HARRY POTTER AND THE GREAT CONSERVATION LAWS
2.1 THE TAXONOMY OF FANTASY
The “physics of fantasy” seems like an oxymoron: by definition, fantasy doesn’t concern itself with science but with magic. However, a lot of fantasy writers follow in the tradition of science fiction writers in trying to set up consistent rules by which their fantasy worlds operate. This is because many fantasy writers are science fiction writers as well. It is almost a universal trait: those who write quasi-realistic science fiction will also write quasi-realistic, rules-based fantasy; those who don’t generally won’t set up rules by which magic works.
Among the former is Ursula K. Le Guin, whose Earthsea trilogy has long descriptions of the “rule of names” underlying all magic. Her books include several lectures by magicians on exactly how this works. Many writers have found her works compelling enough to copy her rules in their own stories. Others base their magic rules on outdated scientific or philosophical ideas, as Heinlein did in his novella, Magic, Incorporated. Magicians in that book use the “laws” of similarity and so forth, to perform their magic. Randall Garrett in his Lord Darcy stories writes of a world in which magic (following these laws) has developed instead of science. The stories are full of descriptions of how the magic works and is used in solving crimes.
The popular writer J. K. Rowling in her Harry Potter novels does not attempt to have the magic in her books follow any known laws of science. Please don’t misunderstand me: I love her books, but not for any attempt on her part to be consistent in how magic works. This is why this section of the book is called “Potter Physics”: her body of work contains innumerable examples of magic being used in ways that violate physical law and are also internally inconsistent. She belongs solidly to the second class of writers.
These rules of magic don’t have to follow known laws of physics—indeed, they can’t, or else we’d call them science fiction instead of fantasy. The laws that many rules-based fantasy writers choose to keep are typically the most fundamental of the physical laws: the great conservation principles—the conservation of mass, energy, and momentum—and the second law of thermodynamics.
2.2 TRANSFIGURATION AND THE CONSERVATION OF MASS
Harry spun around. Professor Moody was limping down the marble staircase. His wand was out and it was pointing at a pure white ferret, which was shivering on the stone-flagged floor, exactly where Malfoy had been standing.
—J. K. ROWLING, HARRY POTTER AND THE GOBLET OF FIRE
The issue of shape-changing in the world of the Harry Potter novels is vexing. Since the 1800s one of the principal ideas of science has been the conservation of mass: the total mass in a closed system cannot change. In turning Draco Malfoy into a ferret, what did Professor Moody do with the rest of his mass? If we assume that Draco at age fifteen had a mass of about 60 kg and a small ferret has a mass of about 2 kg, where did he stash the other 58 kg? This is an issue for most fantasy writers when dealing with shape-changers. In Swan Lake, when Odette is changed into a swan by the evil wizard Rothbart, what happened to the rest of her? One can imagine some sort of weird biological process by which flesh morphs from one animal form to another, but where does the excess go?
A number of fantasy writers have dealt with this issue head-on. In Poul Anderson’s Operation Chaos, the hero is a werewolf who explicitly states in the course of the book that his mass is the same in both states. This is OK for a werewolf, as the average adult wolf has about the same weight as a very light adult male, but it raises problems later on when the hero meets a weretiger. In human form, the magic user must maintain a weight of nearly 400 pounds simply to make a fairly small tiger. This entails health problems and severe psychological stress. In Niven’s story “What Good Is a Glass Dagger?,” the hero implicitly invokes the principle of conservation of mass when he says that he doesn’t look too overweight as a human, but as a wolf he’d look ten years’ pregnant.
