Gravity in Noncommutative Geometry

Please enable javascript in your browser.

Karimi Abadchi, Mohammad Javad | 2011

611 Viewed

Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 44253 (02)
University: Sharif University of Technology
Advisor(s): Esfahani Zadeh, Mostafa
Abstract:
In this thesis, the aim is to stating some of the noncommutative geometry(NCG) applications in formulating and describing the gravity. So, in the first chapter, we will work on some of metric aspects of NCG. In this chapter, we will review elements of the spin geometry, specifically we will define the Dirac operator which is the most important object in this thesis. Then, we will try to find counterparts of some of Riemannian geometric notions like integral and distance in the NCG framework. We will define some notions such as noncommutative infinitesimals, Dixmier trace, Wodzicki residue, noncommutative integral and spectral triples. Then, in the second chapter, we will work on some of the NCG applications in the formulating of gravity. In this chapter, at first, we will state the relationship between Dirac operator, Wodzicki residue and gravity. Second, we will state the spectral action principle and finally, we will compute the spectral action for Robertson-Walker metrics which are one of the most important cosmological models, based on the new method invented by A.Connes and A.H.Chamseddine
Keywords:
Dirac Operator ; Dixmier Trace ; Wodsicki Residue ; Hilbert-Einstein Action ; Spectral Action ; Robertson-Walker Metric