Shadow of the regular Bardeen black holes and comparison of the motion of photons and neutrinos
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Abstract
The aim of the present research is the analysis of the photon motion in the regular spacetimes arising as solutions of the Einstein gravity coupled with a nonlinear electrodynamics (NED). The photons no longer follow the null geodesic of the background spacetime, but the null geodesics of an effective geometry where the electromagnetic nonlinearity is directly reflected in addition to the spacetime geometry. Motion of photons is compared to the motion of neutrinos that are not directly affected by the nonlinearities of a nonMaxwellian electromagnetic field, and follow null geodesics of the background spacetime. We determine shadows of the regular Bardeen black holes, representing a special solution of the general relativity coupled with NED related to a magnetic charge, both for photons and neutrinos, and compare them to the shadow of the related Reissner–Nordstrom black holes. We demonstrate that the direct NED effects give clear signature of the presence of the regular black holes, on the level going up to \(20\%\) that is detectable by recent observational techniques. We also demonstrate strong influence of the NED effects on deflection angle of photons moving in the Bardeen spacetimes, and on the time delay of the motion of photons and neutrinos in vicinity of the black hole horizon.
1 Introduction
In recent years, the Reissner–Nordstrom (RN) black holes are frequently studied in the astrophysical context. In its standard form governed by the Einstein–Maxwell theory with an electric charge as the source of the electromagnetic field, they are often applied as a model explaining the GRBs [38, 40, 56] – of course, there is a criticism of the strong electric field paradigm [37] implying that existence of stable astrophysical black holes with a significant electric charge seems to be improbable. However, this argument is not relevant in the case of dyonic black holes carrying magnetic charge, as in the field of such black holes the vanishing of charge due to accretion of oppositely electrically charged matter is irrelevant. If the magnetic charge is influencing only uncharged matter through the accordingly modified spacetime geometry, its role is the same as of the electric charge [30, 46].
The interest in the RN black hole geometry occurs also due to the solutions with a tidal charge that arise in the multidimensional braneworld Randall – Sundrum approach [21] – possible astrophysical consequences of tidally charged black holes or naked singularities can be found in [1, 16, 31, 32, 49, 54]). We recall that the braneworld tidal charge reflects curvature of the additional dimension in the braneworld models, being formally equivalent to the electric charge squired in the standard models; however, the tidal charge may take both positive and negative values, and currently there are no strong observational constraints on the tidal charge value, contrary to the case of the electric charge. Of course, the tidal charge is not interacting with the electric (magnetic) charges, but it can influence the photon motion, modifying accordingly the optical effect in vicinity of the black hole [41, 42, 54], or the oscillatory motion of Keplerian disks orbiting tidally charged black holes [49, 51]. Further, the RN black holes could be mimicked also by the phantom fields [23], or by the Horndeski gravity [7].
Any form of the RN black hole solutions can be reflected in the shape of its shadow, if the black hole stands between a shining source and a distant observer. It was suggested to apply the RN (or more general) black hole model with a tidal charge for the black hole at the Galactic Center, in particular to consider the gravitational lensing for such an object [4, 13, 14, 15, 41, 42, 48, 63, 65]. The BH shadows were studied also in the case of more complex rotating spacetimes, starting from the pure Kerr BH spacetime [12], or its extension to the Kerr naked singularity spacetimes [8, 29, 45, 50, 52], the KerrNUT spacetime [28] or Kerrde Sitter spacetime [2, 20, 26, 27, 33, 47, 55, 62], or rotating regular black hole spacetimes [3].
On the other hand, theories predicting so called regular black holes (and related “nohorizon” strong gravity objects), lacking any physical singularity with diverging curvature of the spacetime, attracted strong attention in recent years. There is several such theories, e.g., those of the noncommutative gravity [10, 58], but strongest attention is devoted to the models based on the standard general relativity combined with a nonlinear electrodynamics (NED), where a variety of approaches to the NED has been developed and discussed. For the first time, the regular black holes were proposed in connection to magnetic charges by Bardeen [11]. Later their substance has been reflected in detailed studies of their relation to NED [5, 6, 9, 17, 18, 19, 39, 43, 53, 57, 59, 60]. Generic regular NED black holes were introduced in [25, 61]. Along with the static and spherically symmetric nonrotating solutions, generalizations to the rotating spacetimes were derived and studied extensively [34, 59]. It is thus of crucial importance to look for clear observational signatures of the regular black holes.
Clearly, the most relevant signatures of the NED regular black holes are related to the fact that the motion of uncharged matter, e.g., in Keplerian disks, is related purely to the spacetime geometry of these regular black holes, while motion of photons is not, being related to some effective geometry modified by the NED effects additional to those reflected in the structure of the spacetime. It is thus natural to test this effective geometry – first in the simplest, but very significant and in principle measurable effects on the extension of the black hole shadow.
We start our study concentrating attention on the Bardeen spacetimes, where both the magnetic and electric charges could be introduced, but mainly the case of the magnetic charge could be relevant [18]. We compare the effectivegeometry predicted regular black hole shadow related to the photon motion, to the shadow related to the neutrinos considered as massless particles that is determined by the spacetime structure purely; for completeness, we make comparison also to the shadow related to the RN black holes with the same value of the magnetic (electric) charge as those of the regular Bardeen black hole. We demonstrate that the differences are quite significant.
For comparison we consider also the notion of the deflection angle, and the related issue of the time delay of photons and (massless) neutrinos, or general uncharged ultrarelativistic particles, while they are crossing the strong gravity region of a regular Bardeen black hole. We again demonstrate possibility of effects giving clear signatures of NED effects in the regular black hole backgrounds. We consider also a simple situation where both the source and the observer are located oppositely to the BH at the same distance from the BH centre, e.g., on an radiating ring, and determine the time necessary for properly chosen photon and neutrino to reach the observer, if they are radiated simultaneously. There will be a difference of the time interval of the photon and the neutrino, both due to the different trajectory and different metric coefficients of the regular and effective spacetimes. The time difference could be quite significant even for the stellarmass black holes, and very large for the supermassive black holes, as the time delay dimensional factor is linear in the black hole mass parameter.
2 Geometry and effective geometry

