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0  Front page of the original article

0.1  Abstract

     We review higher-dimensional unified theories from the general relativity, rather than the particle physics side. Three distinct approaches to the subject are identified and contrasted: compactified, projective and noncompactified. We discuss the cosmological and astrophysical implications of extra dimensions, and conclude that none of the three approaches can be ruled out on observational grounds at the present time.

1  Introduction

     Kaluza's ^[1] achievement was to show that five-dimensional general relativity contains both Einstein's four-dimensional theory of gravity and Maxwell's theory of electromagnetism. He however imposed a somewhat artificial restriction (the cylinder condition) on the coordinates, essentially barring the fifth one a priori from making a direct appearance in the laws of physics. Klein's ^[2] contribution was to make this restriction less artificial by suggesting a plausible physical basis for it in compactification of the fifth dimension. This idea was enthusiastically received by unified-field theorists, and when the time came to include the strong and weak forces by extending Kaluza's mechanism to higher dimensions, it was assumed that these too would be compact. This line of thinking has led through eleven-dimensional supergravity theories in the 1980s to the current favorite contenders for a possible "theory of everything," ten-dimensional superstrings.

     We review the field of Kaluza-Klein gravity, concentrating on the general relativity, rather than particle physics side of the subject. (For the latter there are already many excellent books ^[3]-[8] and review articles ^[9]-[14] available.) We also aim to re-examine the field to some extent, as it seems to us that the cart of compactification has in some ways gotten ahead of the horse of unification. Kaluza unified not only gravity and electromagnetism, but also matter and geometry, for the photon appeared in four dimensions as a manifestation of empty five-dimensional spacetime. Modern Kaluza-Klein theories, by contrast, routinely require the addition of explicit "higher-dimensional matter" fields in order to achieve successful compactification (among other things). Are they necessary? Yes, if extra coordinates must be real, lengthlike and compact. There are, however, higher-dimensional unified field theories which require none of these things: projective theories ^[15]-[18], in which extra coordinates are not physically real; and noncompactified theories ^[19]-[26], in which they are not necessarily lengthlike or compact. These theories receive special attention in our report.

     We begin in § 2 with a historical overview of higher-dimensional theories of gravity. In § 3 we review Kaluza's original mechanism, emphasizing what to us are its three principal features. The three main approaches to higher-dimensional unification since Kaluza - compactified, projective and noncompactified - are reviewed in § 4, § 5 and § 6 respectively. We note that each one modifies or sacrifices at least one of the key features of Kaluza's theory, and discuss the implications. The all-important question of experimental constraints is addressed in § 7 and § 8, which deal respectively with cosmological and astrophysical effects of extra dimensions. None of the three above-mentioned approaches can be ruled out on observational grounds at the present time. Conclusions and prospects for further work are summarized in § 9.

2  Historical Overview

2.1  Higher Dimensions

     The world of everyday experience is three-dimensional. But why should this be so? The question goes back at least to Kepler ^[27], who speculated that the threefold nature of the Holy Trinity might be responsible. More recent arguments have involved the stability of planetary orbits and atomic ground states, the use of wave propagation for information transmission, the fundamental constants of nature, and the anthropic principle ^[28], as well as wormhole effects ^[29], the cosmological constant ^[30], certain "geometry-free" considerations ^[31], string theories ^[32], and nucleation probabilities in quantum cosmology ^[33]. All these lines of reasoning converge on the same conclusion: that, in agreement with common intuition, space is composed of three macroscopic spatial dimensions x¹, x² and x³.

     Nevertheless, the temptation to tinker with the dimensionality of nature has proved irresistible to physicists over the years. The main reason for this is that phenomena which require very different explanations in three-dimensional space can often be shown to be manifestations of simpler theories in higher-dimensional manifolds. But how can this idea be reconciled with the observed three-dimensionality of space? If there are additional coordinates, why does physics appear to be independent of them?

     It is useful to keep in mind that the new coordinates need not necessarily be lengthlike (in the sense of being measured in meters, say), or even spacelike (in regard to their metric signature). A concrete example which violates both of these expectations was introduced in 1909 by Minkowski ^[34], who showed that the successes of Maxwell's unified electromagnetic theory and Einstein's special relativity could be understood geometrically if time, along with space, were considered part of a four-dimensional spacetime manifold via x⁰ = ict. Many of the abovementioned arguments against more than three dimensions were circumvented by the fact that the fourth coordinate did not mark distance. And the reason that physics had appeared three-dimensional for so long was because of the large size of the dimension-transposing parameter c, which meant that the effects of "mixing" space and time coordinates (ie., length contraction, time dilation) appeared only at very high speeds.

2.2  Kaluza-Klein Theory

     Inspired by the close ties between Minkowski's four-dimensional spacetime and Maxwell's unification of electricity and magnetism, Nordström ^[35] in 1914 and (independently) Kaluza ^[1] in 1921 were the first to try unifying gravity with electromagnetism in a theory of five dimensions (x⁰ through x⁴). Both men then faced the question: why had no fifth dimension been observed in nature? In Minkowski's time, there had already been experimental phenomena (namely, electromagnetic ones) whose invariance with respect to Lorentz transformations could be interpreted as four-dimensional coordinate invariance. No such observations pointed to a fifth dimension. Nordström and Kaluza therefore avoided the question and simply demanded that all derivatives with respect to x⁴ vanish. In other words, physics was to take place - for as-yet unknown reasons - on a four-dimensional hypersurface in a five-dimensional universe (Kaluza's "cylinder condition").

     With this assumption, each was successful in obtaining the field equations of both electromagnetism and gravity from a single five-dimensional theory. Nordström, working as he was before general relativity, assumed a scalar gravitational potential; while Kaluza used Einstein's tensor potential. Specifically, Kaluza demonstrated that general relativity, when interpreted as a five-dimensional theory in vacuum (ie., ⁵G_AB = 0, with A, B running over 0, 1, 2, 3, 4), contained four-dimensional general relativity in the presence of an electromagnetic field (ie., ⁴G_αβ = ⁴T^EM_αβ, with α, β running over 0, 1, 2, 3), together with Maxwell's laws of electromagnetism. (There was also a Klein-Gordon equation for a massless scalar field, but this was not appreciated - and was in fact suppressed - by Kaluza at the time.) All subsequent attempts at higher-dimensional unification spring from this remarkable result.

     Various modifications of Kaluza's five-dimensional scheme, including Klein's idea ^[2],[36] of compactifying the extra dimension (which we will discuss in a moment) were suggested by Einstein, Jordan, Bergmann, and a few others ^[37]-[43] over the years, but it was not extended to more than five dimensions until theories of the strong and weak nuclear interactions were developed. The obvious question was whether these new forces could be unified with gravity and electromagnetism by the same method.

     The key to achieving this lay in the concept of gauge invariance, which was coming to be recognized as underlying all the interactions of physics. Electrodynamics, for example, could be "derived" by imposing local U(1) gauge-invariance on a free-particle Lagrangian. From the gauge-invariant point of view, Kaluza's feat in extracting electromagnetism from five-dimensional gravity was no longer so surprising: it worked, in effect, because U(1) gauge-invariance had been "added onto" Einstein's equations in the guise of invariance with respect to coordinate transformations along the fifth dimension. In other words, gauge symmetry had been "explained" as a geometric symmetry of spacetime. The electromagnetic field then appeared as a vector "gauge field" in four dimensions. It was natural - though not simple - to extend this insight to groups with more complicated symmetry. De Witt ^[44] in 1963 was the first to suggest incorporating the non-Abelian SU(2) gauge group of Yang and Mills into a Kaluza-Klein theory of (4 + d) dimensions. A minimum of three extra dimensions were required. This problem was picked up by others ^[45]-[47] and solved completely by the time of Cho & Freund ^[48],[49] in 1975.

2.3  Approaches to Higher-Dimensional Unification

     We emphasize here three key features of all the models discussed so far:

     (i) They embody Einstein's vision ^[50]-[52] of nature as pure geometry. (This idea can be traced in nonmathematical form at least to Clifford in 1876 ^[53], and there are hints of it as far back as the Indian Vedas, according to Wheeler and others ^[54].) The electromagnetic and Yang-Mills fields, as well as the gravitational field, are completely contained in the higher-dimensional Einstein tensor ^(4+d)G_AB; that is, in the metric and its derivatives. No explicit energy-momentum tensor ^(4+d)T_AB is needed.

     (ii) They are minimal extensions of general relativity in the sense that there is no modification to the mathematical structure of Einstein's theory. The only change is that tensor indices run over 0 to (3 + d) instead of 0 to 3.

     (iii) They are a priori cylindrical. No mechanism is suggested to explain why physics depends on the first four coordinates, but not on the extra ones.

     The first two of these are agreeable from the point of view of elegance and simplicity. The third, however, appears contrived to modern eyes. In the effort to repair this defect, higher-dimensional unified theory has evolved in three more or less independent directions since the time of Kaluza. Each one sacrifices or modifies one of the features (i) to (iii) above.

     Firstly, it has been proposed that extra dimensions do not appear in physics because they are compactified and unobservable on experimentally accessible energy scales. This approach has been successful in many ways, and is the dominant paradigm in higher-dimensional unification (recent reviews include many excellent books ^[3]-[8] and articles ^[9]-[14]). If one wants to unify more than just gravity and electromagnetism in this way, however, it seems that one has in practice to abandon Einstein's goal of geometrizing physics, at least in the sense of (i) above.

     A second way to sweep the extra dimensions out of sight is to regard them as mathematical artifacts of a more complicated underlying theory, sacrificing (ii) above. This can be done, for example, if one replaces the classical (affine) geometry underlying Einstein's general relativity with projective geometry (see for reviews ^[15]-[18]). "Extra dimensions" then become visual aids which may or may not help us understand the underlying mathematics of nature, but which do not correspond to physical coordinates.