This doesn’t bother most fantasy writers, perhaps because it would impose too strong constraints on many fantasy stories if writers stuck to the conservation of mass. For example, when trapped by the sword-wielding barbarian, the beautiful sorceress can’t turn herself into a dove and fly away. Instead, she’d have to turn into an ostrich and kick him in the ribs. (Actually, that would make a good story.) Thinking about where the mass goes leads to headaches. Einstein tells us mass is convertible into energy at a “cost” of 9 × 1016 J/kg. If somehow the mass turns into energy, then even a very small imbalance makes a very big boom. (The Hiroshima bomb blast was about the equivalent of 1 gram of matter converted into energy.) Whenever Professor McGonagall transforms herself from a human into a cat, she ought to release as much energy as all of the atom bomb tests ever done, all at the same time. And where does she get the extra mass when she turns back into a human?
There doesn’t seem to be any good answer to this problem. If mass isn’t conserved, maybe it is sloughed off somewhere during the transformation (yuck) or stored in some extra dimension or something. It is a vexation. You have even more problems if you are transforming material from one element to another: it can be done, but with difficulty. Consider Medusa’s problem of turning people into stone. People are made up mostly of carbon and water, whereas stone is mostly silicon. Carbon has atomic number 6 and atomic mass 12, and has six protons, six electrons, and six neutrons. Silicon has atomic number 14 and atomic mass 28, with 14 protons, 14 electrons, and 14 neutrons. You can’t turn one elementary particle into another willy-nilly: to convert the carbon in a living body into silicon, you have to provide the extra particles somehow. Perhaps Medusa bombards her victims with high-energy particles? And if she can do that, why bother turning them into stone at all?
2.3 DISAPPARITION AND THE CONSERVATION OF MOMENTUM
Morris got bugeyed. “You can teleport?”
“Not from a speeding car,” I said with reflexive fear. “That’s death. I’d keep the velocity.”
—LARRY NIVEN, “THE FOURTH PROFESSION”
In the Harry Potter novels, the power of disapparition is used commonly. This is the ability to vanish from one spot and reappear instantly in another. In other works this maneuver is more commonly referred to as teleportation. In Star Trek the transporter is used for this purpose. Issues of conservation of energy plague teleportation in a similar manner to shape-changing. In disapparating, is Harry being converted to energy, zapped off somewhere at the speed of light, and converted back? If so, this is an awful lot of energy to manage: 9 × 1016 J for every kilogram we teleport. If even 1% of the energy isn’t contained somehow, we have the equivalent energy of an H-bomb going off. We also have all the other problems from the last section: turning 80 kg or so of matter into “pure energy” violates all sorts of conservation laws, and also involves a rather bad issue related to entropy. Or murder, if you’d rather think of it that way. A human being is a very complicated structure, and by transforming the body into pure energy, as far as I can tell, you are killing the person. Bringing a person back to life after doing this by reconstituting him or her elsewhere is implausible.
Tearing someone apart here and putting that person back together there seems pretty hard. Another way in which authors have justified teleportation involves ideas of quantum mechanics. Larry Niven mentions this in his essay “The Theory and Practice of Teleportation” [178]. Quantum mechanical tunneling is a lot like teleportation: a particle, like an electron, goes from one side of a barrier to the other side without moving through the intervening space. Well and good. It works for electrons, why not for people?
To explore this idea, I am going to invoke the famous Heisenberg uncertainty principle. Most of the physics we’ve discussed so far has been classical: we’ve assumed that objects follow well-defined trajectories. In essence, weve assumed that we can know exactly where they are and how fast they move at all times. This isn’t true. Very fundamental ideas of physics tell us that
where Δx is the uncertainty in the position of the object, Δp is the uncertainty in the momentum (= Mv), and h is Planck’s constant, which has a metric value of 6.626 × 10−34 J-s (joule-seconds). This inequality means that we can’t measure the position or speed of anything with arbitrary precision. Because we are taking the product, making the uncertainty in one smaller makes the uncertainty in the other larger. Unfortunately, h is really small. This is why quantum mechanical effects are important only for atoms or subatomic particles, at least under most circumstances. There are some fascinating exceptions, however.
