Time turner from the Harry Potter series (and from the Eliezer Yudkowsky's venerable HPMoR fanfic) is a very useful device if you have some unfinished business in the recent past, like attending an extra class or saving a friend from a certain death. However, General Relativity has a few words to say about them, and they are not very flattering. I will only address one issue here: Energy conservation. TL;DR: if you use a time turner to vanish into the past, those around you will see you blown to tiny bits of Merlin-knows-what, quickly disappearing from view. When you appear in the past, this explosion appears in reverse.
Yes, the Time Turners as described violate conservation of energy. Something is happening that doesn't comply with Reality!Physics. Harry notices this the first time he encounters a witch. From Chapter 2:
"You turned into a cat! A SMALL cat! You violated Conservation of Energy! That's not just an arbitrary rule, it's implied by the form of the quantum Hamiltonian! Rejecting it destroys unitarity and then you get FTL signalling! And cats are COMPLICATED! A human mind can't just visualise a whole cat's anatomy and, and all the cat biochemistry, and what about the neurology? How can you go on thinking using a cat-sized brain?"
Professor McGonagall's lips were twitching harder now. "Magic."
"Magic isn't enough to do that! You'd have to be a god!"
Professor McGonagall blinked. "That's the first time I've ever been called that."
A blur was coming over Harry's vision, as his brain started to comprehend what had just broken. The whole idea of a unified universe with mathematically regular laws, that was what had been flushed down the toilet; the whole notion of physics. Three thousand years of resolving big complicated things into smaller pieces, discovering that the music of the planets was the same tune as a falling apple, finding that the true laws were perfectly universal and had no exceptions anywhere and took the form of simple maths governing the smallest parts, not to mention that the mind was the brain and the brain was made of neurons, a brain was what a person was -
And then a woman turned into a cat, so much for all that.
In both the case of Transmogrification and Time Turning either Conservation of Energy is part of a theory which just is not describing reality or there is some mass redistribution that Harry just hasn't detected. For example, conversion of excess mass into some kind of Dark Matter that exists in the MoR!Universe but neither wizards or physicists have studied yet.
Given the possibility of a Transmogrification mechanism and either any capability for sending information back in time or the presence of an excessively powerful predictor prior to the 'earliest' end of the time jump, Time Turning (or the appearance thereof to all observers) doesn't introduce any additional insanity.
Also don't forget the other side of the time turner action: Hermione appearing out of thin air just before walking into her extra class. The above process has to happen in reverse: an amount of matter equivalent to her mass has to travel inwards out of nowhere and coalesce into a person. Where did this matter come from?
Presumably the same place that the matter came from when MoR!McGonnagal!cat turned back into MoR!McGonnagal!human---either as yet undetected local conversion or from Magic.
How did it form before collapsing into a person?
Tricky question, which generalises to the general problem of determining what all of this 'magic' stuff reduces to or is based on. Simulation, advanced deception or extremely powerful technology from an overwhelmingly superior civilization or agent seem to be the obvious hypotheses. (That is, conditioning on the observations being accurate and not the result of insanity. Eliminating that as the most plausible hypothesis would take a heck of a lot of evidence.)
How did it know that it would need to time its arrival into a certain point perfectly with whatever time turner will have been set to?
Similar problem and (at least some) additional evidence for there being a powerful agent at work in one of the various ways that could occur.
That's some hard-core magic right there. Also, suck it, the Second law of Thermodynamics.
Indeed. In fact one could even go so far as to include that as part of the description of the genres of Science Fiction and Fantasy.
This post is a bit of entertainment for scientifically inclined Harry Potter fans.
Time turner from the Harry Potter series (and from the Eliezer Yudkowsky's venerable HPMoR fanfic) is a very useful device if you have some unfinished business in the recent past, like attending an extra class or saving a friend from a certain death. However, General Relativity has a few words to say about them, and they are not very flattering. I will only address one issue here: Energy conservation. TL;DR: if you use a time turner to vanish into the past, those around you will see you blown to tiny bits of Merlin-knows-what, quickly disappearing from view. When you appear in the past, this explosion appears in reverse.
Before we get to the time turners, however, let us consider an aside.
Let us start with a common question: if the Sun stop shining this instant, when would we notice? The common answer: it takes light 8.5 minutes to travel the distance of 150,000,000 km between the Sun and the Earth, so that's how long it will take. This glosses over the issue of what does "this instant" mean exactly at two different points in space, which is not so trivial given the relativity of simultaneity in Special Relativity. It is easily patched up, however, once we fix a global frame of reference. The Cosmic Microwave Background (CMB) is a natural one to use, and both the Earth and the Sun travel with a negligible fraction of the speed of light relative to the CMB. Anyway, the answer is still very close to 8.5 min.
