In several papers (An: https://arxiv.org/abs/1703.00118, equation (1.5), as well as papers by Klainerman, Rodnianski, and Luk on singularity formation in GR), one introduces a frame $e_a$ to spheres $S_{u, \bar u }$ of a double null foliation. They then write $e^a$ but don't define this; so my question is, what does this notation mean?
Since the $a$ is not a spacetime coordinate index, but rather a label, it is not the usual "raising an index" of general relativity, to my knowledge. I would guess it still means the same thing, in that $e^a$ is the covector which is the metric dual to $e_a$, but I was wondering if someone could confirm this, or give a reference where this is mentioned explicitly? The other likely option I can think of is that $e^a$ is the "linear dual" to $e_a$, i.e. the covector which evaluates to 1 on $e_a$ and zero on $e_b$ for $b \neq a$. Or perhaps another option is that it is just $e_a$, and we have raised an index so that a sum is implicit in the expression $e^a e_a (R)$ where $R$ is a scalar function.