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I have only watched the movies and I am currently reading The Fellowship of the Ring (haven't finished it yet, I am at the Bridge in Khazad-Dum).

A number of times we see characters such as Gandalf, Aragorn or Glorfindel describe how lethal a foe the Nine are. I also read that the Nine ride horses taken in tribute from lands such as Rohan and they only bother with the black cloaks to give some form to their nothingness when they go out in the world of the living.

But the moment Frodo crossed the Ford at Bruinen (and the subsequent flooding of the river), we see the same characters talking about how the Nine will not bother them for some time because apparently they have lost their horses. It seems bizarre to me that such terrible an enemy be dismayed by mere loss of mounts. I understand that the Nine cannot see in the light (but their horses can) yet that doesn't seem to really hamper them as they utilise other senses to track their target.

So why do the people assembled at Rivendell seem to think that just because the Nine have lost their mounts, the threat is somehow temporarily over? They know the flood couldn't have killed or hurt the Nazgul. They know that they have no physical form to begin with that might have been hurt in the flooding. So what gives?

The Ringwraiths couldn't fly on their own or teleport. To get from point A (near Rivendell) to point B (Mordor) they had to travel every foot in between. In LotR, they walk, ride horseback and ride flying mounts. With their horses gone, their only option was to get new transport or walk back to Mordor -- a very long way!

'You cannot destroy Ringwraiths like that,' said Gandalf. 'The power of their master is in them, and they stand or fall by him. We hope that they were all unhorsed and unmasked, and so made for a while less dangerous;

But getting horses would not be easy. To start with, they're in a wilderness with no towns, farmers or herders. But secondly, animals do not react well to the presence of the ringwraiths and it's not clear that a stolen horse would carry them without extensive training.

'Be­cause they are real horses; just as the black robes are real robes that they wear to give shape to their noth­ing­ness when they have deal­ings with the liv­ing.’

‘Then why do these black horses en­dure such rid­ers? All other an­i­mals are ter­ri­fied when they draw near, even the elf-horse of Glo­rfindel..."

‘Be­cause these horses are born and bred to the ser­vice of the Dark Lord in Mor­dor. Not all his ser­vants and chat­tels are wraiths!'

So they were for a time out of the game -- much to the benefit of the Fellowship!

Note also that the Nine could not attack the Fellowship in Rivendell -- Elrond and elven lords like Glorfindel were far too powerful. Nor, even if they knew that the Fellowship would eventually set off, could they lurk around Rivendell waiting, since Elrond sent out scouts and the Nazgul clould not hope to withstand a mounted posse including Gandalf, Elrond, Glorfindel and Aragorn.

But most of all, they had no reason to. Neither Sauron nor (as far as we are told) any of his minions guessed that they would seek to destroy the Ring. The "logical" decision would be to hunker down in Rivendell -- the strongest, best-defended point left in Middle-Earth -- and guard the Ring. Sauron needed that news more than anything.

So why do the people assembled at Rivendell seem to think that just because the Nine have lost their mounts, the threat is somehow temporarily over? They know the flood couldn't have killed or hurt the Nazgul. They know that they have no physical form to begin with that might have been hurt in the flooding. So what gives?

"What gives" is the Bruinen flow was enhanced by "First Age" elven magic. The water did hurt them a great deal. And even before the spell was uttered the Nine waited on the far bank unsure about crossing and only did so once they were taunted.

To see why the Nine were hesitant you must understand that there is a longstanding antipathy between Morgoth and his servants (Sauron is a chief servant of Morgoth and The Nine are chief servants of Sauron) and the Powers that reside in the West, the Valar. Besides Manwë (first of the Valar), there is Ulmo (second among them) and Ulmo in particular distrusted Melkor/Morgoth from the very first. Ulmo's realm and his power are manifest in the ocean deep and great rivers of Middle Earth.

Elrond's spellcraft (or Arwen's in the movie) enhanced the waters of the Bruinen, killed the horses of the Nazgûl, and unmade their forms ("unmasked" to use Gandalf's description) and their wraiths returned to their master at Barad-dûr. So, though they weren't killed (can you kill undead?), they were most decidedly hurt by the floodwaters and not merely unhorsed.

As a final note:
Key aspects of Glorfindel's character are merged with Arwen's in Peter Jackson's movie. Glorfindel, having lived in Aman, brings the light of Aman with him wherever he goes in Middle Earth. We see this when Arwen first shows up on Asfaloth as the party is encamped under the Trolls and in the blessing Arwen gives to the dying Frodo after crossing the Loudwater. Arwen also seems to have been given spellcraft akin to Elrond's during this scene in the movie. In the book it is Elrond, presumably enhancing his spellcraft with Vilya, that causes the flood on the river (given a flourish by Gandalf), which deranged the horses and perhaps the Nine riders, and finally drowning them all as they were swept away.