Tensor Invariance and Coordinate Variance

[vanhees71]

<This thread is a spin-off from another discussion. Cp. https://www.physicsforums.com/threads/wedge-product.914621/#post-5762138>

Also again, be warned about this sloppy notation of indizes. You should put the prime on the symbol (or in addition to the symbol). Otherwise the equations don't make sense strictly speaking (I know that some unfortunate textbooks use this very dangerous notation). Also make sure that both the "vertical and horizontal" placement of the indices is accurate. For a 2nd-rank tensor the transformation law should be written as
$$T_{\mu \nu}'={\Lambda^{\rho}}_{\mu} {\Lambda^{\sigma}}_{\nu} T^{\rho \sigma}.$$
Concerning your question, Orodruin has given you the right hint. The tensor components are all numbers, and thus the product is the usual commutative product of real numbers!

 

[Orodruin]

Also again, be warned about this sloppy notation of indizes.

I actually strongly disagree with this. I prefer priming the indices of the components rather than the components themselves. This underlines the fact that it is the components, not the tensor itself, that are coordinate dependent. The tensor $T$ does not depend on the coordinate system, if you prime $T$ you make it seem as if the tensor is coordinate dependent and the entire point is that it is not.

 

[vanhees71]

But $T_{\mu \nu}$ and $T_{\mu'\nu'}$ denote the same numbers, but they change when referring to different bases. If you want to have the primes on the indices, I've no problems with that, but still you should also write one on the symbol:
$$T_{\mu' \nu'}'={\Lambda^{\mu}}_{\mu'} {\Lambda^{\nu}}_{\nu'} T_{\mu \nu}.$$
Of course you have [corrected now twice :-((]
$$T_{\mu \nu} b^{\mu} \otimes b^{\nu}=T_{\mu' \nu'}' b^{\prime \mu'} \otimes b^{\prime \nu'},$$
where $b^{\mu}$ and $b^{\prime \mu'}$ are the bases co-vectors to which the components belong, respectively.

 

[Zag]

I agree with vanshees71. What remains invariant under a general coordinate transformation are not the components, but the the full object denoted in this case by $$T_{\mu\nu}\hat{\mathrm{e}}^{\mu}⊗\hat{\mathrm{e}}^{\nu}$$.

 

[Orodruin]

But $T_{\mu \nu}$ and $T_{\mu'\nu'}$ denote the same numbers

Just if you define them to denote the same number. I disagree that this is necessarily the case. The coordinate system is implied by what indices you put in, $T_{11}$ is not the same component as $T_{1'1'}$. If you have a coordinate system with named coordinates, such as $x, y, z$ in Euclidean coordinates and $r, \theta, \varphi$ in spherical coordinates, $T_{xx}$ is clearly a different from $T_{rr}$. Would you put a prime on one of those as well and in that case, which one do you choose? In my view, it is much more elegant to let the indices tell you what the coordinate system is. My point is that it is definitely possible to have the indices refer to the appropriate coordinate system while keeping the symbol for the tensor itself, in this case $T$ prime free to underline the fact that the tensor itself does not depend on your choice of coordinates.

Of course you have [corrected]
$$T_{\mu \nu} b^{\mu} \otimes b^{\nu}=T_{\mu'}' b^{\prime \mu'} \otimes b^{\prime \nu'},$$
where $b^{\mu}$ and $b^{\prime \mu'}$ are the bases co-vectors to which the components belong, respectively.

I would write this as (inserting the missing second index)
$$
T = T_{\mu\nu} b^\mu \otimes b^\nu = T_{\mu'\nu'} b'^{\mu'} \otimes b'^{\nu'}
$$
with the indices $\mu$ and $\mu'$ running over different sets of coordinates. The only time when this can be misunderstood is if you are dealing with a tensor density rather than a tensor, since the tensor density does depend on the coordinate system. My point is that this keeps the symbol for the tensor, i.e., $T$ prime free and the indices - which are what refers to something coordinate dependent - is what carries the information on what coordinate system is being used.

I agree with vanshees71. What remains invariant under a general coordinate transformation are not the components, but the the full object denoted in this case by $$T_{\mu\nu}\hat{\mathrm{e}}^{\mu}⊗\hat{\mathrm{e}}^{\nu}$$.

This is exactly the reason why I dislike putting the prime on the $T$ as I said in post #4. The tensor $T$ itself is coordinate independent and so it is just confusing to put a prime on the symbol that refers to the tensor and less confusing to put them on the indices, which is what refers to a coordinate system.

Edit: Let me underline that I do not mind that people like putting primes on their $T$s, it is just a matter of notation. My point is that it is not inconsistent to put the primes on the indices only as long as primed and unprimed indices run over different sets. When it comes to using numbers as indices, they can be distinguished by using $1,2,\ldots$ for unprimed and $1',2',\ldots$ for primed indices.

 

[vanhees71]

$\mu$ and $\nu$ as well as $\mu'$ and $\nu'$ run over the numbers $0$ to $3$ (if we talk about relativistic tensors). There is no number $1'$ that's distinguished from the number $1$! When I started to learn tensor calculus from some textbook (I forgot which one), I got completely confused by this sloppy notation. There's a suble balance between physicists' pragmatic sloppyness and confusion!

 

[Orodruin]

$\mu$ and $\nu$ as well as $\mu'$ and $\nu'$ run over the numbers $0$ to $3$ (if we talk about relativistic tensors). There is no number $1'$ that's distinguished from the number $1$!

What sets of values your indices run over is completely up to you to define. For me, unprimed indices run over 1 through N and primed over 1' through N' and this solves the issue of not knowing from the indices what coordinate system is intended. Of course, if you define $\mu$ and $\mu'$ to run over the same set of indices, there will be a confusion, but in the general setting (not restricting to Cartesian or Minkowski coordinates). I think it is equally confusing to use the same set of indices for different coordinate systems. The usual thing to do is to use the coordinate names as indices and then it is explicitly different. You don't write $g_{11}$ when you mean $g_{rr}$ in Schwarzschild coordinates, do you? Do you put a prime on the metric when you go to a different coordinate system, e.g., Kruskal-Szekeres coordinates? To be consistent with your notation, you should, so which system gets the prime and which does not? Which system gets the double prime?

When I started to learn tensor calculus from some textbook (I forgot which one), I got completely confused by this sloppy notation. There's a suble balance between physicists' pragmatic sloppyness and confusion!

Again, it is not necessarily sloppiness as long as you make sure that there is no room for confusion with the components in different coordinate systems being referred to by different sets of indices.

Actually, one of the more common student misconceptions that I encounter is the notion that tensors depend on the coordinate system and I believe that putting a prime on the components reinforces this misconception. To me, this is clearly obstructing students from a proper understanding of what a tensor is.