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In this paper we obtain the geodesic equations of motion of a test particle (charged particle and photon) in the Kerr-Newman de/anti de Sitter black hole by using the Hamilton-Jacobi equation. We determine the positions of the inner, outer and cosmological horizons of the black hole. In terms of the effective potentials, the trajectory of the test particle within the inner horizon is studied. It appears that there are stable circular orbits of a charged particle and photon within the inner horizon and that the combined effect of the charge and rotation of the Kerr-Newman de/anti de Sitter black hole and the coupling between the charge of the test particle and the electromagnetic field of the black hole may account for this.

The solutions of Einstein-Maxwell equations in the Kerr-Newman de/anti de Sitter space-time in the presence of the cosmological constant, _{a} supernova with high red shift parameter Z ≤ 1 in the framework of the inflationary cosmology [

According to Chandrasekhar [

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 [

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

where

with the normalizing condition,

Effecting the variation of the action, one obtains

where

limits as the end points are fixed. According to the principle of least action,

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

where

are the canonical momenta. Since H does not depend explicitly on

In the standard Boyer-Lindquist coordinates

where the functions

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

The corresponding nonzero contravariant components

The geometrical properties of the metric element given by Equation (9) can be analyzed by taking into account the limiting conditions:

finds

where

sitter space-time with the flipping of the sign of Λ. When

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

whose solution takes the form,

where

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

It may also be verified that the condition

The Kerr-Newman de/anti de Sitter metric (9) have singularities at

The four roots are:

in which

Three out of the four roots of Equation (27), have physical interpretations as follows:

For the negative cosmological constant

1)

2)

3)

For the positive cosmological constant

1)

2)

3)

In the Boyer-Lindquist coordinates, the stationary limit surfaces (SLS) of Kerr-Newman de/anti de Sitter black

hole are obtainable by setting the roots

By taking

in which

For each radial horizon defined by (27) there is an associated stationary limit surface (SLS) defined by (32). Both the hyper surfaces given by Equations (27) and (32) coincide at

We discuss the non rotating charged black hole:

and

As the roots of the Equations (37) and (38) we obtain the following the pairs of values for E and L:

where

and

The photon orbit is obtained in the case of the ultra relativistic limit for very massive particle energy

From (25) and (26), we find:

and

On solving for b=L/E, we get the following expression as roots of (41) and (42):

Using these values of b_{1} and b_{2} given by (43) and (44) back into (41) and (42) we get a pair of values for q:

The condition for stability

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.