Marginally Bound Circular Orbit Research Articles

Null geodesics and QNMs in the field of regular black holes

In this paper, we analyze the null geodesics of regular black holes (BHs). A detailed analysis of geodesic structure, both null geodesics and timelike geodesics, has been investigated for the said BH. As an application of null geodesics, we calculate the radius of photon sphere and gravitational bending of light. We also study the shadow of the BH spacetime. Moreover, we determine the relation between radius of photon sphere [Formula: see text] and the shadow observed by a distance observer. Furthermore, we discuss the effect of various parameters on the radius of shadow [Formula: see text]. Also, we compute the angle of deflection for the photons as a physical application of null-circular geodesics. We find the relation between null geodesics and quasinormal mode (QNM) frequency in the eikonal approximation by computing the Lyapunov exponent. It is also shown that (in the eikonal limit) the QNMs of BHs are governed by the parameter of null-circular geodesics. The real part of QNMs frequency determines the angular frequency, whereas the imaginary part determines the instability timescale of the circular orbit. Next, we study the massless scalar perturbations and analyze the effective potential graphically. Massive scalar perturbations are also discussed. As an application of timelike geodesics, we compute the innermost stable circular orbit (ISCO) and marginally bound circular orbit (MBCO) of the regular BHs which are closely related to the BH accretion disk theory. In the appendix, we calculate the relation between angular frequency and Lyapunov exponent for null-circular geodesics.

Lyapunov exponent, ISCO and Kolmogorov\u2013Senai entropy for Kerr\u2013Kiselev black hole

Geodesic motion has significant characteristics of space-time. We calculate the principle Lyapunov exponent (LE), which is the inverse of the instability timescale associated with this geodesics and Kolmogorov–Senai (KS) entropy for our rotating Kerr–Kiselev (KK) black hole. We have investigate the existence of stable/unstable equatorial circular orbits via LE and KS entropy for time-like and null circular geodesics. We have shown that both LE and KS entropy can be written in terms of the radial equation of innermost stable circular orbit (ISCO) for time-like circular orbit. Also, we computed the equation marginally bound circular orbit, which gives the radius (smallest real root) of marginally bound circular orbit (MBCO). We found that the null circular geodesics has larger angular frequency than time-like circular geodesics (Q_o > Q_{sigma }). Thus, null-circular geodesics provides the fastest way to circulate KK black holes. Further, it is also to be noted that null circular geodesics has shortest orbital period (T_{photon}< T_{ISCO}) among the all possible circular geodesics. Even null circular geodesics traverses fastest than any stable time-like circular geodesics other than the ISCO.

Circular Geodesics, Paczyński-Witta Potential and QNMs in the Eikonal Limit for Ayón-Beato-García Black Hole

We investigate the comprehensive geodesic structure of a spherically symmetric, static charged regular Ayón-Beato and García black hole (BH). We derive the equation of innermost stable circular orbit (ISCO), marginally bound circular orbit (MBCO) and circular photon orbit (CPO) of said BH, which are most relevant to BH accretion disk theory. Using time-like geodesic properties, we derive Paczyński-Witta potential form for this BH which are very relevant to determine the general relativistic effects on the accretion disk. We show that at a certain radius (For example in case of Reissner-Nordstrøm (RN) BH, r ∗ = Q 2 M ), there exists zero angular momentum (ZAM) orbits due to the repulsive gravity. We also show that in the eikonal approximation, the real and imaginary parts of the quasi normal modes (QNM) of the regular BHs can be expressed as in terms of the frequency of the BH and the instability time scale of the unstable null circular geodesics. Moreover, we study the Bañados, Silk and West effect for this BH. We show that the center-of-mass (CM) energy of colliding neutral test particles near the infinite red-shift surface of the regular BHs have the finite energy. In the Appendix section, we have discussed the possibility of a regular ABG BH can act as particle accelerators when two charged test particles of different energies are colliding and approaching to the horizon of the BH provided that one of charged test particle has a critical value of charge.

Circular geodesics in tidal-charged black hole

We study the existence and stability criteria for circular geodesics of spherically symmetric tidal-charged black hole (BH). We investigate in details the equatorial causal geodesics of the tidal-charged BH in comparison with spherically symmetric Reissner–Nordström BH spacetime. We particularly focused on both the null circular geodesics and time-like circular geodesics. Using the effective potential diagram, we have compared the geodesic structure between two spacetimes. We have derived the ISCO (innermost stable circular orbit), MBCO (marginally bound circular orbit) and CPO (circular photon orbit) for both the spacetimes. Moreover, we have derived the quasi-normal modes[Formula: see text]QNMs[Formula: see text] frequency in the eikonal limit for both the spacetimes via Lyapunov exponent. In the appendix section, we have shown that a spherically symmetric tidal-charged BH can act as particle accelerators with ultra-high center-of-mass (CM) energy in the limiting case of maximal BH tidal-charge[Formula: see text] and it is possible when two neutral particles are colliding near the horizon.

