Leaving the theater after watching The Hobbit will put you in one of two moods: ecstatic at the thought of the new 'ship you watched peek its head out at the start of the franchise or confused about how you feel about the new 'ship you watched peek its head out at the start of the franchise. The Hobbit: The Battle of the Fire Armies might not have been the start of the romantic relationship between Galadriel and Gandalf, but it certainly did nothing to sink that 'ship in the end. I know that some of you are now staring at me going, "Say what?! Gandalf in love? What?!" Well, that certainly seems to be the way the films are playing it. However, do Gandalf and Galadriel have a romance in the Tolkien books? Sorry, guys, but the answer there is no.
In Battle of the Fire Armies, Galadriel and Gandalf pick up their subtle relationship right where it left off in An Unexpected Journey. When Gandalf goes to Rivendale to enlist the help of the elves and Galadriel promises to come to his aid whenever he requests it. She simultaneously gives Gandalf a pretty touching face stroke that would have launched a thousand fanfictions if there weren't already a significant base rooting for the two to get together ever since reading the books.
The Battle of the Fire Armies has Galadriel keeping her promise and coming to help Gandalf after he encounters Sauron in The Desolation of Smaug, but then their "relationship" progresses a step further. And by "further," I mean Galadriel carries Gandalf like a damsel in distress and, later, Gandalf stares longingly at her before asking her to come with him and calling her "my lady" rather tenderly. (She doesn't, because she's got to kick Sauron's butt instead.) I mean, really, the only way you can't hear the romantic music playing gently in the background of these moments is if you actually read the books or saw the original trilogy and are incapable of thinking of Gandalf as a particularly romantic character.
Well, you don't have to adjust your world view too much. Gandalf and Galadriel never get together in the books. The films had their chance to pull a Neville Longbottom and Luna Lovegood moment from out of their hats (and out of the fandom) with Gandalf and Galadriel, but this was a missed opportunity that only fanfiction can fix. While the relationship between Gandalf and Galadriel is expanded in the books (as are most things), the one thing the books and films have in common is that that relationship is always 100 percent platonic.
However, there's not necessarily anything wrong with that. It's clear that Gandalf and Galadriel have an unbreakable bond that transcends petty labels like "platonic" or "romantic." They're loyal to one another, they understand one another, and they're there for one another when it counts. There's a beauty in their relationship as it is, no matter how you choose to define it, and you can certainly find evidence to support both conclusions in the books. (Except the truly definitive evidence, like them making out and riding off into the sunset together. Although they do leave Middle Earth together with Frodo, Celeborn, Bilbo, Elrond, and possibly Shadowfax, so make of that what you will.)
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