Can general relativity be constructed with differential forms?

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Discussion Overview

The discussion centers on whether general relativity can be constructed using differential forms, exploring theoretical frameworks and references in mathematical physics. Participants share various texts and approaches related to this topic.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants assert that general relativity can be constructed with differential forms, referencing Takahara's book as a source.
  • Others express difficulty in locating Takahara's work in physics or mathematics, suggesting Nakahara's text instead.
  • A participant mentions Nakahara's book, asking if it specifically presents general relativity in terms of differential p-forms rather than just differential equations.
  • One participant claims that general relativity can indeed be expressed using bundle-valued differential forms and exterior covariant derivatives.
  • Another participant references N. Straumann's book, noting that it employs the tetrad formalism and includes a section on differential geometry.
  • It is pointed out that Straumann's work does not use fibre bundles but includes a footnote for readers familiar with them.
  • A participant recommends a review article by Eguchi, Gilkey, and Hanson, highlighting its relevance to the discussion.
  • Another participant mentions W. Thirring's textbook, which is written entirely in the exterior calculus format, as a relevant resource.

Areas of Agreement / Disagreement

Participants generally agree that general relativity can be constructed using differential forms, but there is no consensus on the specific texts or methods to be used, leading to multiple competing views and references.

Contextual Notes

Some limitations include the lack of clarity on specific definitions of differential forms in the context of general relativity and the varying approaches to the subject matter across different texts.

Phrak
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Can general relativity be constructed with differential forms?
 
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Yes, it can. See for example Takahara's book on mathematical physics.
 
bigubau said:
Yes, it can. See for example Takahara's book on mathematical physics.

I'm coming up empty on a search for Takahara in the categories of physics or mathematics.
 
Last edited:
Phrak said:
I'm coming up empty on a search for Takahara in the categories of physics or mathematics.

Try Nakahara.
 
Daverz said:
Try Nakahara.

On Amazon, I found the text Geometry, Topology and Physics, Second Edition (Graduate Student Series in Physics), M. Nakahara.

To be clear, does Nakahara express general relativity in terms of differential p-forms rather than simply differential equations?

thanks, in advance.
 
Phrak said:
Can general relativity be constructed with differential forms?

It can. With bundle-valued differential forms and bundle connections. You will need to use exterior covariant derivative.
 
Of course you can, try N. Straumann's book: General Relativity with applications to astrophysics. In some place around chapter 2, he work with the tetrad formlism and he have an entire part of Differential Geometry.

Greetings!
 
Straumann does it without fibre bundles, but on p. 111 has a footnote addressed to readers familiar with fibre bundles, so that they will recognize what he is doing using the language and the tools that are familiar to them.
 
Last edited:
I hope I don't mispell their names, but if you have access to Physics Reports, there's a famous review article by Eguchi, Gilkey and Hanson that I would also recommend for your attention.
 
  • #10
"Gravitation, gauge theories and differential geometry", Tohru Eguchi, Peter B. Gilkey, Andrew J Hanson, Physics Reports 66 no 6 (1968) 213-393. Section 3 deals with Riemannian manifolds.
 
  • #11
Yes; Here is a textbook on field theory and entirely written in the exterior calculus format :
W. Thirring, A course in Mathematical Physics, Dynamical systems and Fields (Springer)
 

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