Circular orbits in the Taub–NUT and massless Taub–NUT spacetime

In this work, we study the equatorial causal geodesics of the Taub–NUT (TN) spacetime in comparison with massless TN spacetime. We emphasized both on the null circular geodesics and time-like circular geodesics. From the effective potential diagram of null and time-like geodesics, we differentiate the geodesics structure between TN spacetime and massless TN spacetime. It has been shown that there is a key role of the NUT parameter to changes the shape of pattern of the potential well in the NUT spacetime in comparison with massless NUT spacetime. We compared the innermost stable circular orbit (ISCO), marginally bound circular orbit (MBCO) and circular photon orbit (CPO) of the said spacetime with graphically in comparison with massless cases. Moreover, we compute the radius of ISCO, MBCO and CPO for extreme TN black hole (BH). Interestingly, we show that these three radii coincides with the Killing horizon, i.e. the null geodesic generators of the horizon. Finally in Appendix A, we compute the center-of-mass (CM) energy for TN BH and massless TN BH. We show that in both cases, the CM energy is finite. For extreme NUT BH, we found that the diverging nature of CM energy. First, we have observed that a non-asymptotic flat, spherically symmetric and stationary extreme BH showing such feature.

Stability analysis and quasinormal modes of Reissner–Nordstrøm space-time via Lyapunov exponent

We explicitly derive the proper-time (τ) principal Lyapunov exponent (λ p ) and coordinate-time (t) principal Lyapunov exponent (λ c ) for Reissner–Nordstrøm (RN) black hole (BH). We also compute their ratio. For RN space-time, it is shown that the ratio is $({\lambda _{p}}/{\lambda _{c}})={r_{0}}/{\sqrt {{r_{0}^{2}}-3Mr_{0}+2Q^{2}}}$ for time-like circular geodesics and for Schwarzschild BH, it is $({\lambda _{p}}/{\lambda _{c}})={\sqrt {r_{0}}}/{\sqrt {r_{0}-3M}}$ . We further show that their ratio λ p /λ c may vary from orbit to orbit. For instance, for Schwarzschild BH at the innermost stable circular orbit (ISCO), the ratio is $({\lambda _{p}}/{\lambda _{c}})|_{r_{\text {ISCO}}=6M}=\sqrt {2}$ and at marginally bound circular orbit (MBCO) the ratio is calculated to be $({\lambda _{p}}/{\lambda _{c}})|_{r_{\mathrm {m}\mathrm {b}}=4M}=2$ . Similarly, for extremal RN BH, the ratio at ISCO is $({\lambda _{p}}/{\lambda _{c}})|_{r_{\text {ISCO}}=4M}={2\sqrt {2}}/{\sqrt {3}}$ . We also further analyse the geodesic stability via this exponent. By evaluating the Lyapunov exponent, it is shown that in the eikonal limit, the real and imaginary parts of the quasinormal modes of RN BH is given by the frequency and instability time-scale of the unstable null circular geodesics.

Horizon straddling ISCOs in spherically symmetric string black holes

The causal geodesics in the equatorial plane of a static extremal charged black holes in heterotic string theory is examined with regard to their geodesic stability and compared with similar geodesics in the nonextremal situation. Extremization of the effective potential for timelike and null circular geodesics implies that in the extremal limit, the radius of ISCO (Inner-most Stable Circular Orbit) (r ISCO ), circular photon orbit (CPO) (r ph ) and marginally bound circular orbit (MBCO) (r mb ) are coincident with the event horizon (r hor ) i.e. r ISCO = r ph = r mb = r hor = 2M. Since the proper radial distance on a constant time slice both in Schwarzschild and Painlevé–Gullstrand (PG) coordinates become zero, thus these three orbits indeed coincide with the null geodesic generators of the event horizon. This strange behavior is quite different from the static, spherically symmetric extremal Reissner–Nordstrøm (RN) BH.

Charged dilation black holes as particle accelerators

We examine the possibility of arbitrarily high energy in the center-of-mass (CM) frame of colliding neutral particles in the vicinity of the horizon of a charged dilation black hole (BH). We show that it is possible to achieve the infinite energy in the background of the dilation black hole without fine-tuning of the angular momentum parameter. It is found that the CM energy (Ecm) of collisions of particles near the infinite red-shift surface of the extreme dilation BHs are arbitrarily large while the non-extreme charged dilation BHs have the finite energy. We have also compared the Ecm at the horizon with the ISCO (Innermost Stable Circular Orbit) and MBCO (Marginally Bound Circular Orbit) for extremal Reissner–Nordstrøm (RN) BH and Schwarzschild BH. We find that for extreme RN BH the inequality becomes Ecm|r+>Ecm|rmb>Ecm|rISCO i.e. Ecm|r+=M:Ecmrmb=3+52M:Ecm|rISCO=4M=∞:3.23:2.6. While for Schwarzschild BH the ratio of CM energy is Ecm|r+=2M:Ecm|rmb=4M:Ecm|rISCO=6M=5:2:133. Also for Gibbons–Maeda–Garfinkle–Horowitz–Strominger (GMGHS) BHs the ratio is being Ecm|r+=2M:Ecm|rmb=2M:Ecm|rISCO=2M=∞:∞:∞.