I’ve been reading a bit about modern Kaluza-Klein-like theories of gravity, as advocated (among others) by the people of the ‘Space-Time-Matter consortium‘. Although i don’t know yet whether i buy all their arguments (and i actually have some reservations, more on them later), these ideas are so elegant and simple that i’d like to share a bit of what i’ve learnt so far. Thus, this post is not an endorsement of the physics behind five-dimensional theories, but rather an exposition (hopefully understandable to non-specialists) of what i like about them .
As you surely know, multi-dimensional space-times are nowadays routine to string theorists (who use models of up to 26 dimensions, ten and eleven being also popular choices), but i’ve found little, if any, motivation to take them seriously; surely because of lack of study on my part, but also because i’ve never read an argument making them feasible from a physical point of view. Extra dimensions are needed in string theory to have consistent supersymmetry, and to avoid divergences of the vacuum zero-point field or get a massless photon. Now, that may be a perfectly physical and intuitive motivation for some of you (and one can even argue that the dimensionality of the world is being derived from the theory), but definitely not for me: my rusty intuition says that obtaining up to 22 extra-dimensions and having to hurriedly sweep them under the rug (er, i mean, compactify them) is a strong hint to go look for better ways . More akin to my old-fashioned ways is the path followed by Paul Wessan: let’s just add an extra dimension, take it at face value and see if it makes any sense. Of course, the idea of a fifth dimension does not come out of the blue, so let me start with a bit of motivation.
Wood versus Marble
General Relativity singles out gravitational forces by converting them into pure geometry. Gravitation is no longer an interaction comparable to electromagnetic, strong or weak forces, but, rather, the manifestation of the structure of space-time . As a consequence, the relativistic world is split in two quite dissimilar parts: geometric entities (represented by the space-time metric or, if you prefer, the connection vierbein) and energy/matter/fields (represented by the stress-energy tensor). These are by no means worlds apart; on the contrary, they’re deeply interconnected via Einstein’s equations:
G = k T …….. (1)
On the left hand side, one encounters magnitudes related exclusively to the geometry of space-time: the Einstein’s tensor G is constructed from the metric g and its derivatives (up to second order), which describe the curvature of space-time (G is constructed from the Ricci tensor, which in turns derives from Riemann’s, or curvature, tensor R). On the right, we find the world of matter, energy and fields: the stress-energy tensor T describes the pressure, stresses, density of energy and its flux, i.e., how matter moves through space-time (if you’re new to metrics, tensors and GR, see this nice page for a quick, conceptual introduction, or this one, with beautiful ASCII art). Energy influences geometry, which influences energy, which influences energy, and so on and so forth, in a highly non-linear way. But we have still two differentiated kinds of stuff, which Einstein was fond of calling marble and wood. He spent many years trying to get rid of wood and to find a unified theory containing only geometric entities; five-dimensional theories try to push forward towards that ivory tower.
Before jumping to a fifth dimension we can get a glimpse of how geometry can play a role similar to that of matter fields by taking a look at the famous cosmological constant. The Einstein tensor above is carefully constructed to ensure that its covariant divergence vanishes; as a consequence, matter fields satisfy a generalised conservation equation, i.e., the relativistic version of classical energy, momentum or charge conservation. But one can add a term proportional to the metric to the left hand side of (1) while still preserving this desirable property: the constant of proportionality is the cosmological constant. Introducing it, and considering the case of empty space (vanishing T), we can rewrite Einstein’s equations as
G + L g = 0 …….. (2a)
Everything’s marble here, but we can rewrite the equation above passing the cosmological term to the right hand side
G = -L g …….. (2b)
Compare with (1): Lg plays the role of a stress-energy tensor, made up of marble, which can be seen as representing the energy associated with vacuum. Admittedly, it’s a peculiar form of energy, because it represents a negative pressure, but it can be used to explain (more or less) dark energy and the acceleration of the universe’s expansion. But what about other fields, like the electromagnetic field? Can we recast them using geometric quantities? As long as we stick to four dimensions, we have no spare geometric terms at hand. Enter the fifth dimension.
