Why did Harry and Hermione arrive in the Entrance Hall after having used the Time-Turner in the hospital wing?

Question

In The Prisoner of Azkaban, after the events in the Shrieking Shack, Harry, Ron and Hermione are in the hospital wing. Dumbledore suggests they (Hermione & Harry) use the Time Turner to save more than one innocent, and leaves them. Hermione throws the chain around her and Harry's necks, turns the hourglass, and they arrive, three hours earlier... in the entrance hall.

I know there's not much information about how Time-Turners work, but I thought they only made you travel into time, not into space. Why did they arrive in the entrance hall and not where they were standing, in the hospital wing?

Furthermore, if there is in fact a space-travelling part in this, why the entrance hall? Was it Hermione's choice (it's not implied, and next to Hagrid's might have been a better choice)? Was it the Time-Turner's choice? Something else?

The quote is quite long, I just included a part of the beginning of the chapter showing they were in the hospital wing, then the part where they use the Time-Turner:

"For heaven's sake!" said Madam Pomfrey hysterically. "Is this a hospital wing or not? Headmaster, I must insist --"

[...]

"Here --"
She had thrown the chain around his neck too.
"Ready?" she said breathlessly.
"What are we doing?" Harry said, completely lost.
Hermione turned the hourglass over three times.
The dark ward dissolved. Harry had the sensation that he was flying very fast, backward. A blur of colors and shapes rushed past him, his ears were pounding, he tried to yell but couldn't hear his own voice --
And then he felt solid ground beneath his feet, and everything came into focus again --
He was standing next to Hermione in the deserted entrance hall and a stream of golden sunlight was falling across the paved floor from the open front doors.
(Prisoner of Azkaban, Chapter 21, Hermione's secret - emphasis mine)

Answer 1

Every indication is that using the Time-Turner moves you in space to wherever you were at the time you've just travelled to.

Arguably, this makes Time-Turners more risky than they need to be since the risk of running into yourself increases astronomically if you appear nearby your younger self. Yet this seems to be what happens. In other words, they work as you suggest in both time and space.

The reason why they end up in the Entrance Hall is given straight after the passage that's quoted in the question.

He was standing next to Hermione in the deserted Entrance Hall and a stream of golden sunlight was falling across the paved floor from the open front doors..."In here!" Hermione seized Harry's arm and dragged him across the hall to the door of a broom cupboard.; she opened it, pushed him inside amongst the buckets and mops; followed him in, then slammed the door behind him..."Shh! Listen! Someone's coming! I think - I think it might be us! Hermione had her ear pressed against the cupboard door. "Footsteps across the hall...yes, I think it's us going down to Hagrid's!"

Prisoner of Azkaban, Chapter 21, Hermione's Secret

The fact that Harry and Hermione walked through the Entrance Hall moments afterwards wasn't a coincidence. It happened to be where they were exactly three hours before they went back in time. Presumably if you went to the other side of the country on, say, the Knight bus and then used the Time-Turner then you'd find yourself back in your original location.

This does rather leave it to chance that the location that you're time-travelling into will be deserted so that others don't see you emerging from thin air. And, as I say, the risk that you run into yourself is pretty high. The book doesn't address how Hermione has avoided startling other people by popping into mid-air when using the Time-Turner throughout the year. It has her disappearing when Harry and Ron were sure she was behind them and it has Hermione emerging from a staircase that Harry and Ron know she couldn't feasibly have come down. Nowhere does it show Hermione actually using the Time-Turner, except in the section that I quoted.

Answer 2

Think of it as a safety mechanism: you're traveling backwards in time, which means that presumably you were in a safe location earlier (since you survived being there before) and would, ostensibly, only choose to travel to a time in which it would be safe for you to materialize. If you remained in the same location you were in when you time traveled, you would have no idea what might be (might have been) going on in that location when you arrived.

The spell also seems to have a provision to move you just slightly away from your initial location, so that you don't appear literally on top of yourself; presumably it has buffers that select a safe (read: unobserved) spot in the immediate vicinity for you to pop-up, rather than actually just appearing out of nowhere (as she does in the movies, alas). Perhaps the Time Turners are tied to a sort of Homenum Revelio Charm that nudges the traveler away from any wix in the immediate area?

At any rate, I think if I were going to time travel, I would prefer doing it by materializing in my previous location (where I would know what was happening) than in the one I was in at the present/future moment where I'd have no idea what was waiting for me!

Answer 3

Because Harry Potter Time Turners work like Google Maps or the TARDIS from Doctor Who, it's never accurate and either by plot or by pot luck you could end up anywhere.