     The third approach to the problem of explaining exact cylindricity is to consider the possibility that it may not necessarily be exact, relaxing (iii) above. That is, one takes the new coordinates at face value, allowing physics to depend on them in principle ^[19]-[26]. This dependence presumably appears in regimes that have not yet been well-probed by experiment - much as the relevance of Minkowski's fourth dimension to mechanics was not apparent at non-relativistic speeds. When dependence on the extra dimensions is included, one finds that the five-dimensional Einstein equations ⁵R_AB = 0 contain the four-dimensional ones ⁴G_αβ = ⁴T_αβ with a general energy-momentum tensor ⁴T_αβ instead of just the electromagnetic one ⁴T^EM_αβ.

2.4  The Compactified Approach

     Klein showed in 1926 ^[2],[36] that Kaluza's cylinder condition would arise naturally if the fifth coordinate had (1) a circular topology, in which case physical fields would depend on it only periodically, and could be Fourier-expanded; and (2) a small enough ("compactified") scale, in which case the energies of all Fourier modes above the ground state could be made so high as to be unobservable (except - as we now add - possibly in the very early universe). Physics would thus be effectively independent of Kaluza's fifth dimension, as desired. As a bonus, it seemed early on that the expansion of the electromagnetic field into Fourier modes could in principle explain the quantization of electric charge. (This aspect of the theory has had to be abandoned, however, as the charge-to-mass ratio of the higher modes did not match that of any known particles. Nowadays elementary charges are identified with the ground state Fourier modes only, and their small mass is attributed to spontaneous symmetry-breaking.)

     The scheme was not perfect; one still needed to explain why extra dimensions differed so markedly in topology and scale from the familiar spacetime ones. Their size in particular had to be extremely small (below the attometer (1 am = 10⁻¹⁸ m) scale, according to current experiment ^[55]). There was also the question of how to interpret a new scalar field which appeared in the theory. These difficulties have, however, proved manageable. Scalar fields are not as threatening as they once appeared; one now just assumes that they are too massive to have been observed. And an entire industry has grown up around the study of compactification mechanisms and the topology of compact spaces.

     In fact, Klein's strategy of compactifying extra dimensions has come to dominate higher-dimensional unified physics, leading in recent years to new fields like eleven-dimensional supergravity and ten-dimensional superstring theory. We will survey these developments in this section, and make contact with many of them throughout this report, but it is not our purpose to review them exhaustively. For this the reader is directed to the books ^[3]-[8], and review articles ^[9]-[14] mentioned already. Our goal here is to take a broad view, comparing and contrasting the various approaches to higher-dimensional gravity, and focusing in particular on those which have received less critical attention in the literature. A semantic note: while the term "Kaluza-Klein theory" ought, strictly speaking, to apply only to models which assume both cylindricity and compactified dimensions, we follow popular usage and apply the term to any higher-dimensional unified theory of gravity in which the extra dimensions are regarded as real, whether compactified or not. When distinguishing between these two, we will refer in the latter case to "noncompactified Kaluza-Klein theories," though this is to some extent a contradiction in terms.

2.5  Compactification Mechanisms

     A difficulty with compactification is that one cannot impose it indiscriminately on whichever dimensions one likes - the combination of macroscopic four-dimensional spacetime plus the compactified extra-dimensional space must be a solution of the higher-dimensional Einstein field equations. In particular, one should be able to recover a "ground state" solution consisting of four-dimensional Minkowski space plus a d-dimensional compact manifold. Although this is straightforward when d = 1 (Klein's case), the same thing is not true in higher-dimensional theories like that of Cho & Freund, where the compact spaces are in general curved ^{[7],[56]-[59]}. The consequences of ignoring this inconsistency in "Kaluza-Klein ansatz" have been emphasized by Duff et al. ^{[11],[12],[14],[60]}. This and related problems have even led Cho ^[61]-[64] to call for the abandonment of Klein's "zero modes approximation" as a means of dimensional reduction.

     In general, however, spacetime can still be coaxed into compactifying in the desired manner - at the cost of altering the higher-dimensional vacuum Einstein equations, either by incorporating torsion ^[65]-[68], adding higher-derivative terms (eg., R²) onto the Einstein action ^[69], or - last but not least - adding an explicit higher-dimensional energy-momentum tensor to the theory. If chosen judiciously, this last will induce "spontaneous compactification" of the extra dimensions, as first demonstrated by Cremmer & Scherk ^[70],[71]. This approach, though, sacrifices Einstein and Kaluza's dream ^[50]-[54] of a purely geometrical unified theory of nature. Rather than explaining the "base wood" of four-dimensional matter and forces as manifestations of the "pure marble" of geometry in higher dimensions, one has essentially been driven to invent new kinds of wood. Weinberg ^[72] has likened this situation to the fable of "stone soup," in which a miraculous stew, allegedly made out of rocks, turns out on deeper investigation to be made from rocks plus various kinds of vegetables, meat and spices.

     In spite of this aesthetic drawback, however, the idea of spontaneous compactification gained rapid acceptance ^[73]-[78] and has become the standard way to reconcile extra dimensions with the observed four-dimensionality of spacetime in Kaluza-Klein theory (see Bailin & Love ^[13] for a review). An important variation is that of Candelas & Weinberg ^[79],[80], who showed that the quantum Casimir energy of massless higher-dimensional fields, when combined with a higher-dimensional cosmological constant, can also compactify the extra dimensions in a satisfactory way. Unfortunately, some 10⁴ − 10⁵ matter fields are required.

2.6  D = 11 Supergravity

     One way to make the addition "by hand" of extra matter fields more natural was to make the theory supersymmetric (ie., to match up every boson with an as-yet undetected fermionic "superpartner" and vice versa). The reason for this is that the (compactified) Kaluza-Klein programme of "explaining" gauge symmetries as (restricted) higher-dimensional spacetime symmetries can only give rise to four-dimensional gauge bosons. If the theory is to include fermionic fields, as required by supersymmetry, then these fields at least must be put in by hand. (This limitation may not necessarily apply to noncompactified Kaluza-Klein theories, in which the modest dependence on extra coordinates - subject to experimental constraints - gives the Einstein equations a rich enough structure that matter of a very general kind can be "induced" in the four-dimensional universe by pure geometry in higher dimensions. In five dimensions, for example, one can obtain not only photons, the gauge bosons of electromagnetism, but also dustlike, vacuum, or "stiff" matter.)

     Supersymmetric gravity ("supergravity") began life as a four-dimensional theory in 1976 ^[81],[82], but quickly made the jump to higher dimensions ("Kaluza-Klein supergravity"). It was particularly successful in D = 11, for three principal reasons. First, Nahm ^[83] showed that eleven was the maximum number of dimensions consistent with a single graviton (and an upper limit of two on particle spin). This was followed by Witten's proof ^[84] that eleven was also the minimum number of dimensions required for a Kaluza-Klein theory to unify all the forces in the standard model of particle physics (ie., to contain the gauge groups of the strong (SU(3)) and electroweak (SU(2) ×U(1)) interactions). The combination of supersymmetry with Kaluza-Klein theory thus appeared to uniquely fix the dimensionality of spacetime. Secondly, whereas in lower dimensions one had to choose between several possible configurations for the extra matter fields, Cremmer, Julia & Scherk ^[85] demonstrated in 1978 that in D = 11 exactly one choice was consistent with the requirements of supersymmetry (in particular, that there be equal numbers of Bose and Fermi degrees of freedom). In other words, while a higher-dimensional energy-momentum tensor was still required, its form at least appeared less contrived. Finally, Freund & Rubin ^[86] showed in 1980 that compactification of the D = 11 model could occur in only two ways: to seven or four compact dimensions, leaving four (or seven, respectively) macroscopic ones. Not only did eleven-dimensional spacetime appear to be uniquely favoured for unification, but it also split perfectly to produce the observed four-dimensional world. (The other possibility, of a macroscopic seven-dimensional world, could unfortunately not be ruled out, and in fact at least one such model was explicitly constructed as well ^[87].) Buoyed by these successes, eleven-dimensional supergravity appeared set by the mid 1980s as a leading candidate for the hoped-for "theory of everything" (see ^{[8],[12],[88]} for reviews, and ^[89] for an extensive collection of papers. A nontechnical introduction is given in ^[90].)

     A number of blemishes, however - one aesthetic and three more practical - have dampened this initial enthusiasm. Firstly, the compact manifolds originally envisioned by Witten ^[84] (those containing the standard model) turned out not to generate quarks or leptons, and to be incompatible with supersymmetry ^[8],[13]. Their most successful replacements are the 7-sphere and the "squashed" 7-sphere ^[12], described respectively by the symmetry groups SO(8) and SO(5) ×SU(2). These groups, however, unfortunately do not contain the minimum symmetry requirements of the standard model (SU(3) ×SU(2) ×U(1)). This is commonly rectified by adding more matter fields, the "composite gauge fields" ^[91], to the eleven-dimensional Lagrangian. Secondly, it is very difficult to build chirality (necessary for a realistic fermion model) into an eleven-dimensional theory ^[84],[92]. A variety of remedies have been proposed for this, including the ubiquitous additional higher-dimensional gauge fields ^[73],[77], noncompact internal manifolds ^[93]-[99], and extensions of Riemannian geometry ^[100]-[102]. Thirdly, D = 11 supergravity theory is marred by a large cosmological constant in four dimensions, which is difficult to remove, even by fine-tuning ^[7],[8]. Finally, quantization of the theory leads inevitably to anomalies ^[89].

     Some of these difficulties can be eased by descending to ten dimensions: chirality is easier to obtain ^[92], and many of the anomalies disappear ^[103]. However, the introduction of chiral fermions leads to new kinds of anomalies. And the primary benefit of the D = 11 theory - its uniqueness - is lost, since ten dimensions are not specially favoured, and the higher-energy theory does not break down naturally into four macroscopic and six compact dimensions. (One can still find solutions in which this happens, but there is no reason why they should be preferred.) In fact, most D = 10 supergravity models not only require ad hoc higher-dimensional matter fields to ensure proper compactification, but entirely ignore gauge fields arising from the Kaluza-Klein mechanism (ie., from symmetries of the compact manifold), so that all the gauge fields are effectively put into the theory by hand ^[8]. Kaluza's original aim of explaining forces in geometrical terms is thus abandoned completely.

