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Asghar Qadir

Publications and source records attributed to Asghar Qadir.

At least 55 records · Page 3Linked to original sources
arXiv2009

Approximate Noether Symmetries of the Geodesic Equations for the Charged-Kerr Spacetime and Rescaling of Energy

Using approximate symmetry methods for differential equations we have investigated the exact and approximate symmetries of a Lagrangian for the geodesic equations in the Kerr spacetime. Taking Minkowski spacetime as the exact case, it is shown that the symmetry algebra of the Lagrangian is 17 dimensional. This algebra is related to the 15 dimensional Lie algebra of conformal isometries of Minkowski spacetime. First introducing spin angular momentum per unit mass as a small parameter we consider first-order approximate symmetries of the Kerr metric as a first perturbation of the Schwarzschild metric. We then consider the second-order approximate symmetries of the Kerr metric as a second perturbation of the Minkowski metric. The approximate symmetries are recovered for these spacetimes and there are no non-trivial approximate symmetries. A rescaling of the arc length parameter for consistency of the trivial second-order approximate symmetries of the geodesic equations indicates that the energy in the charged-Kerr metric has to be rescaled and the rescaling factor is $r$-dependent. This rescaling factor is compared with that for the Reissner-Nordström metric.

gr-qc↗
arXiv2008

Weyl collineations that are not curvature collineations

Though the Weyl tensor is a linear combination of the curvature tensor, Ricci tensor and Ricci scalar, it does not have all and only the Lie symmetries of these tensors since it is possible, in principle, that "asymmetries cancel". Here we investigate if, when and how the symmetries can be different. It is found that we can obtain a metric with a finite dimensional Lie algebra of Weyl symmetries that properly contains the Lie algebra of curvature symmetries. There is no example found for the converse requirement. It is speculated that there may be a fundamental reason for this lack of "duality".

gr-qc↗
arXiv2008

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 black hole. Here 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 limit. This condition also challenges the validity of the cosmic censorship conjecture since a naked singularity is eventually produced as magnitude of charge increases compared to mass of black hole.

astro-ph↗
arXiv2008

Comments on "Charged particle dynamics in the field of slowly rotating compact star"

B.M. Mirza \cite{1} presented a solution of coupled Einstein-Maxwell equations for a slowly rotating neutron star; however his derivations had some errors and implicit assumptions that rendered the solution invalid. We point out the errors and present a mathematically consistent solution. The resulting solution is also physically consistent as it remains finite in the no rotation limit, whereas Mirza's solution diverges for zero rotation.

astro-ph↗
arXiv2007

Second-Order Approximate Symmetries of the Geodesic Equations for the Reissner-Nordström Metric and Re-Scaling of Energy of a Test Particle

Following the use of approximate symmetries for the Schwarzschild spacetime by A.H. Kara, F.M. Mahomed and A. Qadir (Nonlinear Dynam., to appear), we have investigated the exact and approximate symmetries of the system of geodesic equations for the Reissner-Nordström spacetime (RN). For this purpose we are forced to use second order approximate symmetries. It is shown that in the second-order approximation, energy must be rescaled for the RN metric. The implications of this rescaling are discussed.

gr-qc↗
arXiv2007

Geometric Linearization of Ordinary Differential Equations

The linearizability of differential equations was first considered by Lie for scalar second order semi-linear ordinary differential equations. Since then there has been considerable work done on the algebraic classification of linearizable equations and even on systems of equations. However, little has been done in the way of providing explicit criteria to determine their linearizability. Using the connection between isometries and symmetries of the system of geodesic equations criteria were established for second order quadratically and cubically semi-linear equations and for systems of equations. The connection was proved for maximally symmetric spaces and a conjecture was put forward for other cases. Here the criteria are briefly reviewed and the conjecture is proved.

math.CA↗
arXiv2007

Invariant Linearization Criteria for Systems of Cubically Semi-Linear Second-Order Ordinary Differential Equations

Invariant linearization criteria of square systems of second-order quadratically semi-linear 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. One is when the system is at most cubic in the first derivatives. We solve this branch of the linearization problem by point transformations in the case of a square sytem of two second-order ODEs. Necessary and sufficient conditions for linearization 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 re-derivation of Lie's conditions for a single second-order ODE and sheds light on more recent results on 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 utilising 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.

