Spherical Photon Orbits Research Articles

Cuspy and fractured black hole shadows in a toy model with axisymmetry

Cuspy shadow was first reported for hairy rotating black holes, whose metrics deviate significantly from the Kerr one. The non-smooth edge of the shadow is attributed to a transition between different branches of unstable but bounded orbits, known as the fundamental photon orbits, which end up at the light rings. In searching for a minimal theoretical setup to reproduce such a salient feature, in this work, we devise a toy model with axisymmetry, a slowly rotating Kerr black hole enveloped by a thin slowly rotating dark matter shell. Despite its simplicity, we show rich structures regarding fundamental photon orbits explicitly in such a system. We observe two disconnected branches of unstable spherical photon orbits, and the jump between them gives rise to a pair of cusps in the resultant black hole shadow. Besides the cuspy shadow, we explore other intriguing phenomena when the Maxwell construction cannot be established. We find that it is possible to have an incomplete arc of Einstein rings and a “fractured” shadow. The potential astrophysical significance of the corresponding findings is addressed.

Observability of spherical photon orbits in near-extremal Kerr black holes

We investigate the spherical photon orbits in near-extremal Kerr spacetimes. We show that the spherical photon orbits with impact parameters in a finite range converge on the event horizon. Furthermore, we demonstrate that the Weyl curvature near the horizon does not generate the shear of a congruence of such light rays. Because of this property, a series of images produced by the light orbiting around a near-extremal Kerr black hole several times can be observable.

Modification of photon trapping orbits as a diagnostic of non-Kerr spacetimes

Photon circular orbits, an extreme case of light deflection, are among the hallmarks of black holes and are known to play a central role in a variety of phenomena related to these extreme objects. The very existence of such orbits when motion is not confined in the equatorial plane, i.e. spherical orbits, is indeed a special property of the separable Kerr metric and may not occur, for instance, in the spacetime of other more speculative ultracompact objects. In this paper we consider a general stationary-axisymmetric spacetime and examine under what circumstances spherical or more general, variable-radius, `spheroidal' non-equatorial photon orbits may exist with the ultimate goal of using the modifications -- or even loss -- of photon trapping orbits as a telltale of non-Kerr physics. In addressing this issue, we first derive a general necessary condition for the existence of spherical/spheroidal orbits and then go on to study photon trapping orbits in a variety of known non-Kerr metrics (Johannsen, Johanssen-Psaltis, and Hartle-Thorne). The first of these is an example of a separable spacetime which supports Kerr-like spherical photon orbits. A more detailed analysis reveals a deeper connection between the presence of spherical orbits and the separability of a metric (that is, the existence of a third integral of motion). Specifically, a spacetime that does not admit spherical orbits in any coordinates is necessarily non-separable. The other two spacetimes considered here exhibit a clear non-Kerr behaviour by having spherical photon orbits replaced by spheroidal ones. More importantly, subject to the degree of deviation from Kerr, equatorial photon rings give place to non-equatorial ones with an accompanying loss of low-inclination spheroidal orbits. The implications of these results for the electromagnetic and gravitational wave signature of non-Kerr objects are briefly discussed.

Completing characterization of photon orbits in Kerr and Kerr-Newman metrics

Recently, several new characteristics have been introduced to describe null geodesic structure of stationary spacetimes, such as photon regions (PR) and transversely trapping surfaces (TTS). The former are three-dimensional domains confining the spherical photon orbits, while the latter are closed two-surface of spherical topology which fill other regions called TTR. It is argued that in generic stationary axisymmetric spacetime it is natural to consider also the non-closed TTSs of the geometry of spherical cups, satisfying the same conditions ("partial" TTS or PTTS), which fill the three-dimensional regions, PTTR. We then show that PR, TTR and PTTR together with the corresponding anti-trapping regions constitute the complete set of regions filling the entire three-space (where timelike surfaces are defined) of Kerr-like spacetimes. This construction provides a novel optical description of such spacetimes without recurring to explicit solution of the geodesic equations. Applying this analysis to Kerr-Newman metrics (including the overspinning ones) we reveal four different optical types for different sets of the rotation and charge parameters. To illustrate their properties we extend Synge analysis of photon escape in the Schwarzschild metric to stationary spacetimes and construct density graphs describing escape of photons from all the above regions.