2.7  D = 10 Superstring Theory

     A breakthrough in solving the uniqueness and anomaly problems of D = 10 theory occured when Green & Schwarz ^[104] and Gross et al. ^[105] showed that there were two, and only two ten-dimensional supergravity models in which all anomalies could be made to vanish: those based on the groups SO(32) and E₈ ×E₈ respectively. Once again, extra terms (known as Chapline-Manton terms) had to be added to the higher-dimensional Lagrangian ^[8]. This time, however, the addition was not completely arbitrary; the extra terms were those which would appear anyway if the theory were a low-energy approximation to certain kinds of superstring theory.

     The state of the art in compactified Kaluza-Klein theory, then, has shifted from supergravity theories to superstring theories, the main significance of the former now being as low-energy limits of the latter ^[89]. Superstrings (supersymmetric generalizations of strings) avoid the generic prediction of tachyons that plagued the first string theories ^[106], but retain their best features, especially the possibility of an anomaly-free path to quantum gravity ^[107]. In fact, their many virtues make them the current favorite contender for a "theory of everything" ^[108]. Connections have recently been made between certain superstring states and extreme black holes ^[14], and it has even been argued that superstrings can help resolve the long-standing black hole information paradox ^[109].

     Something of a uniqueness problem has persisted for D = 10 superstrings in that the groups SO(32) and E₈ ×E₈ admit five different string theories between them. But this difficulty has recently been addressed by Witten ^[376], who showed that it is possible to view these five theories as aspects of a single underlying theory, now known as M-theory (for "Membrane") ^[377]. The low-energy limit of this new theory, furthermore, turns out to be D = 11 supergravity! So it appears that the preferred dimensionality of spacetime in compactified Kaluza-Klein theory may be switching back to eleven.

     Perhaps the biggest obstacle to a wider acceptance of these theories is the difficulty of extracting clear-cut physical predictions from them. String theory is "promising ... ," one worker has said, "... and promising, and promising" ^[110]. M-theory, which (unlike superstring theory) is not perturbative, is even more opaque; Witten has suggested that the "M" might equally well stand for "Magic" or "Mystery" at present ^[378]. We will not consider these interesting developments further here, directing the reader instead to the superstring reviews in ^{[8],[111],[112]} (a nontechnical account may be found in ^[113]), or the recent reviews of M-theory in ^[378],[379].

2.8  The Projective Approach

     Compactification of extra dimensions is not the only way to explain Kaluza's cylinder condition. Another, less well-known approach goes back to Veblen & Hoffmann ^[114] in 1931. These authors showed that the fifth dimension could be "absorbed into" ordinary four-dimensional spacetime if one replaced the classical (affine) tensors of general relativity with projective ones. Rather than being regarded as new coordinates, the extra dimensions were effectively demoted to visual aids. Because they were not physically real, there was no need to explain why they were not observed. The price for this resolution of the problem was that one had to alter the geometrical foundation of Einstein's theory. This idea received attention from Jordan, Pauli and several others over the years ^{[39],[115]-[122]}. Early versions of the theory ran afoul of experimental constraints on the Brans-Dicke parameter ω and had to be ruled out as untenable ^[15]. However the projective approach has been revived in at least two new formulations. That of Lessner ^[15] assigns the scalar field a purely microscopic meaning; this has interesting consequences for elementary particles ^[123]-[126]. The other, due to Schmutzer ^[16]-[18], endows the vacuum with a special kind of higher-dimensional matter, the "non-geometrizable substrate," thereby sacrificing Einstein's dream of reducing physics to geometry. This theory, however, does make a number of testable predictions ^{[17],[127]-[129]} which are so far compatible with observation.

2.9  The Noncompactified Approach

     An alternative to both the compactified and projective approaches is to take the extra dimensions at face value, without necessarily compactifying them, and assume that nature is only approximately independent of them - much as it was on Minkowski's fourth coordinate at nonrelativistic speeds. In other words, one avoids having to explain why cylindricity should be exact by relaxing it in principle. Of course, the question remains as to why nature should be so nearly cylindrical in practice. If the extra dimensions are lengthlike, then one might try to answer this by supposing that particles are trapped near a four-dimensional hypersurface by a potential well. Ideas of this kind have been around since at least 1962 ^[130]; for recent discussions see ^[131]-[134].

     Confining potentials are not, however, an obvious improvement over compactification mechanisms in terms of economy of thought. An alternative is to take Minkowski's example more literally and entertain the idea that extra dimensions, like time, might not necessarily be lengthlike. In this case the explanation for the near-cylindricity of nature is to be found in the physical interpretation of the extra coordinates; ie., in the values of the dimension-transposing parameters (like c) needed to give them units of length. The first such proposal of which we are aware is the 1983 "space-time-mass" theory of Wesson ^[135], who suggested that a fifth dimension might be associated with rest mass via x⁴ = Gm/c². The chief effect of this new coordinate on four-dimensional physics was that particle rest mass, usually assumed to be constant, varied with time. The variation was, however, small and quite consistent with experiment. This model has been studied in some detail, particularly with regard to its consequences for astrophysics and cosmology, by Wesson ^{[19],[136]-[139]} and others ^{[140]-[148],[149]-[158],[159]-[163]}, and has been extended to more than five dimensions by Fukui ^[164],[165], with the constants ħ and e playing roles analogous to c and G.

     Variable gravity theories are, of course, not new. What is new in the models just described - and what is important about noncompactified Kaluza-Klein theory in principle - is not so much the particular physical interpretation one attaches to the new coordinates, but the bare fact that physics is allowed to depend on them at all. It is clearly of interest to study the higher-dimensional Einstein equations with a general dependence on the extra coordinates; ie., without any preconceived notions as to their physical meaning. A pioneering effort in this direction was made in 1986 by Yoshimura ^[166], who however considered only the case where the d-dimensional part of the (4 + d)-space could depend on the new coordinates. The general theory, in which any part of the metric can depend on the fifth coordinate, has been explored recently by Wesson and others ^[19]-[26], and its implications for cosmology ^{[167]-[170],[171]-[179]} and astrophysics ^{[180]-[185],[186]-[194]} have become the focus of a growing research effort. As this branch of Kaluza-Klein theory has not yet been reviewed in a comprehensive manner, we propose to devote special attention to it in this report. Our intention is to compare and contrast this branch of the subject with other ones, however, so we will make frequent contact with the compactified and (to a lesser extent) projective theories.

3  The Kaluza Mechanism

     Kaluza unified electromagnetism with gravity by applying Einstein's general theory of relativity to a five-, rather than four-dimensional spacetime manifold. In what follows, we consider generalizations of his procedure that may be new to some readers, so it will be advantageous to briefly review the mathematics and underlying assumptions here.

3.1  Matter from Geometry

     The Einstein equations in five dimensions with no five-dimensional energy-momentum tensor are:

or, equivalently:

where [^G]_AB ≡ [^R]_AB − [^R] [^g]_AB/2 is the Einstein tensor, [^R]_AB and [^R] = [^g]_AB [^R]^AB are the five-dimensional Ricci tensor and scalar respectively, and [^g]_AB is the five-dimensional metric tensor. (Throughout this report capital Latin indices A, B, ... run over 0, 1, 2, 3, 4, and five-dimensional quantities are denoted by hats.) These equations can be derived by varying a five-dimensional version of the usual Einstein action:

with respect to the five-dimensional metric, where y = x⁴ represents the new (fifth) coordinate and [^G] is a "five-dimensional gravitational constant."

     The absence of matter sources in these equations reflects what we have emphasized as Kaluza's first key assumption (i), inspired by Einstein: that the universe in higher dimensions is empty. The idea is to explain matter (in four dimensions) as a manifestation of pure geometry (in higher ones). If, instead, one introduced new kinds of higher-dimensional matter, then one would have gained little in economy of thought. One would, so to speak, be getting Weinberg's "stone soup" ^[72] from a can.

3.2  A Minimal Extension of General Relativity

     The five-dimensional Ricci tensor and Christoffel symbols are defined in terms of the metric exactly as in four dimensions:

     Note that, aside from the fact that tensor indices run over 0-4 instead of 0-3, all is exactly as it was in Einstein's theory. We have emphasized this as the second key feature (ii) of Kaluza's approach to unification.

     Everything now depends on one's choice for the form of the five-dimensional metric. In general, one identifies the αβ-part of [^g]_AB with g_αβ (the four-dimensional metric tensor), the α4-part with A_α (the electromagnetic potential), and the 44-part with φ (a scalar field). A convenient way to parametrize things is as follows:

where we have scaled the electromagnetic potential A_α by a constant κ in order to get the right multiplicative factors in the action later on. (Throughout this report, Greek indices α, β, ... run over 0, 1, 2, 3, and small Latin indices a, b, ... run over 1, 2, 3. The four-dimensional metric signature is taken to be (+ - - -), and we work in units such that c = 1. In addition, for convenience and accord with other work, we set ħ = 1 in § 3, and G = 1 in in § 7 and § 8.)

3.3  The Cylinder Condition

     If one then applies the third key feature (iii) of Kaluza's theory (the cylinder condition), which means dropping all derivatives with respect to the fifth coordinate, then one finds, using the metric (5) and the definitions (4), that the αβ-, α4-, and 44-components of the five-dimensional field equation (2) reduce respectively to the following field equations ^[15],[41] in four dimensions:

where G_αβ ≡ R_αβ − R g_αβ/2 is the Einstein tensor, T^EM_αβ ≡ g_αβ F_γδ F^γδ/4 − F^γ_α F_βγ is the electromagnetic energy-momentum tensor, and F_αβ ≡ ∂_α A_β − ∂_β A_α. There are a total of 10 + 4 + 1 = 15 equations, as expected since there are fifteen independent elements in the five-dimensional metric (5).

