

Title of Thesis
RICCI COLLINEATIONS OF CYLINDRICALLY SYMMETRIC STATIC SPACETIMES 
Author(s)
Khalid Saifullah 
Institute/University/Department Details
Department of Mathematics/ QuaidiAzam University Islamabad 
Session

Subject
Mathematics 
Number of Pages
187 
Keywords (Extracted from title, table of contents and abstract of thesis)
ricci collineations, cylindrically symmetric static spacetimes, einstein field equations, spacetimes, killing vectors, homothetic motions, ricci collineations, ricci tensor 
Abstract 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 nondegenerate 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 nondegenerate Ricci tensor have dimension ranging from 3 to 10 excluding 8 and 9. For the degenerate case the Lie algebras are mostly infinite dimensional. However, cases of the algebra of dimensions 3, 4, 5 and 10 have also been found. The comparison of the RCs with the KVs and HMs has given rise to numerous interesting cases of proper (or nonisometric) RCs. Corresponding to each Lie algebra there arise differential constraints (mostly nonlinear) 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.






S. No. 
Chapter 
Title of the Chapters 
Page 
Size (KB) 





1 
0 
Contents 

112.55 KB 





2 
1 
Introduction 
4 
285.3 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 Spacetimes 
15 
285.3 KB 

2.1 
The Ricci Collineation Equations 
15 

2.2 
Ricci Collineations For The NonDegenerate Ricci Tensor 
16 

2.3 
Ricci Collineations For The Degenerate Ricci Tensor 
49 





4 
3 
Lie Algebras Of The Ricci Collineations 
72 
250.8 KB 

3.1 
Lie Algebras Of RCS For The NonDegenerate Ricci Tensor 
72 

3.2 
Lie Algebras RCS For The Degenerate Ricci Tensor 
87 





5 
4 
Metrics With Isometries And Ricci Collineations 
91 
218.79 KB 

4.1 
Metr4ics With The NonDegenerate Ricci Tensor 
91 

4.2 
Metrics With The Degenerate Ricci Tensor 
98 





6 
5 
Conclusion 
115 
1267.07 KB 

5.1 
A Symmetries In General Relativity 
115 

5.2 
Ricci Collineations For The NonDegenerate Ricci Tensor 
119 

5.3 
Ricci Collineations For The Degenerate Ricci Tensor 
170 





