5D Kaluza-Klein Theories - a Brief Review

5D Kaluza-Klein theories - a brief review

Andr´eMorgado Patr´ıcio, no 67898 Departamento de F´ısica, Instituto Superior T´ecnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

(Dated: 21 de Novembro de 2013) We review Kaluza-Klein theory in five dimensions from the General Relativity side. Kaluza’s original idea is examined and two distinct approaches to the subject are presented and contrasted: compactified and noncompactified theories. We also discuss some cosmological implications of the noncompactified theory at the end of this paper.

ˆ ˆ 1 ˆ I. INTRODUCTION where GAB ≡ RAB − 2 gÂBR is the Einstein tensor. Inspired by the ties between Minkowki’s 4D space- For the metric, we identify the αβ-part ofg ÂB with time and Maxwell’s EM unification, Nordström[3] in 1914 gαβ, the α4-part as the electromagnetic potential Aα and and Kaluza[4] in 1921 showed that 5D general relativ- gˆ44 with a scalar field φ, parametrizing it as follows: ity contains both Einstein’s 4D gravity and Maxwell’s g + κ2φ2A A κφ2A EM. However, they imposed an artificial restriction of no gˆ (x, y) = αβ α β α , (4) AB κφ2A φ2 dependence on the fifth coordinate (cylinder condition). β Klein[5], in 1926, suggested a physical basis to avoid this where κ is a multiplicative factor. If we identify√ it in problem in the compactification of the fifth dimension, terms of the 4D gravitational constant by κ = 4 πG, idea now used in higher-dimensional generalisations to then, using the metric4 and applying the cylinder con- include weak and strong interactions. dition, the action3 contains three components: The models to be discussed have three key features: Z √ R φ2F F αβ 2∂αφ∂ φ (i) matter in 4D is a manifestation of pure geometry S = − d4x −gφ + αβ + α , (5) in 5D, i.e., no energy-momentum tensor is needed; 16πG 4 3κ2φ2 (ii) the higher-dimensional theory is a minimal exten- where the 4D gravitational constant G is defined in terms sion of general relativity, i.e., there is no modifica- of its 5D counterpart Gˆ by G = G/ˆ R dy. tion to the mathematical structure of the theory; From this action, the field equations3 reduce to the (iii) Physics only depends on the first four coordinates following field equations in terms of 4D quantities[1]: (cylinder condition). κ2φ2 1 G = T EM − [∇ (∂ φ) − g φ] , (6) There are three approaches to study higher- αβ 2 αβ φ α β αβ dimensional theories, each one sacrifices one of the key α α ∂ φ features: compactified, projective and noncompactified ∇ Fαβ = −3 Fαβ, (7) theories. Here we will review the general relativity as- φ κ2φ3 pects of the compactified and noncompactified ones. φ = F F αβ, (8) 4 αβ A. Remarks on notation and conventions where T EM is the EM energy-momentum tensor Throughout this report, capital Latin indices A, B, ... αβ run over 0,1,2,3,4 and Greek indices α, β, ... over 0,1,2,3. γδ EM gαβFγδF γ Five dimensional quantities are denoted by hats and the Tαβ ≡ − Fα Fβγ ,Fαβ ≡ ∂αAβ − ∂βAα. (9) new fifth coordinate will be denoted by y = x4. 4 The four-dimensional metric signature is taken to be This ability is the central “miracle” of Kaluza-Klein (+ − −−) and we work in a system of units with c = 1. theory: 4D matter (EM radiation) has arised purely from We also set ~ = G = 1 in the cosmology section. the geometry of empty 5D space. Two important cases are worth of mention: φ = constant and Aα = 0. II. 5D KALUZA MECHANISM For φ = constant, the field equations are just Einstein From feature (ii) of Kaluza’s approach, the Ricci tensor and Maxwell equations: ˆ ˆ ˆC RAB, Ricci scalar R and Christoffel symbols ΓAB are 2 EM α Gαβ = 8πGφ Tαβ , ∇ Fαβ = 0 . (10) Rˆ = ∂ ΓˆC − ∂ ΓˆC + ΓˆC ΓˆD − ΓˆC ΓˆD , AB C AB B AC AB CD AD BC This result was obtained by Kaluza and Klein, who ˆ AB ˆ R =g ˆ RAB, (1) set φ = 1. However, it isn’t consistent with eq.8 if αβ ˆC 1 CD FαβF 6= 0, fact first noted by Jordan[8] and Thiry[9]. ΓAB = gˆ (∂AgˆDB + ∂BgˆDA − ∂DgÂB) , 2 The restriction Aα = 0 is a physical one. In fact, with the cylinder condition we are working in a special coor- whereg ˆ is the five-dimensional metric tensor. AB dinate system, so that we can’t set A = 0 by a special The action and field equations are a five-dimensional α choice of coordinates because this theory is no longer in- version of the usual 4D Einstein action and equations variant to general coordinate transformations. Then, the −1 Z metric4 becomes block diagonal and we get that the ac- S = Rˆp−gdˆ 4xdy, (2) 16πGˆ tion is the special case ω = 0 of a Brans-Dicke action[10]: ˆ ˆ δS = 0 → GAB = 0 ⇔ RAB = 0, (3) Z Rφ ∂αφ∂ φ S = − d4x + ω α . (11) 16πG φ Another interesting point to analyze is the possibility In fact, from eq. 20, we have of rescaling the metric and put the gravitational part of √ m = |n|/(rpφ), (23) the action5 in canonical form: −gR/16πG. This can n 2 −1 19 be done by making the redefinition φ → φ and then which gives for the electron mass m1 ∼ lpl ∼ 10 GeV, carrying out the conformal rescaling of the metric much less than the experimental value me ≈ 0.5 MeV. This problem can be avoided by doing three things[1]: gˆ → gˆ0 = Ω2gˆ , Ω2 = φ−1/3, (12) AB AB AB (1) identify observed light particles with n = 0 modes; (2) which leads to the rescaled canonical action[2] invoke the mechanism of spontaneous symmetry break- ! ing to give them masses; (3) go to higher dimensions to Z R0 φF 0 F 0αβ ∂0αφ∂0 φ S0 = − d4xp−g0 + αβ + α . (13) solve the problem of n = 0 modes with nonzero charges, 16πG 4 6κ2φ2 because the particles are no longer singlets of the gauge group corresponding to the ground state. The Brans-Dicke case is√ obtained if Aα = 0. In terms of the ”dilaton” σ ≡ lnφ/( 3κ), the action is then that IV. NONCOMPACTIFIED THEORIES for a minimally coupled scalar field with no potential: In this aproach, we don’t restrict the topology and Z R0 ∂0ασ∂0 σ S0 = − d4xp−g0 + α . (14) scale of the fifth dimension in an attempt to satisfy ex- 16πG 2 actly the cylinder condition. Instead, we stay with the III. COMPACTIFIED THEORIES idea that the new coordinates are physical. The theory is then invariant to general 5D coordinate To avoid the cylinder condition, Klein assumed that transformations and we can choose coordinates such that the extra spatial dimension compactifies to the topology Aα = 0. The 5D metric tensor4 is then block diagonal 1 of a circle S of very small radius r. We are then left g 0 gˆ (x, y) = αβ , (24) with a residual 4D coordinate invariance and an abelian AB 0 εφ2 gauge invariance associated with transformations of the coordinate of S1. Thus, space has topology R4 × S1. where ε is a factor which allows a timelike as well as 2 To see this, consider the transformation spacelike signature for the fifth dimension (ε = 1). The procedure to obtain the field equations is exactly y → y0 = y + f(x), (15) the same as in Kaluza’s Mechanism, except that now we ∂xC ∂xD keep derivatives with respect to the fifth coordinate. Re- gˆ → gˆ0 = gˆ . (16) AB AB ∂x0A ∂x0B CD quiring the usual Einstein 4D equations hold, Then, for the particular transformation 15, 1 8πGTαβ = Rαβ − Rgαβ, (25) 0 2 Aα → A = Aα + ∂αf(x). (17) α this procedure produces the 4D field equations 26 to 28: Thus we see that the transformation 15 induces an abelian gauge transformation on A . ∇ (∂αφ) ε ∂4φ ∂4g α 8πG T = β − αβ − ∂ (∂ g ) αβ 2 4 4 αβ To see how Aα couples to matter, we introduce in the φ 2φ φ ˆ " γδ # theory the simplest 5D matter, a scalar field ψ(x, y): ε γδ g ∂4gγδ ∂4gαβ − g ∂4gαγ ∂4g − Z 2φ2 βδ 2 4 p A ˆ ˆ S ˆ = − d xdy −g∂ˆ ψ∂Aψ. (18) ψ ε h gαβ γδ γδ 2i − ∂4g ∂4g + (g ∂4g ) ; (26) 2φ2 4 γδ γδ Because of the S1 topology of the fifth dimension, β β 1 βγ β γ ˆ ˆ ˆ ∇β Pα = 0 ,Pα ≡ √ g ∂4gγα − δα g ∂4gγ ; (27) ψ(x, y + 2πr) = ψ(x, y), from which we can expand ψ: 2 gˆ44 αβ αβ 2 αβ n=+∞ ∂4g ∂4g g ∂ g ∂4φ g ∂4g X εφ2φ = − αβ − 4 αβ + αβ . (28) ψˆ(x, y) = ψˆ(n)einy/r. (19) 4 2 2φ n=−∞ The first equations allow us to interpret 4D matter Putting this expansion into the action 18, one finds described by Tαβ as a manifestation of pure geometry in the 5D world, which has been termed the ”induced- Z 2 ˆ(n) 2 # X 4 √ h α (n) (n) n ψ S = − dyd x −g D ψˆ D ψˆ − , (20) matter interpretation”of Kaluza-Klein theory. ψˆ α φ r2 n The interpretation of eqs. 27-28 is not so straightfor- α inκAα ward. We could conjecture that these equations describe where Dα ≡ ∂ + r is an EM covariant derivative. Comparing this with the usual minimal coupling of the microscopic properties of matter[1]. In particular, if gauge field Aα, we find that the nth Fourier mode of the Pαβ = k(mivαvβ + mggαβ) , (29) scalar field ψˆ carries a quantized charge √ where k is a constant, mi and mg are definitions for −1/2 p qn = nκφ /r = n 16πG/(r φ) , (21) the inertial and gravitational mass of a particle in the √ induced-matter fluid, and vα ≡ dxα/ds is its 4-velocity, where we have divided Aα by φ to get the correct EM contribution in 13. If we identify e = q and assume that then eq. 27 describes the 4D geodesic equation. √ √ 1 Similarly, we can give eq. 28 a deep meaning if we r φ ∼ lpl = G, we get as a corolary the correct order of magnitude for the fine structure constant, identify it with the Klein-Gordon equation: √ √ 2 2 2 2φ = m φ . (30) α ≡ q1/4π ∼ ( 16πG/ G) /4π = 4, (22) which would have made compactified 5D Kaluza-Klein The particle mass is then dependent explicitly on the theoy very attractive. However, the masses of the scalar metric, which allows us to interpret this noncompactified modes are not compatible with this. theory as a realization of Mach’s principle. 2 V. COSMOLOGY As a final example, consider now a curved (k 6= 0) version of the line element in eq. 33: Compactified Kaluza-Klein cosmology requires the introduction of 5D matter in order to describe non-EM dr2 dsˆ2 = eν dt2 − eω + r2dΩ2 − eµ dψ2 . (36) matter in 4D, being thus characterized by competing ex- 1 − kr2 ˆ pressions for the energy-momentum tensor TAB in 5D. ˆ By contrast, in noncompactified cosmology one has only An interesting solution[13] to RAB = 0 is obtained if the economical assumption that the universe in higher- we assume the metric coefficients have the form ˆ dimensions is empty, i.e., TAB = 0. We will thus only eν ≡ L2(t − λψ), eω ≡ M 2(t − λψ), eµ ≡ N 2(t − λψ), (37) review noncompactified cosmology here, in particular the homogeneous and isotropic nonstatic case. where L, M ans N are wavelike functions of the argument We begin with the general spatially isotropic and ho- (t − λψ), with λ acting as a wave number. Aassuming mogeneous 5D line element M = L(3γ+1), with γ a new constant, then one obtains dsˆ2 = eν dt2 − eω dr2 + r2dΩ2 + εeµdψ2 , (31) for the perfect fluid: 3ζ2λ2 ρ = , p = γρ . (38) where dΩ2 ≡ dθ2 + sin2 θ dφ2, ε has the same function 8πL3+3γ as in eq. 24, t, r, θ and φ have their usual meanings and ψ is the fifth coordinate. If we consider the case With this solution, we can describe either the matter- of dependence on t only, this metric is just a general- dominated era (γ = 0) or the radiation-dominated era ization of a flat homogeneous and isotropic Friedmann- (γ = 1/3). In particular, since the scale factor dependes Lemaˆıtre-Robertson-Walker (FLRW) cosmology. If we on ψ as well as t, observers in hypersurfaces with different consider also dependence of ψ and assume separable so- values of ψ would disagree on the time elapsed since the lutions, then we are able to find solutions of the Einstein big bang, i.e., on the age of the universe. equations RÂB = 0. One of particular interest[12] is a solution with ε = −1 that can be written in the form VI. CONCLUSIONS 2 2 2 2 2 2 2 2 2 2 α t 2 dsˆ = ψ dt − t α ψ (1−α) dr + r dΩ − dψ , (32) (1 − α)2 Kaluza’s mechanism unifies electromagnetism and gravity simply by letting the indices run over five val- and reduces on spacetime sections (dψ = 0) to the usual ues. Other interactions can be included by including new k = 0 (FLRW) metric: extra dimensions. We have seen that there are three versions of Kaluza- 2 2 2 2 2 2 ds = dt − R (t) dr + r dΩ . (33) Klein theory, which give different interpretations to the fifth dimension but have similar observational conse- Assuming this induced matter is a perfect fluid quences: compactification, projective geometry and non- α α α compactification. In particular, we have seen that the Tβ = (ρ + p) u uβ − p δβ , (34) noncompactified approach allows us to describe a wide variety of equations of state. where uα is the 4-velocity of the fluid elements, we obtain Some challenges of the theory are: (1) search for new from eq. 26, ρ = T 0 +T 1 −T 2 and p = −T 2, the following 0 1 2 2 exact solutions; (2) more work on quantization; (3) in- expressions for density and pressure: vestigate the physical meaning of the fifth dimension. 3 2α By now, no consistent theory has been found, but more ρ = , p = − 1 ρ . (35) and more physicists believe that postulating extra space 8πGα2ψ2 t2 3 dimensions is the right way to obtain the unification. These allow us to describe a wide variety of equations of state: for α = 2, a radiation-dominated universe; a dust-filled one if α = 3/2; one inflationary if 0 < α < 1.

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