Bardeen black hole spacetimes – specific charge \(q_m/M<0.7698\). In the black hole spacetimes there is only one unstable null circular geodesic located above the outer horizon, and no null circular geodesic located under the inner horizon.

Bardeen nohorizon spacetimes containing null circular geodesics – \(0.7698<q_m/M<0.8586\). There exists an inner stable null circular geodesic and an outer unstable null circular geodesics in such spacetimes.

Bardeen nohorizon spacetimes having no null circular geodesics – \(q_m/M>0.8586\). In such spacetimes a finite maximum of the deflection angle exists for geodesics approaching the centre of the spacetime [43].
Due to the nonlinearity of the electromagnetic field, photons do not follow null geodesics of the Bardeen spacetime. Novello have shown that the photon trajectory is governed by an effective geometry directly reflecting nonlinear electrodynamic effects [17, 22]. In the Appendix, we present an alternative derivation of the effective geometry, using the shortwave approximation; this method leads to the same results as the Novello approach.
3 Equations of motion
In the regular black hole backgrounds with charges related to a NED, the motion of photons is not governed by the spacetime geometry, but by an effective geometry directly reflecting the nonlinearities of the electrodynamics, as shown here in the Appendix. Of course, motion of the other uncharged particles is determined solely by the spacetime geometry. As we would like to test the complete effects of the NED under consideration, we have to study both the motion of photons governed also by the direct nonlinear interaction with the black hole charges, and of the other uncharged particles, here assumed to be simply neutrinos (as they could be effectively considered as zeromass particles, but we can consider any kind of extremely relativistic uncharged particles) that are influenced by the charge parameter only through its influence on the spacetime structure. Therefore, we first summarise the equations of motion of the massless particles in the Bardeen spacetime geometry, and in the related Bardeen effective geometry, presenting their derivation in the Appendix.
3.1 Motion governed by the spacetime geometry
3.2 Motion governed by the effective geometry
4 Circular orbits and the black hole shadows
The circular motion in the Bardeen spacetimes was studied in [43, 53], for both the black hole and nohorizon cases; for the Reissner–Nördström spacetimes, both black hole and naked singularity cases were studied in [48]. We can thus directly use the results of these works, concentrating attention on the motion in the effective geometry.
We can recall that the separation between the Reissner–Nördström black holes and naked singularities corresponds to the specific charge \(q_{RN}/M=1\). The circular null geodesics exist in the Reissner–Nördström spacetimes with specific charge \(q_{RN}/M\le \sqrt{9/8}\) – there is one unstable circular null geodesic for the black hole spacetimes with \(q\le 1\), and an inner stable, and outer unstable circular geodesic in naked singularity spacetimes with \(q \in (1< q < \sqrt{9/8})\). In Reissner–Nördström naked singularity spacetimes with \(q > \sqrt{9/8}\) no circular null geodesics exist [48].
The separation of the blackhole and nohorizon Bardeen spacetimes corresponds to the specific magnetic charge \(q_m/M=0.7698\). The circular null geodesics exist in the Bardeen spacetimes with specific charge \(q_m/M \le 0.85865\), and for Bardeen hohorizon spacetimes with \(q_m/M \in [0.7698, 0.85865]\), even two circular null geodesics exist, the inner being stable, the outer being unstable [53]. No circular null geodesics exist in the nohorizon spacetimes with \(q_m/M > 0.85865\).
4.1 Circular null geodesics of the effective geometry and the spacetime geometry
There is a clear distinction between the circular orbits of neutrinos and photons. There are two neutrino circular orbits for a given specific charge q from the interval \(q \in [0.7698, 0.85865]\). For \(q<0.7698\), there is only one neutrino circular orbit. In the case of the photon circular orbits in the effective geometry, there is no such limit. The photon circular orbits (governed by the effective geometry) thus exist in all the Bardeen spacetimes, contrary to the circular null geodesics of the spacetime, relevant for the other ultrarelativistic particles.
4.2 Shadow of the Bardeen BHs
The apparent angular diameter of two astrophysical blackholes, designated Sgr A* and M87, is given for three models of spacetime, Schwarzschild, and R–N and effBardeen with charge parameter \(q=0.1\)
BH spacetime  \(\theta _{BHSgr A*}\ (\upmu \text {arcsec})\)  \(\theta _{BHM87}\ (\upmu \text {arcsec})\) 

