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Chapter 2
Induced-Matter Theory

2.1  Introduction

     We import from the preceding chapter two important and connected ideas. First, as realized by several workers, the so-called fundamental constants like c, G and h have as their main purpose the transposition of physical dimensions. Thus, a mass can be regarded as a length; and physical quantities such as density and pressure can be regarded as having the same dimensions as the geometrical quantities that figure in general relativity. Second, physical quantities should be given a geometric interpretation, as envisaged by many people through time, including Einstein who wished to transmute the "base wood" of physics to the "marble" of geometry. An early attempt at this was made by Kaluza and Klein, who extended general relativity from 4 to 5 dimensions, but also applied severe restrictions to the geometry (the conditions of cylindricity and compactification). In this chapter, we will draw together results which have appeared in recent years which show that it is possible to interpret most properties of matter as the result of 5D Riemannian geometry, where however the latter allows dependence on the fifth coordinate and does not make assumptions about the topology of the fifth dimension.

     This induced-matter theory has seen most work in 3 areas: (a) The case of uniform cosmological models is easiest to treat because of the high degree of symmetry involved, and is very instructive. (b) The soliton case is more complicated, but important because 5D solitons are the analogs of isolated 4D masses, and the 5D class of soliton solutions contains the unique 4D Schwarzschild solution. (c) The case of neutral matter can be treated quite generally, and lays the foundation for many applications where electromagnetic effects are not involved. After an outline of geometric feasibility (Section 2.2) we will give the main theoretical results in each of the aforementioned areas (Sections 2.3, 2.4, 2.5). We defer the main observational implications to later chapters. Our conclusion (Section 2.6) will be that one extra dimension is enough to explain the phenomenological properties of classical matter.

     From here on, we will absorb the fundamental constants c, G and h via a choice of units that renders their magnitudes unity. We will use the metric signature with diagonal = (+ − − − ±), where the last choice will be seen to depend on the physical application and not cause any problem with causality. Also, we will label 5D quantities with upper-case Latin letters (A = 0−4) and 4D quantities with lower-case Greek letters (α = 0−3). If there is a chance of confusion between the 4D part of a 5D quantity and the 4D quantity as conventionally defined, we will use a hat to denote the former and the straight symbol to denote the latter.

2.2  A 5D embedding for 4D matter

     The 5D field equations for apparent vacuum in terms of the Ricci tensor are

     Equivalently, in terms of the 5D Ricci scalar and the 5D Einstein tensor G_AB ≡ R_AB − Rg_AB/2, they are

     By contrast, the 4D field equations with matter are given by Einstein's relations of general relativity:

     The central thesis of induced-matter theory is that (2.3) are a subset of (2.2) with an effective or induced 4D energy-momentum tensor T_αβ which contains the classical properties of matter.

     That this is so will become apparent below when we treat several cases suggested by physics. However, it is also possible to approach the subject through algebra; and while results in the latter field were subsequent, they are general and can be summarized here. Thus, it is a direct consequence of a little-known theorem by Campbell that any analytic N-dimensional Riemannian manifold can be locally embedded in an (N+1)-dimensional Ricci-flat (R_AB = 0) Riemannian manifold (Romero, Tavakol and Zalaletdinov 1996). This is of great importance for establishing the generality of the proposal that 4D field equations with sources can be locally embedded in 5D field equations without sources. And it can be used to study lower-dimensional (N < 4) gravity, which may be easier to quantize than general relativity (Rippl, Romero and Tavakol 1995). It can also be employed to find new classes of 5D solutions (Lidsey et al. 1997). Some of the latter have the remarkable property that they are 5D flat but contain 4D subspaces that are curved and correspond to known physical situations (Wesson 1994; Abolghasem, Coley and McManus 1996). The latter do not, though, include the 4D Schwarszchild solution, which can only be embedded in a flat manifold with N ≥ 6 (Schouten and Struik 1921; Tangherlini 1963; for general results in embeddings see Campbell 1926; Eisenhart 1949; Kramer et al. 1980). However, the principle is clear: curved 4D physics can be embedded in curved or flat 5D geometry, and we proceed to study 3 prime cases of this.

