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Non-Tensors in GR

by thehangedman
Tags: nontensors
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thehangedman
#1
Dec13-11, 10:35 PM
P: 69
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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atyy
#2
Dec13-11, 11:53 PM
Sci Advisor
P: 8,781
The Levi-Civita connection can be extended to spinors (Winitzki, section 7.3.2).
haushofer
#3
Dec14-11, 09:16 AM
Sci Advisor
P: 913
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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