What math do I need to know to understand General Relativity

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Discussion Overview

The discussion centers around the mathematical prerequisites necessary for understanding General Relativity (GR). Participants explore various mathematical concepts and resources that could aid in grasping GR, including differential geometry, calculus, and linear algebra, as well as the relevance of special relativity and Lagrangian mechanics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • Some participants suggest that differential geometry is essential for understanding GR, along with its prerequisites.
  • Calculus and its foundational concepts are mentioned as necessary for approaching GR.
  • Linear algebra and differential equations are discussed, with some participants questioning their necessity for GR.
  • Several books are recommended, including "Exploring Black Holes" by Wheeler/Taylor and "Hartle's book," which are noted for their accessibility and pedagogical approach.
  • One participant expresses a desire to understand Wald's book, indicating a need for a solid mathematical foundation.
  • There are inquiries about the depth of topology knowledge required for GR, with some participants admitting limited understanding of topology.
  • Participants discuss the importance of special relativity and classical gravitation as prerequisites for GR.
  • Some participants mention that while topology and differential geometry may not be immediately necessary, they are important for a deeper understanding of GR.

Areas of Agreement / Disagreement

There is no consensus on the exact mathematical prerequisites for understanding GR, as participants express varying opinions on the necessity of different mathematical fields and resources. Some agree on the importance of differential geometry and calculus, while others highlight the role of special relativity and Lagrangian mechanics.

Contextual Notes

Participants express uncertainty about their own mathematical backgrounds and the depth of knowledge required for GR. There are references to various levels of understanding in topology and differential geometry, indicating that the discussion is influenced by individual experiences and educational backgrounds.

Who May Find This Useful

This discussion may be useful for students and individuals interested in pursuing General Relativity, particularly those seeking guidance on the necessary mathematical foundations and resources for their studies.

  • #31
As the general math background for physics, V.I. Arnold's Mathematical Methods of Classical Mechanics is the best.
 
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  • #32
Felix Quintana said:
I'm a 16 year old whose summer goal is two understand general relativity, but I'm lost on what math to have to understand it, I understand topological spaces and a topological manifold. but then it becomes more complicated math, and I know I simply don't understand because of the mathematics.

Calculus, basic linear(matrix) algebra, some differential equation familiarity, tensor algebra and analysis. Tensors are a part of differential geometry and are absolutely essential for understanding GR. Shaum's Tensor Calculus may be helpful, not really knowing more details. It includes needed linear algebra, and some other topics like tensor fields on manifolds. Its an outline, but should enable you to reach your goal. Don't be too discouraged. Einstein needed help on some math before he could complete his theory. :headbang:
 
  • #33
For SR try The Feynman lectures, vol. 1.
 
  • #34
This might seem strange but the most understandable book that I read was "The Large Scale Structure of Space-Time" by Hawkins and Ellis. Of course this was after I had gone through several other books that left an aura of mystery about a lot of things. In my mind that book nailed down a lot! Now I can read "Gravity" (which sort self reflects because it's weight certainly proves Gravity) without a hitch; and the details are just that details.
 
  • #35
micromass said:
Also, can you explain us why people care about the Hausdorff property? Can you explain why we care about compactness? Why do we let manifolds be second countable?

Sorry, but I want to gauge your topology knowledge.

I too am trying understand GR, and have read most of MTW. I can't recall any mention of the Hausdorff property in that tome. How is it necessary to understand curvature?

And to the OP, you can download Schutz for free from this site. It is a very good intro to GR.
 
  • #36
Kevin McHugh said:
I too am trying understand GR, and have read most of MTW. I can't recall any mention of the Hausdorff property in that tome. How is it necessary to understand curvature?

A spacetime is by definition Hausdorff. So it is already needed for the very definition of what we're working with. If MTW doesn't need the Hausdorff property, then MTW is just not a rigorous book. That's ok, I'm not saying that physics books need to be mathematically rigorous. But the OP mentioned Wald, and Wald definitely is rigorous (and does cover the Hausdorff property).
 
  • #37
micromass said:
A spacetime is by definition Hausdorff. So it is already needed for the very definition of what we're working with. If MTW doesn't need the Hausdorff property, then MTW is just not a rigorous book. That's ok, I'm not saying that physics books need to be mathematically rigorous. But the OP mentioned Wald, and Wald definitely is rigorous (and does cover the Hausdorff property).

:cool: Thanks for that micromass. You do learn something new every day.
 

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