameters are Schwarzschild ($r_{+}=2$, $M=1$), Reissner-Nordström ($r_{+}=1.997,M=1.055$), and RegMax background ($r_{+}=2.006$, $M=1.0417$) and effective ($r_{+}=1.9011$, $M=0.9970$). (c) Equal ISCO radius $r_{\mathrm{ISCO}}=6$ for Schwarzschild ($r_{+}=2$, $r_c=3$, $M=1$), Reissner-Nordström ($r_{+}=1.9977$, $r_c=3.0186$, $M=1.0614$), and RegMax background ($r_{+}=2.0063$, $r_c=3.0191$, $M=1.0478$) and effective ($r_{+}=1.9981$, $r_c=3.1463$, $M=1.0439$). (d) Equal mass $M=1$ for: Schwarzschild ($r_{+}=2$, $r_c=3$), Reissner-Nordström ($r_{+}=1.8660$, $r_c=2.8229$), and RegMax ($r_{+}=1.9073$) for background ($r_c=2.8708$) and effective ($r_c=3.0094$).

Figure 6

Deflection angle for Schwarzschild. The deflection angle $\beta (r_0)$ is calculated using formula (57) for the Schwarzschild solution with $M=1$, i.e., $r_c=3$.

Figure 7

Deflection angle for RegMax black hole. The difference $\beta -{\beta }_{\mathrm{Sch}}$, where ${\beta }_{\mathrm{Sch}}$ is for reference plotted in Fig. 6, is displayed for background metric (blue), effective metric (red), and Reissner-Nordström (dashed black) solutions with parameters fixed so that $r_c=3$, $Q=1/2$, and $\alpha =1/\sqrt{2}$. Other parameters for the situation where $r_c=3$ are presented in Fig. 5.

Figure 8

Gravitational redshift: Prograde emission. In this picture, we display gravitational redshifts of photons emitted tangentially and in the direction of motion of a massive test particle moving on a circular orbit and observed by a static observer at infinity as they depend on emitter’s radius $r_e$. The red curve corresponds to the RegMax ($M=1,Q=0.95,\alpha =1/\sqrt{2}$) effective metric, the blue curve to the RegMax background metric, solid black to Schwarzschild ($M=1$), and dashed black to the Reissner-Nordström ($M=1,Q=0.95$). Evidently, the kinematic blueshift, due to the relativistic Doppler effect of the light being emitted in the direction of the orbiting test particle outcompetes the effect of pure gravitational redshift. The point in the red line where $z=-1$ corresponds to the radius $r_s\approx 2.9597$ where the photon following the effective metric appears static to the emitter and ${\omega }_e$ changes sign. The blue curve terminates at the background photon sphere radius $r_c^{\mathrm{bcg}}$, while the red curve terminates at the effective photon sphere radius, $r_c^{\mathrm{eff}}$.

Figure 9

Gravitational redshift: Retrograde emission. We display the gravitational redshift of a photon emitted in the direction opposite to the direction of motion of a massive test particle on a circular orbit, as observed by a static observer at infinity, as a function of emitter’s position $r_e$. The legend and the parameters of black holes are identical to Fig. 8.

Figure 10

Gravitational redshift: Radial motion. This picture shows the gravitational redshifts $z$ of a photon emitted radially outward and observed at infinity, displayed as function of static emitter position $r_e$ for the two RegMax metrics (shown in blue), fixing $M=1$, $\alpha =1/\sqrt{2}$, and $Q=0.9$. These two curves coincide, as radial geodesics are not affected by the birefringence (cf. Fig. 4). For comparison we also show the Schwarzschild case ($M=1$) in solid black, and the Reissner-Nordström case ($M=1$, $Q=0.9$) in dashed black. The vertical asymptotes mark the black hole horizons, as expected.

Figure 11

Black hole shadow: Comparison. We display the shadow radius as a function of electric charge $Q$, for various black holes with $M=1/2$ (and $\alpha =0.75$). Namely, the results for background (blue line) and effective (red line) RegMax metrics are compared to the Reissner-Nordström case (dashed black line). On a given RegMax curve, to the left, we start with the Schwarzschild-like branch of black holes. As $Q$ increases, there is a transition to the Reissner-Nordstrom-like branch, which eventually terminates at $Q$ corresponding to an extremal black hole.

Figure 12

Black hole shadow: Mass dependence. In this picture we display the radii of the black hole shadows as functions of black hole charge for various black hole masses and fixed $\alpha =0.75$. Results for the background metric are represented by blue curves, results for the effective metric by red curves. From bottom to top, masses are $M=1/3$ (thin curves), $M=1/2$ (dashed), $M=4/5$ (dotted), and $M=1$ (dot dashed). Similar to the previous figure, charge $Q$ goes from zero to its maximal value corresponding to extremal black hole.

