“What is the fifth dimension – is it love?”
I have been asked this question in wholly unironic sincerity. While that query may have required a particularly deep breath and appeal to my reserves of patience, there are others in a similar vein which require a rotation of mental perspective to process.
“What is the fifth dimension – is it mass?”
Now we are at least within the realm of physics, but still asking a question built on assumptions far wide of the mark. Curiously enough the question, though blindly aimed, almost hits the nail on the head – like tendentious claims that modern physics can be found predicted in Buddhist philosophy or Lucretian metaphysics (or even Newtonian corpuscular optics), it is typically a case of lucky guess rather than deep insight. It is some of that insight that I’d like to provide.
The thinking that inspires this sort of question should be of interest to all those interested in science education and communication, because it betrays a key misunderstanding many people have when it comes to special relativity – the nature of four-dimensional space-time. Public awareness of Einstein’s work is such that many will have heard of the term “space-time” and they may know that “time is the fourth dimension”, but what does this mean? Evidently it appears to connote a random conjoining of disparate concepts into one amalgam that must make sense to those funny boffins with their impenetrable equations, but the motivation and rules governing such a marriage are evidently something of a mystery. If we are allowed to throw time into the mix, why not some other random entity? Mass? Temperature? pH level? Love?
Clearly this is a subject in need of some clarification.
What is a dimension?
The other misconception the question at the top betrays is that there is some meaningful numbering system to dimensions. The more adventurous inquisitor does occasionally ask me what the seventh or eighth dimension is, as if I’m going to answer “rapidity and refulgence” or something similar. Part of this confusion lies in the way the lay person often seems to think of dimensions, namely that they pertain to the solidity and shape of objects and images. We observe that paintings and traditional film images are “2D”, with the two dimensions of length and breadth, while sculptures and new-fangled/eyeball-hurting stereoscopic movies have the third dimension of depth. There is clearly no physical content to this distinction of dimensions, though: to anyone looking at a sculpture from another side, depth becomes breadth and vice versa.
To the physicist, dimensions are a property of a space, not the objects within it – specifically, if a point within a space requires N numbers to specify its exact position, that space can be said to have N dimensions. The surface of the earth is two-dimensional, in that we can specify any point with two numbers: longitude and latitude. Any one point within a swimming pool can be pinpointed by its distance from one end, its distance from the side and its height above the bottom: it is a three-dimensional space. Equivalently, the number of dimensions is the number of perpendicular directions in which it is possible to move.
More generally, the world we inhabit is three-dimensional, as should be fairly familiar – and because it is also rotationally symmetric (that is, there is no direction which has any particular distinction from any other direction), there is no sense in which we can enumerate those dimensions and “this one is the first”.
This was the picture until 1907.
Prior to Albert Einstein and Henri Poincaré, it was generally thought that time was a feature of the universe wholly separate and independent of the events that took place within it: however we might measure time, a universal, objective clock would be ticking away the seconds. With the work of Einstein and others, however, this proved not to be the case. Not only did one’s measurement of time depend on one’s state of motion, this dependence was relative: differences in measurement between observers could be described, but never compared to any objective measure of time, for it simply did not exist. It took Hermann Minkowski to point out that the interplay between space and time, the transformations between the two which described the distortions engendered by relative motion, were analogous to ordinary spatial rotations, so long as time was included as an extra dimension. This feature is important and highlights the reason why time is singled out to join the space party.
So what of extra dimensions beyond time and the simple three spatial dimensions we observe? Though people, often those of a rather dreary mystical bent, may talk of things existing “in another dimension”, as we have seen from the preceding discussion, objects and processes occur within a particular space, of which dimensions are a property; they are not the space itself (so dimension should not be used as a synonym for universe or, god forbid, astral plane). If we do in fact live in a universe with more than three spatial dimensions, it must necessarily be the case that extra degrees of freedom exist, ones which are not appreciable to our senses. The question then arises: why do we only perceive three spatial dimensions?
One option is a so-called “braneworld scenario”. In this particular set-up, our world would be a 3-dimensional subspace of a larger universe, a “brane” floating through a larger space with any number of extra dimensions, a bit like Phantom Zone in the Superman movies (in that case a 2-dimensional brane, floating through a 3-dimensional universe). This particular set-up has the potential for interesting physics – for example, it is the nature of gravity that it acts over all space, while other forces may be confined to lower-dimensional branes or subspaces. In other words, our entire observable world could be confined to a brane, but gravity would inevitable spread out, and this would have the effect that it is weakened, potentially explaining why gravity appears to be more than a trillion trillion trillion times weaker than any other force.