3.4  The Case φ = constant

     If the scalar field φ is constant throughout spacetime, then the first two of eqs. (6) are just the Einstein and Maxwell equations:

where we have identified the scaling parameter κ in terms of the gravitational constant G (in four dimensions) by:

     This is the result originally obtained by Kaluza and Klein, who set φ = 1. (The same thing has done by some subsequent authors employing "special coordinate systems" ^[9],[195].) The condition φ = constant is, however, only consistent with the the third of the field equations (6) when F_αβ F^αβ = 0, as was first pointed out by Jordan ^[39],[40] and Thiry ^[41]. The fact that this took twenty years to be acknowledged is a measure of the deep suspicion with which scalar fields were viewed in the first half of this century.

     Nowadays the same derivation is usually written in variational language. Using the metric (5) and the definitions (4), and invoking the cylinder condition not only to drop derivatives with respect to y, but also to pull ∫dy out of the action integral, one finds that eq. (3) contains three components ^[10]:

where G is defined in terms of its five-dimensional counterpart [^G] by:

and where we have used equation (8) to bring the factor of 16 πG inside the integral. As before, if one takes φ = constant, then the first two components of this action are just the Einstein-Maxwell action for gravity and electromagnetic radiation (scaled by factors of φ). The third component is the action for a massless Klein-Gordon scalar field.

     The fact that the action (3) leads to (9), or - equivalently - that the sourceless field equations (2) lead to (6) with source matter, constitutes the central miracle of Kaluza-Klein theory. Four-dimensional matter (electromagnetic radiation, at least) has been shown to arise purely from the geometry of empty five-dimensional spacetime. The goal of all subsequent Kaluza-Klein theories has been to extend this success to other kinds of matter.

3.5  The Case A_α = 0: Brans-Dicke Theory

     If one does not set φ = constant, then Kaluza's five-dimensional theory contains besides electromagnetic effects a Brans-Dicke-type scalar field theory, as becomes clear when one considers the case in which the electromagnetic potentials vanish, A_α = 0. Without the cylinder condition, this would be no more than a choice of coordinates, and would not entail any loss of algebraic generality. (It would be exactly analogous to the common procedure in ordinary electrodynamics of choosing four-space coordinates in which either the electric or magnetic field disappears.) With the cylinder condition, however, we are effectively working in a special set of coordinates, so that the theory is no longer invariant with respect to general (ie., five-dimensional) coordinate transformations. The restriction A_α = 0 is, therefore, a physical and not merely mathematical one, and restricts us to the "graviton-scalar sector" of the theory.

     This is acceptable in some contexts - in homogeneous and isotropic situations, for example, where off-diagonal metric coefficients would pick out preferred directions; or in early-universe models which are dynamically dominated by scalar fields. Neglecting the A_α-fields, then, eq. (5) becomes:

     With this metric, the field equations (2), and Kaluza's assumptions (i) - (iii) as before, the action (3) reduces to:

     This is the special case ω = 0 of the Brans-Dicke action ^[196]:

where ω is the dimensionless Brans-Dicke constant and the term S_m refers to the action associated with any matter fields which may be coupled to the metric or scalar field.

     The value of ω is of course constrained to be greater than ∼500 by observation ^[197], so this simple model is certainly not viable, in the present era at least. One can however evade this limit by adding a nonzero potential V(φ) to the above action, as in extended inflation ^[198] and other theories ^[199],[200]; or by allowing the Brans-Dicke parameter ω to vary as a function of φ, as in hyperextended ^[201] and other inflationary models ^[202].

3.6  Conformal Rescaling

     One can also re-formulate the problem by carrying out a Weyl, or conformal rescaling of the metric tensor. Conformal factors have begun to appear frequently in papers on Kaluza-Klein theory, but they have as yet received little attention in reviews of the subject, so we will discuss them briefly here, referring the reader to the literature for details.

     The extra factor of φ in the action (9) above implies that, strictly speaking, the scalar field would have to be constant throughout spacetime ^[10],[13] in order for the gravitational part of the action to be in canonical form. Some authors ^[9],[195] have in fact set it equal to one by definition, though this is of course not a generally covariant procedure. The offending factor can however be removed by conformally rescaling the five-dimensional metric:

where Ω² > 0 is the conformal (or Weyl) factor, a function of the first four coordinates only (assuming Kaluza's cylinder condition). This is one step removed from the simplest possible realization of Kaluza's idea. In compactified and projective theories, however, there can be no physical objection to such a procedure since it takes place "in higher dimensions" which are not accessible to observation. Questions only arise in the process of dimensional reduction; ie., in interpreting the "real," four-dimensional quantities in terms of the rescaled five-dimensional ones.

     The four-dimensional metric tensor is rescaled by the same factor as the five-dimensional one (g_αβ → g′_αβ = Ω² g_αβ), and this has the following effect on the four-dimensional Ricci scalar ^[203]:

     A convenient parametrization is obtained by making the trivial redefinition φ² → φ and then introducing the conformal factor Ω² = φ^−1/3, so that the five-dimensional metric reads:

     The same procedure as before then leads ^{[8],[12],[13]} to the following conformally rescaled action instead of eq. (9) above:

where primed quantities refer to the rescaled metric (ie., ∂′^α φ = g′^αβ ∂_β φ), and G and κ are defined as before. The gravitational part of the action then has the conventional form, as desired.

     The Brans-Dicke case, obtained by putting A_α = 0 in the metric, is also modified by the presence of the conformal factor. One finds (again making the redefinition φ² → φ and using Ω² = φ^−1/3) that the action (12) becomes ^[13]:

     In terms of the "dilaton" field σ ≡ ln φ/(√{}3κ), this action can be written:

which is the canonical action for a minimally coupled scalar field with no potential ^[204].

3.7  Conformal Ambiguity

     The question of conformal ambiguity arises when we ask, "Which is the real four-dimensional metric (ie., the one responsible for Einstein's gravity) - the original g_αβ, or the rescaled g′_αβ?" The issue was already raised at least as far back as 1955 by Pauli ^[119]. (The rescaled metric is sometimes referred to in the literature as the "Pauli metric," as opposed to the unrescaled "Jordan metric.") Most authors have worked with the traditional (unrescaled) metric, if indeed they have troubled themselves over the matter at all ^[205]. Others ^[206],[207] have considered the interesting idea of coupling visible matter (including that involved in the classical tests of general relativity) to the Jordan metric, but allowing dark matter to couple to a rescaled Pauli metric. In recent years, a variety of new arguments have been advanced in favor of regarding the rescaled metric as the true "Einstein metric" for all types of matter in compactified Kaluza-Klein theory. The following paragraph is intended as a brief review of these; many are discussed more thoroughly in ^[205].

     The first use of conformal rescaling to pick out "physical fields" was in certain ten-dimensional supergravity ^[208] and superstring ^[209] models of the early 1980s. It then appeared in work on the quantum aspects of Kaluza-Klein theory ^[210], and on the stability of compactified Kaluza-Klein cosmologies ^[211],[212]. In these papers it was asserted that conformal ambiguity affected the physics at the quantum but not the classical level. This was supported by a demonstration ^[213] that the mass of a five-dimensional Kaluza-Klein monopole was invariant with respect to conformal rescaling, although it was speculated in this paper that the addition of matter fields would complicate the situation. Cho ^[207] confirmed this suspicion by showing explicitly that the conformal invariance of the Brans-Dicke action (13) would be broken for S_m = 0. This resulted in different matter couplings to the metric for different conformal factors, which would manifest themselves as "fifth force"-type violations of the weak equivalence principle ^[214]. He argued in addition that only one conformal factor - the factor φ^−1/3 used above - could allow one to properly interpret the metric as a massless spin-two graviton ^[207]; and moreover that without this factor the kinetic energy of the scalar field would be unbounded from below, making the theory unstable ^[61]. This last point has also been emphasized by Sokolowski and others ^{[205],[215],[216]}. (Note that the conformal factor √{ϕ} used by these authors is the same as the one discussed above; the exponent depends on whether one rescales the five-, or only the four-dimensional metric. The scalar ϕ is related to φ simply by ϕ = φ^−3/2.) It has also been claimed that conformal rescaling is necessary in scale-invariant Kaluza-Klein cosmology ^[217] if one is to properly interpret the effective four-dimensional Friedmann-Robertson-Walker scale factor. A recent discussion of conformal ambiguity in compactified Kaluza-Klein theory is found in ^[218].

     There is also something much like a conformal rescaling of coordinates in projective Kaluza-Klein theory, notably in the work of Schmutzer after 1980 ^{[16],[17],[127],[128]}, where it is introduced in order to eliminate unwanted second-order scalar field terms from the generalized gravitational field equations. In noncompactified Kaluza-Klein theory, by contrast, there has been no discussion of conformal rescaling. This is largely because the extra dimensions are regarded as physical (if not necessarily lengthlike or timelike). The five-dimensional metric, in effect, becomes accessible (in principle) to observation, and conformally transforming it at will may no longer be so innocuous. We will not consider the issue further in this report; interested readers are directed to Sokolowski's paper ^[205].

4  Compactified Theories

     So far we have introduced Kaluza's theory, with its cylinder condition, but have deliberately postponed discussion of compactification because we wish to emphasize that it is logically distinct from cylindricity, and in particular that it is only one mechanism by which to explain the apparently four-dimensional nature of the world. We now turn to compactified Kaluza-Klein theory, but keep our discussion short as this subject has been thoroughly reviewed elsewhere (^{[3]-[7],[8]-[14],[195]}).

4.1  Klein's Compactification Mechanism

     The somewhat contrived nature of Kaluza's assumption, that a fifth dimension exists but that no physical quantities depend upon it, has struck generations of unified field theorists as inadequate. Klein arrived on the scene during the tremendous excitement surrounding the birth of quantum theory, and perhaps not surprisingly had the idea ^[2],[36] of explaining the lack of dependence by making the extra dimension very small. (The story that this was suggested to him on hearing a colleague address him by his last name has, so far as we know, no basis in historical fact.)

