Tensors and General Relativity

In summary: I think it's safe to say that you don't need to be an expert in tensors to understand the math behind general relativity. However, if you want to understand the math in more depth, more foundational texts will make heavier demands on your understanding.
  • #1
pierce15
315
2
Hello all,

I will preface this post with an apology for not putting it in the math/science learning materials section. This would have been the best place to post my question, but for some reason I can't post there.

My question is the following: what depth of understanding must I have of tensors in order to start going through the math behind general relativity? Also, if anyone knows of any particularly good books for learning about tensors, feel free to share them. (I bought this book: https://www.amazon.com/dp/0486658406/?tag=pfamazon01-20, but I'm having a terribly difficult time following it; I strongly agree with the reviewer who wrote that he got lost in the gallimaufry of summations).

Edit: sorry for typos
 
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  • #2
Most texts on General Relativity (eg Carroll's Spacetime and Geometry) don't assume a knowledge of tensors and instead introduce the topic themselves.

The knowledge of tensors you need to start doing GR is very, very simple. If you know what a linear map is you're halfway there. Carroll on spends a few pages on the subject, and IMO that's all you really need. It's not necessary to buy a whole book on the subject.
 
  • #3
Thanks for the reply. I'll look into that textbook
 
  • #4
piercebeatz said:
My question is the following: what depth of understanding must I have of tensors in order to start going through the math behind general relativity?

This depends on how rigorous the GR textbook you want to use actually is and how deep into the foundations of GR you want to go-more foundational topics and more rigorous textbooks will make heavier prerequisite demands of the reader.

For an introduction to GR, say at the level of Hartle, Schutz, or Carroll (Carroll is slightly higher level than Schutz) you don't need to worry much (if at all) about the math in the text, you can just jump right in. Hartle is my most favorite introductory GR text with Schutz coming in a very close second. However none of the three aforementioned texts go very deep into the foundations of GR and certainly none of them are mathematically rigorous.
 
  • #6
Well if that book by itself doesn't introduce tensor algebra and tensor calculus (i.e. it presupposes knowledge of them) then you can always make use of the numerous GR lecture notes available online that do introduce these tools in the context of GR. Here's a personal favorite of mine: http://www.physics.uoguelph.ca/~poisson/research/agr.pdf (it's not rigorous but it covers all the necessary tools and concepts).
 

1. What is a tensor in the context of general relativity?

A tensor is a mathematical object that represents physical quantities such as mass, energy, momentum, and electromagnetic fields in the theory of general relativity. Tensors are used to describe the curvature of space-time and how matter and energy interact within it.

2. How is a tensor different from a vector or a matrix?

A tensor is a more general concept than a vector or a matrix. While vectors have magnitude and direction, and matrices have multiple rows and columns, tensors can have an arbitrary number of indices and can represent transformations between different coordinate systems. In general relativity, tensors are used to describe the curvature of space-time and how it is affected by matter and energy.

3. How are tensors used in the equations of general relativity?

Tensors are used extensively in the equations of general relativity, particularly in Einstein's field equations. These equations describe the relationship between the curvature of space-time and the distribution of matter and energy within it. Tensors are also used in the geodesic equation, which describes the motion of particles in curved space-time.

4. Can tensors be visualized or represented graphically?

While tensors themselves cannot be visualized or represented graphically, their components can be represented using diagrams known as tensor diagrams. These diagrams can help illustrate the relationships between different components of a tensor and how they transform under different coordinate systems.

5. How does the concept of tensors relate to the theory of general relativity?

The concept of tensors is central to the theory of general relativity. Tensors are used to mathematically describe the curvature of space-time and how it is affected by matter and energy. Without tensors, it would be impossible to accurately describe the complex relationships between space, time, and matter in Einstein's theory of general relativity.

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