Is there a convention for horizontally shifting indices in tensor notation?

[JohnBell5713]

Hi,

I'm teaching myself tensor analysis and am worried about a notational device I can't find any explanation of (I'm primarily using the Jeevanjee and Renteln texts).

Given that the contravariant/covariant indices of a (1,1) tensor correspond to the row/column indices of its matrix representation, what is indicated by horizontally shifting one index with respect to the other? Is this notationally redundant, or is some extra information I'm missing being encoded here? Given that this convention also applies to (n,m) tensors and even the Kronecker delta, I want to clear this up before proceeding further.

John

 

[Fredrik]

If you intend to use the metric to raise and lower indices, then you also need to keep track of the horizontal positions. Special relativity is a good example:

The components of the metric in an inertial coordinate system are denoted by $\eta_{\mu\nu}$. The matrix with $\eta_{\mu\nu}$ on row $\mu$, column $\nu$ is denoted by $\eta$. The number on row $\mu$, column $\nu$ of $\eta^{-1}$ is denoted by $\eta^{\mu\nu}$. For most other linear transformations M on $\mathbb R^4$, the number on row $\mu$, column $\nu$ of the corresponding matrix is denoted by $M^\mu{}_\nu$.

A Lorentz transformation is a linear operator $\Lambda$ such that the corresponding matrix (also denoted by $\Lambda$) satisfies $\Lambda^T\eta\Lambda=\eta$. This implies that $\Lambda^{-1}=\eta^{-1}\Lambda^T\eta$. So we have
$$ (\Lambda^{-1})^\mu{}_\nu = (\eta^{-1}\Lambda^T\eta)^\mu{}_\nu =\eta^{\mu\rho}(\Lambda^T)^{\rho}{}_\sigma \eta_{\sigma\nu} =\eta^{\mu\rho}\Lambda^{\sigma}{}_\rho \eta_{\sigma\nu}=\Lambda_\nu{}^\mu.
$$