2.3  The cosmological case

     There are many exact solutions known of (2.1) that are of cosmological type, meaning that the metric resembles that of Robertson-Walker and the dynamics is governed by equations like those of Friedmann (see Section 1.3). However, most of these do not involve dependence on the extra coordinate l and are from the induced-matter viewpoint very restricted. Thus while we will use one of these solutions below, we will concentrate on the much more significant solutions of Ponce de Leon (1988). He found several classes of exact solutions of (2.1) whose metrics are separable and reduce to the standard 4D RW ones on the hypersurfaces l = constants. The induced matter and other properties associated with the most physical class of Ponce de Leon solutions were worked out by Wesson (1992a). Since then, many other cosmological solutions and their associated matter properties have been derived by various workers (see, e.g., Chatterjee and Sil 1993; Chatterjee, Panigrahi and Banerjee 1994; Liu and Wesson 1994; Liu and Mashhoon 1995; Billyard and Wesson 1996). In what follows, we will illustrate the transition from the 5D equations (2.1), (2.2) for apparent vacuum to the 4D equations (2.3) with matter, by using simple but realistic solutions.

     It is convenient to consider a 5D metric with interval given by

     Here the time coordinate x⁰ = t and the space coordinates x¹²³ = r θφ(dΩ² ≡ dθ² + sin² θdφ²) have been augmented by the new coordinate x⁴ = l. The metric coefficients ν, ω, and μ will depend in general on both t and l, partial derivatives with respect to which will be denoted by an overdot and an asterisk, respectively. Components of the Einstein tensor in mixed form are:

     These are 5D components. We wish to match the terms in (2.5) with the components of the usual 4D perfect-fluid energy-momentum tensor. This is T_αβ = (p + ρ) μ_α μ_β − p g_αβ, where μ^α ≡ dx^α/ds, and for our case has components T⁰₀ = ρ, T¹₁ = −p for the density and pressure. Following the philosophy outlined above, we simply identify the new terms (due to the fifth dimension) in G₀⁰ with ρ, and the new terms in G₁¹ with p. Then, collecting terms which depend on the new metric coefficient μ or derivatives with respect to the new coordinate l, we define

     These are suggested identifications for 4D properties of matter in terms of 5D properties of geometry.

     To see if they make physical sense to this point, we combine (2.6) and (2.5) with the field equations G^A_B = 0 of (2.2). There comes

     We see from the first of these that ρ must be positive; and from the third that p could in principle be negative, as needed in classical descriptions of particle production in quantum field theory (see, e.g., Brout, Englert and Gunzig 1978; Guth 1981; and Section 1.3). To make further progress, however, we need explicit solutions of the field equations.

     A simple solution of (2.1) or (2.2) that is well known but does not depend on l has ν = 0, ω = logt, μ = − logt in (2.4), which now reads

     This has a shrinking fifth dimension, and from (2.6) or (2.7) density and pressure given by 8 πρ = 3/4t², 8 πp = 1/4t². If these are combined to form the gravitational density (ρ+ 3p) and the proper radial distance R ≡ e^ω/2 r is introduced, then the mass of a portion of the fluid is M = 4 πR³ (ρ+ 3p)/3. The field equations then ensure that [R\ddot] = − M / R² is obtained as usual for the law of motion. Similarly, the usual first law of thermodynamics is recovered by writing dE + pdV = 0 as (ρe^{3 ω/ 2})^* + p(e^{3 ω/ 2})^* = 0 (E = energy, V = 3D volume; see Wesson 1992a). The equation of state of the fluid described by (2.8) is of course the p = ρ/ 3 typical of radiation.

     To go beyond radiation, we use one of the classes of solutions to (2.1) or (2.2) due to Ponce de Leon (1988). With a redefinition of constants appropriate to the induced-matter theory, it has e^ν = l², e^ω = t^2/α l^2/(1−α), e^μ = α² (1 − α)⁻² t² in (2.4), which now reads

     This has a growing fifth dimension, and density and pressure which depend on the one assignable constant α. From (2.6) or (2.7) they are