     Klein assumed that the fifth coordinate was to be a lengthlike one (like the first three), and assigned it two properties: (1) a circular topology (S¹); and (2) a small scale. Under property (1), any quantity f(x,y) (where x = (x⁰, x¹, x², x³) and y = x⁴) becomes periodic; f(x,y) = f(x, y + 2 πr) where r is the scale parameter or "radius" of the fifth dimension. Therefore all the fields can be Fourier-expanded:

where the superscript ⁽ⁿ⁾ refers to the nth Fourier mode. Thanks to quantum theory, these modes carry a momentum in the y-direction of the order |n |/ r. This is where property (2) comes in: if r is small enough, then the y-momenta of even the n = 1 modes will be so large as to put them beyond the reach of experiment. Hence only the n = 0 modes, which are independent of y, will be observable, as required in Kaluza's theory.

     How big could the scale size r of a fourth spatial dimension be? The strongest constraints have come from high-energy particle physics, which probes increasingly higher mass scales and correspondingly smaller length scales (the Compton wavelength of massive modes is of the order M⁻¹). Experiments of this kind ^[55] presently constrain r to be less than an attometer in size (1 am = 10⁻¹⁸ m). Theorists often set r equal to the Planck length l_pl ∼10⁻³⁵ m, which is both a natural value and small enough to guarantee that the mass of any n = 0 Fourier modes lies beyond the Planck mass m_pl ∼10¹⁹ GeV.

     In general, one identifies Kaluza's five-dimensional metric (5) with the full (Fourier-expanded) metric [^g]_AB (x,y), higher modes and all. One then makes what is known in compactified theory as the "Kaluza-Klein ansatz," which consists in discarding all massive (n ≠ 0) Fourier modes, as justified above. In the five-dimensional case, the Kaluza-Klein ansatz amounts to simply dropping the y-dependency of g_αβ, A_α, and φ, giving the effective four-dimensional "low-energy" theory of the graviton g⁽⁰⁾_αβ, photon A⁽⁰⁾_α and scalar φ⁽⁰⁾. For higher dimensions, though, the relationship between the full metric and the metric obtained with the "Kaluza-Klein ansatz" is more complicated, as has been emphasized by Duff et al. ^[11],[12]. These authors also stress the difference between these two metrics and a third important metric in Kaluza-Klein theory, the ground state metric 〈[^g]_AB 〉 which is the vacuum expectation value of the full metric [^g]_AB (x,y), and determines the topology of the compact space. In the five-dimensional case described above, which is topologically M⁴ ×S¹, this looks like:

where η_αβ is the four-dimensional Minkowski space metric.

4.2  Quantization of Charge

     The expansion of fields into Fourier modes suggests a possible mechanism to explain charge quantization, and it is interesting to see what became of this idea ^[8]. One begins by introducing five-dimensional matter into the theory, leaving aside for the moment questions as to what this would correspond to physically. The simplest kind of matter is a massless five-dimensional scalar field [^(ψ)] (x,y). Its action would have a kinetic part only:

     The field can be expanded like those in eq. (20):

     When this expansion is put into the action (22), one finds (using eq. (16)) the following result ^[8], analogous to eq. (9):

     From this action one can read off both the charge and mass of the scalar modes [^(ψ)]⁽ⁿ⁾. Comparison with the minimal coupling rule ∂_α → ∂_α + ieA_α of quantum electrodynamics (where e is the electron charge) shows that in this theory the nth Fourier mode of the scalar field [^(ψ)] also carries a quantized charge:

where we have normalized the definition of A_α in the action (17) by dividing out the factor (φ∫dy)^1/2, and made use of the definitions (8) and (10) for κ and G respectively. As a corollary to this result one can also come close to predicting the value of the fine structure constant, simply by identifying the charge q₁ of the first Fourier mode with the electron charge e. Taking r √{φ} to be on the order of the Planck length l_pl = √G, one has:

     (An improved determination of r √{φ} would presumably hit closer to the mark.) The possibility of thus explaining an otherwise "fundamental constant" would have made compactified five-dimensional Kaluza-Klein theory very attractive.

     However, the masses of the scalar modes are not at all compatible with these ideas. These are given by the square root of the coefficient of the [^(ψ)]⁽ⁿ⁾²-term:

     If r √{φ} ∼l_pl as we assumed, then the electron mass m₁ (corresponding to the first Fourier mode) would be l⁻¹_pl; ie., the Planck mass m_pl ∼10¹⁹ GeV, rather than 0.5 MeV. This discrepancy of some twenty-two orders of magnitude between theory and observation played a large role in the abandonment of five-dimensional Kaluza-Klein theory.

     In modern compactified theories, one avoids this problem by doing three things ^[8]: (1) identifying observed (light) particles like the electron with the n = 0, rather than the higher modes of the Fourier expansion above. From eq. (27), these particles therefore have zero mass at the level of the field equations. However, one then invokes: (2) the mechanism of spontaneous symmetry-breaking to bestow on them them the modest masses required by observation. From eq. (25) above, there is also the problem of explaining how the n = 0 modes can have nonzero charge (or, more generally, nonzero couplings to the gauge fields). This is solved by: (3) going to higher dimensions, where massless particles are no longer "singlets of the gauge group" corresponding to the ground state (eg., the 44-part of the metric (21) above). We look at this procedure briefly in the next section.

     The other way to avoid the problems of compactified five-dimensional Kaluza-Klein theory is, of course, to look at projective theories, or indeed to loosen the restriction of compactification on the fifth dimension altogether. These approaches probably mean giving up the ready-made explanation for charge quantization described above.

4.3  Extension to Higher Dimensions

     The key to extending the Kaluza-Klein formalism to strong and weak nuclear interactions lies in recognizing that electromagnetism has been effectively incorporated into general relativity by adding U(1) local gauge invariance to the theory, in the form of local coordinate invariance with respect to y = x⁴. Assuming the extra coordinate has a circular topology and a small scale, the theory is invariant under transformations:

where x stands for the four-space coordinates x⁰, x¹, x², x³. With the aid of the usual tensor transformation law (in five dimensions):

one then finds that the only change to the metric (5) is:

which is just a U(1) local gauge transformation. In other words the theory is locally U(1) gauge invariant. It is thus not surprising that electromagnetism could be contained in five-dimensional general relativity.

     To extend the same approach to more complicated symmetry groups, one goes to higher dimensions ^[8]. The metric corresponding to the "Kaluza-Klein ansatz" (n = 0 modes only) can be written (cf. eq. (5)):

where [g\tilde]_μν is the metric of the d-dimensional compact space. Indices μ, ν, ... run from 1 to d, while A, B, ... run from 0 to (3 + d), and α, β, ... run from 0 to 3 as usual. The K^μ_i are a set of n linearly independent Killing vectors for the compact manifold (i = 1, ..., n). Analogously to eq. (28) one then assumes that the theory is locally invariant under transformations:

where the εⁱ(x) are a set of n infinitesimal parameters. Because Killing vectors by definition satisfy:

the transformation law (29) leaves the [g\tilde]_μν-part of the metric untouched, and the only effect on eq. (31) is:

which is a local gauge transformation whose gauge group is the isometry group (G, say) of the compact manifold. Thus one might hope that higher-dimensional general relativity could contain any gauge theory.

     The larger symmetry of the higher-dimensional mechanism also allows for nonzero couplings of the n = 0 modes to the gauge fields; ie., for "charged" massless particles (which, as we saw, was impossible in the five-dimensional case). Massless scalar fields φ_a(x) in the adjoint representation of the gauge group, for example, can be introduced ^[8] via:

and these in general have nonzero couplings to the gauge fields because the K^μ_a (y) are not covariantly constant.

4.4  Higher-Dimensional Matter

     It is crucial to realize, however, that the above "ansatz" metric (31) does not satisfy Einstein's equations in 4 + d dimensions unless the Killing vectors are independent of {y}, the extra coordinates ^[8] - ie., unless the compact manifold is flat ^[7]. The ground state metric (cf. eq. (21)) is:

     The vacuum Einstein equations are [^R]_AB − [^R] [^g]_AB/2 = 0. Since 〈[^g]_αβ 〉 = η_αβ is flat, [^R]_αβ = 0. Therefore, from the αβ-components of the field equations, [^R] must vanish. But then the μν-parts of the same equations imply that [^R]_μν = 0; ie., that [g\tilde]_μν must also be flat. In what is perhaps a symptom of the split that has developed since Klein between the particle physics and general relativity sides of higher-dimensional unification research, early workers tended to ignore this "consistency problem" ^[11],[12], and placed no restrictions on the compact manifold while continuing to use the metric (31). Recently Cho ^[61]-[64] has raised related questions about whether the "zero modes" might not become massive (and {y}-dependent) in the event of spontaneous symmetry-breaking, and has even suggested "kicking away the ladder" of Klein's Fourier modes entirely, basing dimensional reduction a priori on isometry instead.

     It is now widely recognized ^[13] that conventional compactification of d extra spatial dimensions (where d > 1) requires either (1) explicit higher-dimensional matter terms, which can induce "spontaneous compactification" by imposing constant curvature on the compact manifold ^[70],[71]; or (2) other modifications of the higher-dimensional theory, such as the inclusion of torsion ^[65]-[68] or higher-derivative (eg. R²) terms ^[69]. Most higher-dimensional compactified Kaluza-Klein theories rely on higher-dimensional matter of one kind or another. For example, in Freund-Rubin compactification ^[86], which is the basis of eleven-dimensional supergravity, one introduces a third-rank antisymmetric tensor field [^A]_BCD with field strength:

and free action given by:

     The effect of this ^[13] is to add an explicit energy-momentum tensor to the right-hand side of the higher-dimensional Einstein equations (1):

     The matter fields required to achieve compactification are not the end of the story, however. Others are in general needed if the theory is to contain the full gauge group of the standard model (including strong and weak interactions). Witten ^[84] has shown that this requires a theory of at least eleven dimensions (including the four macroscopic ones). While there are an infinite number of compact seven-dimensional manifolds whose isometry groups G ⊃ SU(3) ×SU(2) ×U(1), none of them give rise to realistic quark and lepton representations ^[8],[13]. It is possible to obtain quarks and leptons from other manifolds such as the 7-sphere and the "squashed" 7-sphere ^[12]. The symmetry groups of these manifolds (SO(8) and SO(5) ×SU(2) respectively) are, however, not large enough to contain the standard model, and additional "composite" matter fields ^[13],[91] are therefore required.

