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Mass. The stuff of stuff. Tough to move about, painful if you run into it, and a tendency for big things to pull together. Recently I’ve been pondering mass, and given the stunning work on the Higgs boson that’s going on at the moment, I thought I might share some thoughts.

So: mass. It seems a pretty intuitive concept. Push (or pull) something, and the heavier it is the more it resists the force you apply. This is encapsulated in physics in Newton’s second law ($F=ma$) and is fundamental to classical mechanics. It’s so intuitive and well-established in physics that its easy to forget that although Newton’s laws are astoundingly accurate, useful, and powerful they don’t say anything about what mass actually is.

One thing its definitely not, is easy to explain or well-understood. For one thing, there are two concepts of mass in physics. There’s the mass I just mentioned, Inertial mass which is an object’s resistance to changes in its motion, but there is also Gravitational mass, which is the property that causes massive objects to attract each other – this is the stuff of both Newton’s law of gravitation and Einstein’s general relativity. If you think about it, these are in fact quite different things: one is the response of an object to being shoved around, the other is its tendency to attract other masses. In strict terms these are to completely different interactions: if I push something I’m interacting with it via the electromagnetic force (yes, I am. electro-chemical processes in my brain and muscles move my hands (or whatever) to an object an push against it. On the smallest scale this is mediated by the electrons in the molecules of my skin cells repelling the electrons in the atoms at the surface of whatever I’m pushing), whereas gravity is usually thought o be a separate fundamental interaction. General relativity models it as the object curving the space it sits in. Two completely different forces acting in completely different ways.

Looked at from this perspective, it almost seems to be a coincidence that both of these concepts have the same name – why call them both mass? Aren’t we conflating two different things? So here’s something worth pondering: if we measure the inertial mass of an object and then measure the gravitational mass of the same object they are exactly the same. And that, if you think about it, is pretty odd. There have been some astonishingly accurate comparisons of inertial and gravitational mass over the years, and none of them have ever detected any measurable difference between them.

So, enter Einstein. The usual explanation of the equality of inertial and gravitational mass is the Principle of Equivalence. Einstein realised that inertial and gravitational mass are connected via acceleration. Imagine you’re in space, floating, weightless. You’re not experiencing any forces, including gravity. Now imagine you’re in a lift in a very, very tall building. When the lift is stationary you feel gravity. You also feel other little forces when the lift speeds up or slows down – a little heavier when the lift starts to move upwards, a little lighter when the lift slows before stopping at a floor. Now imagine that the lift cable snaps – you fall, until you hit the ground. Being in the lift you don’t feel the air rushing around you and there are no windows so you can’t see the walls zooming past or the floor shooting upwards, encompassing your imminent doom. This means you don’t feel any forces – you, the floor, the walls, the air around you are all being accelerated in the same way towards the ground so while you are falling you are weightless, just like you were in space. Now imaging that the building is so tall that you never hit the ground – you can’t tell the difference between free-fall and the absence of gravity. This is the principle of equivalence – an “inertial frame” (the one in space) is identical to one in freefall. In general relativity, the free-falling one is thought of as moving along a line on a curved spacetime. These two things are equivalent.

So there you have it: inertial and gravitational masses are the same because of the principle of equivalence. Except… well, the clue’s in the name, really – the principle of equivalence is just that, a principle. It works, but it doesn’t tell you WHY these things are equivalent, just that the theory works out if they are. It puts me in mind of a quote about wave-particle duality in quantum mechanics: they are completely different things which we think of as being the same. It is as if we saw a rabbit sitting in a tree. This would be pretty unusual, but is completely explained if just think of the rabbit as being a cat, in which case we’d understand its behaviour quite well. The two masses are equivalent because the theory tells them to be. This is clearly not the end of the story.

But none of this is anything too remarkable – mass and mass, gravity and acceleration — all pretty unremarkable. I congratulate you for persevering this far. The really interesting stuff comes when you start thinking about quantum mechanics and another famous Einsteinian concept-  the equivalence of energy and mass, $E=mc^2$.

