Invariant Tensors in GR and SR

In summary, Carroll's claim that the only invariant tensors in SR are the Kronecker delta, the Levi-Civita tensor, and the metric tensor is backed up by some lecture notes and MathWorld entries. However, the proof is adapted to GR and needs to be completed.
  • #1
velapis
3
0
Hello all, this is my first post on this forum, though I have been perusing it for a while.

I am currently re-reading through Carroll's text on SR and there is a curious comment on p24 that intrigues me. Carroll says that the *only* tensors in SR which are invariant are the Kronecker delta, the Levi-Civita tensor, and the metric tensor. In GR, the latter is no longer invariant so there are only two invariant tensors.

Carroll says "we won't prove it" but I'm dying to see the proof, which I've spent a few hours trying to derive. The internet has also been little help, aside from this topic which doesn't really give an answer:
https://www.physicsforums.com/showthread.php?t=344626

So how do we show that there are no other invariant tensors?

Thanks!
 
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  • #2
Strictly speaking, I think, even the Levi-Civita tensor is not invariant in GR... More generally, the only tensors that can be invariant in GR are the ones where the number of upper indices is equal to the number of lower indices, as can be seen by applying a simple transformation of the form x^i -> c*x^i.
 
  • #3
Well, the Levi-Civita symbol is tensor density, which is artificially made into a tensor by adding a square root of det(g) term to it.

Maybe ignoring such artificial examples would help in answering the question?
 
  • #4
Progress at last: it turns out that tensors whose components are the same in all coordinate systems are called "isotropic tensors".

This is helpful since MathWorld has an entry which lists them, and I found some lecture notes which (partially) show that the list is exhaustive:

http://www.ig.utexas.edu/people/students/classes/spring02/geo391/Lecture1.pdf [Broken]

The proof needs to be adapted to GR but I'm working on it...
 
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1. What is an invariant tensor?

An invariant tensor is a mathematical object that remains unchanged under a transformation. In the context of general relativity (GR) and special relativity (SR), it refers to a tensor that has the same value in all reference frames.

2. Why are invariant tensors important in GR and SR?

Invariant tensors are important in GR and SR because they allow us to describe physical quantities that are independent of the observer's reference frame. This is essential in formulating the laws of physics, which should be the same for all observers.

3. How do invariant tensors relate to the principle of covariance?

The principle of covariance states that the laws of physics should have the same form in all reference frames. Invariant tensors play a crucial role in this principle as they are the mathematical objects that allow us to express physical laws in a covariant form, meaning they remain unchanged under coordinate transformations.

4. Can you give an example of an invariant tensor in GR or SR?

One example of an invariant tensor in GR and SR is the metric tensor. It describes the geometry of spacetime and is the same in all reference frames. Another example is the electromagnetic field tensor, which is invariant under Lorentz transformations in SR.

5. How are invariant tensors used in the theory of general relativity?

Invariant tensors are used extensively in the theory of general relativity to describe the curvature of spacetime and the Einstein field equations. They also play a crucial role in defining physical quantities, such as energy and momentum, in a covariant manner. Without invariant tensors, the mathematical framework of GR would not be possible.

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