Now another, deceptively similar question: if the Sun disappears this instant, how long before the Earth will stop orbiting the point where it used to be? The common answer: gravity travels with the speed of light, so also 8.5 min. This answer is obvious, simple and wrong. Yes, dead wrong. Why? because static gravity is not like light, it's more like electric field, only worse.
Let's first think of how you would make the Sun disappear. Maybe it turned into a black hole? Well, this would not really mean disappearance of gravity, the mass of the black hole will still be that of the Sun, and the Earth will happily (or unhappily, as the case may be) continue orbiting the Sun's corpse. So, in this case the answer is "it won't stop orbiting".
OK, so black hole was a bad example. How about a wormhole instead? You know, the evil Vogon-like aliens need to clear the room for a hyperspace bypass, and they build a wormhole from far away and suck all the matter in the Sun through it out of the way. What would happen then? There are a couple of hints: one is that from outside a wormhole is indistinguishable from a black hole, and the other is the Gauss Law. Both hints lead one to the same answer: just like with turning the Sun into a black hole, there is very little gravitational effect on the surrounding space. The rest of the now ex-Solar system will continue merrily on its way around the point where our Sun used to be.
An aside for those curious about the Gauss Law argument. The law in its integral form states that the flux of the gravitational field inward through any closed surface encompassing the Sun is proportional to the Sun's mass. To change the field, you need to remove some mass from inside this imaginary surface, by having it physically cross the surface. This last point may not be obvious, but it follows from General Relativity. Specifically, the Einstein's most misunderstood theory says that the spacetime curvature is determined by the (past and present) distribution of matter in spacetime. There are some exceptions, like the fixed-mass spherical objects, such as black holes and wormholes, which contains no matter, and gravitational radiation, which can carry away energy. But if you take a spherical object like the Sun and try to calculate what happens if you decrease its mass, General Relativity tells you that this mass has gone outward from the Sun in all directions in some form. It is not fussy about the form, as long as just the right amount of mass/energy has gone out.
Let me repeat for those who skipped the above paragraph: if you take the Sun and decrease its mass, the only way it can happen if this mass leaves the Sun outward and disappears into space. This happens all the time, of course, the Sun constantly loses its mass through radiation and solar wind, or in more drastic cases through Supernova explosions. Effects like this propagate no faster than light, of course. So they take forever to propagate all the way to infinity.
Now, back to the time turners. Hermione Granger might be but a small if incredibly studious girl, but she still has mass. If you were to peek at her using a time turner and disappear, her mass, small though it may be, still has to go some place, just like the disappearing Sun's mass had to go some place. The options are few: she can blow into tiny pieces flying past you, or disappear in a flash of brilliant light (and it takes a lot of light to carry away 50kg, what's with E=mc^2) . Basically, it will not be a pretty sight. What cannot happen is her simply vanishing, with no ill effects whatsoever. Well, it cannot happen if we are willing to keep Relativity around. Maybe we don't have to, what's with a certain deputy mistress turning into a cat and back, probably instantly changing her mass, with no ill effects on her or her surroundings. But if you give up on General Relativity, quite a few things will unravel, like all four Newton's laws.
Also don't forget the other side of the time turner action: Hermione appearing out of thin air just before walking into her extra class. The above process has to happen in reverse: an amount of matter equivalent to her mass has to travel inwards out of nowhere and coalesce into a person. Where did this matter come from? How did it form before collapsing into a person? How did it know that it would need to time its arrival into a certain point perfectly with whatever time turner will have been set to? That's some hard-core magic right there. Also, suck it, the Second law of Thermodynamics.
So, let me summarize: mass cannot just disappear, it has to spread out. mass cannot just appear, it has to coalesce. Thus time turners cannot be used inconspicuously, everyone around would be well aware of one's use, assuming they survive it. Actually, it probably cannot be used at all without breaking General Relativity and/or Thermodynamics. But hey, that's what magic is for.
EDIT: this post currently sits at -2 karma with 6 downvotes. I'd appreciate if any of the people who thought "I want less of this" explicate their logic to me, so I can do better next time.
EDIT2: OK, no one replied to my request... I'm guessing that some of you guys just quietly hate me :)