In the third Harry Potter movie, why did they save Buckbeak if he is already alive?

Question

Why did they go back in time to save Buckbeak if their future selves already saved Buckbeak?

Answer 1

Time travel is weird and hard to understand, but here goes.

Their future selves only saved Buckbeak because their past selves went back in time in order to do so. The Harry and Hermione who saved Buckbeak are the same Harry and Hermione who went back in time to save him, just at a different point in their own time streams.

If they hadn't gone back in time, their future selves wouldn't have been there in the past and Buckbeak wouldn't have been saved after all. They had to go back in time, so that what had already happened would actually happen. Forget causality: as soon as you introduce time travel, there's no longer any such thing as causality.

Answer 2

Potter and Hermione didn't time travel to save Buckbeak primarily. They went to save Sirius,

“Potter,’ said Madam Pomfrey soothingly, ‘it’s all right. They’ve got Black. He’s locked away upstairs. The Dementors will be performing the Kiss any moment now-'

_{-Harry Potter And The Prisoner of Azkaban}

As Dumbledore had no chance to save Sirius and that is the reason he insisted on Harry and Hermione to change the events by travelling back in time.

“Listen to me, Harry. It is too late, you understand me? You must see that Professor Snape’s version of events is far more convincing than yours.”

“What we need,’ said Dumbledore slowly, and his light-blue eyes moved from Harry to Hermione, ‘is more time.’ ‘But –’ Hermione began. And then her eyes became very round. ‘OH!'

_{-Harry Potter And The Prisoner of Azkaban}

If Harry and Hermione somehow came to know that Buckbeak and Sirius are safe, even then they need to travel to past and save them. Also, they have an extra task to let their past themselves know that they saved Sirius without actually meeting them.

Answer 3

Buckbeak was only alive because Harry saved him in the future.

The only reason Buckbeak was alive is because Harry had saved him in the future. Something similar happens when Harry sees a Patronus being cast to save him and Sirius. At the time he sees it, he doesn’t understand where it came from, and thinks it might be his dad.

“With every ounce of strength he could muster, Harry raised his head a few inches and saw an animal amidst the light, galloping away across the lake. Eyes blurred with sweat, Harry tried to make out what it was … it was bright as a unicorn. Fighting to stay conscious, Harry watched it canter to a halt as it reached the opposite shore. For a moment, Harry saw, by its brightness, somebody welcoming it back … raising his hand to pat it … someone who looked strangely familiar … but it couldn’t be …”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 20 (The Dementors’ Kiss)}}

However, it was actually Harry casting the Patronus in the future, and Harry still had to actually cast the Patronus in the future for it to save Sirius in the past.

“And then it hit him – he understood. He hadn’t seen his father – he had seen himself –

Harry flung himself out from behind the bush and pulled out his wand. ‘EXPECTO PATRONUM!’ he yelled.

And out of the end of his wand burst, not a shapeless cloud of mist, but a blinding, dazzling, silver animal.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 21 (Hermione’s Secret)}}

He explains to Hermione that he realized he had actually saw himself, and knowing he’d done it had given him the confidence he needed to cast a full Patronus.

“Did anyone see you?’

‘Yes, haven’t you been listening? I saw me but I thought I was my dad! It’s OK!’

‘Harry, I can’t believe it – you conjured up a Patronus that drove away all those Dementors! That’s very, very advanced magic …’

‘I knew I could do it this time,’ said Harry, ‘because I’d already done it … Does that make sense?”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 21 (Hermione’s Secret)}}

There was no point in the timeline where Harry hadn’t cast the Patronus, but he still actually had to cast it. The same thing would have happened with Buckbeak - he was always alive because Harry saved him, but Harry’d still have to actually go back and save him for him to be saved. They thought Buckbeak had been killed when they heard the executioner’s axe swing and hit something.

“There was a jumble of indistinct male voices, a silence and then, without warning, the unmistakeable swish and thud of an axe.

Hermione swayed on the spot.

‘They did it!’ she whispered to Harry. ‘I d-don’t believe it – they did it!”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 16 (Professor Trelawney’s Prediction)}}

But he’d always been saved by them - they heard the executioner swinging the axe into the fence.

“There was a swishing noise, and the thud of an axe. The executioner seemed to have swung it into the fence in anger. And then came the howling, and this time they could hear Hagrid’s words through his sobs.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 21 (Hermione’s Secret)}}

But none of this would have happened if they didn’t go back and save Buckbeak, because Buckbeak was only gone because Harry and Hermione went back and untied him.

Answer 4

Dumbledore was a smart man

Dumbledore knew that Buckbeak had escaped but he presumably did not have the foggiest idea how. It's not like there was a plethora of people in Hogwarts who would have, or could have, untied Buckbeak precisely in the minute between when the executioner confirmed that Buckbeak was there and when he came out to do the execution:

“No!” said Hermione. “If we steal him now, those Committee people will think Hagrid set him free!

We’ve got to wait until they’ve seen he’s tied outside!”

“That’s going to give us about sixty seconds,” said Harry. This was starting to seem impossible.

Dumbledore, being Dumbledore, tried to come up with a theory to explain what had happened. As we find earlier in Prisoner of Azkaban:

“Have you any theory as to how he got in, Professor?” asked Snape.

Harry raised his head very slightly off his arms to free his other ear.

“Many, Severus, each of them as unlikely as the next.”

When Dumbledore heard the story from Sirius, Harry, and Hermione, he decided that they might be able to use time travel to save Sirius. Once he decided on time travel he must have realized that that could be the perfectly reasonable explanation for how Buckbeak escaped – someone from the future who knew exactly what was going on had freed Buckbeak during the one minute window.

Based on this possibility, Dumbledore told Harry and Hermione to save Buckbeak as well (he didn't actually tell them directly; he hinted it to them).

The cool part is, though, that even if Dumbledore turns out to be wrong about Buckbeak being saved by time-travelers, there is not really anything to lose by telling Harry and Hermione to save him. Either Harry and Hermione will save him, or he will get saved some other way and Harry and Hermione will see how he got saved. Dumbledore knows that Buckbeak will definitely get saved – otherwise Buckbeak would have already been dead. So either he is providing the method of saving, or he is dispatching witnesses to find out what happened. Either way he wins.

Essentially, Dumbledore took a well-founded guess, that had no downside, and everything turned out great.