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

We study causal geodesics in the equatorial plane of the extremal Kerr-Taub-NUT spacetime, focusing on the Innermost Stable Circular Orbit (ISCO),and compare its behaviour with extant results for the ISCO in the extremal Kerr spacetime. Calculation of the radii of the direct ISCO, its Kepler frequency, and rotational velocity show that the ISCO coincides with the horizon in the exactly extremal situation. We also study geodesics in the strong {\it non}-extremal limit, i.e., in the limit of 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.

geodesics are able to avoid issues involving geodesic incompleteness exhibited in the spacetimes for other polar planes. Since there is a good deal of discussion of these issues elsewhere (see [2] for a competent review), we prefer to avoid them and concentrate instead on other issues of interest.
The paper is organized as follows : in section II we discuss the Kerr-Taub-NUT metric and its geodesics on the equatorial plane of precisely extremal spacetime. We discuss ISCOs and other astronomical observables for the null geodesics and timelike geodesics in the section III and IV, respectively. In each section of sections III and IV, there are two subsections in which we derive the radii of ISCOs and some other important astronomical observables (like L, E, v (φ) , Ω) for Taub-NUT and massless Taub-NUT spacetimes as the special cases of Kerr-Taub-NUT spacetimes. We end in section V with a summary and an outlook for future work.
At first, we want to study the geodesic motions in the equatorial plane of the Kerr- Taub where, From the above metric we can easily derive the velocity components (for θ = π 2 andθ = 0) of the massive test particle [17] where, We may set without loss of generality, k = −1 for timelike geodesics = 0 for nulllike geodesics (10) L is specific angular momentum or angular momentum per unit mass of the test particle and, E is to be interpreted as the specific energy or energy per unit mass of the test particle for the timelike geodesics k = −1.

III. ISCOS FOR THE NULL GEODESICS IN KERR-TAUB-NUT SPACETIMES
In this section, we derive the radii of ISCOs and other important astronomical quantities for null geodesics (k = 0) in the two spacetimes, one is Kerr-Taub-NUT spacetimes and another is Taub-NUT spacetimes.
As we have noted, k = 0 for null geodesics and the radial eqn. (7) becomeṡ It will be more convenient to distinguish the geodesics by the impact parameter rather than L.
We first observe the geodesics with the impact parameter Thus, in this case, equations (5), (6) and (11) reduce tȯ and, The radial co-ordinate is described uniformly with respect to the affine parameter while the equations governing t and φ are The solutions of these equations are These solutions exhibit the characteristic behaviors of t and φ of tending to ±∞ as the horizons at r + and r − are approached. The coordinate φ, like the coordinate t, is not a 'good' coordinate for describing what really happens with respect to a co-moving observer: a trajectory approaching the horizon (at r + or r − ) will spiral round the spacetime an infinite number of times even as it will take an infinite coordinate time t to cross the horizon and neither will be experience of the co-moving observer. The null geodesics which are described by eqn. (17), are the members of the principal null congruences and these are confined in the equatorial plane.
In general it is clear that there is a critical value of the impact parameter D = D c for which the geodesic equations allow an unstable circular orbit of radius r c . In case of D < D c , only one kind of orbits are possible: it will be arriving from infinity and cross both of the horizons and terminate into the singularity. For D > D c , we can get two types of orbits: (a) arriving from infinity, have perihelion distances greater than r c , terminate into the singularity at r = 0 and θ = π 2 (b) arriving from infinity, have aphelion distances less than r c , terminate into the singularity at r = 0 and θ = π 2 . For n = 0, we can recover the results in the case of Kerr geometry (see eqn.(77) of [14]). The equations determining the radius (r c ) of the stable circular 'photon orbit' are (eqn. 11) and Substituting D = L E in the above equations (20,21), we get from eqn. Letting and substituting it in eqn. (22) we get So, the critical value of the impact parameter D c in KTN spacetime is To get the value of r c we have to take eqn. (20) which reduces to Substituting the value of (neglecting the higher order terms of r c , we mean r c < n and r c < M ) in the eqn.(26) we get the equation to determine the radius (r c ) of the stable circular photon orbit: where the lower index 'M' of a particular parameter represents that the parameter is divided by 'M' (such as, a M = a M ) For r c < n we get the solution of r c as where the values of U, V, W are given in the footnote 1 . Here, the expression of r cM is the radius of the ISCO as it is the smallest positive real root of the ISCO equation(28).
