What physical meaning do the Christoffel symbols (of GR) have?  by Dale Gray  Quora Question Review
This document contains a review of the answer by Steve Baker on the question in Quora: "What physical meaning do the Christoffel symbols (of GR) have?"
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https://www.quora.com/WhatphysicalmeaningdotheChristoffelsymbolsofGRhave
 The text in italics is copied from the article.
 Immediate followed by some comments
Contents
Reflection
1. Answer Review

Christoffel symbols have no physical meaning, in and of themselves. In other words, Christoffel symbols do not represent physical quantities.

That is an important conclusion about physics in general.

Only scalar, vector, or tensor quantities have physical meaning in the sense of representing physical quantities, and Christoffel symbols are not the components of tensors, despite the fact that they have indices.

Okay.

However, Christoffel symbols do occur in some tensor quantities. For example, the components of the Riemann curvature tensor are constructed from Christoffel symbols and their derivatives. Christoffel symbols also occur in covariant derivatives of vector and tensor quantities, which are also tensors.
There is a context in which Christoffel symbols have a “geometric” meaning as a local representation of something called a connection on the bundle of linear frames of the spacetime manifold.

THis raises the question: What is the physicasl meaning of the Riemann curvature tensor



Christoffel symbols do not just occur in General Relativity.
They occur wherever it is necessary or convenient to consider curvilinear coordinates. In General Relativity it is necessary to consider curvilinear coordinate over extended regions of spacetime.
Because gravitation is due to curvature of spacetime, there are no rectangular coordinate systems in the large; only in infinitesimal regions, and then in approximation only.

This raises an important practical problem: In order to explain that a circle is round you need a square (or something rectangular). The same with anything that is curved. That means in order to explain that something is curved, you need something that is not curved.
My interpretation as such that the concept curvature (of spacetime) does not explain anything. It is even worse by adding the concept spacetime it makes the concept gravitation even more difficult.



The principle of general covariance, incorporated in General relativity, requires that the laws of physics shall have the same mathematical form in all coordinate systems.

What have the laws of physics to do with coordinate systems? Why do you need coordinate systems (plural) in the first place? What are the laws of physics.

This is accomplished by writing the equations which describe physical phenomena as tensor equations.


The equations of physics are usually differential equation, so differentiation of vector and tensor quantities must take into account the fact that the components of these quantities are different in different coordinate systems.
This is so because the basis vectors for curvilinear coordinates, unlike the case of rectangular coordinates, are functions of position.

How are these functions calculated? How are these positions measured?
This are very important practical questions?

What is needed is a differential calculus which is form invariant under change of coordinates, we could call it an absolute differential calculus.
In fact that was a name originally given to the subject we now call tensor analysis.





















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Reflection 1  Question Review
The question "What physical meaning do the Christoffel symbols (of GR) have?" is important in related to the issue how important mathematics is in order to understand the physical reality and or the processes in the universe.
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Created: 1 June 2023
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