Mass is a long-standing problem in the quantum world. On the one hand there are ongoing efforts to unify the four forces and construct a quantum description of gravity. I’m no expert here, but given that this has been THE problem in theoretical physics for the better part of 40 years and we’re still very far from testable experimental predictions, it’s safe to say that this is hard. We’ve got string theory, we’ve got quantum loop gravity, we’ve got extra dimensions and whorls in spacetime, and unfortunately we’ve got serious difficulties with diverges and suggestions based around the compactification of multiple higher dimensions. Heady stuff.

Then there’s inertial mass in the quantum world. This is the stuff that’s been making headlines of late with the likely discovery of the Higgs Boson. The Higgs mechanism is a hypothetic answer to what mass actually is. The idea is that empty space is filled with a thing called the Higgs Field. This is like a sticky soup of virtual particles which resist changes in its motion. A good analogy for this is a ping-pong ball on a string in a bucket of water (no, really). Forget for a moment that ping-pong balls float, imaging that the ball is in the waterand you pull it with the string. The water resists the motion of the ball and makes it feels heavier. It would be worse if it were in treacle. This is what the Higgs field does – it resists changes to the ball’s motion and causes an effect that’s a lot like mass.

The Higgs Boson is the particle tht ediates the interaction between the Higgs Field and particles placed in it, and you might be tempted to think that it’s got inertil mass nailed, but you’d be wrong. The Higgs field explains the mass of elementary particles, like electrons and quarks. As we all know, protons and neutrons are made of three quarks, so you’d think that their mass would be about three times the size of a quark. Quantum Mechanics being what it is, this isn’t the case. Protons and neutrons are actually about 500 times more massive than their constituent quarks. The remainder is made up from the binding energy of the quarks (the energy of the bonds connecting them together). There’s a lot of energy in there, and its contribution to the mass can be calculated via none other than $E=mc^2$ and gives a very accurate estimate of the mass of protons and neutrons.

There’s just one snag here: this mass isn’t coupled to the Higgs Field. The Higgs interaction couples to the Electroweak force, but quarks are bound via the Strong force. $E=mc^2$ tells us the amount of mass in there, but not the reason why this energy causes resistance to changes in its motion. We also don’t know the relationship between Higgs mass and gravitational mass or how all this might relate to the principal of equivalence. Put simply, we don’t know what 98% of the mass in atoms is or how it connects to gravitation. Hence my suggestion that we don’t really understand mass.

And this is also born out by a current crisis in theoretical physics. It’s easy enough to state: what would you see and what would happen to you if you fell into a black hole (and specifically, what happens when you cross the event horizon)? Until recently, the answer from most physicists would have been that you wouldn’t really notice – you be passing through empty space and the principle of equivalence lets us know that this would feel like… floating in space. After a long time you’d notice that your feet were falling faster than your head (assuming you were falling feet first) and after a very long time the difference in force would eventually tear you apart. Ow. Nasty way to go.

More recently, though, cracks have emerged in this story. They have to do with the nature of the vacuum — classically, there’s nothing there, but in the quantum world it’s a writhing, foaming sea of pair of virtual particles spontaneously coming into being and then annihilating (this is a consequence of the uncertainty principle. A completely empy vacuum would have known energy (zero) forever, which is forbidden).

At the event horizon of a black hole one of the pair will fall into the black hole, and the other will zip off to infinity. This is Hawking Radiation, the fleeing particles steal a bit of energy from the black hole and cause it to shrink slightly. Eventually it causes an isolated black hole to vanish completely.

The problem is that it doesn’t stop there. The information about the particle that falls into the back hole is destroyed when it falls in — there’s no way of discovering what the particle was by looking at the radiation coming from the black hole itself. This is a problem because a fundamental principle of Quantum Field Theory is that information is never lost — you can always recover it. A potential way out is to imagine that the pair of particles are entangled, and the one that zips off to infinity tells us about the one that falls in, but this leads to a lot of energy being released when the pair separate.