Figure 13

$F-T$ diagram: Free energy of the RegMax black hole is displayed for fixed $Q=1$ and fixed $P=0.023$. Depending on the value of $\alpha$, we observe two Reissner-Nordström-like (blue) and Schwarzschild-like (red) regimes. In the red regime, represented by a red curve for $\alpha =0.4<{\alpha }_c$. the free energy features a temperature gap, two branches of black holes, and a simple cusp; there is only one stable black hole phase corresponding to the lower branch of large black holes. The marginal case, $\alpha ={\alpha }_c$, is displayed by a black curve, for which the upper branch of (small) black holes has a termination point. Slightly above ${\alpha }_c$, we have two stable black hole phases (of large and small black holes) separated by a first-order phase transition, associated with the characteristic swallowtail behavior shown by a solid blue curve for $\alpha =0.76$. As $\alpha$ increases further, the swallowtail diminishes, and eventually terminates at a critical point, where a second-order phase transition occurs, see blue dashed curve for $\alpha =0.82$. For even higher values of $\alpha$, one cannot distinguish between the two phases and the free energy is smooth, as shown by a blue dot-dashed curve for $\alpha =1$.

Figure 14

$\lambda -T$ diagram: Timelike geodesics. Displayed curves correspond to $\alpha =0.4$, $\alpha =1/\sqrt{2}$, $\alpha =0.76$, $\alpha =0.82$, and $\alpha =1$, from right to left. Cusps in the $F-T$ diagram are here represented by single bends, the presence of first-order phase transitions by double-bent curves and the second-order phase transition by an inflection point in the $T=T(\lambda )$ behavior. Note that the quickly dropping branch (of large black holes) terminates at $\lambda =0$, corresponding to finite $r_{+}$. We have set $L=50$, and as in Fig. 13, $Q=1$ and $P=0.023$.

Figure 15

Effect of $L$ on $\lambda -T$ diagram. Here, we display the effect of various choices of $L$ on the shape of $\lambda -T$ diagram for a representative (critical point) case of $\alpha \approx 0.82$. Namely, $L=10$ corresponds to solid curve, $L=50$ to dashed curve, and $L=100$ to dot-dashed curve. As obvious, the critical temperature remains the same irrespective of which $L$ we choose; the choice of $L$ only affects where our curves finally go down.

Figure 16

Critical point Lyapunov exponent: dependence on $L$. For timelike geodesics (black curve), the value of the Lyapunov characteristic exponent $\lambda$ at the critical point, $T=T_c$, depends on the angular momentum $L$. In particular, note that an unstable circular timelike geodesic only exists for particles with $L$ greater than some minimum value, cf. the behavior of the corresponding effective potential in Fig. 2. The dependence on $L$ of course disappears for null effective (red curve) or null background (blue curve) geodesics, cf. Fig. 3. The plot is shown for values $Q=1$, $\alpha \approx 1$, and $P=P_c\approx 0.01$.

Figure 17

$\lambda -T$ diagram: effective null geodesics. While the $\lambda -T$ dependence for null effective geodesics seems to capture all interesting thermodynamic behavior, it is rather complicated. In particular, for $\alpha ={\alpha }_c$, we develop a hook (black curve) and for slightly higher $\alpha$, there is a self-intersection (solid blue curve). In this plot, we have set $Q=1$, $l=2.3$, and displayed $\alpha =0.5$ (dashed red), $\alpha =0.7$ (solid red), $\alpha =1/\sqrt{2}$ (black), $\alpha =0.708$ (solid blue), $\alpha =0.75$ (thin blue), $\alpha =0.822$ (dashed blue), and $\alpha =1$ (dot-dashed blue). Although it is not obvious, solid and dashed blue curves display both cusps of the swallowtail; the corresponding first-order phase transition temperature is, however, greater than $T(\lambda =0)$.

Figure 18

$\lambda -T$ diagram: Null background geodesics. The values of ${\alpha }^{\prime }s$ and coloring are the same as in Fig. 14. Since (for large black holes) the curves do not go down but continue to large temperatures, in this case we can clearly identify all thermodynamic features, including the critical point (inflection), first-order phase transition (triple-valued curves), and the cusp behavior.

Figure 19

$W-T$ diagram: Here, we display the grand-canonical free energy $W$ as a function of temperature $T$ for $\alpha =1$ and $P=0.01$, and various values of $\varphi$: $\varphi =0.5$ (solid red), $\varphi =0.67$ (dashed red), $\varphi =\frac{1}{\sqrt{2}}$ (thin black), $\varphi =0.725$ (dashed black), $\varphi =0.76$ (solid black), and 0.85 (solid blue). Similar to the canonical ensemble, the diagram features cusps, swallowtails, a termination point, or a critical point. When interpreting the corresponding phase diagram, however, one also has to take into account the possibility of the radiation phase, characterized by $W\approx 0$, see [26] for details.

Figure 20

Grand-canonical ensemble: $\lambda -T$ diagram for background null geodesics. The figure is displayed for the same values of parameters and with the same coloring as in Fig. 19. We clearly see the presence of the critical point (solid black), as well as the first-order phase transition (triple valued dashed black), and minimal temperature cusps.

Figure 21

Grand-canonical ensemble: $\lambda -T$ diagram for effective null geodesics. The values of parameters and the coloring is the same as in Fig. 19 except for an additional dot-dashed black line with $\varphi =0.7072$ to show the intersection behavior near ${\varphi }_c$.

Figure 22

Is there a Maxwell’s equal area law? Focusing on the null background metric $\lambda -T$ diagram, we display the first-order phase transition coexistence curve (shown in black) in the grand canonical ensemble corresponding to $\varphi =0.76,\alpha =1$, and various pressures; $P=0.001$ (red curve), $P=0.0055$ (green), and $P\approx 0.0111$ corresponding to the critical point (blue). The presence of some kind of Maxwell’s equal area law for Lyapunov exponents does not seem obvious from this figure.