It is known that spherically symmetric static spacetimes admit a foliation by at hypersurfaces. S... more It is known that spherically symmetric static spacetimes admit a foliation by at hypersurfaces. Such foliations have explicitly been constructed for some spacetimes, using different approaches, but none of them have proved or even discussed the uniqueness of these foliations. The issue of uniqueness becomes more important due to suitability of at foliations for studying black hole physics. Here at spherically symmetric spacelike hy- persurfaces are obtained by a direct method. It is found that spherically symmetric static spacetimes admit at spherically symmetric hypersurfaces, and that these hypersurfaces are unique up to translation under the time- like Killing vector. This result guarantees the uniqueness of at spherically symmetric foliations for such spacetimes.
Foliation and the First Law of Black Hole Thermodynamics
Chinese Physics Letters, 2011
Title: Foliation and the First Law of Black Hole Thermodynamics. Authors: Azad, A. Siddiqui; Syed... more Title: Foliation and the First Law of Black Hole Thermodynamics. Authors: Azad, A. Siddiqui; Syed, Muhammad Jawwad Riaz; Akbar, M. Affiliation: AA(Department of Basic Sciences and Humanities, EME College, National University ...
It is known that spherically symmetric spacetimes admit flat spacelike foliations. We point out a... more It is known that spherically symmetric spacetimes admit flat spacelike foliations. We point out a simple method of seeing this result via the Hamiltonian constraints of general relativity. The method yields explicit formulas for the extrinsic curvatures of the slicings.
Standard application of similarity method to find solutions of PDEs mostly results in reduction t... more Standard application of similarity method to find solutions of PDEs mostly results in reduction to ODEs which are not easily integrable in terms of elementary or tabulated functions. Such situations usually demand solving reduced ODEs numerically. However, there are no systematic procedures available to utilize these numerical solutions of reduced ODE to obtain the solution of original PDE. A practical and tractable approach is proposed to deal with such situations and is applied to obtain approximate similarity solutions to different cases of an initial-boundary value problem of unsteady gas flow through a semi-infinite porous medium.
It is pointed out that the claim made by Joshi and Joshi [1], has not been rigorously demonstrate... more It is pointed out that the claim made by Joshi and Joshi [1], has not been rigorously demonstrated by them. A simpler and more correct proof
Abstract: It is generally assumed that naked singularities must be physically excluded, as they c... more Abstract: 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
The Cosmic Censorship Hypothesis and the Naked Reissner-Nordstrom Singularity
Penrose's cosmic censorship hypothesis excludes the physical existence of naked singularitie... more Penrose's cosmic censorship hypothesis excludes the physical existence of naked singularities, as they could otherwise introduce unpredictable influences in their future null cones. In this paper, we have analyzed timelike geodesics for a naked Reissner-Nordstrom singularity. It is found that the singularity is effectively clothed by its own repulsive nature, making the hypothesis redundant in this case as it provides "censorship without censorship".
This paper introduces a new approach to tackle the boundary value problem representing magnetohyd... more This paper introduces a new approach to tackle the boundary value problem representing magnetohydrodynamics flow developed by a rotating disk. The problem comprises of a coupled and non-linear differential system which appears difficult to be computed exactly. A contemporary MATLAB's package bvp4c is used to solve the problem numerically and then, with the help of the numerical solution, a closed-form solution is constructed via a systematic method. All computations are made for a specific value of the magnetic interaction parameter. Residuals of governing equations are plotted which ascertain that the obtained closed-form solution matches closely with the exact solution. INDEX TERMS Closed-form solution, Dirichlet conditions, residual analysis.
A class of nonstatic exact solutions of the Einstein–Maxwell field equations
International Journal of Modern Physics D
In this paper, we present a class of spherically symmetric nonstatic exact solutions of Einstein–... more In this paper, we present a class of spherically symmetric nonstatic exact solutions of Einstein–Maxwell’s field equations. We have assumed the pressure distribution to be anisotropic and have taken ansatz on two of the gravitational potentials. The solutions admit negative values of the radial pressure. We have also discussed criteria on the physical variables to be acceptable.
A study of classical time-like geodesics in a naked Reissner–Nordström singularity background is ... more A study of classical time-like geodesics in a naked Reissner–Nordström singularity background is presented. Regarding the electron as naked a singularity, the size of the clothed singularity 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 quantum theory is discussed.
Foliation of the Schwarzschild and Reissner–Nordström Space–Times by Flat Space-Like Hypersurfaces
International Journal of Modern Physics D, 2006
It is known that spherically symmetric space–times admit flat space-like foliations. In this pape... more It is known that spherically symmetric space–times admit flat space-like foliations. In this paper a simple procedure for the complete foliation of the Schwarzschild space–time by flat space-like hypersurfaces is developed, using the fact that unforced geodesics are orthogonal to such hypersurfaces. The method is then extended to obtain a complete foliation of the Reissner–Nordström space–time by such hypersurfaces. In this case, as there is a barrier beyond which the geodesics do not go, a complete foliation is obtained by analytically continuing the hypersurfaces beyond the barrier. A duality between imaginary time-like geodesics and space-like hypersurfaces is noted, which also provides complete foliation — but the hypersurfaces are not flat. The features of our foliation are discussed.
We have obtained a new class of solutions for the Einstein–Maxwell field equations for static sph... more We have obtained a new class of solutions for the Einstein–Maxwell field equations for static spherically symmetric space–times by considering the negative anisotropic pressures, which represents a potential model of a dark energy star. We take the equation of state pr = −ρ, where pr is the radial pressure and ρ is the density. We have also checked that for these solutions metric coefficients, mass density, radial pressure, transverse pressure, electric field, and current density are well defined for suitable values of the parameters involved in the solution. These exact solutions can be used to develop models of dark energy stellar interiors satisfying all physical constraints except for the causality condition, which cannot be satisfied for the equation of state considered here, and which is arguably not an applicable physical constraint for dark matter–energy.
Though popular presentations give the Schwarzschild singularity as a point, it is known that it i... more Though popular presentations give the Schwarzschild singularity as a point, it is known that it is spacelike and not timelike. Thus, it has a "length" and is not a "point." In fact, its length is necessarily infinite. It has been proven that the proper length of the Qadir–Wheeler suture model goes to infinity,1 while its proper volume shrinks to zero, and the asymptotic behavior of the length and volume has been calculated. That model consists of two Friedmann sections connected by a Schwarzschild "suture." The question arises whether a similar analysis could provide the asymptotic behavior of the Schwarzschild black hole near the singularity. It is proven here that, unlike the behavior for the suture model, for the Schwarzschild essential singularity Δs ~ K1/3 ln K and V ~ K-1 ln K, where K is the mean extrinsic curvature, or the York time.
The eigenvalues of a graph are the eigenvalues of its adjacency matrix. The energy of a graph is ... more The eigenvalues of a graph are the eigenvalues of its adjacency matrix. The energy of a graph is the sum of absolute values of its eigenvalues. Recently, the concept of energy of graphs is extended to digraphs. Minimal and maximal energy among n-vertex unicyclic digraphs is known, where n 2. In this paper, we address the problem of finding minimal and maximal energy among n-vertex bicyclic digraphs which contain vertex-disjoint directed cycles, where n 4 .
It is known that spherically symmetric static spacetimes admit a foliation by at hypersurfaces. S... more It is known that spherically symmetric static spacetimes admit a foliation by at hypersurfaces. Such foliations have explicitly been constructed for some spacetimes, using different approaches, but none of them have proved or even discussed the uniqueness of these foliations. The issue of uniqueness becomes more important due to suitability of at foliations for studying black hole physics. Here at spherically symmetric spacelike hy- persurfaces are obtained by a direct method. It is found that spherically symmetric static spacetimes admit at spherically symmetric hypersurfaces, and that these hypersurfaces are unique up to translation under the time- like Killing vector. This result guarantees the uniqueness of at spherically symmetric foliations for such spacetimes.
Foliation and the First Law of Black Hole Thermodynamics
Chinese Physics Letters, 2011
Title: Foliation and the First Law of Black Hole Thermodynamics. Authors: Azad, A. Siddiqui; Syed... more Title: Foliation and the First Law of Black Hole Thermodynamics. Authors: Azad, A. Siddiqui; Syed, Muhammad Jawwad Riaz; Akbar, M. Affiliation: AA(Department of Basic Sciences and Humanities, EME College, National University ...
It is known that spherically symmetric spacetimes admit flat spacelike foliations. We point out a... more It is known that spherically symmetric spacetimes admit flat spacelike foliations. We point out a simple method of seeing this result via the Hamiltonian constraints of general relativity. The method yields explicit formulas for the extrinsic curvatures of the slicings.
Standard application of similarity method to find solutions of PDEs mostly results in reduction t... more Standard application of similarity method to find solutions of PDEs mostly results in reduction to ODEs which are not easily integrable in terms of elementary or tabulated functions. Such situations usually demand solving reduced ODEs numerically. However, there are no systematic procedures available to utilize these numerical solutions of reduced ODE to obtain the solution of original PDE. A practical and tractable approach is proposed to deal with such situations and is applied to obtain approximate similarity solutions to different cases of an initial-boundary value problem of unsteady gas flow through a semi-infinite porous medium.
It is pointed out that the claim made by Joshi and Joshi [1], has not been rigorously demonstrate... more It is pointed out that the claim made by Joshi and Joshi [1], has not been rigorously demonstrated by them. A simpler and more correct proof
Abstract: It is generally assumed that naked singularities must be physically excluded, as they c... more Abstract: 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
The Cosmic Censorship Hypothesis and the Naked Reissner-Nordstrom Singularity
Penrose's cosmic censorship hypothesis excludes the physical existence of naked singularitie... more Penrose's cosmic censorship hypothesis excludes the physical existence of naked singularities, as they could otherwise introduce unpredictable influences in their future null cones. In this paper, we have analyzed timelike geodesics for a naked Reissner-Nordstrom singularity. It is found that the singularity is effectively clothed by its own repulsive nature, making the hypothesis redundant in this case as it provides "censorship without censorship".
This paper introduces a new approach to tackle the boundary value problem representing magnetohyd... more This paper introduces a new approach to tackle the boundary value problem representing magnetohydrodynamics flow developed by a rotating disk. The problem comprises of a coupled and non-linear differential system which appears difficult to be computed exactly. A contemporary MATLAB's package bvp4c is used to solve the problem numerically and then, with the help of the numerical solution, a closed-form solution is constructed via a systematic method. All computations are made for a specific value of the magnetic interaction parameter. Residuals of governing equations are plotted which ascertain that the obtained closed-form solution matches closely with the exact solution. INDEX TERMS Closed-form solution, Dirichlet conditions, residual analysis.
A class of nonstatic exact solutions of the Einstein–Maxwell field equations
International Journal of Modern Physics D
In this paper, we present a class of spherically symmetric nonstatic exact solutions of Einstein–... more In this paper, we present a class of spherically symmetric nonstatic exact solutions of Einstein–Maxwell’s field equations. We have assumed the pressure distribution to be anisotropic and have taken ansatz on two of the gravitational potentials. The solutions admit negative values of the radial pressure. We have also discussed criteria on the physical variables to be acceptable.
A study of classical time-like geodesics in a naked Reissner–Nordström singularity background is ... more A study of classical time-like geodesics in a naked Reissner–Nordström singularity background is presented. Regarding the electron as naked a singularity, the size of the clothed singularity 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 quantum theory is discussed.
Foliation of the Schwarzschild and Reissner–Nordström Space–Times by Flat Space-Like Hypersurfaces
International Journal of Modern Physics D, 2006
It is known that spherically symmetric space–times admit flat space-like foliations. In this pape... more It is known that spherically symmetric space–times admit flat space-like foliations. In this paper a simple procedure for the complete foliation of the Schwarzschild space–time by flat space-like hypersurfaces is developed, using the fact that unforced geodesics are orthogonal to such hypersurfaces. The method is then extended to obtain a complete foliation of the Reissner–Nordström space–time by such hypersurfaces. In this case, as there is a barrier beyond which the geodesics do not go, a complete foliation is obtained by analytically continuing the hypersurfaces beyond the barrier. A duality between imaginary time-like geodesics and space-like hypersurfaces is noted, which also provides complete foliation — but the hypersurfaces are not flat. The features of our foliation are discussed.
We have obtained a new class of solutions for the Einstein–Maxwell field equations for static sph... more We have obtained a new class of solutions for the Einstein–Maxwell field equations for static spherically symmetric space–times by considering the negative anisotropic pressures, which represents a potential model of a dark energy star. We take the equation of state pr = −ρ, where pr is the radial pressure and ρ is the density. We have also checked that for these solutions metric coefficients, mass density, radial pressure, transverse pressure, electric field, and current density are well defined for suitable values of the parameters involved in the solution. These exact solutions can be used to develop models of dark energy stellar interiors satisfying all physical constraints except for the causality condition, which cannot be satisfied for the equation of state considered here, and which is arguably not an applicable physical constraint for dark matter–energy.
Though popular presentations give the Schwarzschild singularity as a point, it is known that it i... more Though popular presentations give the Schwarzschild singularity as a point, it is known that it is spacelike and not timelike. Thus, it has a "length" and is not a "point." In fact, its length is necessarily infinite. It has been proven that the proper length of the Qadir–Wheeler suture model goes to infinity,1 while its proper volume shrinks to zero, and the asymptotic behavior of the length and volume has been calculated. That model consists of two Friedmann sections connected by a Schwarzschild "suture." The question arises whether a similar analysis could provide the asymptotic behavior of the Schwarzschild black hole near the singularity. It is proven here that, unlike the behavior for the suture model, for the Schwarzschild essential singularity Δs ~ K1/3 ln K and V ~ K-1 ln K, where K is the mean extrinsic curvature, or the York time.
The eigenvalues of a graph are the eigenvalues of its adjacency matrix. The energy of a graph is ... more The eigenvalues of a graph are the eigenvalues of its adjacency matrix. The energy of a graph is the sum of absolute values of its eigenvalues. Recently, the concept of energy of graphs is extended to digraphs. Minimal and maximal energy among n-vertex unicyclic digraphs is known, where n 2. In this paper, we address the problem of finding minimal and maximal energy among n-vertex bicyclic digraphs which contain vertex-disjoint directed cycles, where n 4 .
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