Extremal case: We know that in KTN spacetime, the horizons are at where r + and r − define the event horizon and Cauchy horizon, respectively. In case of extremal KTN spacetime, So, the horizon of extremal KTN spacetime is at Now, we can substitute the value of eqn. (22) in eqn. (20) (also take a 2 = M 2 + n 2 ) and get the equation for determining the radius (r c ) of the stable circular photon orbit is Interestingly, the solution of this sixth order non-trivial equation is This is the radius of the ISCO as it is the smallest positive real root of the ISCO equation(33). It means that the radius of the ISCO (r c ) coincides with the horizon in extremal KTN spacetime for null geodesics. This is the same thing which happens in extremal Kerr spacetime also. In extremal Kerr spacetime, the direct ISCO coincides with the horizon at r = M for null geodesic. Now, we can calculate the critical value of the impact parameter in extremal KTN spacetime: The physical significance of the impact parameter D c and the ISCO have already discussed in detail. Now, we can give our attention to the Taub-NUT spacetime which is quite interesting.

A. ISCOs in Taub-NUT spacetimes
The Taub-NUT spacetime is a stationary and spherically symmetric vacuum solution of Einstein equation [18]. Now, we are discussing a very special case in where we set a = 0, the angular momentum of the spacetime vanishes. We note that for a = 0, the primary metric (eqn. 1) of Kerr-Taub-NUT spacetime reduce to the Taub-NUT metric in which the constant set as C = 0. If we take the Taub-NUT metric in more general form, it would be where, C is an arbitrary real constant. If we take C = 0 for the above metric, the form of Taub-NUT metric is same as the metric (1) of 'Kerr-Taub-NUT spacetimes with a = 0'. Physically, C = 0 leads to the only possibility for the NUT solutions to have a finite total angular momentum [19]. After noting the above points, we write the line element of the Taub-NUT spacetimes at the equator as To determine the radius (r cn ) of the stable circular photon orbit, we put a = 0 in eqn.(33) and get the following Solving the above equation we get, or, This is the radius of the ISCO as it is the smallest positive real root of the ISCO equation(37). For n M = n M = 0, we can recover r c = 3M of Schwarzschild spacetime. We know that the position of event horizon in Taub-NUT spacetime is: (40) or, It seems that the position of the ISCO (r cn ) could be in timelike, nulllike or spacelike surfaces depending on the value of NUT charge (n M ) in Taub  The NUT spacetime, for the mass M = 0 is also well-defined (see, for example, appendix of [20]). So, we can also determine the radius (r 0cn ) of the stable circular photon orbit for massless Taub-NUT spacetimes. This turns out to be We note that the horizons of massless Taub-NUT spacetime are at r = ±n [15]. So, the ISCO of this spacetime for null geodesic is always outside the horizon, we mean in the timelike surface. Here, it should also be highlighted that in the case of Taub-NUT spacetime the horizons are at as a = 0. We can see that one horizon is located at r + > 2M and the other at −|n| < r − < 0. Thus there are mainly two important features of the Taub-NUT and massless Taub-NUT spacetime, one is that the second horizon is at a negative value of the radial Schwarzschild coordinate and another is that the both of the spacetimes possess the quasi-regular singularity [21]. But, the presence of the Kerr parameter (a) leads to the naked singularity in non-extremal KTN spacetime as well as non-extremal Kerr spacetime. In these cases, both the horizons are located at imaginary distances. This means that we can observe the singularity from the infinity. So, this is called as a naked singularity. But, in the Taub-NUT spacetime the absence of Kerr parameter never leads to any of the horizons at imaginary distance. So, there could not be any naked singularity in Taub-NUT and massless Taub-NUT spacetime. The absence of the Kerr parameter also leads to the non-existence of any extremal case for Taub-NUT spacetime as the horizons of Taub-NUT spacetime cannot act as the actual horizons like the horizons of our other well-known spacetimes (such as Kerr, Reissner-Nordström geometry). In the extremal KTN spacetime, the horizon is located at r = M which is same as the Kerr spacetime. We note that the NUT charge has no any effect to determine the ISCO in extremal KTN spacetime as the Kerr parameter takes it highest value a = √ M 2 + n 2 but the ISCO radius (eqn.29) in non-extremal KTN spacetime depends on the NUT charge.

