# Christoffel Symbols - Gauge Fields

• tunafish
In summary, two questions were asked: 1) What kind of gauge group is used in general relativity? and 2) What are the generators of the Lie algebra in a gauge theory? The first question was answered by saying that the gauge group is the Lorentz group, and the generators of the Lie algebra are the vierbeins. The second question was answered by saying that the generators of the Lie algebra are the components of the covariant derivative in terms of the vierbeins.
tunafish
Hi everyone! Two question for you ():

1) I know that General relativity may also be seen as a gauge theory, but which kind of gauge group is used there??

2) In the gauge theory wiew the Christoffel symbols $\Gamma^{\alpha}_{\mu\kappa}$ in the covariant derivative $$\nabla_{\mu}\vec{U}=\left(\frac{\partial U^{\alpha}}{\partial x^{\mu}}+U^{\kappa}\Gamma^{\alpha}_{\mu\kappa} \right)\vec{e}_{\alpha}$$ takes the role of the gauge fields, and so I (should) be able to express them in function of the generators of the Lie algebra, but what kind of Lie algebra am I supposed to use? And what are its generators??Thanks for your help! (first post!)

Hi, and welcome to PF!
The clearest way to see GR as a gauge theory is to formulate it in terms of an orthonormal basis of vectors $e_\alpha$, satisfying $g(e_\alpha,e_\beta)=\eta_{\alpha\beta}$. [The components $e_\alpha^\mu$ are sometimes called vierbeins. I use Greek letters from the start of the alphabet to denote the basis vectors and components in terms of them, and from the middle of the alphabet to denote a coordinate basis.]

There is then a gauge symmetry in that we can take a Lorentz transformation independently at each point in spacetime. The gauge group is the Lorentz group.

The covariant derivative can then be expressed as
$$\nabla_\mu U=\nabla_\mu (U^\alpha e_\alpha)=(\partial_\mu U^\alpha)e_\alpha+U^\alpha \nabla_\mu e_\alpha=(\partial_\mu U^\alpha+\Gamma^\alpha_{\mu\beta}U^\beta)e_\alpha$$

Notice that there are a mixture of indices from the coordinates and the orthonormal basis in the Christoffel symbols: the $\alpha,\beta$ indices are very much like the internal, Lie algebra indices in an ordinary gauge theory, and Gamma the gauge fields.

Thanks for your time henry_m, I didn't expect a rensponse so fast!

You obviously say that since the coordinate transformation (an consequentely the change of basis they induce on the manifold) does not affect any phenomena (physic is the same for all observer) the lorentzian group, the group of all the coordinate transormation is the gauge group for GR.

It makes sense: a coordinate transformation doesn't affect the lagrangian, the symmetry is rigid in a minkowsky spacetime and local on a curved manifold, so I need to define a covariant derivative (which "connects" all the tangent spaces by defining when a vector is parallel transported), which has the form I've written above.

In this wiew the Christoffel symbols are effectively gauge field.

All I've written above is what i thought to be true since a few days ago, when I red that the gauge group for GR is SL (of which I've never heard before), which has something to do whit spinors.

How is that?

## 1. What are Christoffel symbols and how are they related to gauge fields?

Christoffel symbols are mathematical objects that represent the curvature of a space. They are used in differential geometry to describe the behavior of vector fields on a manifold. Gauge fields are a type of vector field that play a crucial role in gauge theories, which are used to describe the interactions between elementary particles. Christoffel symbols are used in the definition of gauge fields and play a key role in understanding their behavior.

## 2. Why are Christoffel symbols important in theoretical physics?

Christoffel symbols are important in theoretical physics because they allow us to describe the curvature of space and the behavior of gauge fields. Gauge fields are essential in understanding the fundamental forces of nature, such as electromagnetism and the strong and weak nuclear forces. By using Christoffel symbols, we can better understand the underlying geometry of space and how it relates to the behavior of gauge fields.

## 3. How are Christoffel symbols calculated?

Christoffel symbols are calculated using the metric tensor, which describes the geometric properties of a space. The metric tensor is used to calculate the Riemann curvature tensor, which then allows us to calculate the Christoffel symbols. This involves taking derivatives of the metric tensor and performing some algebraic operations. The resulting Christoffel symbols can then be used to calculate the behavior of gauge fields.

## 4. Are Christoffel symbols used in any real-world applications?

Yes, Christoffel symbols are used in many real-world applications, particularly in the field of theoretical physics. They are an essential tool in understanding gauge theories, which are used to describe the fundamental forces of nature. They are also used in general relativity to describe the behavior of gravity. Additionally, Christoffel symbols have applications in other fields such as computer graphics and robotics.

## 5. Can you explain the difference between covariant and contravariant Christoffel symbols?

Covariant and contravariant Christoffel symbols are two different ways of representing the same mathematical object. The difference lies in the way they transform under a change of coordinates. Covariant symbols are used to describe the behavior of objects that are measured using a fixed coordinate system, while contravariant symbols are used to describe the behavior of objects that are measured using a moving coordinate system. In other words, covariant symbols are used to describe the behavior of objects from an observer's point of view, while contravariant symbols are used to describe the behavior of objects from the object's point of view.

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