Tensor Densities: Coordinate Independent Definition

[dEdt]

Is there a coordinate independent/geometric definition of a tensor density?

 

[bcrowell]

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, $v\rightarrow\alpha v$. 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.

 

[dEdt]

Okay, follow up question: how would I show that the charge 4-current is a tensor density?

 

[bcrowell]

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

 

[sardar]

Yes, there is a coordinate independent/geometric definition of a tensor density. A tensor density is a mathematical object that behaves like a tensor, but with the added property that it transforms under a change of coordinates in a specific way. In other words, a tensor density is a tensor that is weighted by a certain power of the determinant of the transformation matrix between two coordinate systems.

Geometrically, this can be understood as a tensor density being a tensor field that is defined on a manifold, and its components at each point on the manifold are multiplied by a certain power of the determinant of the Jacobian matrix of the coordinate transformation. This allows the tensor density to have a well-defined behavior under coordinate transformations, making it coordinate independent.

In summary, a tensor density can be defined as a tensor field with a specific transformation law under coordinate changes, making it a coordinate independent object. This definition is important in many areas of physics and mathematics, such as differential geometry and general relativity.