     The presence of l here may appear puzzling at first, but the coordinates are of course arbitrary and the proper time is T ≡ lt. (Alternatively, the presence of x⁴ = l depends on whether we consider the pure 4D metric or the 4D part of the 5D metric.) In terms of this, 8 πρ = 3 / α² T² and 8 πp = (2 α− 3) / α² T². For α = 3/2, 8 πρ = 4 / 3 T² and p = 0. While for α = 2, 8 πρ = 3 / 4 T² and 8 πp = 1 / 4 T². The former is identical to the 4D Einstein-de Sitter model for the late universe with dust. The latter is identical to the 4D standard model for the early universe with radiation or highly relativistic particles. (The coincidence of the properties of matter for this model with α = 2 and the previous model does not necessarily imply that they are the same, since similar matter can belong to different solutions even in 4D.) As before, the usual forms of the law of motion and the first law of thermodynamics are recovered, provided we use the effective gravitational density of matter defined by the combination ρ+ 3 p (see Section 1.3; these laws are recoverable generally for metrics with form (2.4) using the proper time T ≡ e^{ν/ 2} t and the proper distance R ≡ e^ω/2 r). The equation of state of the fluid described by (2.9) is p = (2 α/ 3 − 1) ρ, and so generally describes isothermal matter.

     Other properties of (2.9) were studied by Wesson (1992a, 1994), including the sizes of horizons and the nature of the extra coordinate l. We defer further discussion of this model, because here we are mainly dealing with theoretical aspects of induced-matter theory. However, we note two things. First, the solutions (2.8) and (2.9) describe in general photons with zero rest mass and particles with finite rest mass, respectively; and the fact that the former does not depend on l whereas the latter does, gives us a first inkling that l is related to mass (see later). Second, the solution (2.9) gives an excellent description of matter in the late and early universe from the big-bang perspective of physics in 4D; but it has a somewhat amazing property from the perspective of geometry in 5D. Thus consider a coordinate transformation from t, r, l to T, R, L specified by

     Then (as may be verified by computer) the Ponce de Leon metric (2.9) in standard form becomes

     This means that our universe can either be viewed as a 4D spacetime curved by matter or as a 5D flat space that is empty.

2.4  The soliton case

     There is a class of exact solutions of (2.1) which has been rediscovered several times during the history of Kaluza-Klein theory. The metric is static, spherically-symmetric in ordinary (3D) space, and independent of the fifth coordinate. (There are many of these solutions rather than one because Birkhoff's theorem does not apply in its conventional form in 5D.) The solutions have been interpreted as describing magnetic monopoles (Sorkin 1983), massive objects of which some called solitons have no gravitational effect (Gross and Perry 1983), and black holes (Davidson and Owen 1985). The first usage is questionable, because the 4D Schwarzschild solution is a special member of the class and gravitational in nature, and magnetic monopoles are in any case conspicuous by their absence in the real world. The last usage is misleading, because all but the Schwarzschild-like member of the class lack event horizons of the conventional sort. The middle usage can be extended, since in the induced-matter picture we will see that these solutions represent stable, extended objects (Wesson 1992b). Thus even though the word is over-worked in physics, we regard these 5D 1-body solutions as representing objects called solitons.

     A particularly simple member of the soliton class was rediscovered by Chatterjee (1990). It is instructive to start with this, because it has been analyzed in standard or Schwarzschild-like coordinates (as opposed to the isotropic coordinates used below). Thus the 5D metric has an interval given by

     Here the constant A would normally be identified in gravitational applications via the r → ∞ limit as √A = M_*, the mass of an object like a star at the centre of ordinary space. However, as mentioned above, it cannot be assumed that (2.13) is a black hole. Indeed, the first metric coefficient in (2.13) goes to zero only for r tending to zero. In other words, the event horizon in the coordinates of (2.13) shrinks to a point at the centre of ordinary space. (This is not altered by a Killing-vector prescription for horizons and different sets of coordinates: see Wesson and Ponce de Leon 1984.) To see what (2.13) actually represents, let us use the induced-matter approach. It is helpful to consider a metric we will come back to later, namely

     This includes (2.13) if we assume that the metric coefficients ν, λ, R, μ depend only on the radius and not on the time or extra coordinate. Then following the same procedure as in the preceding section, we obtain the components of the induced 4D energy-momentum tensor:

     Here a prime denotes the partial derivative with respect to the radius. Other components are zero, and of course T³₃ = T²₂ because of spherical symmetry. The components (2.15) which define the properties of matter depend on derivatives of μ, that is upon the geometry of the fifth dimension. However, matter and geometry are unified via the field equations G^A_B = 0 of (2.2), and we can use these to rewrite (2.15) in a more algebraically convenient form:

