Studies of Bertrands Theorem and Geodesic Congruences in Bertrand Spacetimes

Abstract

This thesis addresses five different problems about representations of Bertrand s newlinetheorem (BT) and Bertrand Spacetime (BST). Bertrand theorem has diverted from newlineclassical mechanics to general relativity, which is now celebrating BST. BT have many newlinemethods of proof by different physicist, one of the chapters we discussed. BST has newlinetwo different metrics, which is BST type-I (BST-I) and BST type-II (BST-II). In this newlineresearch, we have taken BST as our celebrity and performed. newlineBertrand s theorem is one of the landmark results in the context of the central newlineforce problem in mechanics. It leads to the conditions for closed bound orbits. This newlinepedagogical article intends to motivate the teacher as well as the student to engage in newlinea formal proof of the theorem. We overview the basics on central forces and the Abel newlineintegral. Thereafter, we obtain the condition for closed orbits. We provide the reader newlinewith a broad panoroma of areas/topics where Bertrand s theorem arises, including newlineinteresting scenarios in the regime of Einstein s general theory of relativity. newlineA new approach to general relativity is Newman Penrose (NP) formalism. A null newlinetetrad is chosen from several geometrical groups as the basis for tetrad formalism. This newlinemethod is commonly utilised in both analytical and numerical analyses of Einstein s newlineequations. The spinor relative association is the rational use of explicit complex straight newlineemulsions of Ricci rotation coefficients present in the product. In the general situation, newlinewe developed a null tetrad, with specific attention paid to Bertrand spacetimes metric newlinewith all the exceptional cases. newlineA class of BST-I from the perspective of geodesic deformations and energy newlineconditions is explored. The geodesics and ESR (expansion, shear, and rotation) newlineparameters used in the Raychaudhuri equation and geodesic deformations are newlineprincipally considered. The Ricci scalar and the energy-momentum tensor are newlinecomputed to study the energy conditions. The values that the spacetime parameters newlinecan take are constrained by the energy requiremen