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Covectors not identical with 1-forms?

  1. Apr 8, 2012 #1
    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"?
     
  2. jcsd
  3. Apr 8, 2012 #2
    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 ]
     
  4. Apr 8, 2012 #3

    Hurkyl

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    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.
     
  5. Apr 8, 2012 #4
    I will ponder this a while. Thanks, guys.
     
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