Did Darth Vader astral-project a solid image of himself in the cave on Dagobah the same way that Luke did on the planet Crait?
No. There are plenty of reasons why Vader could not have been projecting himself into the cave. Here are a few:
Yoda encouraged Luke to enter the cave without his weapons, which would have put Luke in grave danger had Vader actually been projecting himself into the cave:
Yoda: That place... is strong with the dark side of the Force. A domain of evil it is. In you must go.
Luke: What's in there?
Yoda: Only what you take with you...Your weapons... you will not need them.
Star Wars Episode V: The Empire Strikes Back script
- Vader would not have lost if he'd actually been fighting Luke.
- It would make no sense for Vader to cause his projection to show Luke's face under his mask.
- Projection is a difficult skill to learn but Vader would have had no need to spend time learning it since the Empire controls the galaxy and Vader can go pretty much wherever he wants/needs.
- Presumably Vader would need to know where Luke was to project himself there...but if that's the case why not go to Dagobah with his fleet to capture Luke? The entire plot to capture Luke by enticing him to save his friends on Cloud City would have been pointless.
I don't think so. Vader was busy at the time chasing the Millenium Falcon. He also had no idea Luke was on Degoba, let alone training with Yoda. During their later duel we see that he assumed Obi-wan had trained him:
Vader: Obi-Wan has taught you well. You have controlled your fear... now release your anger.
So what then? An alternative theory for Darth Vader's presence in the cave is that the whole cave bit was a test of Luke's control, which Luke failed. This is referred to as Luke is preparing to leave for cloud city:
Luke: But I can help them! I feel the Force!
Ben: But you cannot control it. This is a dangerous time for you, when you will be tempted by the dark side of the Force.
Yoda: "Yes, yes. To Obi-Wan you listen. The cave. Remember your failing in the cave"
Luke: "Yes, but I've learnt so much since then"
Yoda said the only thing in there is what Luke took with him, so assuming Yoda was correct, there was only Luke and his weapons in the cave. So who was he fighting? It must have been himself. This is further supported by the fact that when he slices off the facemask he sees his own face looking back at him.
The first time I saw this I assumed the face thing was because he was the son of Darth Vader, but that doesn't fit well because he didn't know that then. It seems much more likely that Luke failed to control his fears and he projected them in the form of the scariest thing he knew - Darth Vader.
Who knows what the evil in the cave was. I saw in one of the books someone suggest Yoda had fought a dark side jedi there when he first arrived on Degoba following the loss of the clone wars and was shielded from detection by the remnant dark side of the force left by his defeated adversary. But even that was conjecture on the part of an actual character! It does seem likely that whatever the evil was, it fed and amplified Luke's own fear to the point where he either hallucinated or conjured up some kind of physical manifestation and fought it. Either way, he should have trusted Yoda and gone in there confident of his ability to figure out whatever the cave threw at him.
Yoda: Your weapons... you will not need them.
By ignoring Yoda and taking his weapons, he was showing fear right from the start and therefore destined to fail the test.