IV. ISCOS FOR THE TIMELIKE GEODESICS IN KERR-TAUB-NUT SPACETIMES
For timelike geodesics, equations (5) and (6) forφ andṫ remain unchanged; but equation (7) is reduced to The circular and associated orbits: We now turn to consideration of the radial eqn.(44) in general. With the reciprocal radius u(= 1 r ) as the independent variable and substituting L − aE = x, the equation takes the form Like the Schwarzschild, Reissner-Nordström and Kerr geometries, the circular orbits play an important role in the classification of the orbits. Besides, they are useful for some special features of the spacetimes, after all; the reason for studying the geodesics. When the values of L and E are completely arbitrary, the quartic polynomial of right hand side of eqn.(45) will have a triple root. The conditions for the occurrence of the triple root from the above eqn.(45) we get, and Equations (46) and (47) can be combined to give and By eliminating E between these equations we get the following quadratic eqn. for x The discriminant " 1 4 (b 2 − 4ac)" of this equation is where, The solution of the eqn(50) is where, We verify that, So, the solution of the x takes the simple form, It will appear presently that the upper sign in the foregoing eqn. applies to the retrograde orbits, while the lower sign applies to the direct orbits. We adhere to this convention in this whole section. Substituting the value of x in eqn(48), we get and the value of L which is associated with E is As the manner of derivation makes it explicit, E and L given by eqn. (58) and (59) are Energy and Angular momentum per unit mass, of a particle describing a circular orbit of reciprocal radius u. The angular velocity Ω follows from the equation Substituting the values of x, L and E in the above equation we get the angular velocity of a chargeless massive test particle which is moving in a particular orbit of radius R(= 1 u ) in Kerr-Taub-NUT spacetime : Generally, Ω KT N is also called as the Kepler frequency in KTN spacetime. The Time period (T ) of a massive chargeless test particle which is rotating in a orbit of radius R(= 1 u ) can also be determined from the Kepler frequency by the simple relation between Ω and T : For n = 0, we can easily get the Kepler frequency in Kerr spacetime: It is already noted that the upper sign is applicable for the retrograde orbits and the lower sign is applicable for the direct orbits. The Rotational velocity v (φ) of a chargeless massive test particle could be determined by the following equation: where, ψ, ν and ω is defined from the general axisymmetric metric [14] ds 2 = −e 2ν (dt) 2 + e 2ψ (dφ − ωdt) 2 + e 2µ2 (dx 2 ) 2 + e 2µ3 (dx 3 ) 2 (65) In the above metric ψ, ν, ω, µ 2 and µ 3 are the functions of x 2 and x 3 . In our case (for Kerr- Taub-NUT  spacetimes), Now, substituting the above values in the equation(64), we get the Rotational velocity of a chargeless massive test particle which is moving in a particular orbit of radius R(= 1 u ) in Kerr-Taub-NUT spacetime : Here, a point, considered as describing a circular orbit (with the proper circumference πe ψ ) with an angular velocity Ω (also called Kepler frequency) in the chosen coordinate frame, will be assigned an angular velocity, in the local inertial frame. Accordingly, a point which is considered as at rest in the local inertial frame (i.e., velocity components u (1) = u (2) = u (3) = 0), will be assigned an angular velocity ω in the coordinate frame. On this account the non-vanishing of ω is said to describe as 'dragging of inertial frame'. In weak gravity regime where J is the angular momentum of the spacetime. For exact calculation of ω in strong gravity regime, see Chakraborty and Majumdar [15].
We note that the Effective potential expression of a massive test particle plays many interesting roles in Gravitational Physics and also in Astrophysics. Here, we need this potential at this moment to determine the radius of the ISCO in KTN spacetime. We know that where the effective potential governing the radial motion is For n = 0, the effective potential reduces to eqn. (15.20) of [4]. This is applicable to Kerr geometry. An important difference is that the potentials are energy and angular momentum dependent in stationary spacetimes, i.e., Kerr, Kerr-Taub-NUT etc. This is not happened in the static spacetimes, i.e, Schwarzschild, Reissner-Nordström etc. In the static spacetimes the Effective potential is only depends on radial coordinate r, not depends on L and E. This difference is arisen due to the involvement of 'rotational motion' in stationary spacetimes. For example, particles that fall from infinity rotating in the same direction as the spacetime (positive values of L) move in a different effective potential than initially counter-rotating particles (negative values of L). These differences reflect, in part, the rotational frame dragging of the spinning spacetime and the test particles are dragged by its rotation. Many interesting properties of the orbits of particles in the equatorial plane could be explored with the radial equation(71) and the equations of the other components (which are already discussed) of the four velocity. We could calculate the radii of circular orbits, the shape of bound orbits etc. These are all different, depending upon whether the particle is rotating with the black hole (corotating) or in the opposite direction (counterrotating). For instance, in the geometry of an extremal Kerr black hole (a = M ), there is a corotating stable circular particle orbit at r = M (direct ISCO) and a counterrotating stable circular orbit at r = 9M (retrograde ISCO). However, as we already mentioned, for an introductory discussion it seems appropriate not to all these interesting properties but rather to focus on the one property which is most important for Astrophysics, mainly for Accretion mechanism--the binding energy of the innermost stable circular particle orbit (ISCO).