For the past century physicists have performed experiments in which quantum mechanical effects have manifested themselves on macroscopic scales. Two older examples of large-scale quantum behavior are superconductivity and superfluidity. At low temperatures, atoms in liquid helium behave in some ways as if they were a single atom, with no individual identity of their own. This superfluid state is characterized by almost no resistance to motion, helium liquid “climbing” out of containers, and weird quantum mechanical vortices occurring. Unfortunately, this superfluid state happens only when you can get the temperature below about 2 K above absolute zero.
Superconductivity is a similar low-temperature effect in which the resistance of metals to electrons flowing through them drops abruptly to zero at temperatures a few degrees above absolute zero. This happens more or less because the electrons in the metal find themselves in a superfluid state. In the early 1980s physicists found examples of substances that became superconductors at high temperatures. However, “high” is a relative term, meaning around the temperature of liquid nitrogen, 77 K. This is still about 200°C below room temperature.
In 1995, Carl Weiman and Eric Cornell at the University of Colorado at Boulder led a team that created the first Bose-Einstein condensate in atoms of rubidium. Wolfgang Ketterle’s team at MIT achieved this a short time later. A Bose-Einstein condensate (BEC) is a group of atoms cooled to such a low temperature that their quantum mechanical uncertainty is so large that one cannot tell one individual atom apart from another one, even in principle. (Superfluid helium is like a BEC in some ways, but is much more complicated.) To get this state, the teams used a combination of lasers and other techniques to achieve temperatures of about 200 billionths of a degree above absolute zero!
In a BEC, the position of an atom is completely uncertain within a region a few hundred micro-meters across. This is roughly the diameter of a human hair. This doesn’t sound big, but by comparison, the size of an individual atom is only about an angstrom, or 100,000 times smaller than that. It’s a good start.
There is an effect called quantum teleportation in which the information about a quantum mechanical system can be transmitted from one quantum system to another. It has only been performed on systems of a few atoms at a time so far. As the webcomic xkcd points out, it isn’t the same as “real” teleportation [172]. Unfortunately, there is another quantum mechanical effect called the “no-clone” theorem, which proves that it is impossible to create a perfect copy of a quantum mechanical system [255].
The common feature is that all of these effects occur at really low temperatures. In recent years physicists have been able to take large macroscopic objects and “cool” them to the point that quantum mechanical effects are important. (I put “cool” in quotation marks because the techniques that are used don’t involve cryogenics or any of the classical methods used to cool large objects.) This is a long way from teleporting objects, but it is a start. The main examples are the mirrors used in LIGO, the detector designed to detect gravity waves from objects such as colliding black holes. The LIGO mirrors are “cooled” using lasers and electromechanical techniques so that their motion is limited only by quantum mechanical uncertainty. This is because they are being used to detect gravitational waves, which are so weak that gravity waves from two colliding black holes will make the mirrors move by a distance 100,000 times smaller than an atomic nucleus!
If we could only make h larger somehow, we might be able to build a practical teleporter. Unfortunately, two things stand in the way of doing this. First, h is a fundamental constant of nature. No one has any idea of how to change its value, let alone whether this is even possible. Second, even if we could change h, small changes in its value radically change the laws of chemistry. Changing its value by only 1% or 2% would probably make life impossible.
The idea of locally changing the value of Planck’s constant has been used by Tim Powers in some of his urban fantasy novels, most notably in Last Call and On Stranger Tides, although not for teleportation [195][196]. In the latter book, an eerie scene takes place at the Fountain of Youth. Dr. Hurwood, the book’s villain, states that the “uncertainty” is polarized there: the ground has none, while the air’s quantum uncertainty is huge, to the point that a shadowy personality can answer questions from it. Powers is an author who likes to play around with pseudoscientific ideas like this in fantasy settings, which gives his works a uniquely creepy vibe.
Teleportation certainly seems like fantasy, in that I don’t see any means of teleporting large objects short of magic. It violates too many laws of physics.