The usual alternative solution is known as “compactification”. In this scenario, the missing auxiliary dimensions are believed to be exceedingly small. To understand what is meant by this, consider an old computer game, of the sort where moving a character off one side of the screen causes them to magically appear back on the other side. It is as if that direction is shaped like a circle, so that moving along it causes you to loop back on yourself an unlimited number of times. If up-and-down on the screen doesn’t have this feature, it is equivalent to moving around on the surface of a cylinder. If both directions have this property, like the old Nokia phone game Snake, then the equivalent surface is that of a doughnut (or torus). The idea of compactification is that the universe is like this – all the other invisible dimensions are genuinely there, but they double back on themselves and are curled up so tight that we do not notice them. Our eyes are far too large to discern these extra directions to space, and experiments cannot uncover them either, for reasons which are fundamentally quantum and lie in the famous Heisenberg Uncertainty Principle. Low energies, such as the ones we typically encounter, correspond to large uncertainties in position, so if any dimension is curled up on a scale smaller than that uncertainty, we have no means of detecting it – that is, unless we pump a bit more energy into the situation.
This feature opens up the possibility of a world of interesting physics to be uncovered by cranking up the energy of our particle accelerators, as is currently happening at the Large Hadron Collider at CERN. If we are lucky (and to be honest, we’d have to be very lucky), extra dimensions would be curled up at scales just large enough to be detected by the LHC, the most powerful particle probe we have. Customarily, compactified dimensions are thought to be many many orders of magnitude smaller than anything we have so far probed. Surprisingly, however, in the braneworld scenario, an extra dimension could be as large as a human hair and still have escaped detection (they could even be infinitely large in some exotic scenarios).
Notably, one phenomenon having compactified dimensions would produce would be the appearance of a whole new raft of heavy particles – in fact, we’d find a new heavy particle for every low-mass particle thus far discovered. At high energy, the low-mass particles gain the ability to move along the extra dimensions – we would not see that extra momentum directly, but it would appear to our four-dimensional eyes as adding to the mass of the particle. In other words, an experiment would appear to feature particles identical in every way to ordinary electrons, protons, etc. but each would just be a bit heavier. In a peculiar and roundabout way, and given a generous interpretation, the extra dimension does indeed correspond to mass!
One other intriguing possibility is that a new realm of gravity is opened up by probing the smallest distances. Above the distance scale of compactification, the presence of extra dimensions is immaterial to gravity, but as they come into play below that scale, they affect the behaviour of gravity, making it stronger than it otherwise would be. This would, in principle, make it possible to form microscopic black holes at a particle collider like the LHC. It is, sadly, not a particularly realistic prospect and certainly isn’t anything to be feared – in the unlikely event of any black holes forming, they would quickly evaporate via a process called Hawking Radiation, without ever posing a threat to our existence.
It’s all very well discussing what extra dimensions mean and why we might not observe them, but why on earth should we believe they exist? I have already alluded to some answers (the weakness of gravity, otherwise known as the Hierarchy Problem), but the first answer goes back to the earliest days of general relativity and the work of Theodor Kaluza and Oskar Klein on the implications of combining modern physics with a fifth, compactified dimension and when he did he discovered something extraordinary: gravity in five dimensions would, to our four-dimensional eyes, resolve into ordinary gravity and electromagnetism. The two known (at the time) fundamental forces could be perfectly synthesised into one simple phenomenon. In this interpretation, electrically-charged particles would simply be those moving in the fifth dimension: positively charged particles move one way, negatively charged particles the other way. Unfortunately, part of the success of the theory also necessitates its failures, making predictions that cannot be married to observation, but these ideas have subsequently been reborn in later years in String Theory.
String theory, a topic for a larger separate post, is a theory (or a rather a theoretical framework) developed in the last quarter of the twentieth century as a candidate “theory of everything”, a single framework with the potential to account for all known physical phenomena. In this theory, particles are replaced by tiny vibrating loops, called strings. A crucial ingredient of this particular theory is that it inevitably requires an extra six spatial dimensions, arranged in various baroque shapes and configurations (M-theory, the successor to string theory demands an extra seven!). While much work goes into figuring out how stringy phenomena might be observed at LHC-level energies, it is generally held that the predictions of string theory lie far beyond the reach of modern experiment, so we are unlikely to discover these hidden dimensions any time soon. One curious feature of string theory is that a curled-up dimension cannot be smaller than the size of the strings. Trying to describe physics with dimensions any smaller than the string scale just leads to a world identical to one with much larger dimensions. The world of the minute is bizarre indeed.
In the end it is worth emphasising that extra dimensions are not invented by the physicists for the sheer hell of it, because they are bored with ordinary reality or because they do their work when stoned. Extra dimensions are in some cases demanded naturally by attempts to solve particular physical problems, and sometimes just investigating their implications opens up fascinating and unexpected avenues of research. I personally cannot imagine any discovery, with all its implications, that would be more awesome than a discovery of this kind. I can only hope nature likes the idea as much as I do.