Physical interpretation of order of tensor indices

Using positional index notation with tensors is common. For example, the following simple equation from Carroll's Spacetime and Geometry text (eq. 3.146): $$R = R^\mu_{\,\,\mu} = g^{\mu\nu}R_{\mu\nu}$$ Where the Ricci scalar summation, $R^\mu_{\,\,\mu}$ uses positional offsets on the upper and lower indices.

So, I think I understand the usage and motivation for this positional offset. I am curious and thus the puzzle as to why so many texts (almost all of the current raft of GR texts) totally ignore any type of explanation for the use of positional index notation. And, some texts ignore it altogether -- that is, not using positional notation at all.

Apparently, authors seem to believe that this notation should be obvious (and, in some ways it is) but I am puzzled by the lack of explanation. Is there a formal definition of the motivation and "need" for positional index notation?

• Are you asking why $R$ is used instead of $R^{\mu}_{\,\,\mu}$? The reason is this: $R^{\mu}_{\,\,\mu} = R^{0}_{\,\,0} + R^{1}_{\,\,1} + .... + R^{n}_{\,\,n}$. All these are scalars so they are brought together in a single term, just how you can set $a + b + c$ equal to another number $d$. This results in your equations having the bare minimum of indices needed for obvious reasons. – Constandinos Damalas Oct 24 '14 at 16:49
• Why some indices are up and some down for example? – Constandinos Damalas Oct 24 '14 at 17:00
• @PhotonicBoom, I guess my question was not as clear as I thought it was. No, I understand upper and lower index -- traditionally referred to as Contravariant (upper) and Covariant (lower). I also understand dual vector spaces, convectors, and other aspects of tensors and notation. I am specifically referring to the slot positional notation as where the upper and lower index are in columnar positions with spacing to maintain these slot positions. This is the reason I chose this equation on the Ricci Scalar as an example in that it used such offsets in the $R_\mu_{\,\,\mu}$ summation term. – K7PEH Oct 24 '14 at 17:04
• i think the answer is to make clear which of the 2 indices has been made contra-variant or co-variant (since generaly the 2 indices $\mu$ and $\nu$ may play different parts). This if i understand correctly, positional notation havent heard it – Nikos M. Oct 24 '14 at 17:11
• OP writes (v1): So, I think I understand the usage and motivation for this positional offset. I am curious and thus the puzzle as to why so many texts (almost all of the current raft of GR texts) totally ignore any type of explanation for the use of positional index notation. It seems OP understands the notational issue and is only asking a primarily opinion-based question about the author's choice of presentation. – Qmechanic Oct 25 '14 at 19:37

On a two-index tensor, swapping the two indices is equivalent to transposing a matrix.

You may not see many authors spending a lot of effort on this issue simply because an awful lot of the tensors we deal with are symmetric. This includes the metric, Ricci tensor, Einstein tensor, and stress-energy tensor. Therefore there is no special interest in discussing transposition. Sometimes there is some physical interest in understanding why these tensors must be symmetric, e.g., understanding why the stress-energy tensor is symmetric leads to ideas about torsion.

We also run into some tensors that are antisymmetric. Here the physical interpretation of swapping indices is generally something to do with choosing an orientation. I think texts that discuss antisymmetric tensors do usually give this interpretation. E.g., the electromagnetic field tensor is antisymmetric, and this relates to the right-hand rule for magnetic forces.

To make an analogy, real numbers can be negative or positive. A physics textbook will not give a general physical interpretation of what is meant by a negative number, because there is none. But it will usually give an interpretation in specific physical cases, e.g., negative velocities or a negative temperature on the Celsius scale.

• The horizontal position of indices is important for a tensor that is not totally symmetric, e.g., the EM field strength $F_{\mu\nu}$ or the Riemann curvature tensor $R_{\mu\nu\lambda\kappa}$, etc, in order to properly identify which indices get raised or lowered.
• As usual, be prepared that different authors use different conventions and notations. E.g. different authors order the indices of the Riemann curvature tensor $R_{\mu\nu\lambda\kappa}$ differently.