How does abstract index notation work? What do the the indices represent? I know the Lorentz transformation tensor in arbitrary direction, so if you want to use a specific tensor in an example, that would be a good one.
We mathematicians tend to prefer to think of tensors in coordinate-free notation. And we make fun of physicists for all the index nonsense, and Einstein summation conventions. So, Penrose, trained as a mathematician, came up with the abstract index notation as a sort of compromise between the index notation and the coordinate-free notation. But, I think it's kind of just formalizing the way physicists were thinking of it all along.In Wald's book he write:
"Thus, the distinction between the index notation and the component notation is
much more one of spirit (i.e., how one thinks of the quantities appearing) than of
substance (i.e., the physical form the equations take)."
So,don't worry it much .