Matrix notation for Lorentz transformations

[decerto]

I'm having some confusion with index notation and how it works with contravariance/covariance.

(v_{new})^i=\frac{\partial (x_{new})^i}{\partial (x_{old})^j}(v_{old})^j

(v_{new})^i=J^i_{\ j}(v_{old})^j


(v_{new})_i=\frac{\partial (x_{old})^j}{\partial (x_{new})^i}(v_{old})_j

(v_{new})_i=(J^{-1})^j_{\ i}(v_{old})_j

So these are the standard rules for transforming contra and covariant vectors.
Now if we want to convert this into a matrix equation is there an exact set of rules with regards index position?

For example for the covariant transformation I can transpose the matrix which swaps the index order(Not sure how this makes sense) and this gives the right answer if we treat the covariant vectors as columns.

Or I can move the (v_{old})_j to the right of the J inverse and treat it as a row vector and this gives the right answer and I don't need to even consider what a transpose is in this interpretation.

Now both of these interpretations give the correct answers but they seem to have different meanings for upper vs lower and horizontal order.

Is there a best way to think about this, which way makes the most sense in terms of raising/lowering with metric tensors and transforming higher order tensors?

 

[Fredrik]

I suggest the following rules:

1. If the metric tensor is denoted by $g$, the matrix with $g_{ij}$ on row i, column j is denoted by $g$ as well.

2. The component on row i, column j of the matrix $g^{-1}$ is denoted by $g^{ij}$.

3. For most other matrices X, the element on row i, column j is denoted by $X^i{}_j$.

4. For n×1 matrices v, you don't write the column index. In other words, you write $v^i$ instead of $v^i{}_1$.

5. For 1×n matrices v, if you use them at all (it's probably best if you don't), you can't drop the row index from the notation. The notation $v_i$ is already reserved for $g_{ij}v^j=g_{ij}v^j{}_1$, so it can't be used for $v^1{}_i$.

6. In products, you transpose matrices if you have to, to ensure that each index that's summed over appears once upstairs and once downstairs.​

Example: A Lorentz transformation is linear function $\Lambda:\mathbb R^4\to\mathbb R^4$ such that $\Lambda^T\eta\Lambda=\eta$. The component on row $\mu$, column $\nu$ of this equation is
$$\eta_{\mu\nu}=(\Lambda^T)^\mu{}_\rho \eta_{\rho\sigma}\Lambda^\sigma{}_\nu =\Lambda^\rho{}_\mu\eta_{\rho\sigma}\Lambda^\sigma{}_\nu.$$ The intermediate step is usually not written out, because $\rho$ appears twice downstairs.

Note that the horizontal positions of the indices are important, because of weird things like this:
$$(\Lambda^{-1})^\mu{}_\nu = (\eta^{-1}\Lambda^T\eta)^\mu{}_\nu =\eta^{\mu\rho} \Lambda^\sigma{}_\rho \eta_{\sigma\nu} =\Lambda_\nu{}^\mu.$$