Invariant Tensors in GR and SR

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Discussion Overview

The discussion revolves around the nature of invariant tensors in special relativity (SR) and general relativity (GR), specifically focusing on claims made in Carroll's text regarding which tensors are invariant and the implications of these claims. Participants explore the definitions and properties of invariant tensors, as well as the challenges in proving the existence of such tensors in these frameworks.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that Carroll claims the only invariant tensors in SR are the Kronecker delta, the Levi-Civita tensor, and the metric tensor, but in GR, the metric tensor is no longer invariant, leaving only the Kronecker delta and Levi-Civita tensor as invariant tensors.
  • Another participant argues that even the Levi-Civita tensor is not invariant in GR and suggests that invariant tensors in GR must have an equal number of upper and lower indices.
  • A different participant points out that the Levi-Civita symbol is a tensor density, which is modified to become a tensor by including a square root of the determinant of the metric, implying that it may not be a straightforward example of an invariant tensor.
  • One participant introduces the concept of "isotropic tensors," which are defined as tensors whose components remain the same across all coordinate systems, and mentions finding resources that partially address the proof of their exhaustiveness.

Areas of Agreement / Disagreement

Participants express differing views on the invariance of the Levi-Civita tensor in GR, with some asserting it is invariant while others contest this claim. The discussion remains unresolved regarding the complete characterization of invariant tensors in both SR and GR.

Contextual Notes

There are limitations regarding the definitions of invariance and the conditions under which tensors are considered invariant, as well as the dependence on specific transformations. The proof of the exhaustiveness of invariant tensors is noted to require adaptation for GR.

Who May Find This Useful

This discussion may be of interest to students and researchers in theoretical physics, particularly those focusing on the mathematical foundations of relativity and tensor analysis.

velapis
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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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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.
 
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?
 
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

The proof needs to be adapted to GR but I'm working on it...
 
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