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Tensor densities

  1. Jan 27, 2014 #1
    Is there a coordinate independent/geometric definition of a tensor density?
     
  2. jcsd
  3. Jan 27, 2014 #2

    bcrowell

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    You have a set of quantities V={v} that obey the axioms of a vector space. This automatically implies that there is some notion of scaling them, [itex]v\rightarrow\alpha v[/itex]. For comparison, you also have the vector space of infinitesimal displacements T={dx}, i.e., the tangent space. To talk about whether V is a space made of pure tensors or tensor densities, I think you need additional information that tells you how transformations on T relate to transformations on V. For example, if rotations and boosts act the same way on T and V, but a scaling by α on T corresponds to a scaling by α2 on V, then you know that V is a space of tensor densities, not pure tensors. This additional information could be purely geometrical and based on some construction of V's elements out of T's elements (e.g., elements of V could describe areas), in which case that's how you'd know how to correlate the scaling behaviors. Or the comparative behavior under scaling could be fixed by some physical rather than geometrical consideration, e.g., elements of V could describe the mass per unit area in a 2-surface, which has different scaling behavior than the charge per unit area.
     
  4. Jan 28, 2014 #3
    Okay, follow up question: how would I show that the charge 4-current is a tensor density?
     
  5. Jan 28, 2014 #4

    bcrowell

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    Have you already established that current density is a pure vector?
     
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