math.CA↗
arXiv2007

Linearizability Criteria for a Class of Third Order Semi-Linear ODEs

Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations, we extend to the third order by differentiating the second order equation. This yields criteria for linearizability of a class of third order semi-linear ordinary differential equations, which is distinct from the classes available in the literature. Some examples are given and discussed.

math.CA↗
arXiv2007

Constructing a Space from the System of Geodesic Equations

Given a space it is easy to obtain the system of geodesic equations on it. In this paper the inverse problem of reconstructing the space from the geodesic equations is addressed. A procedure is developed for obtaining the metric tensor from the Christoffel symbols. The procedure is extended for determining if a second order quadratically semi-linear system can be expressed as a system of geodesic equations, provided it has terms only quadratic in the first derivative apart from the second derivative term. A computer code has been developed for dealing with larger systems of geodesic equations.

math.DG↗
arXiv2007

Conditional linearizability criteria for a system of third-order ordinary differential equations

We provide linearizability criteria for a class of systems of third-order ordinary differential equations (ODEs) that is cubically semi-linear in the first derivative, by differentiating a system of second-order quadratically semi-linear ODEs and using the original system to replace the second derivative. The procedure developed splits into two cases, those where the coefficients are constant and those where they are variables. Both cases are discussed and examples given.

math.CA↗
arXiv2007

The Momentum Four-Vector in the $eψ$N Formalism and the Angular Momentum Imparted to Test Particles by Gravitational Waves

Since gravitational waves are solutions of Einstein's field equations with a zero stress-energy tensor, the reality of these waves was questioned. To demonstrate it, the momentum imparted to test particles by such waves was evaluated. A closed form expression for this quantity was provided by Qadir and Sharif, using an extension of the pseudo-Newtonian formalism. That formalism carried with it the zero component of the momentum vector, which could not be interpreted as the energy imparted to the test particle. Sharif proposed that it may represent the angular momentum imparted to test particles by gravitational waves. In this paper it is shown that this interpretation is not valid. An alternative explanation has been provided.

gr-qc↗
arXiv2007

Homotheties of Cylindrically Symmetric Static Manifolds and their Global Extension

Cylindrically symmetric static manifolds are classified according to their homotheties and metrics. In each case the homothety vector fields and the corresponding metrics are obtained explicitly by solving the homothety equations. It turns out that these metrics admit homothety groups $H_m$, where $m=4,5,7,11$. This classification is then used to identify the cylindrically symmetric static spaces admitting the local homotheties, which are globally prohibited due to their topological construction. Einstein's field equations are then used to identify the physical nature of the spaces thus obtained.

gr-qc↗
arXiv2007

The Relativistic Generalization of the Gravitational Force for Arbitrary Spacetimes

It has been suggested that re-expressing relativity in terms of forces could provide fresh insights. The formalism developed for this purpose only applied to static, or conformally static, space-times. Here we extend it to arbitrary space-times. It is hoped that this formalism may lead to a workable definition of mass and energy in relativity.

gr-qc↗
arXiv2006

Similarities Between Classical Timelike Geodesics in a Naked Reissner-Nordstrom Singularity Background and the Behaviour of Electrons in Quantum Theory

It is generally assumed that naked singularities must be physically excluded, as they could otherwise introduce unpredictable influences in their future null cones. Considering geodesics for a naked Reissner-Nordstrom singularity, it is found that the singularity is effectively clothed by its repulsive nature. Regarding electron as naked singularity, the size of the clothed singularity (electron) turns out to be classical electro-magnetic radius of the electron, to an observer falling freely from infinity, initially at rest. The size shrinks for an observer falling freely from infinity, with a positive initial velocity. For geodetic parameters corresponding to negative energy there are trapped geodesics. The similarity of this picture with that arising in the Quantum Theory is discussed.

gr-qc↗
arXiv2005

A Representation for the Anyon Integral Function

The Fermi-Dirac and Bose-Einstein particles satisfy corresponding statistical distributions. In the phenomena of charge fractionalization and the fractional quantum Hall effect it is found that particles behave as if they are neither fermions nor bosons. Such particles are called anyons. The integral functions for bosons and fermions are available in the literature. However, there is no anyon integral function available. In this note we propose a pair of functions that interpolate, in some sense, between the gamma function and the zeta function, which we call "gamma-zeta" functions. It is pointed out that this pair of functions very naturally provides a representation of the anyon integral function.

math-ph↗