Covectors not identical with 1-forms?

[pellman]

I came across the following statement in this often-referenced paper on Einstein-Cartan theory (3rd page, right-hand column):

"In a space with torsion it matters whether one considers the potential of the electromagnetic field to be a scalar-valued 1-form or a covector-valued 0-form."

.. and the author then proceeds to list the resulting different behavior of torsion.

However, I am unaware of any difference between scalar-valued 1-forms and covector-valued 0-forms. In a 4d manifold are not both represented by the same four components? Are not both identical to the dual of the tangent vectors?

Perhaps I am not clear on the meaning of these terms. Can anyone here clarify the difference, if any, between a "scalar-valued 1-form" and a "covector-valued 0-form"?

 

[wisvuze]

a "covector-vector valued 0-form" is a function that gives you a linear functional ( or a covector ) for each argument. A scalar valued 1-form is a covector.

So, if f is my 0-form, and L is a covector, f( x ) = L where f( x ) ( v ) = L ( v ) [ where L itself is a scalar valued 1-form, so that L(v) is a scalar ]

 

[Hurkyl]

The trick is you've focused too much on what an object "is" (really, what a set-theoretic representation of the object is), and forgotten to pay attention to how it interacts with other things.

Covector fields, covector-valued 0-forms, 1-forms, and scalar-valued 0-forms are all naturally isomorphic sorts of objects (and maybe literally the same, depending on your choice of realizations of these ideas), but we have different types of objects associated to the different ideas.

For example:

1. The exterior derivative acts on a 1-form to give a 2-form.
2. The connection induced by the exterior derivative acts on a scalar-valued 1-form to give a scalar-valued 2-form.
3. A connection on the cotangent bundle acts on a covector-valued 0-form to give a covector-valued 1-form

While I'm pretty sure 1 and 2 give the essentially same thing, the third may give something very different.

 

[pellman]

I will ponder this a while. Thanks, guys.

 

[blue_raver22]

I would like to address this question by first clarifying the terms used. In mathematics, a 1-form is a type of differential form that assigns a scalar value to each tangent vector at a point on a manifold. This is often represented by a covector, which is a linear map from the tangent space of the manifold to the real numbers. On the other hand, a 0-form is a type of differential form that assigns a scalar value to each point on the manifold. This is often represented by a function.

Therefore, a scalar-valued 1-form and a covector-valued 0-form are not identical, as they operate on different spaces - tangent vectors and points, respectively. In a 4-dimensional manifold, both are represented by four components, but they are not the same.

In the context of Einstein-Cartan theory, the difference between considering the potential of the electromagnetic field as a scalar-valued 1-form or a covector-valued 0-form lies in the behavior of torsion, which is a measure of the curvature of spacetime. This theory suggests that the presence of torsion can affect the behavior of electromagnetic fields, and thus, the choice of representing the potential as a 1-form or a 0-form can lead to different predictions.

In conclusion, while scalar-valued 1-forms and covector-valued 0-forms may have similar representations in terms of components, they operate on different spaces and can lead to different predictions in certain theories. It is important to understand the distinction between these terms in order to accurately interpret and apply mathematical concepts in scientific research.