     Substitutinginto these equations for the Chattejee solution (2.13) gives

     These components obey the equation of state

which is radiation-like. The matter described by (2.17) has a gravitational mass that can be evaluated using the standard 4D expression

where g₄ is the determinant of the 4-metric and dV₃, is a 3D volume element. Using (2.14) and (2.16) this gives

     This is most conveniently evaluated in the coordinates of the Chatterjee solution in the form (2.13). Thus putting R → r and integrating gives

which with the coefficients of (2.13) is

     We see that M_g(∞) = √A, agreeing with the usual metric-based definition of the mass as noted above. However, we also see that M_g(0) = 0, meaning that the gravitational mass goes to zero at the centre. In summary, the Chatterjee soliton (2.13) is a ball of radiation-like matter whose density and pressure fall off very rapidly away from the centre, and whose integrated mass agrees with the conventionai definition only at infinity.

     The above concerned a special case of a broad class of 5D solutions which has been widely studied in forms due to Gross and Perry (1983) and Davidson and Owen (1985). These authors use different terminologies, particularly for two dimensionless constants which enter the solutions. The former use α, β and the latter use κ, ε where the two are related by κ = −1 / β, ε = − β/ α We adopt the latter notation, as it is more suited to the induced-matter approach. In it, positive effective density of matter requires κ > 0, and positive gravitational mass as measured at spatial infinity requires εκ > 0 (see below). Thus, physicality requires that both κ and ε be positive. In terms of these constants the Chatterjee solution we have looked at already has just κ = 1, ε = 1. And the Schwanschild solution we will look at below has ε→ 0, κ→ ∞, εκ→ 1. We now proceed to consider the general class.

     This has usually been discussed with the metric in spatially isotropic form, which we write as

     Then solutions of the apparently empty 5D field equations (2.1) or (2.2) are given by

     Here a is a dimensional constant to do with the source, and the two dimensionless constants are related by a consistency relation derived from the field equations:

     This means that the class is a 2-parameter one, depending on a and one or the other of ε, κ. Also, we noted above that physicality requires that both κ and ε be positive. Now, the surface area of 2-shells around the centre of the 3-geometry varies as (ar − 1)^{1−ε(κ−1)}, and will shrink to zero at r = 1/a provided 1 − ε(κ− 1) > 0. This combined with (2.25) means κ > 0. That is, the centre of the 3-geometry is at r = l/a for physical choices of the parameters (see Billyard, Wesson and Kalligas 1995 for a more extensive discussion). Also, e^ν → 0 for r → 1/a for ε, κ > 0. So as for the Chatterjee case above, the event horizon for the general class shrinks to a point at the centre of ordinary space.

     The properties of the induced matter associated with (2.24) can be worked out following the same procedure as before. The components of the induced 4D energy-momentum tensor are:

     Substituting into these equations for the solutions in the form (2.24) and doing some tedious algebra gives

     These components obey the same equation of state as before, namely (T⁰₀ + T¹₁ + T²₂ + T³₃) = 0. If we average over the 3 spatial directions, this is equivalent to saying that the equation of state is [p] = ρ/ 3. Also as before, we can calculate the standard 4D gravitational mass of a part of the fluid by using (2.19). This with (2.23) gives

which with (2.26) is

     Then with the coefficients of (2.24) we obtain

     This is the gravitational mass of a soliton as a function of (isotropic) radius r, and to be positive as measured at infinity requires that εκ > 0. Since positive density requires that κ > 0 by (2.27) we see that we need both κ > 0 and ε > 0, as we stated above. Then (2.30) shows that M_g(r = 1/a) = 0, meaning again that the gravitational mass goes to zero at the centre. However, the mass as measured at spatial infinity is now 2 εκ/ a and not just 2/a = M_* as it was for the Chatterjee case.

     This is interesting, and should be compared to what we obtain if we substitute parameters corresponding to the Schwarzschild case, namely ε→ 0, κ→ ∞, εκ→ 1. Then (2.30) gives M_g(r) = constant = 2 / a = M_*. And the metric (2.23), (2.24) becomes

     This is just the Schwarzschild solution (in isotropic coordinates) plus a flat and therefore physically innocuous extra dimension. In other words, if we use the conventionally defined 4D gravitational mass as a diagnostic for 5D solitons, we recover the usual 4D Schwarzschild mass exactly.