     Explicit higher-dimensional fields may also be required to incorporate chirality into eleven-dimensional compactified theory ^{[7],[73],[77]} (this is difficult in an odd number of dimensions). Two other schemes by which this might be accomplished are modifications of Riemannian geometry ^[100]-[102] and noncompact internal manifolds ^[93]-[99]. Thus in the D = 11 "chirality problem" one finds again a choice between sacrificing either (i) the equation "Matter=Geometry;" (ii) the geometrical basis of Einstein's theory; or (iii) cylindricity. Compactified theory has in general been characterized by a readiness to drop (i).

     We conclude this section by noting that the situation (with regard to non-geometrized matter) does not improve in ten-dimensional compactified theory; in fact, in many cases a six-dimensional internal manifold with no isometries is used ^[8], which means that all the matter is effectively put in by hand, marking a complete abandonment of the original Kaluza programme. Besides ensuring compactification and making room for fermions, extra terms in the ten-dimensional Lagrangian also play a role in suppressing anomalies. In the two most popular D = 10 theories, for example (those based on the symmetry groups SO(32) ^[104] and E₈ ×E₈ ^[105]), this is accomplished by Chapline-Manton terms ^[8]. To the extent that these terms arise naturally in the low-energy limit of ten-dimensional superstring theory, however, they are less arbitrary than some of the others we have mentioned. There is no doubt that superstrings currently offer, within the context of compactified Kaluza-Klein theory, the best hope for a unified "theory of everything" ^[89],[108]. Whether the compactified approach is the best one remains - as we hope to show in the rest of this report - an open question.

5  Projective Theories

     Projective theories were designed to emulate the successes of Kaluza's five-dimensional theory without the epistemological burden of a real fifth dimension. Early models did this too well: like Kaluza's (with no dependence on the fifth coordinate and no added "higher-dimensional matter" fields) they gave back ω = 0 Brans-Dicke theory when the electromagnetic potentials were switched off. This contradicted time-delay measurements like that of the Martian Viking lander ^[197]. There were other problems as well ^[16]. Modern projective theories ^[15]-[18] attempt to overcome these shortcomings in at least two different ways.

5.1  A Theory of Elementary Particles

     Lessner ^[15] has suggested that, although experiments rule out a macroscopic Brans-Dicke-type scalar field, the theory might still be applicable on microscopic scales, and could be used to describe the internal structure of elementary particles. He begins with the same five-dimensional field equations (2) (now interpreted as projector equations), and obtains the same four-dimensional field equations (6), except that the constant G is replaced by "B," which becomes essentially a free parameter of the theory. A solution (g_αβ, F_αβ, φ) of the field equations is called a "particle" if it satisfies certain conditions on symmetry, positivity and asymptotic behaviour ^[15]. Some of the properties of these particles are explored in ^[123]-[126]. The theory is only applicable to macroscopic phenomena when φ = 1 and the third of the field equations (6) is omitted.

5.2  Projective Unified Field Theory

     Schmutzer has taken an alternate approach since 1980 in his "projective unified field theory" or PUFT ^[16]-[18] by explicitly introducing "non-geometrizable matter" (the so-called "substrate"). In ordinary Kaluza-Klein theory this would correspond to higher-dimensional matter and be represented in the five-dimensional Einstein equations (1) by a nonzero energy-momentum tensor [^T]_AB; in the projective theory one has instead an energy projector [^(θ)]_AB:

     There is also a conformal rescaling of the four-dimensional metric, as mentioned earlier:

where σ is a new scalar field. Eqs. (40) break down, analogously to the five-dimensional ones (2), to the following set of equations in four dimensions:

where T^EM_αβ is the electromagnetic energy-momentum tensor as before, and where there are also two new energy-momentum tensors: the substrate energy tensor θ_αβ = [^(θ)]_αβ and the scalaric energy tensor Σ_αβ defined by:

     The other terms in eqs. (42) are the electric four-current density J_α, the electromagnetic field strength tensor B_αβ, the induction tensor H_αβ = e^3σ B_αβ (the factor e^3σ acts here as a kind of "scalaric dielectricity"), and one more new quantity, the scalaric substrate density:

     The conservation of energy [^(∇)]^A [^(θ)]_AB = 0 implies not only conservation of four-current (∇_α J^α = 0) but also conservation of substrate energy:

     The existence of substrate and scalaric matter in PUFT gives rise to phenomena such as "scalaric polarization" of the vacuum, and violations of the weak equivalence principle for time-dependent scalaric fields. These can be quantified in terms of a "scalarism parameter" γ, defined as the ratio of scalaric substrate density to the density ρ of ordinary matter:

     This number becomes in practice the primary free parameter of the theory, showing up in PUFT-based cosmological models ^[127],[129], equivalence principle-type experiments ^[128], and Solar System tests (perihelion shift, light deflection and time delay) ^[17]. Experimental constraints on the theory take the form of upper limits on the size of γ.

     In comparing these projective theories to the compactified Kaluza-Klein theories of the last section, one could perhaps summarize as follows: Kaluza's unified theory as it stands is an elegant (no higher-dimensional matter) and minimal extension of general relativity, but suffers from the defect of a very contrived-looking cylinder condition. Five-dimensional compactified theory, beginning with Klein, repairs this flaw (and even offers the possibility of explaining charge quantization) but turns out to disagree radically with observation. To overcome this problem within the context of compactified theory, one has to go to higher dimensions and either introduce higher-dimensional matter or higher-derivative terms to the Einstein action, if one wishes to obtain satisfactory compactification. Projective theory offers an alternative way to "explain" the cylinder condition, and can (unlike compactified theory) be formulated in a way that is compatible with experiment using only one extra "dimension." This comes at the price, however, not only of modifying the geometrical foundation of Einstein's theory, but (in Schmutzer's case) of introducing a "non-geometrizable substrate," or (in Lessner's case) of limiting one's ambitions to microscopic phenomena. Overall, the projective approach does not appear to us to be an improvement over compactified theory.

6  Noncompactified Theories

     An alternative is to stay with the idea that the new coordinates are physical, but to generalize the compactified approach by relaxing the cylinder condition ^[19]-[26], instead of restricting the topology and scale of the fifth dimension in an attempt to satisfy it exactly. This means that physical quantities, including in particular those derived from the metric tensor, will depend on the fifth coordinate. In fact it is precisely this dependence which allows one to obtain not only electromagnetic radiation, but matter of a very general kind from geometry via the higher-dimensional field equations. The equations of motion, too, are modified by dependence on extra coordinates. We review these facts in the next few sections.

     Of course, the fifth dimension might also be expected to appear elsewhere in physics, and one of the primary challenges of noncompactified theory is to explain why its effects have not been noticed so far. Why, for example, have experiments such as those mentioned earlier ^[55] been able to restrict the size of any extra dimensions to below the attometer scale? In noncompactified theory, the answer is that extra coordinates are not necessarily lengthlike, as these experiments assume. Following Minkowski's example, one can imagine coordinates of other kinds, scaled by appropriate dimension-transposing parameters (like c) to give them units of length. We review this important issue, and the evidence for the hypothesis that a fifth dimension might be physically related to rest mass, at the end of § 6. For the moment, however, we put off questions of interpretation and begin by simply seeing how far Kaluza's five-dimensional unified field theory can be taken when the cylinder condition is dropped.

6.1  The Metric

     Without cylindricity, there is no reason to compactify the fifth dimension, so this approach is properly called "noncompactified." Noncompact extra dimensions have also been considered in compactified Kaluza-Klein theory by Wetterich and others ^[93]-[99] as a way to bring chiral fermions into the theory and arrange for a vanishing four-dimensional cosmological constant. These authors, however, retain Klein's mechanism of harmonic expansion, which in turn means that the compact manifold must have finite volume. In the fully noncompactified approach we wish to make no a priori assumptions about the nature of the extra-dimensional manifold.

     We begin with the same five-dimensional metric (5) as before, but choose coordinates such that the four components of A_α vanish. Since we are no longer imposing cylindricity on our solutions, this entails no loss of algebraic generality; it is analogous to the common strategy in electromagnetic theory of choosing coordinates such that either the electric or magnetic field vanishes. We also eschew any conformal factor here, preferring to treat the fifth dimension on the same footing as the other four. The five-dimensional metric tensor, then, is:

where we have introduced the factor ε in order to allow a timelike, as well as spacelike signature for the fifth dimension (we require only that ε² = 1).

     Timelike extra dimensions are rarely considered in compactified Kaluza-Klein theory, for several reasons ^[13]: (1) they lead to the wrong sign for the Maxwell action in eq. (9) relative to the Einstein one; and (2) they lead to the wrong sign for the mass m_n of the charged modes in eq. (24); ie., to the prediction of tachyons. The relevance of these two arguments to noncompactified theory may be debated. A third common objection (3) is that additional temporal ^[13] or timelike ^[7],[86] dimensions would lead to closed timelike curves and hence allow causality violation. One should be careful here to discriminate between temporal dimensions, which actually have physical units of time; and timelike ones, which merely have timelike signature. If the physical nature of the fifth coordinate were actually temporal, one could certainly imagine problems with causality. One can, however, transpose units with the proper combination of fundamental constants; changing a temporal one, for instance, into a spatial one with c. With regard to timelike extra dimensions, the situation is also less clear than is sometimes claimed. It has even been argued ^[219] that physics might be quite compatible with closed timelike curves. All in all, it is probably prudent to keep an open mind toward the signature of a physical fifth dimension.

6.2  The Field Equations

     One now follows the same approach as Kaluza, using the same definitions (4) of the five-dimensional Christoffel symbols and Ricci tensor. Now, however, one keeps derivatives with respect to the fifth coordinate x⁴ rather than assuming that they vanish. The resultant expressions for the αβ-, α4- and 44-parts of the five-dimensional Ricci tensor [^R]_AB are ^[20]:

where "^[¯]" is defined as usual (in four dimensions) by ^[¯] φ ≡ g^αβ ∇_β(∂_α φ).

