Inner-most stable circular orbits in extremal and non-extremal Kerr–Taub-NUT spacetimes

The European Physical Journal C, Feb 2014

We study causal geodesics in the equatorial plane of the extremal Kerr–Taub-NUT spacetime, focusing on the inner-most stable circular orbit (ISCO), and we compare its behavior with extant results for the ISCO in the extremal Kerr spacetime. Calculations of the radii of the direct ISCO, its Kepler frequency, and the rotational velocity show that the ISCO coincides with the horizon in the exactly extremal situation. We also study geodesics in the strong non-extremal limit, i.e., in the limit of a vanishing Kerr parameter (i.e., for Taub-NUT and massless Taub-NUT spacetimes as special cases of this spacetime). It is shown that the radius of the direct ISCO increases with NUT charge in Taub-NUT spacetime. As a corollary, it is shown that there is no stable circular orbit in massless NUT spacetimes for timelike geodesics.

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Inner-most stable circular orbits in extremal and non-extremal Kerr–Taub-NUT spacetimes

Chandrachur Chakraborty 0 0 Saha Institute of Nuclear Physics , Kolkata 700064, India We study causal geodesics in the equatorial plane of the extremal Kerr-Taub-NUT spacetime, focusing on the inner-most stable circular orbit (ISCO), and we compare its behavior with extant results for the ISCO in the extremal Kerr spacetime. Calculations of the radii of the direct ISCO, its Kepler frequency, and the rotational velocity show that the ISCO coincides with the horizon in the exactly extremal situation. We also study geodesics in the strong non-extremal limit, i.e., in the limit of a vanishing Kerr parameter (i.e., for Taub-NUT and massless Taub-NUT spacetimes as special cases of this spacetime). It is shown that the radius of the direct ISCO increases with NUT charge in Taub-NUT spacetime. As a corollary, it is shown that there is no stable circular orbit in massless NUT spacetimes for timelike geodesics. - It is perhaps Lynden-Bell and Nouri-Zonoz [1] who are the first to motivate investigation on the observational possibilities for (gravito)magnetic monopoles. It has been claimed that signatures of such spacetimes might be found in the spectra of supernovae, quasars, or active galactic nuclei. The authors of [2] have recently brought this into focus, by a careful and detailed analysis of geodesics in such spacetimes. Note that (gravito)magnetic monopole spacetimes with angular momentum admit relativistic thin accretion disks of a black hole in a galaxy or quasars [3]. This provides a strong motivation for studying geodesics in such spacetimes because they will affect accretion in such spacetimes from massive stars, and might offer novel observational prospects. We know that the marginally stable orbit [also called Inner-most stable circular orbit (ISCO)] plays an important role in the accretion disk theory. That fact is important for spectral analysis of X-ray sources [4,5]. The circular orbits with r > rISCO turn out to be stable, while those with r < rISCO are not. Basically, accretion flows of almost free matter (stresses are insignificant in comparison with gravity or centrifugal effects) resemble almost circular motion for r > rISCO, and almost radial free-fall for r < rISCO. In the case of thin disks, this transition in the character of the flow is expected to produce an effective inner truncation radius in the disk. The exceptional stability of the inner radius of the X-ray binary LMC X-3 [6] provides considerable evidence for such a connection and, hence, for the existence of the ISCO. The transition of the flow at the ISCO may also show up in the observed variability pattern, if variability is modulated by the orbital motion [5]. One may expect that the there will be no variability observed with frequencies > ISCO, i.e., higher than the Keplerian orbital frequency at ISCO, or that the quality factor for variability, Q will significantly drop at ISCO. Several variants of this idea have been discussed in the following references [7,8]. The Taub-NUT geometry [9,10] possesses gravitomagnetic monopoles. Basically, this spacetime is a stationary and spherically symmetric vacuum solution of Einstein equation. As already mentioned, the authors of Ref. [2] have made a complete classification of geodesics in Taub-NUT spacetimes and describe elaborately the full set of orbits for massive test particles. However, there is no specific discussion on the various innermost stable orbits in such spacetimes for null as well as timelike geodesics. This is the gap in the literature which we wish to fill in this paper. Our focus here is the three parameter Taub-NUT version of the Kerr spacetime which has angular momentum, mass, and the NUT parameter (n, the gravitomagnetic monopole strength), and which is a stationary, axisymmetric vacuum solution of the Einstein equation. The geodesics and the orbits of the charged particles in KerrTaub-NUT (KTN) spacetimes have also been discussed by Miller [11]. Abdujabbarov et al. [12] discuss some aspects of these geodesics in KTN spacetime, although the black hole solution remains a bit in doubt. Liu et al. [3] have also obtained the geodesic equations but there are no discussions of the ISCOs in KTN spacetime. We know that ISCO plays many important roles in astrophysics as well as in gravitational physics; hence the strong physical motivation to study them. The presence of a NUT parameter lends the Taub-NUT spacetime a peculiar character and renders the NUT charge into a quasi-topological parameter. For example in the case of maximally rotating Kerr spacetime (extremal Kerr) where we can see that the timelike circular geodesics and null circular geodesics coalesce into a zero energy trajectory. This result and also the geodesics of the extremal Kerr spacetime have been elaborately described in Ref. [13]. They show that the ISCOs of the extremal Kerr spacetime for null geodesics and timelike geodesics coincide on the horizon (at r = M ) which means that the geodesic on the (...truncated)


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Chandrachur Chakraborty. Inner-most stable circular orbits in extremal and non-extremal Kerr–Taub-NUT spacetimes, The European Physical Journal C, 2014, pp. 2759, Volume 74, Issue 2, DOI: 10.1140/epjc/s10052-014-2759-9