Non-Tensors in GR: Affine Connections & Christoffel Symbols

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SUMMARY

The discussion centers on the extension of General Relativity (GR) concepts, specifically affine connections and Christoffel symbols, to non-tensor entities such as spinors. It is established that while GR primarily utilizes tensors, the Levi-Civita connection can be adapted for spinors, as detailed in Winitzki's section 7.3.2. Additionally, the spin connection is crucial for managing fields represented in non-tensorial forms within the Lorentz algebra in tangent spaces. Carroll's notes on GR, particularly chapter 3, provide further insights into these extensions.

PREREQUISITES
  • Understanding of General Relativity concepts, particularly tensors
  • Familiarity with affine connections and their mathematical implications
  • Knowledge of spinors and their role in physics
  • Basic comprehension of the Lorentz algebra and its applications
NEXT STEPS
  • Study the Levi-Civita connection in detail, focusing on its application to spinors
  • Explore the spin connection and its significance in non-tensorial representations
  • Review Winitzki's section 7.3.2 for a deeper understanding of affine connections
  • Examine Carroll's notes on GR, especially chapter 3, for practical examples and applications
USEFUL FOR

This discussion is beneficial for theoretical physicists, mathematicians specializing in differential geometry, and students exploring advanced topics in General Relativity and its applications to non-tensor entities.

thehangedman
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I know that GR deals exclusively with tensors (at least, in every book I have), but how does the same concepts of affine connection etc extend to non-tensor entities? Example would be a spinor, or even a mix of spinor / tensor. Are there different affine connections? Something different yet related (somehow) to Christoffel Symbols?

This is probably more of a pure mathematics question, but not all things in physics are tensors, and I was wondering how GR's ideas extend to these other mathematical entities.
 
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The Levi-Civita connection can be extended to spinors (Winitzki, section 7.3.2).
 
The spin connection handles with fields living in "non-tensorial" representations of the Lorentz algebra in the tangent space. See e.g. Carroll's excellent notes on GR, chapter 3 :)
 

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