Spherical photon orbits in the field of Kerr naked singularities

For the Kerr naked singularity (KNS) spacetimes, we study properties of spherical photon orbits (SPOs) confined to constant Boyer-Lindquist radius r. Some new features of the SPOs are found, having no counterparts in the Kerr black hole (KBH) spacetimes, especially stable orbits that could be pure prograde/retrograde, or with turning point in the azimuthal direction. At r>1 (r<1) the covariant photon energy mathcal{E}> 0 (mathcal{E}< 0), at r=1 there is mathcal{E}= 0. All unstable orbits must have mathcal{E}> 0. It is shown that the polar SPOs can exist only in the spacetimes with dimensionless spin a < 1.7996. Existence of closed SPOs with vanishing total change of the azimuth is demonstrated. Classification of the KNS and KBH spacetimes in dependence on their dimensionless spin a is proposed, considering the properties of the SPOs. For selected types of the KNS spacetimes, typical SPOs are constructed, including the closed paths. It is shown that the stable SPOs intersect the equatorial plane in a region of stable circular orbits of test particles, depending on the spin a. Relevance of this intersection for the Keplerian accretion discs is outlined and observational effects are estimated.

Selected spherical photon orbits around a deformed Kerr black hole

In this paper, we investigate the so-called spherical photon orbits around a deformed Kerr black hole with an extra deformation parameter. The change in the azimuth Δφ and the angle of dragging of nodes per revolution ΔΩ of a complete latitudinal oscillated orbit is calculated analytically. Finally, representative six examples orbits are plotted out to illustrate how spherical photon orbits look like and exhibit some interesting behavior. Especially, these spherical photon orbits are absent in circular orbits and different from the Kerr case.

Spherical photon orbits around a 5D Myers–Perry black hole

We study the motion of bound null geodesics with fixed coordinate radius around a five-dimensional rotating black hole. These spherical photon orbits are not confined to a plane, and can exhibit interesting quasiperiodic behaviour. We provide necessary conditions for the existence of these orbits, and explicitly compute the radii of circular orbits in the equatorial and polar planes. Finally, we plot representative examples of some of the types of possible orbits, commenting on their qualitative features.

Null geodesics and observables around the Kerr–Sen black hole

We investigate the geodesic motion in the background of the Kerr–Sen black hole arising in heterotic string theory. The nature of the effective potential is discussed in the radial, as well as latitudinal, direction. A special class of spherical photon orbits is obtained along with an expression for the turning point for the radial photons. Dependence of photon motion within this class of solution is discussed explicitly in view of the different black hole parameters. We have discussed the allowed regions for the geodesic motion of massless test particles around the Kerr–Sen black hole in a more generalised way by considering non-equatorial motion of the photons into account. The conditions for different types of possible orbits are discussed with specific parameter values along with the corresponding orbit structure. No terminating orbits are possible for photons due to non-zero black hole charge. Observables on the angular plane (viz. bending of light and perihelion precession for massive test particles) are analysed as special cases. We have also calculated the rotation and mass parameters for the Kerr–Sen black hole in terms of the red/blue shifts of the photons in circular and equatorial orbits emitted by the massive test particles which represent stars or other probable sources of photons.

Photon motion in Kerr\u2013de\xa0Sitter\xa0 spacetimes

We study the general motion of photons in the Kerr–de Sitter black-hole and naked singularity spacetimes. The motion is governed by the impact parameters X, related to the axial symmetry of the spacetime, and q, related to its hidden symmetry. Appropriate ‘effective potentials’ governing the latitudinal and radial motion are introduced and their behavior is examined by the ‘Chinese boxes’ technique giving regions allowed for the motion in terms of the impact parameters. Restrictions on the impact parameters X and q are established in dependence on the spacetime parameters M, Lambda , a. The motion can be of orbital type (crossing the equatorial plane, q>0) and vortical type (tied above or below the equatorial plane, q<0). It is shown that for negative values of q, the reality conditions imposed on the latitudinal motion yield stronger constraints on the parameter X than that following from the reality condition of the radial motion, excluding the existence of vortical motion of constant radius. The properties of the spherical photon orbits of the orbital type are determined and used along with the properties of the effective potentials as criteria of classification of the KdS spacetimes according to the properties of the motion of the photon.