Schwarzschild.  62.87  25.07 
R–N  62.77  25.03 
EffBardeen  51.32  20.46 
4.3 Deflection angle
The physically very important scattering effects of photons and neutrinos on the Bardeen background can be represented by the deflection angle that is governed again by the effective geometry in the photon case,and by the spacetime geometry in the neutrino case. Usually, in the spherically symmetric black hole (and even naked singularity) backgrounds, the deflection angle increases with decreasing impact parameter, and it diverges while the impact parameter is approaching the value corresponding to the impact parameter of the null geodesic. However, in the special class of the nohorizon Bardeen spacetime geometry having no circular null geodesic, the deflection angle increases with decreasing impact parameter up to a maximum value, and with continuing decreasing impact parameter it starts to decrease – this effect could be reflected by a creation of so called ghost images of Keplerian disks, as shown in [43].
The deflection angle profiles diverge at the value of \(l=l_c\). The magnitude of the impact parameter corresponding to the circular orbit is higher for neutrinos than for photons as long as \(q_m \le 0.85865\). For this case there exist an impact parameter \(l_i\) such that \(\Delta \phi _\gamma (l_i)=\Delta \phi _\nu (l_i)\). The value of this special impact parameter \(l_i\) decreases with increasing value of the charge parameter \(q_m\). For the values of the charge parameter \(q_m > 0.85865\), there is no neutrino circular orbit (and the ghost images can occur), but there is still the photon circular orbit and divergence of the photon deflection angle. Therefore, the ghost images are forbidden for photon images on the Keplerian disks, being allowed only for the neutrino images. Such a qualitative difference could be again a clear signature of the existence of strongly charged nohorizon Bardeen spacetimes.
5 Timedelays

Starting with \(q_m/M>0.8586\), the Bardeen nohorizon spacetimes having no neutrino circular orbit. We picked value \(q_m/M=1.0\). There is a value of impact parameter \(l_i\) where timedelay curves intersect, i.e., where is \(\Delta t_\gamma (l_i)=\Delta t_\nu (l_i)\). The timedelay for a fixed impact parameter \(l>l_i\) is longer for neutrinos than for photons and for \(l<l_i\) the situation is inverse. This behaviour could be a very specific signature of this class of the Bardeen spacetimes.

In the case of Bardeen nohorizon spacetimes containing two null circular geodesics, \(0.7698<q_m/M<0.8586\), we set \(q_m/M=0.8\). There is no intersection between the curves \(\gamma \) and \(\nu \). The neutrino time delay is always larger than the photon time delay.

