Generating elements and isometric Galois actions over Tate fields

For a prime number $p$, let $T$ be an element of the field $\mathbb{C}_p$, which is the completion of the algebraic closure of the $p$-adic number field $\mathbb{Q}_p$. In this paper, we study the class of generic elements of $\widetilde{\mathbb{Z}_p[T]}$ by relating this class with an algebraic-metric equivalent relation, which is defined by isometric Galois actions of the $p$-adic absolute Galois group.