Why can the affinity connection be arbituarily chosen?

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In summary, an affine connection is a quantity that is defined within a chosen coordinate frame and its components can be arbitrarily chosen within that frame. However, once these components are determined, the components with respect to any other frame are fixed by the transformation law. This is because an affine connection is defined to satisfy certain properties, and choosing arbitrary functions as the Christoffel symbols satisfies these properties. To define parallel displacement, the affine connection can be represented as F=g[ij,k], where [ij,k] is the Christoffel symbol of the first kind, g is the metric, and F is the affine connection.
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Lucien1011
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In covariant derivative, we have a quantity called affinity.
The book says: When defining an affine connection, a coordinate frame must first be selected and the choice of the components of the affinity is then arbitrary within this frame. However, when these are all determined, the component of the affinity with respect to any other frame are completely fixed by the transformation law.
My question is why can the components be arbitrarily chosen?
Thank you very much!
 
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Why not? An affine connection is defined to satisfy certain properties. Choosing arbitary functions as the Christoffel symbols satisfies these properties.
 
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zhentil said:
Why not? An affine connection is defined to satisfy certain properties. Choosing arbitary functions as the Christoffel symbols satisfies these properties.
So what is the definition of affine connection?

Am I correct to say that to define the parallel displacement such that the vector displaced does not change in magnitude and direction, we have to define F=g[ij,k] (ignore the superscript and subscript) as our affine connection?
where [ij,k]is the Christoffel symbol of the first kind and g is the metric and F is the affine connection
 
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1. Why is the affinity connection in scientific experiments chosen arbitrarily?

The affinity connection, also known as the binding strength between molecules, is often chosen arbitrarily in scientific experiments because it allows for easier manipulation and control of the experiment. By choosing an arbitrary value, scientists can simplify their experiments and focus on the specific variables they are investigating.

2. How does choosing an arbitrary affinity connection affect the accuracy of scientific results?

Choosing an arbitrary affinity connection can potentially affect the accuracy of scientific results as it may not accurately represent the real-world scenario. However, scientists often use multiple affinity connections to compare results and ensure the validity of their findings.

3. Can the affinity connection be changed during a scientific experiment?

Yes, the affinity connection can be changed during a scientific experiment. This allows scientists to observe the effects of different affinity connections on their results and draw more accurate conclusions.

4. Are there any limitations to choosing an arbitrary affinity connection in scientific experiments?

One limitation of choosing an arbitrary affinity connection is that it may not accurately reflect the complexity of real-life scenarios. Additionally, it may be difficult to replicate the same results when using different affinity connections.

5. How do scientists determine which affinity connection to use in their experiments?

Scientists often base their decision on the current understanding of the molecules being studied and previous research on their binding strengths. They may also use computer simulations and models to predict the effects of different affinity connections before conducting their experiments.

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