     We assume that there is no "higher-dimensional matter," so the Einstein equations take the form (2), [^R]_AB = 0. The first of eqs. (48) then produces the following expression for the four-dimensional Ricci tensor:

     The second can be written in the form of a conservation law:

where we have defined a new four-tensor by:

     And the third of eqs. (48) takes the form of a scalar wave equation for φ:

     Eqs. (49) - (52) form the basis of five-dimensional noncompactified Kaluza-Klein theory. It only remains to interpret their meaning in four dimensions, and then to apply them to any given physical problem by choosing the appropriate metric [^g]_AB. The rest of § 6 is taken up with interpretation; applications to cosmology and astrophysics are the subjects of § 7 and § 8. We concentrate in this report on the five-dimensional case. The extension to arbitrary dimensions has yet to be investigated in detail, although some aspects of this have recently been discussed by Rippl, Romero & Tavakol ^[24]. (These authors also consider noncompactified lower-dimensional gravity, which might be more easily quantized than Einstein's theory).

6.3  Matter from Geometry

     The best-understood of eqs. (49) - (52) is the first. It allows us to interpret four-dimensional matter as a manifestation of five-dimensional geometry ^[20]. One simply requires that the usual Einstein equations (with matter) hold in four dimensions:

where T_αβ is the matter energy-momentum tensor. Contracting eq. (49) with the metric g^αβ gives (with the help of eq. (52)) the following expression for the four-dimensional Ricci scalar:

     Inserting this result, along with eq. (49), into eq. (53), one finds that:

     Provided we use this expression for T_αβ, the four-dimensional Einstein equations G_αβ = 8 πG T_αβ are automatically contained in the five-dimensional vacuum ones [^G]_AB = 0. The matter described by T_αβ is a manifestation of pure geometry in the higher-dimensional world. This has been termed the "induced-matter interpretation" of Kaluza-Klein theory, and eq. (55) is said to define the energy-momentum tensor of induced matter.

     This tensor satisfies the appropriate requirements: it is symmetric (the first term is a second derivative, while the others are all explicitly symmetric), and reduces to the expected limit when the cylinder condition is re-applied (ie., when all derivatives ∂₄ with respect to the fifth dimension are dropped). In this case, the scalar wave equation (52) becomes just the Klein-Gordon equation for a massless scalar field:

and the contracted energy momentum tensor of the induced matter vanishes:

which implies a radiationlike equation of state (p = ρ/3) for the induced matter, in agreement with earlier work ^[220] based on the cylinder condition. The induced matter in this case consists of photons, the gauge bosons of electromagnetism - exactly the same result obtained by Kaluza. This is the only kind of matter one can obtain in the induced-matter interpretation as long as the cylinder condition is in place. To extend Kaluza's approach to other kinds of matter, it is necessary to do one of two things: (1) go to higher dimensions and add an explicit energy-momentum tensor (or other terms) to the higher-dimensional vacuum field equations (compactified theories in practice involve both these things); or (2) loosen the restriction of cylindricity. In noncompactified theory, which takes the latter course, it turns out that matter described by T_αβ - even in five dimensions - is already general enough to describe many physical systems, including in particular those connected with cosmology and the classical tests of general relativity.

     The interpretation of eqs. (50) and (52) - the α4- and 44- components of the five-dimensional field equations ([^R]_AB = 0) - is not as straightforward as that of eq. (49). The relative simplicity of the conservation equation (50) suggests that there is a deeper physical significance to the four-tensor P^β_α, whose fully covariant form is P_αβ ≡ (∂₄ g_αβ − g_αβ g^γδ ∂₄ g_γδ)/(2 √{[^g]₄₄}). It may be related to more familiar conserved physical quantities, or to the Bianchi identities ^[20].

     Alternatively, it has been conjectured ^[177] that, as the αβ-components of the field equations link geometry with the macroscopic properties of matter, so the α4- and 44-components might describe their microscopic ones. In particular, if one makes the tentative identification:

where k is a constant, m_i and m_g are the (suitably defined) inertial and gravitational mass of a particle in the induced-matter fluid, and υ^α ≡ dx^α/ds is its four-velocity, then the conservation equation (50) turns out to be the four-dimensional geodesic equation (for one class of metrics at least). This is interesting, since equations of motion are usually quite distinct from the field equations. Similarly, using appropriate definitions of particle mass m, one can identify the scalar wave equation (52) with the simplest possible relativistic quantum wave equation, namely the Klein-Gordon equation:

     The relevant expression for particle mass turns out to depend explicitly on the components of the metric, which means that this variant of noncompactified Kaluza-Klein theory is a realization of Mach's Principle ^[251],[252]. These are interesting results, but speculative ones, and we do not discuss them further here. Some other Machian aspects of noncompactified theories have been explored in ^{[25],[150],[178],[179],[184]}.

6.4  The Spherically-Symmetric Case

     To appreciate what the induced-matter energy-momentum tensor (55) means physically, one has to supply a five-dimensional metric [^g]_AB - preferably one specific enough to simplify the mathematics but general enough to be broadly applicable, eg., to both cosmological and one-body problems. We begin here with the general spherically-symmetric five-dimensional line element:

where ε serves the same function as before, t, r, θ and φ have their usual meanings, ψ is the fifth coordinate, and ν, λ, R and μ are, for now, arbitrary functions of r, t and ψ. Denoting derivatives with respect to t by overdots ([( )\dot]), derivatives with respect to r by primes (′), and derivatives with respect to ψ by star superscripts (^*), one finds ^[21] that the energy-momentum tensor (55) of induced matter has the following nonzero components:

     If one then assumes that this induced matter takes the form of a perfect fluid:

where u^α is the four-velocity of the fluid elements, then the density ρ and pressure p can be readily identified ^[21] from the relations ρ = T₀⁰ + T₁¹ − T₂² and p = −T₂². Inserting the expressions (61), one obtains:

     It is immediately apparent that under the restriction of cylindricity (all starred quantities vanish), one can obtain only radiation (p = ρ/3) from Kaluza's mechanism, as noted already.

     With the relaxation of this condition, by contrast, one obtains a very general equation of state. For example, one can split the density and pressure into four components (ρ = ρ_r + ρ_d + ρ_v + ρ_s and p = p_r + p_d + p_v + p_s), where the radiation component obeys p_r = ρ_r/3, the dust-like component obeys p_d = 0, the vacuum component obeys p_v = −ρ_v, and the stiff component obeys p_s = ρ_s. One then finds from eqs. (63) that:

     From the first of these equations, it follows that in a radiationlike universe whose metric coefficients depend only on time, the fifth dimension must contract with time ([(μ)\dot] < 0) if one is to have spatial expansion ([R\dot] > 0) and positive density (ρ_r > 0). Mechanisms of this sort have been used in compactified Kaluza-Klein cosmology to pump entropy into the four-dimensional universe, solving the horizon and flatness problems ^[221]; or indeed to explain why the fifth dimension is compact in the first place ^[222] (see § 7.1). In the noncompactified approach, they no longer have to be assumed a priori, but can be seen to be required by the field equations. From the second of the above equations, meanwhile, it follows that a dustlike universe must have a spacelike fifth dimension (ε = −1 in our convention) in order for its density to be positive (ρ_d > 0). This agrees with the causality argument (§ 6.1).

6.5  The Isotropic and Homogeneous Case

     One can go farther by making additional assumptions about the metric. Suppose the line element (60) is rewritten in spatially isotropic form:

where dΩ² ≡ dθ² + sin² θdφ². If one assumes that ν, ω and μ are separable functions of the variables t, r and ψ, one can obtain specialized solutions to the field equations (2) whose properties of matter, as specified by the energy-momentum tensor (61), agree very closely with those expected from four-dimensional theory.

     Consider first the case of dependence on t only. The metric (65) is then just a five-dimensional generalization of a flat homogeneous and isotropic Friedmann-Robertson-Walker (FRW) cosmology. But in the context of noncompactified Kaluza-Klein theory, one ought also to allow dependence on the extra coordinate ψ. So the general flat five-dimensional cosmological metric, assuming separability, should have:

     Ponce de Leon ^[145] was the first to investigate solutions of the vacuum Einstein equations (2) with this form. Of his eight solutions, one is of special interest because it reduces on hypersurfaces ψ = constant to the spatially flat four-dimensional FRW metric. This solution has ε = −1 and:

and can be written in the form:

where dx² + dy² + dz² = dr² + r² dΩ² are the usual rectangular coordinates, and α is a free parameter of the theory ^[168]. Because this solution reduces on spacetime sections (dψ = 0) to the familiar k = 0 FRW metric:

it can properly be called the generalization of the flat FRW cosmological metric to five dimensions.

     Assuming that cosmological matter behaves like a perfect fluid, one obtains from eqs. (63) the following expressions for density and pressure ^[167]:

     These are consistent with a wide variety of equations of state: a radiation-dominated universe, for example, if α = 2; a dust-filled one if α = 3/2; or an inflationary one if 0 < α < 1. Physical properties of cosmologies based on the metric (68) have been explored in ^[167]-[171], and its implications for the equations of motion (eg., of galaxies) are known ^[172]. Generalizations to k ≠ 0 cosmologies ^[173],[174] and extended (eg., Gauss-Bonnet) theories of gravity ^[175],[176] have been made, and a connection to Mach's principle ^{[25],[177]-[179]} has been identified. These and related issues are reviewed in § 7.

     One other of Ponce de Leon's homogeneous and isotropic solutions ^[145] deserves mention. It has:

and looks like:

     This reduces on spacetime hypersurfaces (ψ = constant) to the de Sitter metric, and Λ ≡ 3 / ψ² is a cosmological constant induced in four-dimensional spacetime by the existence of the fifth coordinate ψ. The equation of state of the "matter" induced in four dimensions is that of the classical de Sitter vacuum, p = − ρ, with ρ = Λ / (8 πG).