     We have carried out a numerical investigation of preceding relations to clarify the status of the Schwarzschild solution (Wesson and Ponce de Leon 1994). The problem is that if we set ε = 0 and εκ = 1 then (2.30) gives M_g(r) = 2/a for all r; but if we keep ε small and let r → 1/a, then (2.30) gives M_g(r = 1/a) = 0 irrespective of ε. Clearly the limit by which one is supposed to recover the Schwanschild solution from the soliton solutions is ambiguous. However, our numerical results show that, from the viewpoint of perturbation analysis at least, the Schwarzschild case is just a highly compressed soliton. We have also looked at other definitions for the mass of a soliton, including the so-called proper mass (which depends on an integral involving only and is badly defined at the centre) and the ADM mass (which depends on a product of field strength and area and is well defined at the centre). To clarify what happens near the centre of a soliton, we have also calculated the geometric scalars for metric (2.23), (2.24). The relevant 5D invariant is the Kretschmann scalar K ≡ R_ABCD R^ABCD, which we have evaluated algebraically and checked by computer. It is

     Taking into account the constraint (2.25), this may be found to diverge for κ > 0 at r = 1/a, showing that there is a geometric singularity at the centre as we have defined the latter. [Note that if we let ε→ 0, κ→ ∞, εκ→ 1 then (2.32) gives K = 192 a¹⁰ r⁶ / (ar + 1)¹² in isotropic coordinates, or K = 48 M²_* / (r^′)⁶ in curvature coordinates where r^′ = r (1 + 1/ar)² and a = 2/M_*. This result agrees formally with the one from Einstein theory, but from our viewpoint no longer has much significance since the point r^′ = 0 or r = −1/a is not part of the manifold, which ends at r = 1/a or r^′ = 2M_*.] The relevant 4D invariant is C ≡ R_αβ R^αβ, which is most easily evaluated algebraically using the field equations. The latter are (2.1) or R_AB = 0, but if there is no dependency on the extra coordinate read just R_αβ = 8 πT_αβ because the 4D Ricci scalar R is zero. Then C = 64 π² [(T⁰₀)² + (T¹₁)² + 2(T²₂)²], and can be evaluated using the components of the energy-momentum tensor (2.27). It is

     This also diverges at r = 1/a, confirming that there is a singularity in the geometry at the centre. In conjunction with the fact that most of them do not have event horizons of the standard sort, this means that technically solitons should be classified as naked singularities.

     Whether or not we can see to the centre of a soliton, practically, is a different question. What was a point mass in 4D general relativity has become a finite object in 5D induced-matter theory. The fluid is `hot', with anisotropic pressure and density that falls off rapidly away from the centre (for large distances it goes as M<sup>2</sup><sub>*</sub>/r<sup>4</sup> where M<sub>*</sub> is the mass as measured at spatial infinity). The Schwarzschild solution is somewhat anomalous, but can be regarded as a soliton where matter is so concentrated towards the centre as to leave most of space empty. In short, solitons are `holes' in the geometry surrounded by induced matter.

2.5  The case of neutral matter

     Kaluza-Klein theory in 5D has traditionally identified the g_{4 α} components of the metric tensor with the potentials A_α of classical electromagnetism (see Section 1.5). These set to zero therefore give in some sense a description of neutral matter. However, any fully covariant 5D theory, such as induced-matter theory, has 5 coordinate degrees of freedom, which used judiciously can lead to considerable algebraic simplification without loss of generality. Therefore, a natural case to study is specified by g_{4 α} = 0, g₄₄ ≠ 0. This removes the explicit electromagnetic potentials and leaves one coordinate degree of freedom over to be used appropriately (e.g., to simplify the equation of motion of a particle). This choice of coordinates or choice of gauge involves g_αβ = g_αβ (x^A), g₄₄ = g₄₄ (x^A) and so is not restricted by the cylinder condition of old Kaluza-Klein theory. It admits fluids consisting of particles with finite or zero rest mass, and thus includes the cases we have studied in Sections 2.3 and 2.4. In the present section, we follow Wesson and Ponce de Leon (1992). Our aims are to give a reasonably self-contained account of the matter gauge and to lay the foundation for later applications.

