Non-Tensors in GR: Affine Connections & Christoffel Symbols

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In summary, the conversation discusses the extension of the concepts of affine connection, Christoffel Symbols, and the Levi-Civita connection to non-tensor entities such as spinors. The spin connection is mentioned as a way to handle fields in non-tensorial representations of the Lorentz algebra. This is a topic that falls under pure mathematics and is related to the ideas of General Relativity.
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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).
 
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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 :)
 

FAQ: Non-Tensors in GR: Affine Connections & Christoffel Symbols

What are affine connections in general relativity?

Affine connections are mathematical objects that describe how to compare vectors at different points in a curved space in general relativity. They account for the fact that the geometry of spacetime is not flat, and are used to define parallel transport, which is necessary for defining the notion of a covariant derivative.

Why are affine connections important in general relativity?

Affine connections play a crucial role in general relativity because they allow us to define the concept of parallel transport, which is necessary for calculating the covariant derivative of a tensor field. The covariant derivative, in turn, is essential for understanding the behavior of objects moving in a curved spacetime, such as particles and light rays.

What are Christoffel symbols and how are they related to affine connections?

Christoffel symbols are a set of numbers that are used to describe the behavior of affine connections in a specific coordinate system. They are derived from the components of the metric tensor and are used to calculate the covariant derivative of a tensor field. The Christoffel symbols are related to the affine connections through a set of equations known as the geodesic equation.

How do affine connections and Christoffel symbols relate to the curvature of spacetime?

Affine connections and Christoffel symbols are used to calculate the curvature of spacetime in general relativity. The Riemann curvature tensor, which describes the curvature of spacetime, can be expressed in terms of the Christoffel symbols. This allows us to understand how the geometry of spacetime is affected by the presence of matter and energy.

Are affine connections and Christoffel symbols the only way to describe the curvature of spacetime?

No, there are other mathematical objects used to describe the curvature of spacetime in general relativity, such as the Ricci tensor and the Einstein tensor. However, affine connections and Christoffel symbols are important tools for calculating these tensors and understanding the behavior of objects in a curved spacetime.

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