The solutions of Einstein-Maxwell equations in the Kerr-Newman de/anti de Sitter space-time in the presence of the cosmological constant, , give rise to the geometrical structure of a Kerr-Newman black hole and its singu- larity. The geometrical structure is asymptotically de sitter at large times when a repulsive cosmological constant, , is considered and may contain a cosmological horizon with a dynamic background and for an attractive cosmological constant., the geometrical structure, is asymptotically anti de Sitter and contains black hole horizons. The recent serious study of the observational and theoretical aspects of a small positive cosmological constant term which may be relevant at the present epoch [1] [2] and the accurate measurements of the anisotropy of the cosmic relic background radiation and the observational analysis of type I_a supernova with high red shift parameter Z ≤ 1 in the framework of the inflationary cosmology [3] -[6] , suggests that a repulsive cosmological constant Λ > 0 has to be taken seriously for understanding the properties of the presently observed universe. On the other hand, it is recognized that the de/anti de Sitter space-time has an important role in the multidimensional string theory [7] .

According to Chandrasekhar [8] , there exist two general types of particle orbits in the black hole gravitational field: orbits of the first kind, which are completely confined outside the black hole event horizon, and orbits of the second kind, which penetrate inside the black hole. In the case of rotating or charged black hole, there are bound periodic planetary orbits, known as the orbits of the third kind, which neither come out nor terminate at the central singularity [9] . The third kind of bound orbits inside the black hole horizon were discussed by [10] [11] for charged particle in the vicinity of the rotating charged black holes and by [12] for neutral particle.

The geometrical properties of the Kerr-Newman de/anti de sitter space-time with non zero cosmological constant are described by the geodesic equations of motion of a test particle. The motion of a test charged particle in the gravitational field of a charged black hole is fully described by three integrals of motion namely, E, the total particle energy, L_φ, the azimuthal component of the angular momentum and Q, the Carter constant [13] .

In Section 2, we review general geodesic orbits of test particle in the Kerr Newman de/anti de sitter space- time. In Section 3, we discuss the bound stable periodic orbits for a charged particle and photon inside the inner horizon. Section 4 is a brief conclusion. We use the units G = c = 1 throughout the paper.

The equations of motion of a test particle of mass m and charge in the gravitational field of a rotating charg- ed black hole in the Kerr-Newman de/anti Sitter space-time may be determined from the principle of least action with the action defined as [14] ,

where, being the proper time of the test charged particle along geodesics;

is the covariant four vector potential and a dot overhead a symbol denotes differentiation with respect to the parameter. From the action, the Lagrangian is identified as

with the normalizing condition,

Effecting the variation of the action, one obtains

where is the electromagnetic field tensor and. The integrated term vanishes at both

limits as the end points are fixed. According to the principle of least action, for the correct path of motion of the particle. This condition leads to the following equations of motion for the test charged particle in the given field:

The Hamiltonian corresponding to the Lagrangian (2) is found as

where

are the canonical momenta. Since H does not depend explicitly on, the Hamiltonian is a constant of motion. As such, by using the normalization condition, one finds

In the standard Boyer-Lindquist coordinates, the metric describing the Kerr-Newman de/anti de Sitter space-time takes the form,

where the functions and are defined respectively by

On the other hand, the electromagnetic field for the source is given by the required vector potential:

The corresponding nonzero contravariant components of the metric are obtained as

The geometrical properties of the metric element given by Equation (9) can be analyzed by taking into account the limiting conditions: in the metric. By imposing these limiting conditions, one

finds which is actually the flat space-time in the spheroi- dal coordinates under the coordinate transformation:;;

in the ranges of leading to the standard Cartesian form. In the plane, represents a ring disk of radius with properties different from the usual radial coordinate. If we consider the limit, the metric recovers the Reissner-Nordstrom de/anti de Sitter metric which, in standard spherical coordinates, is expressed as:

where. Under the limiting condition., we see the de/anti de

sitter space-time with the flipping of the sign of Λ. When we recover the Schwarzchild de/anti Sitter space-time.

Since the metric is stationary and axisymmetric, it is clear that there exists two Killing vector fields given by

. We find two constants of motion namely, the energy and angular momentum of the test particle corresponding to the dot product of the four-momentum with the Killing vector: and. By (6), the general form of the Hamilton-Jacobi equation is

whose solution takes the form,

where (15)

and

Here Q is the Carter constant. Using the action given in Equation (14), the following differential equations governing the motion of the test particle can be deduced:

where

The functions and serve as the effective potentials defining the motion of the test particle in and -directions [15] . Thus, the study of a test particle in the gravitational field of the Kerr-Newman de/anti de Sitter space-time is reduced to the study of motion of the test particle in the effective potentials and. For a circular orbit for which, the following conditions are satisfied at some radius r:

It may also be verified that the condition is satisfied showing that the circular orbit is stable.

We have shown that charged particles and photons may have stable periodic orbits inside the inner horizon of Kerr-Newman de/anti de Sitter black hole. The interaction between the charge of the particle and the electromagnetic field of the black hole may account for existence of stable circular orbits in the case of a charged parti- cle inside the inner horizon. However, in the case of a photon inside the inner horizon, its stable circular orbit may be due to the combined effect of the rotation and the charge of the black hole.

The authors would like to thank the Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune for providing hospitality and support during the preparation of the paper. Shanjit Heisnam is grateful to the University Grants Commission for providing him financial assistance in the form of UGC-BSR fellowship.