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Title of Thesis

Institute/University/Department Details
Department of Mathematics/ Quaid-i-Azam University Islamabad, Pakistan
Number of Pages
Keywords (Extracted from title, table of contents and abstract of thesis)
cylindrically symmetric static spacetime, ricci collineations, einstein field equations, killing vectors, homothetic motions, lie derivative, ricci tensor

Symmetries are used in general relativity not only to find the exact solutions of the Einstein Field Equations (EFE), but some of them provide invariant bases for the classification of spacetimes as well. Killing vectors (KVs), homothetic motions (HMs) and Ricci collineations (RCs) are some of these symmetries. In this thesis RCs are used to serve both of these purposes. A complete classification of cylindrically symmetric static spacetimes according to their RCs is provided. After the introductory chapter, where a survey of the related literature is given, we have obtained the RCs for the non-degenerate as well as the degenerate Ricci tensor. We have also provided their Lie algebras. It is found that the Lie algebras of the RCs of these spacetimes, for the non-degenerate Ricci tensor have dimension ranging from 3 to 10 excluding 8 and 9. For t.he degenerate case the Lie algebras are mostly infinite dimensional. However, cases of the algebras of dimensions 3, 4, 5 and10 have also been found. The comparison of the RCs with the KVs and HMs has given rise to numerous interesting cases of proper (or non-isometric) RCs. Corresponding to each Lie algebra there arise differential constraints (mostly non-linear) on the metric coefficients. We have solved these constraints to construct examples of metrics which include some exact solutions admitting proper RCs. Their physical interpretation is also given. The classification of plane symmetric static spacetimes emerges as a special case of this classification when the cylinder is unfolded. Some results are summarized in the form of theorems in the concluding chapter.

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1859.4 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
78.38 KB
2 1 Introduction 4
173.92 KB
  1.1 The Einstein Field Equations 6
  1.2 The Lie Derivative 7
  1.3 Symmetries in General Relativity 9
  1.4 Cylindrical Symmetry 13
3 2 Ricci Collineations of Cylindrically Symmetric Static Space Times 15
568.06 KB
  2.1 The Ricci Collineatio Equations 15
  2.2 Ricci Collineation for the Non-Degenerate Ricci Tensor 16
  2.3 Ricci Collineations for the Degenerate Ricci Tensor 49
4 3 Lie Algebras of the Ricci Collineations 72
184.36 KB
  3.1 Lie Algebras of RCs for the Non-Degenerate Ricci Tensor 72
  3.2 Lie Algebras of RCs for the Degenerate Ricci Tensor 87
5 4 Metrics with their Isometrics and Ricci Collineations 91
158.16 KB
  4.1 Metrics with the Non-Degenerate Ricci Tensor 91
  4.2 Metrics with the Degenerate Ricci Tensor 98
6 5 Conclusion 106
915.44 KB
  5.1 Symmetries in General Relativity 115
  5.2 Ricci Collineations for the Non-Degenerate Ricci Tensor 119
  5.3 Ricci Collineations for the Degenerate Ricci Tensor 170