From Christoffel symbols to connection 1-forms

In summary, the conversation discusses the relation between Christoffel symbols and connection 1-forms, and how to find the latter from the former. The importance of using a non-coordinate basis and the definition of the connection 1-form as a matrix-valued one-form are also mentioned. Additionally, sources for further study on the topic are suggested.
  • #1
cosmicstring1
24
0
I have Christoffel symbols for a metric and I want to find the connection 1-forms.

I have the relation:

w(^i j)=Chr(^i j k)*dx(^k)

w: conn. 1-form
Chr: Christoffel symbols

But Christoffel symbols do not share the symmetries of the conn. 1-forms. Do you know any way to make this possible?
 
Physics news on Phys.org
  • #2
Have you studied the relevant chapter in Misner, Thorne, and Wheeler, Gravitation? Or the textbook by Harley Flanders, Differential forms, with applications to the physical sciences? These books offer clear introductions to computing the connection one-forms, as do many others.
 
  • #3
cosmicstring1 said:
I have Christoffel symbols for a metric and I want to find the connection 1-forms.

I have the relation:

w(^i j)=Chr(^i j k)*dx(^k)

w: conn. 1-form
Chr: Christoffel symbols

But Christoffel symbols do not share the symmetries of the conn. 1-forms. Do you know any way to make this possible?
I'm not aware of anything called the connection 1-forms. All I've heard are Chjristoffel symbols of the first and second kind. Are you sure that is not what you're referring to?

Pete
 
  • #4
Your relation is incorrect.

You're going to have to pick a set of orthonormal basis vectors compatible with your metric, then either compute the connection from them, or use them with your Christoffel symbols to get the connection.

This is laid out in detail here:

http://deferentialgeometry.org/#[[Christoffel symbols]]
 
  • #5
cosmicstring1 said:
I have Christoffel symbols for a metric and I want to find the connection 1-forms.

I have the relation:

w(^i j)=Chr(^i j k)*dx(^k)

w: conn. 1-form
Chr: Christoffel symbols

But Christoffel symbols do not share the symmetries of the conn. 1-forms. Do you know any way to make this possible?

Firstly, begin by taking [itex]\{\hat{\theta}^i\}[/itex] to be a non-coordinate basis. This non-coordinate basis is related to the coordinate basis [itex]\{dx^i\}[/itex] by

[tex]\hat{\theta}^i = \theta^i_{\phantom{i}j}dx^j[/tex]

for some matrices [itex]\theta^i_{\phantom{i}j}[/itex]. Then the relationship between the connection one-form and the Christoffel symbols is

[tex]\omega^i_{\phantom{i}j} = \Gamma^i_{\phantom{i}jk}\hat{\theta}^k[/tex]

Thus, if you know the Christoffel symbols and the relationship between the non-coordinate and coordinate bases, the answer to your question is to simply read off the value of the components of the connection one form from the above.

pmb said:
I'm not aware of anything called the connection 1-forms. All I've heard are Chjristoffel symbols of the first and second kind. Are you sure that is not what you're referring to?

Pete

He was perfectly clear when he said he was interested in the components of the connection one-form. This is a basic idea covered in, for example, the first few lectures of any decent differential geometry course.
 
  • #6
pmb_phy said:
I'm not aware of anything called the connection 1-forms.
Consider the frame bundle, which is a principal SO(n)-bundle. (At least, it is for a Riemannian manifold)

If you have any curve, you can take a frame at one endpoint and invoke your connection to parallel transport it to become a frame at the other endpoint. In particular, we get an element of SO(n) detailing how the frame was rotated/twisted/whatever as it was parallel transported along the curve.

If we differentiate (i.e. if we parallel transport by an infinitessimal amount), then to each tangent vector at a point, we get an element of so(n). Since this gives an so(n) value to each tangent vector, and is a linear functional, we call this object a "so(n)-valued one-form".

This so(n)-valued one-form completely determines parallel transport (you integrate it along a curve to get an SO(n) value), thus the term "connection 1-form".
 
  • #7
Your relationship is correct, although i don't like to call them the christoffel symbols, i prefer to use the term connection co-efficients since they rely on the second half of the levi-civita connection to calculate.

your wijk = 1/2( g(ek,[ei,ej]) - g(ei,[ej,ek]) - g(ej, [ek,ei]) )

where ei, ej, ek are the othronormal vectors prescirbed by your metric.
 
  • #8
Chris Hillman said:
Have you studied the relevant chapter in Misner, Thorne, and Wheeler, Gravitation? Or the textbook by Harley Flanders, Differential forms, with applications to the physical sciences? These books offer clear introductions to computing the connection one-forms, as do many others.

Yes, I read that section from MTW but it does not give the answer. I do'nt have Flanders' book.
 
  • #9
cosmicstring1 said:
I have Christoffel symbols for a metric and I want to find the connection 1-forms.

I have the relation:

w(^i j)=Chr(^i j k)*dx(^k)

w: conn. 1-form
Chr: Christoffel symbols

But Christoffel symbols do not share the symmetries of the conn. 1-forms. Do you know any way to make this possible?
What is a connection 1-form. Please defined the term for me.

Thank you

Pete
 
  • #10
pmb_phy said:
What is a connection 1-form. Please defined the term for me.

