I would like to ask about the relative definition of a principal bundle. Let $G$ an algebraic group acting trivially on a scheme $S$. A principal $G$-bundle over $S$ is a a $G$-fibration $P\rightarrow S$ such that there exists an open covering of $S$ by open sets $U$, such that $$P\times_{U}S\rightarrow U$$ is isomorphic to the trivial principal $G$-bundle: $G\times U\rightarrow U$.
Let us consider know an arbitrary $S$-scheme $X\rightarrow S$. What is a principal bundle over $X\rightarrow S$ now?