Gauge-Field Connections vs. Connections in General Relativity [Higher and More Elegant Electrodynamics · Extra 1]

hfwang132@gmail.com (hfwang132) — Tue, 29 Jul 2025 15:08:10 +0800

In a previous article, we said that the electromagnetic field is a connection. This led many readers to think of the connection in general relativity. What are the differences and commonalities between these two kinds of connections?

The electromagnetic connection $A$ is a connection on a principal bundle, whereas the general-relativistic connection $\Gamma$ is a connection on a vector bundle. Their definitions appear to be quite different. Is there a way to relate them?

Are Christoffel Symbols Tensors After All?

hfwang132@gmail.com (hfwang132) — Sun, 27 Apr 2025 23:12:31 +0800

My favorite way to understand this is to regard $\Gamma$ as a connection on the frame bundle. Then, the transformation law of the Christoffel symbols

$\boxed{ \begin{aligned} \bar{\Gamma}^i_{\mu j} = \frac{\partial \bar{x}^i}{\partial x^k}\frac{\partial x^l}{\partial \bar{x}^j} \frac{\partial x^\nu}{\partial \bar{x}^\mu} \Gamma^k_{\nu l} \color{red}{ + \frac{\partial \bar{x}^i}{\partial x^k} \frac{\partial^2 x^k}{\partial x^\mu \partial \bar{x}^j}} \end{aligned} }$

is nothing more than a gauge transformation:

$\boxed{ \bar{A} = g^{-1}A g \color{red}{+ g^{-1} \mathrm{d}g} }$

The extra inhomogeneous term $ g^{-1} \mathrm{d}g$ is inherently part of a gauge transformation.

General Relativity on Haifei's Home

Gauge-Field Connections vs. Connections in General Relativity [Higher and More Elegant Electrodynamics · Extra 1]

Are Christoffel Symbols Tensors After All?