Asghar Qadir

Extended hypergeometric and confluent hypergeometric functions

An extension of the beta function by introducing an extra parameter, which proved to be useful earlier, is applied here to extend the hypergeometric and confluent hypergeometric functions. Since the latter functions contain many of the familiar special functions as sub-cases, this extension is expected to prove to be useful. The object of the present paper is to study this extension and its relationship with the hypergeometric and confluent hypergeometric functions.

Extension of Euler's beta function

An extension of Eulers beta function, analogous to the recent generalization of Eulers gamma function and Riemanns zeta function, for which the usual properties and representation are naturally and simply extended, is introduced. It is proved that the extension is connected to the Macdonald, error and Whittaker functions. In addition, the extended beta distribution is introduced.

General formula for the momentum imparted to test particles in arbitrary spacetimes

Abstract Ehlers and Kundt have provided an approximate procedure to demonstrate that gravitational waves impart momentum to test particles. This was extended to cylindrical gravitational waves by Weber and Wheeler. Here a general, exact, formula for the momentum imparted to test particles in arbitrary spacetimes is presented.

Charged black holes in phantom cosmology

In the classical relativistic regime, the accretion of phantom-like dark energy onto a stationary black hole reduces the mass of the black hole. We have investigated the accretion of phantom energy onto a stationary charged black hole and have determined the condition under which this accretion is possible. This condition restricts the mass-to-charge ratio in a narrow range. This condition also challenges the validity of the cosmic-censorship conjecture since a naked singularity is eventually produced due to accretion of phantom energy onto black hole.

Ricci collineations of static spherically symmetric spacetimes

The Ricci collineations of static spherically symmetric spacetimes are classified and their relationship with isometries is discussed. A general theorem about this relationship is stated and its extension to all spherically symmetric spacetimes is discussed.

INVARIANT LINEARIZATION CRITERIA FOR SYSTEMS OF CUBICALLY NONLINEAR SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS

Invariant linearization criteria for square systems of second-order quadratically nonlinear ordinary differential equations (ODEs) that can be represented as geodesic equations are extended to square systems of ODEs cubically nonlinear in the first derivatives. It is shown that there are two branches for the linearization problem via point transformations for an arbitrary system of second-order ODEs and its reduction to the simplest system. One is when the system is at most cubic in the first derivatives. One obtains the equivalent of the Lie conditions for such systems. We explicitly solve this branch of the linearization problem by point transformations in the case of a square system of two second-order ODEs. Necessary and sufficient conditions for linearization to the simplest system by means of point transformations are given in terms of coefficient functions of the system of two second-order ODEs cubically nonlinear in the first derivatives. A consequence of our geometric approach of projection is a rederivation of Lies linearization conditions for a single second-order ODE and sheds light on more recent results for them. In particular we show here how one can construct point transformations for reduction to the simplest linear equation by going to the higher space and just utilizing the coefficients of the original ODE. We also obtain invariant criteria for the reduction of a linear square system to the simplest system. Moreover these results contain the quadratic case as a special case. Examples are given to illustrate our results.

Collineations of the Ricci tensor

Ricci collineations for the Ricci tensor which is constructed from a general spherically symmetric and static metric are classified for all possibilities of Rab(r) (such that Rab≠0 for a=b). It turns out that the only collineations admitted by this tensor can be ten, six, or four and there does not appear any case in between.

Primordial black holes in phantom cosmology

We investigate the effects of accretion of phantom energy onto primordial black holes. Since Hawking radiation and phantom energy accretion contribute to a decrease of the mass of the black hole, the primordial black hole that would be expected to decay now due to the Hawking process would decay earlier due to the inclusion of the phantom energy. Equivalently, to have the primordial black hole decay now it would have to be more massive initially. We find that the effect of the phantom energy is substantial and the black holes decaying now would be much more massive—over ten orders of magnitude! This effect will be relevant for determining the time of production and hence the number of evaporating black holes expected in a universe accelerating due to phantom energy.

Symmetries of static, spherically symmetric space‐times

In this paper it is shown that reduction from maximal to minimal static, spherical symmetry of a space‐time occurs in only one step reducing the number of independent Killing vector fields from 10 to 4. Maximal symmetry corresponds only to the de Sitter, anti‐de Sitter, and Minkowski metrics, without reference to the Einstein field equations.

The classification of plane symmetric spacetimes by isometries

A complete classification of plane symmetric Lorentzian manifolds according to their additional isometries and metrics (or classes of metrics) is obtained by solving the Killing equations. We obtain all metrics (or classes of metrics), that admit the group of motions Gr (where r=3,4,5,6,7 and 10) containing SO(2)⨷R2, the minimal symmetry inherited by the plane symmetric manifolds.