Ricci Tensor: Understanding the Mathematics & Concepts

[mathsTKK]

Hi everyone, I am new to PF but truly appreciate the kind assistance from all people in this forum.

I am very interested in learning Relativity as I really want to know it essence either in its Physics or Mathematics. I have a little basics on integration and some vector calculus. Amazed by the geometrical visualisation of Einstein Field Equations, I want to know what is the meaning of the equations, mathematically. However, I come across a lot of terms which are aliens to me such as Ricci Tensor, Riemann curvature tensor etc.

So, I wish that I can start by understanding what is a ricci tensor, what does it tell and what is the mathematics behind the expression especially the conceptual background of the term. I would be thankful if someone can help and guide me in learning this great theory.


Thank you for your kind assistance !

 

[homeomorphic]

Maybe the first thing to note is that in the Einstein field equation the tensors that are used are analogous, but not the same as those traditionally used in differential geometry (Riemannian geometry). Of course, differential geometers also sometimes study pseudo-Riemannian geometry which includes the case of the Lorentzian manifolds of general relativity.

On the conceptual side, you can try this introduction.

http://math.ucr.edu/home/baez/einstein/

The best description of Riemannian curvature that I have seen by far is in an appendix of Mathematical Methods of Classical Mechanics by Arnold.