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Author

Zafar Ahsan

Bio: Zafar Ahsan is an academic researcher from Aligarh Muslim University. The author has contributed to research in topics: Lanczos tensor & Einstein tensor. The author has an hindex of 5, co-authored 15 publications receiving 83 citations.

Papers
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Journal ArticleDOI
TL;DR: In this paper, the relativistic significance of concircular curvature tensors has been explored and the existence of Killing and conformal Killing vectors has been established for spacetimes satisfying Einstein field equations.
Abstract: In the differential geometry of certain F-structures, the importance of concircular curvature tensor is very well known. The relativistic significance of this tensor has been explored here. The spacetimes satisfying Einstein field equations and with vanishing concircular curvature tensor are considered and the existence of Killing and conformal Killing vectors have been established for such spacetimes. Perfect fluid spacetimes with vanishing concircular curvature tensor have also been considered. The divergence of concircular curvature tensor is studied in detail and it is seen, among other results, that if the divergence of the concircular tensor is zero and the Ricci tensor is of Codazzi type then the resulting spacetime is of constant curvature. For a perfect fluid spacetime to possess divergence-free concircular curvature tensor, a necessary and sufficient condition has been obtained in terms of Friedmann-Robertson-Walker model.

38 citations

Book
01 Jan 2008
TL;DR: In this article, the Riemann tensors and their algebraic properties are studied, including transformation of coordinates, transformation of coordinate systems, transformation laws, and transformation laws for Christoffel symbols.
Abstract: 1. Tensors and their Algebra 1.1 Introduction 1.2 Transformation of Coordinates 1.3 Summation Convention 1.4 Kronecker Delta 1.5 Scalars, Contravariant and Covariant Vectors 1.6 Tensors of Higher Rank 1.7 Symmetry of Tensors 1.8 Algebra of Tensors, Addition and Subtraction, Equality of Tensors, Inner and Outer Products, Contraction, The Quotient Law 1.9 Irreducible Tensors, Exercises 2. Riemannian Space and Metric Tensor 2.1 Introduction 2.2 The Metric Tensor 2.3 Raising and Lowering of Indices-Associated Tensor 2.4 Vector Magnitude 2.5 Relative and Absolute Tensors 2.6 The Levi-Civita Tensor, Exercises 3. Christoffel Symbols and Covariant Differentiation 3.1 Introduction 3.2 Christoffel Symbols 3.3 Transformation Law for Christoffel Symbols 3.4 Equation of a Geodesic 3.5 Affine Parameter 3.6 Geodesic Coordinate System 3.7 Covariant Differentiation, Covariant Derivatives of Contravariant and Covariant Vectors, Covariant Derivatives of Rank Two Tensors, Covariant Derivatives of Tensors of Higher Rank 3.8 Rules for Covariant Differentiation 3.9 Some Useful Formulas, Divergence of a Vector Field, Gradient of a Scalar and Laplacian, Curl of a Vector Field, Divergence of a Tensor Field 3.10 Intrinsic Derivative-Parallel Transport 3.11 Null Geodesics 3.12 Alternative Derivation of Equation of Geodesic, Exercises 4. The Riemann Curvature Tensor 4.1 Introduction 4.2 The Riemann Curvature Tensor 4.3 Commutation of Covariant Derivatives 4.4 Covariant Form of the Riemann Curvature Tensor 4.5 Properties of the Riemann Curvature Tensor 4.6 Uniqueness of the Riemann Curvature Tensor 4.7 The Number of Algebraically Independent Components of the Riemann Curvature Tensor 4.8 The Ricci Tensor and the Scalar Curvature 4.9 The Einstein Tensor 4.10 The Integrability of the Riemann Tensor and the Flatness of the Space 4.11 The Einstein Spaces 4.12 Curvature of a Riemannian Space 4.13 Spaces of Constant Curvature, Exercises 5. Some Advanced Topics 5.1 Introduction, 5.2 Gewodesic Deviation 5.3 Decomposition of Riemann Curvature Tensor 5.4 Electric and Magnetic Parts of the Riemann and Weyl Tensors 5.5 Classification of Gravitational Fields 5.6 Invariants of the Riemann Curvature Tensor 5.7 Curvature Tensors Involving the Riemann Tensor, Space-matter Tensor, Conharmonic Curvature Tensor 5.8 Lie Derivative 5.9 The Killing Equation 5.10 The Curvature Tensor and Killing Vector, Exercises

14 citations

01 Jan 2010
TL;DR: In this article, a study of conharmonic curvature tensors has been made on the four dimensional spacetime of general relativity and the existence of Killing and confor- mal Killing vectors on such spacetime have been established.
Abstract: The signiflcance of conharmonic curvature tensor is very well known in the difierential geometry of certain F-structures (e.g., complex, almost complex, Kahler, almost Kahler, Hermitian, almost Hermitian structures, etc.). In this paper, a study of conharmonic curvature ten- sor has been made on the four dimensional spacetime of general relativity. The spacetime satisfying Einstein fleld equations and having vanishing conharmonic tensor is considered and the existence of Killing and confor- mal Killing vectors on such spacetime have been established. Perfect ∞uid cosmological models have also been studied.

13 citations

Journal ArticleDOI
TL;DR: In this article, a Lanczos potential for an arbitrary Petrov type D vacuum spacetimes, using the compacted spin coefficient formalism (or GHP-formalism), has been obtained, which leads to a solution of Weyl-Lanczos equations.
Abstract: A Lanczos potential for an arbitrary Petrov type D vacuum spacetimes, using the compacted spin coefficient formalism (or GHP-formalism), has been obtained; which in turn leads to a solution of Weyl-Lanczos equations.

8 citations


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Book
01 Jan 1970

329 citations

Journal ArticleDOI
TL;DR: In this paper, a code based on the nonorthogonal curvilinear coordinates is developed with a collocated grid system generated by the two-boundary method, which is used to compare simulated results for a fin and tube surface with coupled and decoupled solution methods.
Abstract: In the present study, a code based on the nonorthogonal curvilinear coordinates is developed with a collocated grid system generated by the two-boundary method. After validation of the code, it is used to compare simulated results for a fin-and-tube surface with coupled and decoupled solution methods. The results of the coupled method are more agreeable with the test data. Simulation for dimpled and reference plain plate fin-and-tube surfaces are then conducted by the coupled method within a range of inlet velocity from 1.0 m/s to 5 m/s. Results show that at identical pumping power the dimpled fin can enhance heat transfer by 13.8–30.3%. The results show that relative to the reference plain plate fin-and-tube surface, heat transfer rates and pressure drops of the dimpled fin increase by 13.8%–30.3% and 31.6%–56.5% for identical flow rate constraint. For identical pumping power constraint and identical pressure drop constraint, the heat transfer rates increase by 11.0%–25.3% and 9.2%–22.0%, respectively. B...

26 citations

Journal ArticleDOI
TL;DR: The projective curvature tensor is an invariant under geodesic preserving transformations on semi-Riemannian manifolds and possesses different geometric properties than other generalized curvatures as mentioned in this paper.
Abstract: The projective curvature tensor is an invariant under geodesic preserving transformations on semi-Riemannian manifolds. It possesses different geometric properties than other generalized curvature ...

26 citations