When do we use tensor densities rather than tensors?

In summary, tensor densities are used in certain cases instead of tensors, such as when integrating over a region of an n-manifold or in an Einstein Lagrangian. They can be converted to tensors by multiplying by the metric determinant. For more information, references are provided in Appendix B of General Relativity - Wald, Spacetime and Geometry - Carroll, and Introducing Einstein's Relativity - D'Inverno. Additionally, there are resources available online for further understanding of area, volume, and angular momentum.
  • #1
Phrak
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When do we use tensor densities rather than tensors?
 
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  • #2


I can't think of definite points where tensor densities are used over tensors but one point is that when integrating some scalar or tensor over a region of an n - manifold the volume element is actually an anti - symmetric tensor density which is constructed through wedge products like differential forms. Another place I have seen tensor densities is in an Einstein Lagrangian [tex]\L = (-g)^{1/2}R[/tex] where g is the metric determinant and R is the ricci scalar. If I recall correctly you can always turn a tensor density into a tensor by multiplying the density by [tex]g^{W/2}[/tex] where W is the weight of the tensor density you are changing.

If you want check out Appendix B of General Relativity - Wald (he talks about integration and forms but not densities explicitly), Spacetime and Geometry - Carroll (sections 2.8 - 2.10), or Introducing Einstein's Relativity - D'Inverno (sections 7.1 - 7.4).
 
  • #3


For area, volume, and angular momentum:

http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html#Section4.6 [Broken]
 
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  • #4


bcrowell said:
For area, volume, and angular momentum:

http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html#Section4.6 [Broken]

Off topic sorry,

just want to gives props to bcrowell for the free texts. I'm really enjoying them :)
 
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  • #5


Thanks, bcrowell. That clears it up.
 

1. What is the difference between tensors and tensor densities?

Tensors are mathematical objects that describe geometric quantities, such as vectors and matrices, and how they transform under different coordinate systems. Tensor densities, on the other hand, are a more general concept that includes tensors, but also take into account the scaling of coordinate systems. This allows tensor densities to be used in situations where tensors alone are not sufficient.

2. When should I use tensor densities instead of tensors?

Tensor densities are particularly useful in physics and engineering, where the coordinate system may change and the scaling of the system is important. For example, in fluid dynamics, tensor densities are used to describe the velocity field, which varies with position and time, and is affected by the density of the fluid.

3. Are there any other applications for tensor densities?

Yes, tensor densities are also used in differential geometry, where they are used to define geometric objects that are independent of the choice of coordinates. They are also used in general relativity, where they play a crucial role in describing the curvature of spacetime.

4. How do I know when to use a tensor density instead of a tensor?

This depends on the specific problem you are trying to solve. If the coordinate system is fixed and the scaling of the system is not important, then tensors alone may be sufficient. However, if the coordinate system is changing or the scaling is important, then tensor densities are needed.

5. Are there any mathematical properties that differentiate tensors and tensor densities?

Yes, tensors are defined as multilinear maps that transform according to certain rules under a change of coordinates. Tensor densities, on the other hand, are defined as multilinear maps that transform according to a different set of rules that take into account the scaling of the coordinate system. Additionally, the number of independent components of a tensor density is different from that of a tensor of the same rank, which can be an important consideration in certain applications.

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