Tensors and differential geometry

[Qazsxwced]

Hi, I've decided to learn GR myself recently since it's like the "sexy" side of physics. But I'm getting stuck with the tensors notations already. Maybe my math background is just not sufficient enough to do GR.

In general, how do I know that an object is tensorial; for example, objects like Ta|b-Tb|a or Ta||b - Tb||a (| for partial derivatives, || for covariant derivatives and Ta or Tb are just the covariant vector components.) Thanks.

Also, can anyone recommend some good intro books on tensors and differential geometry? Maybe I should learn those concepts before going into GR...

 

[atyy]

Eric Poisson's notes:
http://www.physics.uoguelph.ca/~poisson/research/agr.pdf

Sergei Winitzki's notes:
http://homepages.physik.uni-muenchen.de/~winitzki/T7/GR_course.html

My favourite text is Crampin and Pirani's "Applicable Differential Geometry". Their presentation is absolutely logical, and the don't skip any steps, which means it's a slow read. But when learning on my own I usually get stuck when someone has missed a step or done a quick and dirty proof, so I appreciate their approach.

 

[Entropia]

Hi there,

Tensors and differential geometry are indeed fundamental concepts in general relativity, and it's great that you are interested in learning them. It is common for people to struggle with the notation and concepts of tensors, especially if their math background is not strong enough. However, with practice and patience, you will be able to grasp these concepts.

To answer your question, an object is considered tensorial if it transforms according to certain rules under coordinate transformations. In other words, if the components of the object change in a specific way when the coordinates are changed, then it is a tensor. For example, the components of a covariant vector (Ta) will transform differently from the components of a contravariant vector (Tb) under a coordinate transformation. This transformation property is what makes an object tensorial.

As for recommendations for introductory books on tensors and differential geometry, some good options are "A Geometric Approach to Differential Forms" by David Bachman and "Tensor Analysis on Manifolds" by Richard L. Bishop and Samuel I. Goldberg. It is always helpful to have a good understanding of these concepts before delving into general relativity, but it is not necessary. You can also learn these concepts alongside GR, as long as you are willing to put in the effort to understand the notation and concepts.

I hope this helps and wish you all the best in your learning journey. Keep practicing and don't give up, and you will eventually be able to master tensors and differential geometry. Good luck!