I've read that in general for a modal formula P, a world w and a Kripke frame ⟨W,R⟩
w⊨□P if and only if for every u∈W, if wRu then u⊢P
In case of LTL, being a modal logic, I assumed that the worlds are different moments in time and the relation R is the chain of "arrows" between those moments. However, in this case the semantic of □ in LTL would not correspond to the definition above.
What did I misunderstood?
In LTL, you don't have $\diamond$ and $\Box$, but you have $X$ and $U$, meaning next and until, respectively.
In LTL, every world has a unique successor.
Next is defined as $w \models X \phi$ if $w' \models \phi$, where $w'$ is the successor of $w$.
Until is defined as follows: $w \models \phi U \psi$ if there exists $w''$ such that $w'' \models \psi$ and for every $w'$ between $w$ and $w''$, we have that $w' \models \phi$.
Now, from these modalities, we can construct other modalities, such as globally, $G \phi$, and finally, $F \phi$. It is common to denote
tl;dr The accessibility relation needs to be the reflexive transitive closure of what you had in mind.
Details: Let $P$ be a set of atomic proposition. Write $\Sigma$ for $P$'s powerset. Write $T$ for $\Sigma^{\mathbb{N}}$ (infinite traces).
A Kripke frame $\langle W, R\rangle$ where the meaning of formulas agrees with that of the fragment of LTL using just $P$, boolean connectives, $\square$ and $\diamond$ would consist of $W = T\times\mathbb{N}$ (ie infinite traces with an index) and $R = \{~((\sigma,i),(\sigma,k))~|~\sigma\in T\wedge i\leq k\in\mathbb{N}~\}$.