If we imagine that teleportation really works, however, what do the laws of physics imply about it? Larry Niven was the first writer to discuss the conservation of momentum as it applies to teleporting. The epigraph at the beginning of this section illustrates the point. Let’s say you’re in a moving car, and let’s also say that you’re a wizard in Harry Potter’s London and have passed your disapparition test. You are being driven down the highway at 60 mph and see a friend on the side of the road, so you disapparate out of the car to her side. According to Einstein, all reference frames are equivalent, so in principle you keep the momentum you had from the car and appear by her side traveling at a speed of 60 mph relative to her and the ground. Ouch! In a similar vein, there is a serious issue in “beaming down” to a planet using a Star Trek transporter:
Let’s say that the Enterprise or one of its descendants is in a geosynchronous orbit around the Earth, so that it always stays over the same point on Earth’s equator. You might think that this means that the Enterprise is moving at the same speed as the point Earth, but not so: the Earth makes one full revolution around its axis every 24 hours, so the spaceship must rotate around the Earth in the same amount of time. It moves through a larger circle in the same amount of time as a point on the surface of the Earth, so it must be moving faster. Because the geosynchronous radius is 42,000 km, it is moving faster by a factor of 42,000/6,400 = 6.6, which is the ratio of the radius of the geosynchronous orbit to that of the Earth. Since a point on the Earth’s equator moves at a speed of about 1,000 mph, anyone beaming down to the planet will land with a speed relative to the ground of 6,600 mph − 1,000 mph = 5,600 mph. This could be a problem.
Larry Niven has written some excellent stories that have been collected in the book A Hole in Space, which dealts with teleportation and its scientific and cultural ramifications [180]. In these stories and in the essay “The Theory and Practice of Teleportation,” he discusses whether objects teleporting downhill increase their temperature, the conservation of momentum in teleporting to other latitudes, and similar ideas. We’ll look at these ideas more in the web-based exercises, which can be found at press.princeton.edu/titles/10070.html.
On a side note, there’s a lot of inconsistency in how Star Trek handles the transporter. Larry Niven in his collection All the Myriad Ways and Alfred Bester in his novel The Stars My Destination both noted how good a weapon a teleporter is [178][38]. The Star Trek transporter represents the epitome of this: using it, one can put a bomb anywhere. One can kidnap anyone. In one episode, a transporter malfunction regressed Picard to a ten-year-old child! Think of the ramifications: repeat this somehow and you have the Fountain of Youth. Maybe by changing the settings, you can heal any disease, any injury. And the show never explored any of the ramifications of this!
2.4 REPARO AND THE SECOND LAW OF THERMODYNAMICS
“Would you like my assistance clearing up?” asked Dumbledore politely.
“Please,” said the other.
They stood back to back, the tall thin wizard and the short round one, and waved their wands in one identical sweeping motion.
The furniture flew back to its original places; ornaments reformed in midair, feathers zoomed into their cushions; torn books repaired themselves as they landed upon their shelves … rips, cracks and holes healed everywhere, and the walls wiped themselves clean.
—J. K. ROWLING, HARRY POTTER AND THE HALF-BLOOD PRINCE [197 PP. 64–65]
There’s a scene in one of Jim Butcher’s Dresden Files novels in which Harry Dresden and his friends are being chased by a giant-sized scarecrow through the streets of Chicago [45]. It’s just rained, and Harry needs to blast something in front of him, so he sucks energy out of a water puddle the giant is about to step in to use to fireball whatever is in front of him. The giant slips on the newly formed ice, the obstacle is destroyed and everyone, except the monster, and most physicists, is happy.
What Harry just did violated one of the fundamental principles of physics: the second law of thermodynamics. The second law of thermodynamics says:
In order to make heat flow from an object at a higher temperature to one at a lower temperature, you have to do work.
It certainly doesn’t seem that Harry is doing any work in making the heat flow in the opposite direction it “wants” to go. Harry has violated the law that in any thermal interaction, the entropy of the world must increase.