     We have a 5D interval dS² = g_AB dx^A dx^B where

     Here ε² = 1 and the signature of the scalar part of the metric is left general. (We will see later that there are well-behaved classical solutions of the field equations with ε = +1 as well as the often-assumed ε = −1, and the freedom to choose this may also help with the Euclidean approach to quantum gravity.) The 5D Ricci tensor in terms of the 5D Christoffel symbols is given by

     Here a comma denotes the partial derivative, and below we will use a semicolon to denote the ordinary (4D) covariant derivative. Putting A → α, B → β in (2.35) gives us the 4D part of the 5D quantity. Expanding some summed terms on the r.h.s. by letting C → λ, 4 etc. and rearranging gives

     Part of this is the conventional Ricci tensor that only depends on indices 0123, so

     To evaluate this we need the Christoffel symbols.

     These can be tabulated here in appropriate groups:

     We will use these respectively to evaluate (2.37) above and (2.44), (2.54) below.

     Thus substituting into and expanding some terms in (2.37) gives

     Some of the terms here may be rewritten using

     Where Φ_α ≡ Φ_,α. Then (2.41) gives

     We will use this below when we consider the field equations.

     Returning to (2.35), we put A = 4, B = 4 and expand with C → λ, 4 etc. to obtain

     The Christoffel symbols here are tabulated in (2.39), and cause (2.44) to become

     Some of the terms here may be rewritten using

     Here we have obtained the last line by noting that Φ_{β; λ} = Φ_{β, λ} − Γ^σ_βλ Φ_σ implies

and that (σ^μ_ν)_,μ = 0 implies (g^λσ_,λ + g^βλ g^σμ g_{μβ, λ}) Φ_σ = 0. Putting (2.46) in (2.45) gives lastly

where ^[¯] Φ ≡ g^μν Φ_{μ; ν} defines the 4D curved-space box operator. Equations (2.43) and (2.48) can be used with the 5D field equations (2.1) which we repeat here:

     Then [^R]_αβ = 0 in (2.43) gives

     And R₄₄ = 0 in (2.48) gives

where we have noted that (δ^μ_ν)_,4 = 0 implies g^μβ g^λσ g^*_λβ g^*_μσ + g^{* μσ} g^*_μσ = 0. From (2.50) we can form the 4D Ricci curvature scalar R = g^αβ R_αβ. Eliminating the covariant derivative using (2.51), and again using (δ^μ_ν)_,4 = 0 to eliminate some terms, gives

     With (2.50) and (2.52), we are now in a position to define if we wish an energy-momentum tensor in 4D via 8 πT_αβ ≡ R_αβ − Rg_αβ/2. It is

     Provided we use this energy-momentum tensor, Einstein's 4D field equations (2.3) or G_αβ = 8 πT_αβ will of course be satisfied.

     The mathematical expression (2.53) has good properties. It is a symmetric tensor that has a part which depends on derivatives of Φ with respect to the usual coordinates x⁰¹²³, and a part which depends on derivatives of other metric coefficients with respect to the extra coordinate x⁴. [The first term in (2.53) is implicitly symmetric because it depends on the second partial derivative, while the other terms are explicitly symmetric.] It is also compatible with what is known about the recovery of 4D properties of matter from apparently empty 5D solutions of Kaluza-Klein theory. Thus the cosmological case studied in Section 2.3 agrees with (2.53) and has matter which owes its characteristics largely to the x⁴-dependency of g_αβ in that relation. While the soliton case studied in Section 2.4 agrees with (2.53) and has matter which depends on the first or scalar term in that relation. With (2.53) and preceding relations, the case where there is no dependency on x⁴ becomes transparent. Then (2.51) becomes the scalar wave equation for the extra part of the metric (g^μν Φ_{μ; ν} = 0 with g₄₄ = εΦ²). And (2.53) gives T = T_αβ g^αβ = 0, which implies a radiation-like equation of state. However, in general there must be x⁴-dependence if we are to recover more complex equations of state from solutions of R_AB = 0.