A world without matter
The metric tensor in N dimensions consists of N(N+1)/2 independent functions. Therefore, going to five dimensions buys us 5 more geometric entities to play with and make the cause of, say, electromagnetism. Tensor equations such as those in the previous section do not depend on the number of dimensions: one just gets more components as we go from four to five of them. The trick is to group on the left the terms making up, say, the Einstein tensor in four dimensions, and to put on the right hand side the rest (containing terms arising from the fifth dimension), which can now be reinterpreted as stress-energy arising from geometry. So, if we start with a 5D universe without matter, its corresponding Einstein equations
G = 0 …….. (3)
(where we explicitly denote the dimensionality) a subset of the equations represented by (3) can be algebraically re-arranged as
G = T …….. (4)
where the new tensor T is made up of all the terms appearing in (3) that include the new metric functions associated with the new dimension. Thus, if one happens to life in a 4-dimensional subspace of the 5-dimensional universe, 4-dimensional observations will be naturally interpreted using (4). That is, to these lower dimensional beings, higher-dimensional geometric effects look wooden! But there’s more: as i mentioned, (4) is just a subset of (3) and the rest of equations (involving, again, terms relative to the extra dimension) can be reinterpreted as conservation/evolution equations for our geometrically induced matter fields.
Kaluza and Klein were the first, during the 20s, to apply these ideas, incorporating the electromagnetic potential as 4 of the 5 extra available metric coefficients  (for a quick summary of the maths involved, see this nice page by Viktor Toth , or this excellent paper by Jeroen van Dongen on Einstein’s reaction to Kaluza and Klein’s ideas). By further stipulating that the old 4D ones are independent of the new coordinate, the right hand side of (4), which contains combinations of the 5D metric functions and its derivatives, takes the form of the EM stress-energy tensor. And the additional conservation equations coming from (3) are Maxwell equations. Beautiful as this result is, people felt a need to explain where is the additional spatial dimension hiding . While Kaluza was at first happy with just making the 4D fields of the theory independent of the fifth dimension by prescription, Klein proposed what is still today the most popular way of hiding extra dimensions: compactification. That is the famous hose pipe model, where the fifth dimension is finite and curled into a circle, giving rise to a slim cylinder: the extra dimension is there, but we cannot see it, in the same manner as we would take the hose pipe as a one-dimensional line when seen from a distance. This model has further interesting properties, on account of the periodicity of the additional dimension; for instance, constraining electromagnetism to a circle immediately explains charge quantization, as waves directed along a finite axis can only have discrete frequencies. Unfortunately, the same calculations give a mass for the electron which is wrong by twenty-two orders of magnitude, so that this line of inquiry was soon abandoned.
Compactification lives up in today’s theories of unification in string theory, where, however, the pure marble world idea has been abandoned: we have stress-energy contributions not coming from geometry, but from matter, or string, fields (see for instance Michael Duff’s writings, including his fun Flatland, Modulo 8, or Kaluza-Klein theory in perspective). But there’s a second perspective, namely, to take the extra dimensions at face value, without necessarily compactifying them, and assume that nature is only approximately independent of them. At the same time, we stick to the idea of avoiding non-geometrical quantities, deriving the stress-energy contents of the four-dimensional subspace exclusively from 5D geometry. This is the approach championed by P. Wessan’s Space-Time-Matter (SMT) theory, to which we turn our attention in the following sections.
It’s a small, embedded universe
It is not difficult to prove that any solution to Einstein’s equations (1) in N dimensions can be recast as an empty N+1 dimensional space satisfying the source-less equations (3). This purely mathematical result, known as Campbell’s theorem, allows us to reinterpret our 4D universe with its fields and matter contents as a subspace of a wider, five-dimensional, and empty one. Now, one of the key features of Einstein’s General Relativy is gauge invariance, that is, the fact that all relevant physical quantities and equations are independent of the coordinate system we chose to express them. This invariance is still valid in the whole 5D theory, but when we restrict ourselves to a 4D subspace we loose it: physics in four dimensions depends on how we choose to embed our perceived world in the five dimensional universe. Different embeddings yield different dependencies of 4D fields in the fifth coordinate and, therefore, different physics. Thus, if we start with a flat 5D universe where all test particles move along null geodesics, what we actually perceive in our limited 4D view depends on how our 4D hypersurface is embedded in that flat space-time. While i find this situation a bit disturbing (i have yet to make my mind up as to how disturbing), it is arguably not new: in regular GR accounts of solar system observations or universe models we use routinely very specific coordinate systems to match our measurements with the results of the theory (for instance, we could choose a co-moving coordinate system in our cosmological models and expansion would suddenly disappear).
