Interpretation of relations in varying-domain models of F.O. modal logic

Question

I am studying the book "First Order Modal Logic" By Fitting and Mendelsohn. In their definition of interpretation for varying domain models (def 4.7.3 pg 103), the interpretation of a relation in a world is defined as an n-tuple from the union of all domains.

That is, given a frame $\mathscr{F}=\left\langle\mathscr{G,R,D} \right\rangle$ where $\mathscr{G}$ is the set of worlds, $\mathscr{R}$ is the accesibility relation on $\mathscr{G}$, and $\mathscr{D}$ is the domain function assigning a domain to each world, we define $\mathscr{D(F)=\bigcup\{D(\Gamma)\mid \Gamma \in G\}}$. So an interpretation $\mathscr{I}$ assignes for each $\Gamma \in \mathscr{G}$ and each $n$-ary relation symbol $R$ an $n$-tuple $\mathscr{I}(\Gamma , R)\subset \mathscr{D(F)}^n$.

This interpretation defines the truth of an atomic formula in a world. That is, a model would be a quadruple $\mathscr{M}=\left\langle\mathscr{G,R,D,I} \right\rangle$, and given a valuation $v:Variables \to \mathscr{D(M)}$ we say that an atomic formula $R(x_1,...,x_n)$ is true in a world $\Gamma \in \mathscr{G}$ under $v$ iff $(v(x_1),...,v(x_n))\in \mathscr{I}(\Gamma,R)$

My question is - why do we allow the relation to be from the entire domain of the frame ($\mathscr{D(F)}$) and not only from $\mathscr{D}(\Gamma)$? Why don't we relativise the valuation according to the world we speak of? It strikes me a bit odd that we might say that in a certain world a relation can hold between elements of domains of different worlds, which may not be known in the world we are focusing on.

Is there a text where this different semantics is proposed? Does it even make a difference?

I should say that my motivation is from set theory, where the only relation symbol in the language is $\in$, and the worlds are e.g. countable models inside some big model. So it doesn't make sense to say that a model satisfies $x\in y$ if they are not in it.

Answer 1

The explanation is in page 102 (and EXAMPLE 4.7.5, page 103).

You have to consider some "tricky" cases as the following:

is Napoleon necessarily bald ?

We have to consider "possible worlds" were Napoleon does not exist. Thus, the authors decided, instead of allowing for "partial models", i.e. models were the said sentence has no definite truth-value, to assume

that even though $v(x)$ might not exist in the domain associated with [the world] $\Delta$, it does exist under alternative circumstances we are willing to consider, and consequently talk about $v(x)$ is meaningful.

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Considering your "set theory motivation", we can say that the answer to the question:

is the empty set necessarily empty?

is: Indeed... also if we can imagine some "alternative" universes where there is no empty set at all.