     These field equations have 4 other components we have not so far considered, namely R_{4 α} = 0. This relation by (2.35) expanded is

     The Christoffel symbols here are tabulated in (2.40) and cause (2.54) to become

     Here we have done some algebra using (g₄₄ g⁴⁴)_{, α and 4} = 0 or g^{* 44} g_{44, α} − g⁴⁴_{, α} g^*₄₄ = 0, and (δ^β_λ)_,4 = 0 or g^λν g_μν g^μσ + g^{* λσ} = 0. [We also note in passing that one can use (Γ⁴_{4 α})_{, 4} = (Γ⁴₄₄)_{, α} in (2.54) and obtain an alternative form of (2.55) with the last term replaced by g^μβ_{, α} g^*_μβ/4.] While (2.55) may be useful in other computations, it is helpful for our purpose here to rewrite it as

     Noting that ∂/ ∂x^a = δ^β_α (∂/ ∂x^β) and that −g⁴⁴ g_{44, β} / 2 = √{g₄₄} (∂/ ∂x^β) (1 / √{g₄₄}) allows us to obtain finally

     This form suggests we should introduce the 4-tensor

     The divergence of this is

which when written out in full may be shown to be the same as the r.h.s. of (2.57). The latter therefore reads

     The field equations (2.49) as R_{4 α} = 0 can then be summed up by the relations

     These have the appearance of conservation laws for P^β_α. The fully covariant form and associated scalar for the latter are:

     We will examine these quantities elsewhere, but here we comment that while our starting gauge (2.34) removed the explicit electromagnetic potentials, the field equations R_{4 α} = 0 or (2.60) are of electromagnetic type.

     It is apparent from the working in this section that the starting conditions (2.34) provide a convenient way to split the 5D field equations R_AB = 0 into 3 sets: The 5D equations [^R]_αβ = 0 give a set of equations in the 4D Ricci tensor R_αβ (2.50); the 5D equation R₄₄ = 0 gives a wave-like equation in the scalar potential (2.51); and the 5D equations R_{4 α} = 0 can be expressed as a set of 4D conservation laws (2.60). Along the way we also obtain some other useful relations, notably an expression for the 4D Ricci scalar in terms of the dependency of the 4D metric on the extra coordinate (2.52). However, the physically most relevant expression is an effective or induced 4D energy-momentum tensor (2.53). Another way to express these results is to say that the 15 field equations R_AB = 0 of (2.1) or G_AB = 0 of (2.2) can always be split into 3 sets which make physical sense provided the metric is allowed to depend on the extra coordinate x⁴. These sets consist of 4 conservation equations of electromagnetic type, 1 equation for the scalar field of wave type, and 10 equations for fields and matter of gravitational type. In fact, the last are Einstein's equations (2.3) of general relativity, with matter induced from the extra dimension.

2.6  Conclusion

     The idea of embedding G_αβ = 8 πT_αβ (4D) in R_AB = 0 (5D) is motivated by the wish to explain classical properties of matter rather than merely accepting them as given. In application to the cosmological case it works straightforwardly, and gives back 5D geometric quantities which are identical to the 4D density and pressure (Section 2.3). This is important: what we derive from the 5D equations is not something esoteric but ordinary matter. In application to the soliton or 1-body case, the idea leads to a class of radiation-like solutions which contains as a very special case the Schwarzschild solution (Section 2.4). In general application to neutral matter, the properties of the latter turn out to be intimately connected to x⁴-dependency of the metric (Section 2.5). Induced-matter theory actually admits a wide variety of equations of state (Ponce de Leon and Wesson 1993). But in the matter gauge at least, independence from x⁴ implies radiation-like matter, while dependence on x⁴ implies other kinds of matter.

     The theoretical basis we have demonstrated in this chapter leads naturally to the question of observations, particularly with regard to the solitons. As mentioned above, there is a class of these in 5D rather than the unique Schwarzschild solution of 4D, because Birkhoff's theorem in its conventional form does not apply. Indeed, there are known exact solutions which represent time-dependent solitons (Liu, Wesson and Ponce de Leon 1993; Wesson, Liu and Lim 1993). And there is known an exact solution which is x⁴-dependentand Schwarzschild-like (Mashhoon, Liu and Wesson, 1994). We will return to the latter, where we will find that it implies the same dynamics as in general relativity and so poses no problem. However, there remains the question of the observational status of the standard solitons. This has been investigated by a number of people, most of whom were not working in the induced-matter picture (see Overduin and Wesson 1997). Here, we can regard the soliton as a concentration of matter at the centre of ordinary space, and ask about the motions of test particles at large distances. Specifically, we ask what constraints we can put on the soliton 1-body metric from the classical tests of relativity.

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