At any rate, there’s quite a few interesting ways of embedding our 4D slice in five dimensions providing new perspectives on our (so to speak) every day physics. For instance, the existence of an extra dimension can make big-bang singularities vanish, due to the fact that the geometry-induced stress-energy tensor comes with a time-varying cosmological constant. In a nutshell, the 4D big-bang becomes a coordinate singularity (due to an ill-chosen 4D coordinate system) which is regular in five dimensions. The figure on the left shows such an embedding, discussed in full in this article by Seahra and Wesson (in other models, the big-bang can also be interpreted as a a shock-wave in a curved, if empty, five-dimensional universe). We already see here one of the interesting traits of these theories: since the fifth dimension is not a priory hidden, observables such as the cosmological constant can depend on time.
The fifth force
In vanilla General Relativity, one is often interested in the motion of test particles, that is, particles whose associated stress-energy tensor is so tiny that it does not contribute as a source in the right hand side of (1). These free-falling particles, characterized by their rest mass, follow geodesics of the underlying metric (the curved space generalisation of straight lines in Euclidean space) and serve as a probe of the geometry of space-time. As we know since Galileo, the equation of motion of a free particle in flat space-time simply states that its velocity, v, is constant:
dv/dt = 0
The relativistic generalisation is the geodesic equation, which substitutes the 3-velocity by its four-dimensional version u, whose spatial components are still v and whose temporal component measures proper time s, i.e., time as measured from the point of view of the moving particle. In addition, one has to take into account the curvature of space-time, which plays funny games with parallel transport, and substitute regular derivatives by covariant ones, taking into account the effects of geometry on the velocity vector when it gets moved in a curved background. Thus, one can write our free-falling equation of motion as
Du/ds = 0 …….. (5)
where the capital D is not a typo, but denotes covariant differentiation. We can let our test particle interact with other fields: for instance, if it’s charged an there’s an electromagnetic field in the region of space-time that it traverses, the particle will no longer follow a geodesic, but deviate according to an equation of the form
D(mu)/ds = f …….. (6)
where i have included the constant rest mass (m) and f denotes the force four-vector, which describes the interaction of the test particle with the field (this is just the natural generalisation of Newton’s second law). But let’s go back to our five-dimensional universe. Since there are no matter fields in there, geodesic motion is described by the source-free equation (5), where u is now the five-velocity. We can play again the trick we used with Einstein equations and split the geodesic equation to see how it looks in our 4D embedding. The result is that motion as viewed in the four dimensional slice is not geodesic, but satisfies an equation of the form (6) with a geometry-induced force. Not only that, the rest mass can also depend on the fifth dimension; actually, with an appropriate embedding the mass is the extra-dimension. Another intriguing possibility is that null geodesics in five dimensions (representing causally connected paths) can appear as time-like or even space-like curves in four dimensions, so that apparently unconnected events in 4D are actually causally related in the wider five dimensional universe. In such a model, what we take for two separate particles could be reinterpreted as a single one following a 5D geodesic, making true the old speculation by Feynman/Wheeler according to which there are not many electrons, but a single one travelling in time. Bizarre as all this may sound at first, the math is relatively straightforward and, if nothing else, may serve to hone your differential geometry skills: see for instance this paper by Wesson and Seahra for details.
We encounter a by now familiar theme: the actual form of the fifth force and mass function is gauge dependent, i.e., it varies as we change the coordinate system used to describe the four dimensional subspace. Moreover, they seem to be mutually incompatible: the embedding where you get rid of big-bang singularities is different from the one that tries to reveal the nature of inertia, or (as we mention below) the quantum mechanical aspects of the four-dimensional subspace. But, again, i must concede that in regular GR we express different experimental outcomes using different coordinates systems, so maybe the situation is not as ugly as it seems at first. At any rate, these 5D gauge-dependent models have the redeeming quality of making concrete predictions which are, in principle, measurable, so that we should be able to let experiment decide for us, just as in the old good days.
