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)