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Register a new userMetamorphoses of a photon sphere
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Authors
Andrey A. Shoom
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There are circular planar null geodesics at $r=3M$ around a Schwarzschild black hole of mass $M$. These geodesics form a photon sphere. Null geodesics of the Schwarzschild space-time which do not form the photon sphere are either escape to null infinity or get captured by the black hole. Thus, from the dynamical point of view, the photon sphere represents a smooth basin boundary that separates the basins of escape and capture of the dynamical system governing the null geodesics. Here we consider a Schwarzschild black hole distorted by an external, static, and axisymmetric quadrupolar gravitational field. We study null geodesics around such a black hole and show that the photon sphere transforms into a fractal basin boundary that indicates chaotic behavior of the null geodesics. We calculate the box-counting fractal dimension of the basin boundary and the related uncertainty exponent, which depend on the value of the quadrupole moment.
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It is generally believed that the shadows of either a black hole or naked singularity arise due to photon spheres developing in these spacetimes. Here we propose a new spherically symmetric naked singularity spacetime solution of Einstein equations which has no photon sphere, and we show that the singularity casts a shadow in the absence of the photon sphere. We discuss some novel features of this shadow and the lightlike geodesics in this spacetime. We compare the shadow of the naked singularity here with shadows cast by Schwarzschild black hole and the first type of Joshi-Malafarina-Narayan (JMN1) naked singularity, where for the last two spacetimes the shadow is formed due to the presence of a photon sphere. It is seen, in particular, that the size of shadow of the singularity is considerably smaller than that of a black hole. Our analysis shows that the shadow of this naked singularity is distinguishable from the shadow of a Schwarzschild black hole and the JMN1 naked singularity. These results are useful and important in the context of recent observations of shadow of the M87 galactic center.
From any location outside the event horizon of a black hole there are an infinite number of trajectories for light to an observer. Each of these paths differ in the number of orbits revolved around the black hole and in their proximity to the last photon orbit. With simple numerical and a perturbed analytical solution to the null-geodesic equation of the Schwarzschild black hole we will reaffirm how each additional orbit is a factor $e^{2 pi}$ closer to the black holes optical edge. Consequently, the surface of the black hole and any background light will be mirrored infinitely in exponentially thinner slices around the last photon orbit. Furthermore, the introduced formalism proves how the entire trajectories of light in the strong field limit is prescribed by a diverging and a converging exponential. Lastly, the existence of the exponential family is generalized to the equatorial plane of the Kerr black hole with the exponentials dependence on spin derived. Thereby, proving that the distance between subsequent images increases and decreases for respectively retrograde and prograde images. In the limit of an extremely rotating Kerr black hole no logarithmic divergence exists for prograde trajectories.
In this paper, we extend the study of the relationship between the photon sphere and the thermodynamic phase transition, especially the reentrant phase transition, to this black hole background. According to the number of the thermodynamic critical points, the black hole systems are divided into four cases with different values of Born-Infeld parameter b, where the black hole systems can have no phase transition, reentrant phase transition, or Van der Waals-like phase transition. For these different cases, we obtain the corresponding phase structures in pressure-temperature diagram and temperature-specific volume diagram. The tiny differences between these cases are clearly displayed. On the other hand, the radius rps and the minimal impact parameter ups of the photon sphere are calculated via the effective potential of the radial motion of photons. For different cases, rps and ups are found to have different behaviors. In particular, with the increase of rps or ups, the temperature possesses a decrease-increase-decrease-increase behavior for fixed pressure if there exists the reentrant phase transition. While for fixed temperature, the pressure will show an increase-decrease-increase-decrease behavior instead. These behaviors are quite different from that of the Van der Waals-like phase transition. Near the critical point, the changes of rps and ups among the black hole phase transition confirm an universal critical exponent 12. Therefore, all the results indicate that, for the charged Born-Infeld-AdS black holes, not only the Van der Waals-like phase transition, but also the reentrant phase transition can be reflected through the photon sphere.
Stationary axisymmetric metric describing the exterior field of a rotating, charged sphere endowed with magnetic dipole moment is presented and discussed. It has a remarkably simple multipole structure defined by only four nonzero Hoenselaers-Perjes relativistic moments.
We introduce a non-commutative deformation of the algebra of bipolar spherical harmonics supporting the action of the full Lorentz algebra. Our construction is close in spirit to the one of the non-commutative spherical harmonics associated to the fuzzy sphere and, as such, it leads to a maximal value of the angular momentum. We derive the action of Lorentz boost generators on such non-commutative spherical harmonics and show that it is compatible with the existence of a maximal angular momentum.
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