Besides the book linked above, there’s a quite long list of articles on STM and 5D gravity by Wessan and friends. A very good discussion of all the above topics, and then more, is the review paper Kaluza-Klein Gravity, which also discusses alternative multi-dimensional theories like strings or Randall and Sundrum’s braneworlds. Interestingly, it has been shown that the latter are algebraically equivalent to 5D STM, albeit their physical motivation is quite different (in Randall’s branes, stress-energy is not geometrical, but caused by regular fields constrained to the 4D subspace). Also worth reading are Wessan musings on the connection between 5D physics and 4D indeterminism (see, for instance, Space-Time Uncertainty from Higher-Dimensional Determinism), which i haven’t commented further because i don’t really understand them and, probably as a result, find too speculative (as far as i can see, quantization rules are introduced by hand–more on this when i find time to study it properly!).
All in all, higher-dimensional physics is intriguing and, i’d say, worth investigating further. I don’t have a problem with unseen dimensions, as long as they introduce new physics or provide explanation for ill-understood known phenomena (with due respect to Occam). After all, we do not see time, or atoms for that matter, but use them everyday because they explain in simple terms what we actually see. The theory’s ability to make testable predictions (variable rest-mass, variable cosmological constant, etc.) is undoubtedly a strong incentive to further study, as is its simplicity and (to a geometer’s eyes) elegance. Albeit the latter is somewhat tainted by the dependency of physics on the coordinate system used to describe the 4D embedding. In a sense, the 5D universe is playing the part of an absolute space-time (is that Newton laughing?), not a very fashionable thing these days. But, as Feynman said , the question is not whether we like Nature…
 During his Auckland lectures on QED, Richard Feynman was asked about his opinion on quantum indeterminism. His answer was that the point was not whether he liked or not how nature was, but, rather, how nature actually behaves. Caveat emptor.
 Feynman again, in a quotation that can be taken either to back or to debunk my critique, according to taste:
Let me say something that people who worry about mathematical proofs and inconsistencies seem not to know. There is no way of showing mathematically that a physical conclusion is wrong or inconsistent. All that can be shown is that the mathematical assumptions are wrong. If we find that certain mathematical assumptions lead to a logically inconsistent description of Nature, we change the assumptions, not Nature (Feynman Lectures on Gravitation).
 This is, by the way, one of my difficulties with unification attempts along the lines of QFT, for gravitation is not, in an essential way, a force in the sense of QFT. Therefore, trying to second quantize it in the same way as electroweak or strong interactions has always struck me as misguided in a fundamental way, as long as one takes the lessons of GR at face value. My first gut reaction was to consider QFT on curved space-time much more appropriate, although (for various reasons that i leave for a future post) it can be just a first approximation. If one is to quantize anything related to gravity, if feels more natural to quantize space-time, along the lines of, for instance, Loop Quantum Gravity. That’s also the reason for my being skeptical when people claim that string theory contains GR, meaning that a weak field perturbation of the flat background has to have spin 2.
 Actually, the first attempt at explaining electromagnetism by means of an extra dimension predates General Relativity. In 1914, Gunnard Nordström made such a proposal, in the context of his own theory of gravitation (see also this WikiPedia page for a quite detailed and pedagogical account).
 If you’re interested in CAS systems able to tackle tensor calculus, you may be also interested on Viktor’s work with Maxima, a quite nice free CAS.
 Arguments in favour of three spatial dimensions date back, at least, to Pythagoras:
For, as the Pythagoreans say, the world and all that is in it is determined by the number three, since beginning and middle and end give the number of an ‘all’, and the number they give is the triad. And so, having taken these three from nature as (so to speak) laws of it, we make further use of the number three in the worship of the Gods. Further, we use the terms in practice in this way. Of two things, or men, we say ‘both’, but not ‘all’: three is the first number to which the term ‘all’ has been appropriated. And in this, as we have said, we do but follow the lead which nature gives. (Aristotle, De Caelo.)
Some centuries later, Kepler’s theory explaining the solar system in terms of platonic solids (which unsettlingly reminds me of modern attempts at explaining everything in particle physics in terms of symmetry groups and their representations) depended on a three dimensional world, which Kepler linked to the Holly Trinity (for an entertaining discussion of Kepler’s physics in the context of extra dimensions, see this Physics Teachers article).