About the tensorial nature of the Christoffel symbols, or lack thereof

TL;DR

This paper clarifies why Christoffel symbols are not tensors despite some textbooks referring to them as such, providing insights into the mathematical foundation of metric connections in General Relativity.

Contribution

It offers a detailed analysis explaining the non-tensorial nature of Christoffel symbols, addressing misconceptions in standard textbooks and deepening understanding of metric connections.

Findings

Christoffel symbols do not transform as tensors.

Standard textbooks often incorrectly refer to Christoffel symbols as tensors.

The analysis clarifies the mathematical reasons behind the non-tensorial transformation.

Abstract

From the first approach to General Relativity we learn about the Christoffel symbols, and it is easy to notice that their coordinated components do not transform as a tensor. It is then confusing to find renowned books on the subject that refer to these symbols as if they were indeed a tensor. A prominent example of such a book is "GeneralRelativity" written by Rober Wald, which has been embraced as a textbook on the matter by a large amount of prestigious institutions around the globe. The simplest alternative would be to think that the books just mentioned present a wrong description of the Christoffel symbols, but this is not the case. Understanding the mathematical foundation of why some authors treat the Christoffel symbols as a tensor provides valuable insight about the nature of the metric connection in General Relativity, and that is why in this paper we dedicate the space to the needed analysis. The discussion provided here is aimed to those that have already taken at least one class in General Relativity, hence some of the theoretical tools will be taken as known by the reader.