2.4.1 Entropy Changes
Entropy is a subtle concept in physics. As most of my readers are aware, entropy is a measure of the “disorder” of the world. It is subtle because it is hard to formally define what order and disorder are. In lay terms, an ordered state of the world is one that is less probable than a disordered one. To give an example, let’s say you are a college student. You have just cleaned your dorm room, so your books are all on the bookshelf, the covers are on the bed, blank paper is in the desk drawer, beers are in the fridge. This is an ordered state because it is an improbable one. Joking aside, we can imagine taking each item in the room—textbooks, beer bottles, bedcovers, and so forth—and tossing a die to see where to put everything. If the die lands showing 1, we put the item in the fridge; if it shows 2, we put the item on the bed; if it shows 3, we put the item in the bookshelf, and so on. When we do this, we are likely to find bedsheets on the bookshelf, beers on the bed, and books in the refrigerator. The ordered state is ordered because it is a low-probability one in that sense: tossing the die will far more likely end up with everything in the room thoroughly mixed up.
It is unfortunately a long and hard road to get from the concept of order and probability to heat and temperature. It can be done, but it is beyond the scope of this book. Suffice it to say, if we have a thermal system at absolute temperature T and add an amount of heat Q to the system, the entropy of this thermal system increases by an amount
If we remove the heat, the entropy decreases by the same amount. There’s nothing that forbids the entropy of one part of a thermal system from decreasing, but the second law of thermodynamics tells us that it has to increase by at least as much somewhere else.
The reason why heat spontaneously flows from high temperatures to low temperatures has to do with the denominator in the formula. Because we are dividing by T, you get more entropy adding heat to a cold object than to a hot object. So a hot object in contact with a cold one will “want” to lose heat to the other one. Yes, its own entropy is lowered by losing heat, but the cold object’s entropy increases by a larger amount. The total entropy in the world increases. If the process were to go in reverse, as in the Dresden Files novel, the net entropy would decrease. Let’s try to put some numbers in.
Harry extracts enough energy to freeze a large puddle. Let’s assume that the water in the puddle has a mass of 100 kg. I’m going to simplify things by assuming that the puddle is already at 0°C (= 273 K), as befits a novel set in Chicago in midwinter. It is already at its freezing temperature, although still in a liquid state. All we have to do now is extract energy at constant temperature to freeze it. The latent heat of fusion of water is 3.34 × 105 J/kg. This is the amount of energy we must extract from each kilogram of water at its freezing point to turn it into ice. Therefore Dresden must extract 3.35×107 J from the puddle to freeze it. The entropy of the puddle has decreased by
This energy is put into the fireball. All Ray Bradbury fans know that paper burns at Farenheit 451, or 232°C, or 505 K. Its entropy has increased by
This is less than the entropy leaving the puddle, so, all other things being equal, the entropy of the world has decreased by the difference, or 5.66 × 104 J/K.
Is there a way out? The astute reader has probably been wondering how refrigerators work if entropy must increase. The answer is that refrigerators must generate enough entropy by some means or another to overcome the deficit. Harry must be acting as the refrigeration unit here. Work must flow from Harry into the fireball to make up the deficit. By my calculations, Harry must expend a minimum of 2.8×107 J to make this happen. This is about the amount of energy he would consume in food over the course of three days.
The reparo spell used by Dumbledore and Slughorn is no less problematic. It decreases perceptible disorder; it is easier to see that disorder is lessened than in the Dresden case, but harder to calculate the entropy decrease. We can do a very rough estimate of this. Let’s say there are a certain number of places where we can put any of the things in the room—say, 1,000. We simply imagine dividing up the floor of the room into 1,000 different cells and tossing objects at random into each of them. (Don’t like 1,000? Then divide it up finer if you want. It’ll turn out not to make a huge difference).