     Billyard & Wesson ^[171] have considered generalizations of this solution:

where ω is a frequency, k_i a wave vector, and l measures the size of the extra dimension. The induced-matter equation of state is again p = − ρ, but now with ρ = −3 ω²/(32 πG ψ²). The field equations (2) turn out to require l² = 4 / ω² so the vacuum has positive energy density if the fifth dimension is spacelike. The metric coefficients of ordinary three-space exhibit wave-like behaviour, but the associated medium is an unperturbed de Sitter vacuum - so this solution describes what might be termed "vacuum waves" in Kaluza-Klein theory. (They are not gravitational waves of the conventional sort because three-space is spherically-symmetric.) One might apply this to the inflationary universe scenario; imagining, for example, that ω starts out with real values (corresponding to a vacuum-dominated universe with oscillating three-space coefficients) but later takes on imaginary values (for which the universe enters an expanding de Sitter phase of the usual kind). On this interpretation, the big bang occurs as a (presumably quantum-induced) phase change - as has previously been suggested elsewhere on other grounds ^[223],[224].

6.6  The Static Case

     The metric (65) reduces to another well-known form when the coefficients ν, ω and μ depend only on the radial coordinate r. This is just a five-dimensional generalization of the one-body or Schwarzschild metric, and has been variously interpreted in the literature as describing a magnetic monopole ^[225], "black hole" ^[227], and soliton ^[226] (see § 8.1 for discussion). Again, however, from a noncompactified point of view there is no a priori reason to suppress dependence on ψ, so a general static spherically-symmetric metric, assuming separability, should have:

     Ponce de Leon & Wesson ^[21] searched for two-parameter solutions of the five-dimensional field equations (2) with this form and found only four. (Liu & Wesson ^[193],[194] have recently obtained a three-parameter generalization of this class). The most useful is the one which contains the ordinary four-dimensional Schwarzschild solution as a limiting case. This is the solution with D, E and F constant (= 1 without loss of generality), and is thus identical to the soliton metric just mentioned. The coefficients A, B and C are:

where a is a constant related to the mass of the central body, and ε and k are other parameters (in the notation of ^[227]). Only one of these is strictly a free parameter, as they are related by a consistency relation:

     Written out explicitly, the metric is:

where we have assumed a spacelike fifth coordinate (ε = −1) in agreement with other work. In the limits ε→ 0, k → ∞, and εk → 1 (where a ≡ 2/GM_* and M_* is the mass of the central body), this metric reduces on spacetime sections dψ = 0 to the familiar Schwarzschild metric (in isotropic coordinates):

     It is therefore properly called the generalization of the Schwarzschild metric to five dimensions. Elsewhere in § 6 we will refer to the above values of and k as the "Schwarzschild limit" of the theory.

     Assuming as usual that the induced matter takes the form of a perfect fluid, eqs. (63) give for both solutions the following density and pressure ^[180]:

     The soliton metric (77) thus describes a central mass surrounded by an inhomogeneous cloud of radiation-like matter whose density goes as ∼1 / a² r⁴ at large values of r. (The Schwarzschild limit defined above is the special case where the density and pressure of the cloud are zero; in this case p = 0 = − ρ which is the usual vacuum solution, with its attendant classical tests of general relativity.) The ε² k-term indicates that this combination of the two (related) parameters ε and k may characterize the soliton's energy density ^[181]. (This is somewhat different from the traditional interpretation, in which these parameters are related to its "scalar charge" ^[215],[228].) The equation of state (79) obtained in the induced-matter interpretation differs from the one found by Davidson & Owen ^[227], who used an approach based on Kac-Moody symmetries ^[59] and concluded that p = − ρ/ 3. Density shows the same r-dependence at large distances in both approaches, however, and goes to zero in the same Schwarzschild limit. Both solutions are invariant under a → −a, ε→ − ε, and require k > 0 for positive density. One can define a pressure three-tensor p^a_b in the induced-matter interpretation, using the ab-components of the vacuum field equations (2), and this yields a result ^[180] very similar to the series expression obtained by Davidson & Owen. If one then takes p ≡ p^a_a/3 (as in ^[227]), one gets back exactly the result in eq. (79).

     In Cartesian spatial coordinates the pressure tensor in general contains off-diagonal components, which implies that the matter making up the soliton is a sum of both a material (perfect) fluid and a free electromagnetic field ^[180]. The terms "density" and "pressure" therefore have to be treated with caution. Solitonic matter, in fact, is best be described as a relativistic fluid with anisotropic pressure ^[181]. Anisotropic spherically-symmetric fluids have an energy-momentum tensor given by:

where χ_α is a unit spacelike vector orthogonal to u_α, and p_||, p_⊥ refer to pressure parallel and perpendicular to the radial direction. Assuming that the induced matter (61) takes this form rather than that of the perfect fluid (62); and choosing u^α = (u⁰, 0, 0, 0), χ^α = (0, χ¹, 0, 0), one finds:

with ρ exactly as in eq. (79). These expressions satisfy ρ = p_|| + 2 p_⊥, confirming that the fluid has the nature of radiation. The physical properties of solitons based on the metric (77) have been studied by several authors ^{[180]-[183],[225]-[228]}. Their implications for astrophysics ^[184],[185], the classical tests of general relativity ^[187]-[189], and the equivalence principle ^[190] have been explored, and the class has been extended to time-dependent ^[191],[192] and charged solutions ^[193],[194]. These and related issues are reviewed in § 8.

     We mention for completeness the other three static solutions of the form (74) obtained by Ponce de Leon & Wesson ^[21]. Two of them have A(r), B(r) and C(r) exactly as in eqs. (75), but have F(ψ) an arbitrary function of ψ, with D^*(ψ) ∝ F(ψ) and E(ψ) = constant in the first case, and D(ψ) ∝ E⁻¹(ψ) and E^*(ψ) ∝ F(ψ) / E(ψ) in the second one. The density and pressure for both these solutions is exactly as in eqs. (79) above, except for an added factor of E²(ψ) in the denominators. This is physically innocuous in the first case (E(ψ) = constant) but means in the second one that these attributes of the radiation cloud depend on the extra coordinate. The ψ-dependent components of the field equations place an extra constraint on these both these solutions, restricting the allowed values of the parameters ε and k. The final solution is more interesting, and can be written in the form:

where a is related to the mass of the central object as before, b = −ε/4a, and l is one other independent constant of the system. Although this solution was found by assuming separability in r and ψ, it also satisfies the field equations (2) when a and b are arbitrary functions of ψ. This is intriguing, as it hints at a relationship between the mass of the central object and the fifth coordinate. Another interesting feature of the metric (82) is its induced-matter equation of state, which - unlike that of the other soliton solutions found so far - is not radiationlike, but turns out to be the one discussed by Davidson & Owen ^[227]: p = −ρ/3. This is an unusual form of matter, but has been considered previously in several other contexts ^[229]-[232], largely because it describes matter that does not disturb other objects gravitationally (gravitational or Tolman-Whittaker mass is proportional to 3p + ρ). Thus it might, for instance, be useful in reconciling the extremely high energy densities expected for quantum zero-point fields with the small value observed for Einstein's cosmological constant ^[231].

6.7  General Covariance in Higher Dimensions

     We have reviewed a number of solutions to the spherically-symmetric vacuum field equations in five dimensions. In each case the five-dimensional geometry manifests itself in four dimensions as induced matter, with an associated equation of state. The equation of state, in fact, follows from the field equations in the induced-matter interpretation of Kaluza-Klein theory, rather than having to be supplied separately as in four-dimensional general relativity. In several cases, the physical form of the metric on spacetime hypersurfaces dψ = constant, or the equation of state for the induced matter, is such as to make the solutions useful for testing the predictions of noncompactified theory. The theory is so far not in conflict with any experimental data (see § 7 and § 8).

     However, it is important to keep in mind that physical quantities such as the scalars ρ and p, which are designed to be invariant with respect to four-dimensional coordinate changes x^α → [x\tilde]^α cannot in general stay that way in noncompactified Kaluza-Klein theory, which is invariant with respect to five-dimensional ones x^A → [x\tilde]^A. Any quantities - even those normally thought of as conserved - are vulnerable if they depend on the fifth coordinate x⁴.

     What this means in practice is that density, pressure, and the equation of state in the induced-matter interpretation are to some extent dependent on the coordinates in which one chooses to express them. A search for the correct solution to a (four-dimensional) physical problem is also a search for the appropriate system of (five-dimensional) coordinates. This can perhaps best be illustrated with a series of simple x⁴-dependent coordinate transformations ^[23], beginning with five-dimensional Minkowski space:

     Spacetime sections of this metric are of course four-dimensional Minkowski spaces. If one transforms to primed coordinates:

this metric becomes:

     Spacetime sections (ψ′ = constant) of the new primed metric are static Einstein cosmologies; ie., four-dimensional FRW metrics:

with k = +1 and a constant scale factor R(t′) = ψ′ . One can obtain from the Friedmann equations the value of Einstein's cosmological constant Λ, and (assuming a perfect fluid) expressions for the density and pressure of matter:

     The cosmological constant represents a vacuum energy density ρ_v = Λ / (8 πG) with associated pressure p_v = − ρ_v. So altogether one has:

     The effective equation of state in the four-dimensional spacetime sections of the primed metric (85) is thus that of non-gravitating matter of the kind discussed in § 6.6. (The same result could have been obtained by plugging the metric directly into eqs. (63) for induced-matter density and pressure.) A second coordinate transformation to double-primed coordinates:

puts the metric into the new form:

     Spacetime sections of this double-primed metric are expanding FRW cosmologies, with k = +1 and R(t") = ψ" cosht". The density and pressure of the accompanying perfect fluid, as obtained from eqs. (63), are:

so that the effective equation of state is that of a pure vacuum.

     Of the metrics (83), (85) and (90), which one is the best choice for a description of the real universe? None, of course, since none of them admits spacetime sections with realistic four-dimensional properties. A metric which is adequate to this task is the cosmological one (68). It can be obtained from Minkowski space (83) by transforming from t, r, ψ to [t\tilde], [r\tilde], [(ψ)\tilde] via: