In this paper, we have studied the gravitational baryogenesis of isotropic and homogeneous univer... more In this paper, we have studied the gravitational baryogenesis of isotropic and homogeneous universe in the framework of general relativity. We investigate an exact and new solution of Einstein's field equations for FRW metric. Our solution represents a transitioning model of the universe which was expanding in decelerated mode and it transit in accelerated mode after dominance of cosmological constant Λ. We observe that gravitational baryogenesis occurs in the derived universe and derived baryon entropy ratio is in good agreement with its observational value.
In this work, we perform the Lie symmetry analysis on the Einstein-Maxwell field equations in pla... more In this work, we perform the Lie symmetry analysis on the Einstein-Maxwell field equations in plane symmetric spacetime. Here Lie point symmetries and optimal system of one dimensional subalgebras are determined. The similarity reductions and exact solutions are obtained in connection to the evolution of universe. The present study deals with the electromagnetic energy of inhomogeneous universe where $F_{12}$ is the non-vanishing component of electromagnetic field tensor. To get a deterministic solution, it is assumed that the free gravitational field is Petrov type-II non-degenerate. The electromagnetic field tensor $F_{12}$ is found to be positive and increasing function of time. As a special case, to validate the solution set, we discuss some physical and geometric properties of a specific sub-model.
The present study deals with the inhomogeneous plane symmetric models in scalar-tensor theory of ... more The present study deals with the inhomogeneous plane symmetric models in scalar-tensor theory of gravitation. We used symmetry group analysis method to solve the field equations analytically. A new class of similarity solutions have been obtained by considering the inhomogeneous nature of metric potential. The physical behavior and geometrical aspects of the derived models are also discussed.
Abstract. We consider a curve = (s) in Minkowski 3-space E31 and denote by fT;N;Bg the Frenet fra... more Abstract. We consider a curve = (s) in Minkowski 3-space E31 and denote by fT;N;Bg the Frenet frame of . We say that is a slant helix if there exists a xed direction U of E31 such that the function ⟨N(s); U ⟩ is constant. In this work we give characterizations of slant helices in terms of the curvature and torsion of . Finally, we discuss the tangent and binormal indicatrices of slant curves, proving that they are helices in E31.
In this paper, we derive some new invariant solutions of dark energy models in cylindrically symm... more In this paper, we derive some new invariant solutions of dark energy models in cylindrically symmetric space-time. To quantify the deviation of pressure from isotropy, we introduce three different time dependent skewness parameters along the spatial directions. The matter source consists of dark energy which is minimally interact with perfect fluid. We use symmetry analysis method for solving the non-linear partial differential equations (NLPDEs) which is more powerful than the classical methods of solving NLPDEs. The geometrical and kinematical features of the models and the behaviour of the anisotropy of dark energy, are examined in detail.
Salkowski salkow, one century ago, introduced a family of curves with constant curvature but non-... more Salkowski salkow, one century ago, introduced a family of curves with constant curvature but non-constant torsion (Salkowski curves) and a family of curves with constant torsion but non-constant curvature (anti-Salkowski curves) in Euclidean 3-space ^3. In this paper, we adapt definition of such curves to time-like curves in Minkowski 3-space _1^3. Thereafter, we introduce an explicit parametrization of a time-like Salkowski curves and a time-like Anti-Salkowski curves in Minkowski space _1^3. Also, we characterize them as space curve with constant curvature or constant torsion and whose normal vector makes a constant angle with a fixed line.
Inhomogeneous Bianchi type-I cosmological model with electro-magnetic field based on Lyra geometr... more Inhomogeneous Bianchi type-I cosmological model with electro-magnetic field based on Lyra geometry is investigated. Using separated method, the Einstein field equations have been solved analytically with the aid of Mathematica programm. A new class of exact solutions have been obtained by considering the potentials of metric and displacement field are functions of coordinates t and x. We have assumed that F(12) is the only non-vanishing component of electro-magnetic field tensor F(ij). The Maxwells equations show that F(12) is the function of x alone whereas the magnetic permeability is the function of x and t both. To get the deterministic solution, it has been assumed that the expansion scaler Theta in the model is proportional to the value sigma(11) of the shear tensor sigma(ij). Some physical and geometric properties of the model are also discussed and graphed.
We investigate some new similarity inhomogeneous solutions of anisotropic dark energy and perfect... more We investigate some new similarity inhomogeneous solutions of anisotropic dark energy and perfect fluid in Bianchi type-I space-time. Three different equation of state parameters along the spatial directions are introduced to quantify the deviation of pressure from isotropy. We consider the case when the dark energy is minimally coupled to the perfect fluid as well as direct interaction with it. The Lie symmetry generators that leave the equation invariant are identified and we generate an optimal system of one-dimensional subalgebras. Each element of the optimal system is used to reduce the partial differential equation to an ordinary differential equation which is further analyzed. We solve the Einstein field equations, described by a system of non-linear partial differential equations (NLPDEs), by using the Lie point symmetry analysis method. The geometrical and kinematical features of the models and the behavior of the anisotropy of dark energy, are examined in detail.
In classical differential geometry, the problem of the determination of the position vector of an... more In classical differential geometry, the problem of the determination of the position vector of an arbitrary space curve according to the intrinsic equations κ=κ(s) and τ=τ(s) (where κ and τ are the curvature and torsion of the space curve ψ, respectively) is still open eisenh, lips. However, in the case of a plane curve, helix and general helix, this problem is solved. In this paper, we solved this problem in the case of a slant helix. Also, we applied this method to find the representation of a Salkowski, anti-Salkowski curves and a curve of constant precession, as examples of a slant helices, by means of intrinsic equations.
In this paper, we have searched the existence of the similarity solution for plane symmetric inho... more In this paper, we have searched the existence of the similarity solution for plane symmetric inhomogeneous cosmological models in general relativity. The matter source consists of perfect fluid with proportionality relation between expansion scalar and shear scalar. The isovector fields of Einstein's field equation for the models under consideration are derived. A new class of exact solutions of Einstein's field equation have been obtained for inhomogeneous space-time. The physical behaviors and geometric aspects of the derived models have been discussed in detail.
In this paper, we search the existence of invariant solutions of Bianchi type I space-time in the... more In this paper, we search the existence of invariant solutions of Bianchi type I space-time in the context of f(R,T) gravity. The exact solution of the Einstein's field equations are derived by using Lie point symmetry analysis method that yield two models of invariant universe for symmetries X^(1) and X^(3). The model with symmetries X^(1) begins with big bang singularity while the model with symmetries X^(3) does not favour the big bang singularity. Under this specification, we find out at set of singular and non singular solution of Bianchi type I model which present several other physically valid features within the framework of f(R,T).
We investigate plane symmetric space-time filled with perfect fluid in the C-field cosmology of H... more We investigate plane symmetric space-time filled with perfect fluid in the C-field cosmology of Hoyle and Narlikar. A new class of exact solutions have been obtained by considering the creation field C as a function of time only. To get the deterministic solution, it has been assumed that the rate of creation of matter-energy density is proportional to the strength of the existing C-field energy density. Several physical aspects and geometrical properties of the models are discussed in detail, especially it is shown that some of our solutions of C-field cosmology are free from singularity in contrast to the Big Bang cosmology. A comparative study has been carried out between two models, one singular and the other nonsingular, by contrasting the behaviour of the physical parameters and noted that the model in a unique way represents both the features of the accelerating as well as decelerating Universe depending on the parameters and thus seems provides glimpses of the oscillating or...
In this paper we derive some new invariant solutions of Einstein-Maxwell's field equations for st... more In this paper we derive some new invariant solutions of Einstein-Maxwell's field equations for string fluid as source of matter in cylindrically symmetric space-time with Variable Magnetic Permeability. We also discuss the physical and geometrical properties of the models derived in the paper. The solutions, at least one of them, are interesting physically as they can explain the accelerating as well as singularity free Universe.
In this paper, we have searched the existence of the similarity solution for plane symmetric inho... more In this paper, we have searched the existence of the similarity solution for plane symmetric inhomogeneous cosmological models in general relativity. The matter source consists of perfect fluid with proportionality relation between expansion scalar and shear scalar. The isovector fields of Einstein's field equation for the models under consideration are derived. A new class of exact solutions of Einstein's field equation have been obtained for inhomogeneous space-time. The physical behaviors and geometric aspects of the derived models have been discussed in detail.
We consider a unit speed curve α in Euclidean four-dimensional space E 4 and denote the Frenet fr... more We consider a unit speed curve α in Euclidean four-dimensional space E 4 and denote the Frenet frame by {T,N,B1,B2}. We say that α is a slant helix if its principal normal vector N makes a constant angle with a fixed direction U. In this work we give different characterizations of such curves in terms of their curvatures. MSC: 53C40, 53C50
We introduce the notion of k-type slant helix in Minkowski space E 4 1 . For partially null and p... more We introduce the notion of k-type slant helix in Minkowski space E 4 1 . For partially null and pseudo null curves in E 4 1 , we express some characterizations in terms of their curvature and torsion functions.
Abstract. We consider a unit speed timelike curve α in Minkowski 4-space E41 and denote the Frene... more Abstract. We consider a unit speed timelike curve α in Minkowski 4-space E41 and denote the Frenet frame of α by {T,N,B1,B2}. We say that α is a generalized helix if one of the unit vector fields of the Frenet frame has constant scalar product with a fixed direction U of E41. In this work we study those helices where the function 〈B2, U 〉 is constant and we give different characterizations of such curves.
In this paper, we study the surfaces foliated by ellipses in three dimensional Euclidean space $... more In this paper, we study the surfaces foliated by ellipses in three dimensional Euclidean space $\mathbf{E}^3$. We prove the following results: \textbf{(1)} The surface foliated by an ellipse have constant Gaussian curvature $K$ if and only if the surface is flat, i.e. $K=0$. \textbf{(2)} The surface foliated by an ellipse is a flat if and only if it is a part of generalized cylinder or part of generalized cone.
In this paper, we define a rectifying spacelike curve in the Minkowski space-time $E_1^4$ as a cu... more In this paper, we define a rectifying spacelike curve in the Minkowski space-time $E_1^4$ as a curve whose position vector always lies in orthogonal complement $N^{\bot}$ of its principal normal vector field $N$. In particular, we study the rectifying spacelike curves in $E_1^4$ and characterize such curves in terms of their curvature functions.
In this paper, we focused our attention to find Ricci solitons of spherically symmetric static sp... more In this paper, we focused our attention to find Ricci solitons of spherically symmetric static spacetimes. It is shown that special classes of such spacetime metrics admit shrinking, expanding or s...
In this paper, we have studied the gravitational baryogenesis of isotropic and homogeneous univer... more In this paper, we have studied the gravitational baryogenesis of isotropic and homogeneous universe in the framework of general relativity. We investigate an exact and new solution of Einstein's field equations for FRW metric. Our solution represents a transitioning model of the universe which was expanding in decelerated mode and it transit in accelerated mode after dominance of cosmological constant Λ. We observe that gravitational baryogenesis occurs in the derived universe and derived baryon entropy ratio is in good agreement with its observational value.
In this work, we perform the Lie symmetry analysis on the Einstein-Maxwell field equations in pla... more In this work, we perform the Lie symmetry analysis on the Einstein-Maxwell field equations in plane symmetric spacetime. Here Lie point symmetries and optimal system of one dimensional subalgebras are determined. The similarity reductions and exact solutions are obtained in connection to the evolution of universe. The present study deals with the electromagnetic energy of inhomogeneous universe where $F_{12}$ is the non-vanishing component of electromagnetic field tensor. To get a deterministic solution, it is assumed that the free gravitational field is Petrov type-II non-degenerate. The electromagnetic field tensor $F_{12}$ is found to be positive and increasing function of time. As a special case, to validate the solution set, we discuss some physical and geometric properties of a specific sub-model.
The present study deals with the inhomogeneous plane symmetric models in scalar-tensor theory of ... more The present study deals with the inhomogeneous plane symmetric models in scalar-tensor theory of gravitation. We used symmetry group analysis method to solve the field equations analytically. A new class of similarity solutions have been obtained by considering the inhomogeneous nature of metric potential. The physical behavior and geometrical aspects of the derived models are also discussed.
Abstract. We consider a curve = (s) in Minkowski 3-space E31 and denote by fT;N;Bg the Frenet fra... more Abstract. We consider a curve = (s) in Minkowski 3-space E31 and denote by fT;N;Bg the Frenet frame of . We say that is a slant helix if there exists a xed direction U of E31 such that the function ⟨N(s); U ⟩ is constant. In this work we give characterizations of slant helices in terms of the curvature and torsion of . Finally, we discuss the tangent and binormal indicatrices of slant curves, proving that they are helices in E31.
In this paper, we derive some new invariant solutions of dark energy models in cylindrically symm... more In this paper, we derive some new invariant solutions of dark energy models in cylindrically symmetric space-time. To quantify the deviation of pressure from isotropy, we introduce three different time dependent skewness parameters along the spatial directions. The matter source consists of dark energy which is minimally interact with perfect fluid. We use symmetry analysis method for solving the non-linear partial differential equations (NLPDEs) which is more powerful than the classical methods of solving NLPDEs. The geometrical and kinematical features of the models and the behaviour of the anisotropy of dark energy, are examined in detail.
Salkowski salkow, one century ago, introduced a family of curves with constant curvature but non-... more Salkowski salkow, one century ago, introduced a family of curves with constant curvature but non-constant torsion (Salkowski curves) and a family of curves with constant torsion but non-constant curvature (anti-Salkowski curves) in Euclidean 3-space ^3. In this paper, we adapt definition of such curves to time-like curves in Minkowski 3-space _1^3. Thereafter, we introduce an explicit parametrization of a time-like Salkowski curves and a time-like Anti-Salkowski curves in Minkowski space _1^3. Also, we characterize them as space curve with constant curvature or constant torsion and whose normal vector makes a constant angle with a fixed line.
Inhomogeneous Bianchi type-I cosmological model with electro-magnetic field based on Lyra geometr... more Inhomogeneous Bianchi type-I cosmological model with electro-magnetic field based on Lyra geometry is investigated. Using separated method, the Einstein field equations have been solved analytically with the aid of Mathematica programm. A new class of exact solutions have been obtained by considering the potentials of metric and displacement field are functions of coordinates t and x. We have assumed that F(12) is the only non-vanishing component of electro-magnetic field tensor F(ij). The Maxwells equations show that F(12) is the function of x alone whereas the magnetic permeability is the function of x and t both. To get the deterministic solution, it has been assumed that the expansion scaler Theta in the model is proportional to the value sigma(11) of the shear tensor sigma(ij). Some physical and geometric properties of the model are also discussed and graphed.
We investigate some new similarity inhomogeneous solutions of anisotropic dark energy and perfect... more We investigate some new similarity inhomogeneous solutions of anisotropic dark energy and perfect fluid in Bianchi type-I space-time. Three different equation of state parameters along the spatial directions are introduced to quantify the deviation of pressure from isotropy. We consider the case when the dark energy is minimally coupled to the perfect fluid as well as direct interaction with it. The Lie symmetry generators that leave the equation invariant are identified and we generate an optimal system of one-dimensional subalgebras. Each element of the optimal system is used to reduce the partial differential equation to an ordinary differential equation which is further analyzed. We solve the Einstein field equations, described by a system of non-linear partial differential equations (NLPDEs), by using the Lie point symmetry analysis method. The geometrical and kinematical features of the models and the behavior of the anisotropy of dark energy, are examined in detail.
In classical differential geometry, the problem of the determination of the position vector of an... more In classical differential geometry, the problem of the determination of the position vector of an arbitrary space curve according to the intrinsic equations κ=κ(s) and τ=τ(s) (where κ and τ are the curvature and torsion of the space curve ψ, respectively) is still open eisenh, lips. However, in the case of a plane curve, helix and general helix, this problem is solved. In this paper, we solved this problem in the case of a slant helix. Also, we applied this method to find the representation of a Salkowski, anti-Salkowski curves and a curve of constant precession, as examples of a slant helices, by means of intrinsic equations.
In this paper, we have searched the existence of the similarity solution for plane symmetric inho... more In this paper, we have searched the existence of the similarity solution for plane symmetric inhomogeneous cosmological models in general relativity. The matter source consists of perfect fluid with proportionality relation between expansion scalar and shear scalar. The isovector fields of Einstein's field equation for the models under consideration are derived. A new class of exact solutions of Einstein's field equation have been obtained for inhomogeneous space-time. The physical behaviors and geometric aspects of the derived models have been discussed in detail.
In this paper, we search the existence of invariant solutions of Bianchi type I space-time in the... more In this paper, we search the existence of invariant solutions of Bianchi type I space-time in the context of f(R,T) gravity. The exact solution of the Einstein's field equations are derived by using Lie point symmetry analysis method that yield two models of invariant universe for symmetries X^(1) and X^(3). The model with symmetries X^(1) begins with big bang singularity while the model with symmetries X^(3) does not favour the big bang singularity. Under this specification, we find out at set of singular and non singular solution of Bianchi type I model which present several other physically valid features within the framework of f(R,T).
We investigate plane symmetric space-time filled with perfect fluid in the C-field cosmology of H... more We investigate plane symmetric space-time filled with perfect fluid in the C-field cosmology of Hoyle and Narlikar. A new class of exact solutions have been obtained by considering the creation field C as a function of time only. To get the deterministic solution, it has been assumed that the rate of creation of matter-energy density is proportional to the strength of the existing C-field energy density. Several physical aspects and geometrical properties of the models are discussed in detail, especially it is shown that some of our solutions of C-field cosmology are free from singularity in contrast to the Big Bang cosmology. A comparative study has been carried out between two models, one singular and the other nonsingular, by contrasting the behaviour of the physical parameters and noted that the model in a unique way represents both the features of the accelerating as well as decelerating Universe depending on the parameters and thus seems provides glimpses of the oscillating or...
In this paper we derive some new invariant solutions of Einstein-Maxwell's field equations for st... more In this paper we derive some new invariant solutions of Einstein-Maxwell's field equations for string fluid as source of matter in cylindrically symmetric space-time with Variable Magnetic Permeability. We also discuss the physical and geometrical properties of the models derived in the paper. The solutions, at least one of them, are interesting physically as they can explain the accelerating as well as singularity free Universe.
In this paper, we have searched the existence of the similarity solution for plane symmetric inho... more In this paper, we have searched the existence of the similarity solution for plane symmetric inhomogeneous cosmological models in general relativity. The matter source consists of perfect fluid with proportionality relation between expansion scalar and shear scalar. The isovector fields of Einstein's field equation for the models under consideration are derived. A new class of exact solutions of Einstein's field equation have been obtained for inhomogeneous space-time. The physical behaviors and geometric aspects of the derived models have been discussed in detail.
We consider a unit speed curve α in Euclidean four-dimensional space E 4 and denote the Frenet fr... more We consider a unit speed curve α in Euclidean four-dimensional space E 4 and denote the Frenet frame by {T,N,B1,B2}. We say that α is a slant helix if its principal normal vector N makes a constant angle with a fixed direction U. In this work we give different characterizations of such curves in terms of their curvatures. MSC: 53C40, 53C50
We introduce the notion of k-type slant helix in Minkowski space E 4 1 . For partially null and p... more We introduce the notion of k-type slant helix in Minkowski space E 4 1 . For partially null and pseudo null curves in E 4 1 , we express some characterizations in terms of their curvature and torsion functions.
Abstract. We consider a unit speed timelike curve α in Minkowski 4-space E41 and denote the Frene... more Abstract. We consider a unit speed timelike curve α in Minkowski 4-space E41 and denote the Frenet frame of α by {T,N,B1,B2}. We say that α is a generalized helix if one of the unit vector fields of the Frenet frame has constant scalar product with a fixed direction U of E41. In this work we study those helices where the function 〈B2, U 〉 is constant and we give different characterizations of such curves.
In this paper, we study the surfaces foliated by ellipses in three dimensional Euclidean space $... more In this paper, we study the surfaces foliated by ellipses in three dimensional Euclidean space $\mathbf{E}^3$. We prove the following results: \textbf{(1)} The surface foliated by an ellipse have constant Gaussian curvature $K$ if and only if the surface is flat, i.e. $K=0$. \textbf{(2)} The surface foliated by an ellipse is a flat if and only if it is a part of generalized cylinder or part of generalized cone.
In this paper, we define a rectifying spacelike curve in the Minkowski space-time $E_1^4$ as a cu... more In this paper, we define a rectifying spacelike curve in the Minkowski space-time $E_1^4$ as a curve whose position vector always lies in orthogonal complement $N^{\bot}$ of its principal normal vector field $N$. In particular, we study the rectifying spacelike curves in $E_1^4$ and characterize such curves in terms of their curvature functions.
In this paper, we focused our attention to find Ricci solitons of spherically symmetric static sp... more In this paper, we focused our attention to find Ricci solitons of spherically symmetric static spacetimes. It is shown that special classes of such spacetime metrics admit shrinking, expanding or s...
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Papers by Ahmad T Ali