For a particle to describe a circular orbit at radius r = R, its initial radial velocity must vanish. Imposing this condition we get from eqn.(70) To stay in a circular orbit the radial acceleration must also vanish. Thus, differentiating eqn.(70) with respect to r leads to the condition: Stable orbits are ones for which small radial displacements away from R oscillate about it rather than accelerate away from it. Just as in newtonian mechanics, that is the condition that the effective potential must be a minimum: Equations (72-74) determine the ranges of E, L, R allowed for stable circular orbits in the KTN spacetime. At the ISCO, the one just on the verge of being unstable-(eqn.74) becomes an equality. The last three equations is solved to obtain the values of E, L, R = r ISCO that characterized the orbit. Equations(72) and (73) have already been solved for KTN spacetime at the very beginning of this section as the equations (46) and (47) to obtain the exact expressions of E and L. Now, we obtain the following from eqn.(74) Substituting the values of Ex and x 2 in terms of r = R in the above equation we get, This is the equation to obtain the radius of the ISCO in non-extremal KTN spacetime for timelike geodesic. Solving the above equation, we can determine the radius of inner-most stable circular orbit in the non-extremal Kerr-Taub-NUT spacetime. But, this equation cannot be solved anlytically as it is actually twelfth order equation. So, we determine the radius of ISCO in extremal KTN spacetime. For that we can substitute Interestingly, the solution of the above non-trivial equation is This is the radius of the ISCO as it is the smallest positive real root of the ISCO equation(79). So, in extremal KTN spacetime, the direct ISCO radius does not depend on the value of NUT charge. ISCO radius is completely determined by the ADM mass of the spacetime. We note that the horizon in extremal KTN spacetime is at R = M . So, we can say that R = M , the direct ISCO is on the horizon in extremal KTN spacetime. It is also further observed that the timelike circular geodesics and null circular geodesics coalesce into a single zero energy trajectory (both are on the R = M ) Thus, the geodesic on the horizon must coincide with the principal null geodesic generator. The existence of a timelike circular orbit turning into the null geodesic generator on the event horizon (nulllike) is a peculiar feature of extremal KTN spacetime. It is not only that the energy (E = 0) vanishes on the ISCO in KTN spacetime but angular momentum (L = 0) also vanishes on that. The same thing is also happened in case of Kerr geometry. Substituting n = 0, we can recover the expressions of Ω, v (φ) in Kerr spacetimes. These are already described by Chandrasekhar [14] in details. The ISCO equation in Kerr spacetime is For extremal Kerr spacetime we can put a = M in the above equation and solving the above equation we get, A. ISCOs in Taub-NUT spacetimes To get various useful expressions in Taub-NUT spacetimes, we should first take a = 0. We don't want to reiterate the whole process for Taub-NUT spacetimes. This is the same as in the previous section in which we have done things in detail for KTN spacetimes. The Energy per unit mass (E T N ) and the Angular momentum per unit mass (L T N ) of a chargeless massive test particle which is moving in a particular orbit of radius r = 1 u in Taub where, Now, we can find the angular velocity Ω T N of a chargeless massive test particle which is moving in a particular orbit of radius r = 1 u at Taub-NUT spacetime: For n = 0, we can recover the well-known Kepler frequency for non-rotating star whose geometry is described by Schwarzschild metric. The Kepler frequency which is a very useful parameter in Relativistic Astrophysics, is defined as (substituting n = 0 in eqn. (87)) where, M is the mass of the star, R is the distance of the satellite from the centre of the star and Ω Kep is the uniform angular velocity of the satellite, moving in a circular orbit of radius R around the star. The Rotational velocity v (φ) T N of a chargeless massive test particle which is moving in a particular orbit of radius r = 1 u at Taub-NUT spacetime: