General Relativistic Study of Thin Disk with Radial Flow in Kerr – Newman Geometry

A detailed general relativistic formulation of the thin disk with the radial flow in Kerr – Newman geometry. The effects of rotation through an angular momentum and charge were obtained on pressure, velocity and density for co rotating and counter rotating thin disk.


Introduction
In the study of the charged rotating black hole, the formation of thin disk is one of the most important aspects. We have developed fully general relativistic approach for the dynamics of the thin accretion disk having angular momentum and charge in the Kerr -Newman back ground geometry. When the pressure force is negligible, the disk become a thin, so that confined to the equatorial plane. A solution for thin disk rotating around Kerr -Newman black hole having non zero velocity in the radial direction is obtained. The effect of rotating charge black hole clearly seen by comparing solution with their Kerr and Schwarzschild counterpart.

Mathematical Formulation
The Kerr -Newman (KN) space-time is an exact solution of the Einstein-Maxwell equations that describes the exterior gravitational and electromagnetic field of a rotating charged source with mass , and angular momentum and electric charge . In Boyer -Lindquist coordinate, The KN line element can be written as (Puglies, Daniela, et al. 2013).
Using geometrized units with %& 1 ', and The parameter stands for the angular momentum per unit mass, as measured by a distant observer. The limiting case of the KN metric is the Kerr metric (Kerr. R P, 1963) for 0, The Schwarzschild metric which is recovered for 0, the Reissner-Nordström (RN) space time for 0, and the Minkowski metric for special relativity for 0. The KN spacetime is asymptotically flat and free of curvature singularities outside the region situated very close to the origin of coordinates (Puglies, Daniela, et al. 2013). In particular, the function ∆ vanishes at the radii Become real only if the condition 0 is satisfied. In this case # 1 and # represents the radii of the outer and inner horizon, respectively and the KN solution is interpreted as describing the exterior field of a rotating black hole. In the case 2 , no zero of ∆ exits and the gravitational field correspond to that of a ring singularities situated at (Lynden-Bell. 2004). # * 0, Then for positive in the region # * 2 # < 0 or # 3 < # < # 1 3 with In particular, for a black hole it is # 1 < # < # + a region which is known as the ergosphere where − − * ≥ 0 is satisfied.

Hydrodynamical Equation for Kerr Newman Background Geometry
The equation governing the motion of perfect fluid is described by law of conservation of the energy-momentum tensor Where C is the pressure, Dis the energy density including rest mass energy and E is the four velocity satisfying the normalization relationship E E = 1.

Momentum equation for
Continuity equation The equation of momentum and continuity in terms of 3-velocity I J (Prasanna, AR, 1982 ).

Dynamics of the Disk with Radial Flow in Locally Non-rotating Frame (L. N. R. F)
In LNRF Physical phenomena becomesmore transparent because it cancels out the effect of frame dragging of black hole rotation as observeris chosen to rotate with the black hole. The observer whose world line are# =constant, =constant and = " + * Where " = − With this geometry, introducing a locally nonrotating frame (Kinnersley. W, 1969).

Thin Disk
Confining to the case of cold disk with C = 0 and =¯ for negligible pressure the adiabatic equation and baryon conservation equation become identically zero and left with equation three only (Mishra, K N. and Chakraborty. D K, 1999). Rewriting equations in dimensionless form, In the above ´ and D @ are two constants, for boundary condition V % ' = 0 at # = ∞ and C = 0 at =¯ plane.

Conclusion
The solutions D, V % ' and V %=' carry the signature of the rotation of the charge black hole, the profile for V % ' , V %=' and D for ¸= −0.4. −0.2, 0. +0.2, and +0.4 the velocity V % ' decreases (increases) when is positive (negative ), The density D increase (decrease) while velocity V %=' decrease (increase) for prograde (retrograde) motions as compared in the Schwarzchild case. The motion of the disk increase (decrease) for prograde (retrograde) depending on radial distance ·. In Kerr geometry the solution show the effect of the rotation of the black hole through the term ¸, as changing ¸ to -¸ disks move from co rotating to counter rotating (Mishra, K N. and Chakraborty, D K. 1999 ). In the special case of Schwarzschild geometry.