Thank you

Pete

Well, presumably you're familiar with the idea of using a coordinate basis [itex]\{dx^i\}[/itex] in differential geometry, right? Presumably you're also familiar with the idea that in many practical situations it's a far smarter idea to use a non-coordinate basis [itex]\{\hat{\theta}^i\}[/itex]. Well, if you know about these things it's a simple step to define a matrix-valued one-form [itex][\omega^i_{\phantom{i}j}][/itex] as

[tex]\omega^i_{\phantom{i}j} \equiv\Gamma^i_{\phantom{i}jk}\hat{\theta}^k[/tex]

That's really all there is to it. Once you have defined the matrix-valued connection one-form like this, it's a simple matter to go on and show that a neat and tidy way to express the curvature and torsion associated with a given metric is to use the Cartan structure equations.
 
  • #11
Hi, cosmicstring, you asked:

cosmicstring1 said:
I have Christoffel symbols for a metric and I want to find the connection 1-forms.

I suggested starting from a frame field, a la Cartan, as explained in MTW. I didn't answer the question you asked, I told you where to find the answer to the question you should have asked. You replied:

cosmicstring1 said:
Yes, I read that section from MTW but it does not give the answer.

Sure it does--- look at the sections on computing curvature. This contains what you need to answer the question you asked, as well as showing you a much better way to do all this stuff.

cosmicstring1 said:
But Christoffel symbols do not share the symmetries of the conn. 1-forms. Do you know any way to make this possible?

I don't think that even makes sense!

But there is one small point concerning the symmetry properties of the matrix of connection one-forms which MTW slur over. Flanders doesn't cover it either because he only discusses Riemannian manifolds. Namely, the rule for adjoint is a bit different for Lorentzian vice Riemannian. You need to know this when you use the symmetries of the connection one-forms in order to read off the connection one-forms from the exterior derivatives of the basis one-forms, as suggested by Cartan.

By definition, if [itex]A[/itex] is some linear operator and [itex](\cdot , \cdot)[/itex] is some inner product, the adjoint [itex]A^\ast[/itex] is the linear operator satisfying
[tex] (Ax, y) = (x, A^{\ast} y)[/tex]
for all [itex]x,y[/itex]. If you write this out in matrix notation, with the inner product written [itex] (x,y) = x^T L y[/itex], you obtain
[tex] x^T \, A^T \, L \, y = x^T \, L \, A^{\ast} \, y [/tex]
This holds for all [itex]x,y[/itex], so
[tex] A^\ast = L^{-1} \, A^T \, L[/tex]
Now apply this when [itex] L = \operatorname{diag} (-1,1,1,1)[/itex] to obtain the Lorentzian transpose. Now apply this to your matrix of one-forms when reading off the connection one-forms as suggested by Cartan.

In the past I have gone through all this stuff in great detail--- you can Google for old posts by myself in sci.physics.*
 
Last edited:
  • #12
coalquay404 said:
He was perfectly clear when he said he was interested in the components of the connection one-form. This is a basic idea covered in, for example, the first few lectures of any decent differential geometry course.
I didn't say he wasn't clear. I mentioned my uncertainty about them once before but it now appears that such things exist, hence the new question, i.e. what are they? That was why I was asking him. No textbook that I have on this subject matter even mentions is. Can you please give me an example? Thank you.

Pete
 
  • #13
Never mind. I finally found it defined in Lovelock and Rund. Thanks for the help folks. :smile:

Pete
 

What are Christoffel symbols?

Christoffel symbols are a set of mathematical quantities used in differential geometry to describe the curvature of a manifold. They are named after mathematician Elwin Bruno Christoffel.

What is the significance of Christoffel symbols in understanding connections?

Christoffel symbols are used to calculate the covariant derivative, which is a measure of how a vector changes as it is transported along a certain path on a manifold. This is essential in understanding the concept of connections in differential geometry.

What is the relationship between Christoffel symbols and connection 1-forms?

Christoffel symbols can be used to construct connection 1-forms, which are mathematical objects that describe how vectors change when parallel transported along curves on a manifold. Connection 1-forms are a fundamental tool in studying connections in differential geometry.

Why are connection 1-forms important in physics?

In physics, connection 1-forms are used to describe the curvature and geometry of spacetime in the theory of general relativity. They are also important in gauge theories, which are used to describe fundamental interactions between particles in the standard model of particle physics.

How are Christoffel symbols and connection 1-forms related to the concept of parallel transport?

Christoffel symbols and connection 1-forms are essential in understanding the concept of parallel transport, which is the process of moving a vector along a curve while maintaining its direction. Christoffel symbols and connection 1-forms describe how vectors change when parallel transported along different paths on a manifold.

Similar threads

  • Differential Geometry
Replies
7
Views
4K
  • Differential Geometry
Replies
3
Views
2K
  • Differential Geometry
Replies
15
Views
4K
  • Differential Geometry
Replies
9
Views
3K
  • Differential Geometry
Replies
5
Views
2K
Replies
16
Views
2K
Replies
7
Views
2K
Replies
4
Views
1K
Replies
6
Views
2K
  • Special and General Relativity
Replies
11
Views
1K
Back
Top