And here comes the punchline: General Relativity says you wouldn’t feel much when you fall into a black hole, quantum mechanics says you would meet a wall of fire at the event horizon. Either the principle of equivalence is wrong or the holographic principle is wrong. The two pillars of modern physics are in contradiction: at least one is going to fall, and it’s all linked to the nature of mass and the nature of the vacuum.

So what are people doing to get around this? A surprising amount, it appears.

There re a couple of different approaches, all of which seem to revolve around studying the quantum vacuum itself. Firstly, there are several theories which suggest that the reason why it’s so difficult to combine quantum field theory with general relativity is that gravity isn’t actually a fundametal force at all – it’s an emergent effect from the interaction of particles with the quantum vacuum. General relativity then emerges as an “effective theory” at larger scales. The sort of emergence has been observed in systems like crystals and superconductors and might lead to the origin of gravitational mass.

On the other hand, there an approach that models the quantum vacuum using classical physics (which is apparently quite good at reproducing blackbody curves) which treats inertial mass as another interaction of particles and the vacuum. This time resistance is generated by changes in motion due to exchanging energy with this classical quantum vacuum.

What’s interesting about these is how similar the approaches are: particle interacting with some model of the vacuum. They also both suggest that if we could manipulate the vacuum we could also manipulate mass, which would be very exiting indeed.

There’s also the ongoing problem with Dark Matter and Dark Energy – we don’t know what they are (especially Dark Energy) but we need them to make our models work. Wildly speculating, it’s possible that this is linked to the black hole conundrum.

For an outsider like me, though, I can’t help but think that all this is very reminiscent of the situation in physics at the end of the 19th century. We have a problem that appears to destroy all our well-established physics that we cannot solve. The way out of that one was quantum mechanics, which has done pretty well for itself and lead to new technologies that would have seemed like magic beforehand.

The way out of this will undoubtedly be exciting and revolutionary. Can’t wait to see what people come up with!

I was in a tile shop the other day and it got me thinking about the fourth dimension. What do you mean “what are you talking about?”, I would have thought the connection was obvious. Oh, all right then…

Anyway, I was in a tile shop looking at different tiling patterns. Many of them were very lovely and its impressive to see how a good tile shop can embrace patterns form so many different countries and cultures. This shop had a fine line in British, Spanish, Italian, Moroccan, Arabic, Greek, you name it. Bathroom designs to satisfy even the most demanding and culturally promiscuous time-traveler.

Except after a while you start to notice that all these patterns have something in common: they are all periodic. That is to say, all the patterns repeat. In maths this is known as a translational symmetry – if you slide a repeating pattern around it will eventually line up with the tiles a few feet over. This could be as simple as a checkerboard pattern or something much more complex but typically a tiling that perfectly fills the floor space available repeats (i.e. it doesn’t have any gaps in it) will be repetitive.

There’s actually a whole branch of maths that investigates tilings and symmetries. It’s provable that certain patterns will fill the plane perfectly (like squares or hexagons) and other (like pentagons or octagons) won’t, and that certain combinations of shapes (like octagons and squares) can be used together to perfectly tile the plane. It’s related to the wider study of symmetry and a field called Group Theory, which turns out to be hugely important in physics, cropping up in everything from molecular chemistry and spectroscopy to particle physics. Heady stuff.

Anyway, what’s all this got to do with the 4th dimension. Well, aside from culturally promiscuous time-travel (which I’ll cover in a separate post). The fourth dimension is link to tiling in a rather surprising way. I mentioned that tilings tend to be periodic, but does that mean that EVERY tiling of the plane has to be periodic? The idea is related to a thing called the Domino problem in maths. Specifically, given a set of shapes, is it possible to design an algorithm that will decide if they can tile the plane or not. During the 1960s, a mathematician called Hao Wang suggested that this problem was solvable if all tilings of the plane were in periodic. You’d just need to decide if your set of tiles could be arranged periodically or not and you’d know the answer. Neat, huh?