In the Bardeen black hole case with an unstable null circular geodesic, \(q_m/M<0.7698\), we set \(q_m/M=0.5\). There is no intersection between curves \(\gamma \) and \(\nu \). The neutrino time delay is always larger than the photon time delay.
Timedelays of neutrinos and photons emitter from \((r,\phi )=(100,0)\) and arriving to \((r,\phi )=(100,\pi )\) and to \((r,\phi )=(100,3\pi )\) for three representative values of magnetic charge parameter \(q=0.5\), 0.8, and 1.0
\(q_m\)  

0.5  0.8  1.0  
\(\Delta t_\gamma (\pi )\)  197.457  197.478  197.498 
\(\Delta t_\nu (\pi )\)  211.417  211.408  211.4 
\([\Delta t_\nu \Delta t_\gamma ](\pi )\)  13.96  13.93  13.90 
\(\Delta t_\gamma (3\pi )\)  241.166  241.261  – 
\(\Delta t_\nu (3\pi )\)  248.747  247.049  – 
\([\Delta t_\nu \Delta t_\gamma ](3\pi )\)  7.58  5.79  – 
\(\Delta t_{\gamma RN}(\pi )\)  211.380  211.312  211.248 
\(\Delta t_{\gamma RN}(3\pi )\)  242.813  242.224  243.073 
We can see that the time delay effect can be quite large and easily observable. The time delay of the direct irradiation is almost twice the indirect irradiation, although the photons and neutrinos are orbiting the black hole ones in the case of the indirect irradiation, however, they are orbiting closer to the black hole in comparison with perihelion of the direct trajectory. Clearly, the time delay of the irradiation can be also a significant distinguishing effect (Table 4).
6 Conclusion
List of impact parameters \(l_i\) (\(i=(\gamma ,\nu ,{\gamma }RN)\)) for geodesics connecting observer and emitter at \(r=100\) separated by azimuthal angle \(\phi =\pi \) of particular spacetime
\(q_m\)  0.5  0.8  1.0 

\(l_\gamma \)  28.1592  28.1808  28.2007 
\(l_\nu \)  15.7271  15.6955  15.6662 
\(l_{\gamma RN}\)  15.6593  15.5194  15.3872 
The impact parameters of null geodesics for two representative values of magnetic charge parameter \(q_m\), presented in Fig. 8
\(q_m\)  0.5  0.8 

\(l_\gamma \)  4.23715  4.19502 
\(l_\nu \)  4.96948  4.47621 
\(l_{\gamma RN}\)  4.97496  4.55684 
We have determined shadows of the regular Bardeen black holes by studying the photon motion in the effective geometry governed by the spacetime geometry and the Lagrangian of the NED. Critical impact parameters for photon geodesics separate capture and scattering regions and these parameters characterise shadow sizes (radii). We demonstrate explicitly that the shadows of the black holes have to be by about \(20\%\) smaller in comparison with what can be expected due to the standard spacetime geometry effect, measurable by other massless particles. Similar signatures have been found also in the case of the deflection angle. Moreover, we have shown that the direct effect of the NED represented by the effective Bardeen geometry excludes existence of the ghost photon images predicted in the special nohorizon Bardeen spacetimes with no circular null geodesics of the spacetime geometry [43] – thus the ghost images could be relevant only for the neutrino images of the Keplerian disks, or due to highly ultrarelativistic particles.
Assuming that a regular black hole is located in the Galactic Center, and in the centres of other galaxies (e.g., M87), one can use the results obtained in the present paper to analyse future observational data obtained with advanced observational facilities as the Event Horizon Telescope [24].
Finally, we have demonstrated that the relative time delay effects related to the motion of photons and neutrinos (or general ultrarelativistic particles) in strong gravity regions of the Bardeen spacetimes can be also efficient signatures of the relevance of the NED effect in the Bardeen spacetimes.
In the future research we plan to illustrate the NED phenomena in direct or indirect images of the Keplerian disks, and in the related profiled spectral lines. Extension to the studies of other NED regular black holes is also planed.
Notes
Acknowledgements
The authors acknowledge institutional support of the Faculty of Philosophy and Science of the Silesian University at Opava, and the Albert Einstein Centre for Gravitation and Astrophysics supported by the Czech Science Foundation Grant no. 1437086G.
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