a footnote is a "moving argument" -- it is saved up to be used elsewhere.

when this happens, the contents of the footnote expand, and alphabetical control sequences end up with a space after their text. (in other words, " \infty " ends up as " \infty ".) ordinarily, this doesn't matter. but when a term is written out for sorting into an index, the space does matter, because the index sort is simple ascii, and a string with an extra space in the middle will sort separately.

the space can be avoided by preceding the affected control sequence by \string . ( \protect doesn't work here.) so your index entries in this example can be entered (in the footnote) as

\index{$r_{\string\infty}$}\index{r@$r_{\string\infty}$}

and then they will sort together with the "normal" ones.

(i applaud your making two entries for this expression. that will ensure that someone can actually find the concept in the index.)