Except there’s a snag. What if there are tilings of the plane that aren’t periodic? If those existed the test would fail. In 1966 a non-periodic tiling was found: it used over 20000 different tiles (pretty hard to construct, and all the more impressive for being found before powerful computers were widespread). Then anther with 104 tiles was found. Then another with 40 and another with just 13. Finally in 1974 Roger Penrose found an aperiodic tiling that required only 2 tiles. These are pretty interesting patterns: patterns that cover the plane by don’t (quite) repeat. You can’t slide the pattern around and match it to itself, and it only contains two types of tile! It looks like this:

At first it looks regular, but stare at it for a while and you can see that it doesn’t quite repeat. Also notice that this piece doesn’t form a repeating unit on its own either. It’s a pretty cool thing. What turns out to be extra interesting, though, is that tiling patterns are related to packing problems in 3D, and in particular to the arrangement of atoms in crystals. You could ask the same question here: if I want to pack a load of (spherical) atoms into a 3D space, does the pattern have to be repetitive?

Well, no. crystals are periodic, glasses are disordered arrangements of atoms. More organic forms like wood are the result of nonlinear growth processes and many rocks are made up of mixtures of crystals, glasses and dusts all mixed up together but none of these have the same properties as the Penrose Tiling. There are things that do, though: they’re called quasicrystals, and they have a whole field of research attached to them.

A quasi-crystal is a regular but non-periodic packing of atoms. Using electronic force microscopy you can actually make images of the individual atoms and the surfaces turn out to look a bit like this:

Look familiar? This is actually the surface of an Aluminium-Palladium-Manganese quasicrystal. Looks an awful lot like the Penrose Tiling, don’t you think?

But what about the 4th dimension? Well, it turns out that the best way to think about aperiodic tilings is to thing of them as a 3D slice through a regular 4D lattice. If the slice is parallel to one of the regular planes in 4D, it’s periodic. Any non-parallel slice is aperiodic.

What’s a 3D slice? Think of a line. A line is 1D, and you can cut it with a single point. Similarly, a square is 2D and can be cut with a line, which is 1D. A cube is 3D and can be cut with a plane, which is 2D. There’s a pattern here: an nD volume can be sliced with an (n-1)D object. So a 4D object (don’t worry about picturing it!) can be sliced with a 3D volume. Mathematically this a relatively easy thing to do, and so your 3D quasicrystal is a the 3D slice through a 4D object.

This idea of slicing through a 4D volume is something that crops up in my own work. I do a bit of work with 3D graphics. Here is turns out to be convenient to think of your 3D space containing your graphics as a slice through 4D. Why is that, you ask? Well, in 3D I want to move around and also rotate around a point. A rotation in 3D is written as a 3×3 matrix. In fact you can think of any rotation as the result of three other rotations: one about each of the x, y, and z axes. This is provable using – guess what – Group Theory.

The thing is, that once your 3×3 matrix is full of rotations (any other things like scale factors and sheers) there no room for the translations that move things around. The most natural thing to now is to make the matrix bigger to include extra elements for moving around, and this is equivalent to (guess what) using a fourth dimension. So, weirdly, 3D geometry turns out to be easier to think of in 4D. The 3D graphics in your favourite computer game are in fact a slice through a 4D space, just like the tilings and packings for the quasicrystals. The guy in the tiling shop seemed quite interested in this when I mentioned it, although I did steer clear of projective geometry and quasicrystals at the time. It was Sunday lunchtime.

One more thing before I go. I was talking to Mrs of-Science about this later on that day, and she immediately asked if I was talking about time as the 4th dimension. I’m not – all of this is about a 4th space-like dimension, and is very different from the geometry of relativity’s 4D Spacetime, which is fascinating in a completely different way and not at all like regular Euclidean space. Interestingly, though, you can add a 4th space dimension to general relativity. Kaluza ad Klein did it back in the 30s. It turns out that if you do this you get a unification of gravity and classical electromagnetism, which was the start of all the work on grand unification of the the four fundamental interactions that we hear so much about. So the fourth dimension reveal secrets here too.

Life in 4D is pretty cool, and I fully expect my bathroom to look pretty good too.