The quasi-normal modes of charged scalar fields in Kerr-Newman black hole and its geometric interpretation

It is well-known that there is a geometric correspondence between high-frequency quasi-normal modes (QNMs) and null geodesics (spherical photon orbits). In this paper, we generalize such correspondence to charged scalar field in Kerr-Newman space-time. In our case, the particle and black hole are all charged, so one should consider non-geodesic orbits. Using the WKB approximation, we find that the real part of quasi-normal frequency corresponds to the orbits frequency, the imaginary part of the frequency corresponds to the Lyapunov exponent of these orbits and the eigenvalue of angular equation corresponds to carter constant. From the properties of the imaginary part of quasi-normal frequency of charged massless scalar field, we can still find that the QNMs of charged massless scalar field possess the zero damping modes in extreme Kerr-Newman spacetime under certain condition which has been fixed in this paper.

Spherical null geodesics of rotating Kerr black holes

The non-equatorial spherical null geodesics of rotating Kerr black holes are studied analytically. Unlike the extensively studied equatorial circular orbits whose radii are known analytically, no closed-form formula exists in the literature for the radii of generic (non-equatorial) spherical geodesics. We provide here an approximate formula for the radii rph(a/M;cosi) of these spherical null geodesics, where a/M is the dimensionless angular momentum of the black hole and cosi is an effective inclination angle (with respect to the black-hole equatorial plane) of the orbit. It is well-known that the equatorial circular geodesics of the Kerr spacetime (the prograde and the retrograde orbits with cosi=±1) are characterized by a monotonic dependence of their radii rph(a/M;cosi=±1) on the dimensionless spin-parameter a/M of the black hole. We use here our novel analytical formula to reveal that this well-known property of the equatorial circular geodesics is actually not a generic property of the Kerr spacetime. In particular, we find that counter-rotating spherical null orbits in the range (33−59)/4≲cosi<0 are characterized by a non-monotonic dependence of rph(a/M;cosi=const) on the dimensionless rotation-parameter a/M of the black hole. Furthermore, it is shown that spherical photon orbits of rapidly-rotating black holes are characterized by a critical inclination angle, cosi=4/7, above which the coordinate radii of the orbits approach the black-hole radius in the extremal limit. We prove that this critical inclination angle signals a transition in the physical properties of the spherical null geodesics: in particular, it separates orbits which are characterized by finite proper distances to the black-hole horizon from orbits which are characterized by infinite proper distances to the horizon.

Quasinormal-mode spectrum of Kerr black holes and its geometric interpretation

There is a well-known, intuitive geometric correspondence between high-frequency quasinormal modes of Schwarzschild black holes and null geodesics that reside on the light ring (often called spherical photon orbits): the real part of the mode's frequency relates to the geodesic's orbital frequency, and the imaginary part of the frequency corresponds to the Lyapunov exponent of the orbit. For slowly rotating black holes, the quasinormal mode's real frequency is a linear combination of the orbit's precessional and orbital frequencies, but the correspondence is otherwise unchanged. In this paper, we find a relationship between the quasinormal-mode frequencies of Kerr black holes of arbitrary (astrophysical) spins and general spherical photon orbits, which is analogous to the relationship for slowly rotating holes. To derive this result, we first use the Wentzel-Kramers-Brillouin approximation to compute accurate algebraic expressions for large-$l$ quasinormal-mode frequencies. Comparing our Wentzel-Kramers-Brillouin calculation to the leading-order, geometric-optics approximation to scalar-wave propagation in the Kerr spacetime, we then draw a correspondence between the real parts of the parameters of a quasinormal mode and the conserved quantities of spherical photon orbits. At next-to-leading order in this comparison, we relate the imaginary parts of the quasinormal-mode parameters to coefficients that modify the amplitude of the scalar wave. With this correspondence, we find a geometric interpretation of two features of the quasinormal-mode spectrum of Kerr black holes: First, for Kerr holes rotating near the maximal rate, a large number of modes have nearly zero damping; we connect this characteristic to the fact that a large number of spherical photon orbits approach the horizon in this limit. Second, for black holes of any spins, the frequencies of specific sets of modes are degenerate; we find that this feature arises when the spherical photon orbits corresponding to these modes form closed (as opposed to ergodically winding) curves.