Then let’s say we have 100 objects to distribute around: a few books, crystals from the chandelier, pens, furniture, other items. We take one object and toss it into a random place. There are 1,000 different ways to do this. We take a second object and do the same: there are 1,000 ways to do this. For this calculation I am assuming that we can put more than one object into a given cell. Therefore, there are one million different ways of distributing two separate objects! If N is the number of objects we have and M is the number of ways we can distribute each object, the total number of ways of distributing all the objects, is Ω,
Therefore, there are 1,000100 = 10300 different ways to distribute all of the different objects at random in the room, and only one way to do it correctly. If we were able to toss everything out at random once per second, it would take on average 10300 seconds, or 10292 years, before everything was in its proper place. This is much, much longer than the universe has lasted and much, much longer than the stars will last.
If one digs into the issue, it turns out that the change in the entropy of the situation is given by the formula
where k = 1.38 × 10−23 J/K. To do the calculation I am using the property of logarithms that ln MN = N ln M:
This is a small entropy change compared to what Harry Dresden had to achieve when fighting the monster. The reason why the change is low is because of the small number of objects involved: we only had to sort out 100 things into their proper places. Another way to put it is that it would take Dumbledore and Slughorn a few hours to sort out where everything went to straighten up the room. In Harry Dresden’s encounter, he would need to “sort” hundreds of moles of particles, something that is much, much more difficult.
Rowling ignored a lot of ramifications of the reparo spell. Can it be used to heal injuries? Can it be used to make someone younger? After all, an increase in entropy is linked to the flow of time. We separate the future from the past by looking at the direction in which entropy increases. Time travel is possible in Rowling’s world; is the reparo spell reversing entropy by reversing the flow of time somehow?
I’d like to end this chapter the way I started it, by looking at contrasting philosophies of how magic works in a given author’s world. Superficially, the Dresden chronicles are a dark mirror of the Harry Potter books. There are a number of striking similarities between them, apart from being about wizards named Harry.
• They are set in the modern-day world with magic side-by-side with the mundane.
• The magic worlds are hidden from the “muggles” in each series.
• Both have draconian magic “governments,” the White Council in the Dresden Files and the Ministry of Magic in the Potter books. Neither of the governments has much regard for the civil rights of its respective subjects.
• Both series center on conspiracies to undermine said governments.
• Both series use fake Latin for incantations.
However, Jim Butcher seems to have put more thought into making his system of magic self-consistent than J. K. Rowling has. As I discussed above, the reparo spell involves less of an entropy decrease than Dresden’s spell did, and thus in a physics sense is less implausible. However, to me Dresden’s spell feels less implausible than Dumbledore’s. This is because Jim Butcher established rules for how magic is used throughout his books. In other works, Harry Dresden remarks that the rings and amulets he wears store up kinetic energy over time by robbing a little bit from when he walks. When they are discharged in a fight, they must be recharged. Harry Dresden gets exhausted from too much magic use, which never seems to happen in Rowling’s novels. The plots in the Dresden Files novels are constrained and driven by the limitations on magic.
Rowling’s books don’t work the same way. Even though she writes in Harry Potter and the Sorcerer’s Stone that Harry learns early on that not all of magic is wand-waving and saying a few words, this is pretty much how it works throughout her books. There don’t seem to be any major rules on what is hard to do with magic versus what is easy to do. This is highlighted in the third book by the Ministry of Magic handing over a time machine to a thirteen-year-old girl, for no better reason than to let her take more classes. By contrast, in the Dresden Files novels, time travel into the past is a capital offense. Out of nowhere it is mentioned toward the end of Harry Potter and the Deathly Hallows that one cannot transfigure other items into food; this is probably driven by a plot need, to force the students in hiding in Hogwarts to have a connection to the outside world (i.e., the Hogshead tavern) [206]. I will reiterate that I enjoy the Harry Potter novels, but the purist in me gets annoyed by the lack of any overarching philosophy of what magic can and can’t do. To paraphrase Poul Anderson, designing a magic system is the source of many plot points.