Example:
Let's say I want to reference "the bridge that I ran underneath".
I would presume that the sentence would be:
下に走った橋
I feel like there's a lot of ambiguity here, but I guess that's sometimes the nature of Japanese, and immediate understanding of what part of the relative clause the noun refers to would come with fluency.
Anyway, is this right? What general rules are there for nouns referring to different parts of the relative clause?
Anyway, is this right?
Unfortunately, no. The particle you need is を, not に. 下に would mean downwards.
下を走った橋 should be understandable if there's enough context, but if you worry about ambiguity, it may be better to explicitly add the subject of 走る and say 私が下を走った橋. Although we don't usually say 橋が走る, we do commonly say 道路/川が走る "the street/river runs (e.g., across the town)", and thus 下を走った橋 without any other modifier might also be taken as "a bridge that crosses some low place" by some people.
This ambiguity is an innate restriction of Japanese relative clause, and there is not simple rule to determine the interpretation in one way. See also: How is the subject of this subclause made clear?
