A sectorial approach to Kaluza-Klein theory Terence V. Sewards Sandia Research Center 21 Perdiz Canyon Road Placitas, N.M. 87043 (505) 867-4392 [email protected] 1 Abstract The Kaluza and Klein versions of Kaluza-Klein theory are reviewed and compared. The differences in the field equations of the two theories are related to the transformation properties of the metrics employed. Based on this comparison a modified version of Kaluza’s theory is proposed, in which the different sectors of the metric which describe gravitation, electromagnetism and the scalar field are considered to be partially separate entities, with distinct coordinate transformation properties. In addition, the “cylinder condition” is relaxed, so that it only affects the 4-D sector of the metric. This results in a theory in which the gravitational and electromagnetic fields are restricted to a four- dimensional “brane”, and the scalar field is restricted to the 5-D “bulk”. I. INTRODUCTION Kaluza-Klein (KK) theory seeks to unify two of the fundamental forces of nature, gravitation and electromagnetism. In Kaluza's original paper1, the 4-D theory of general relativity (GR) derived originally by Einstein2 is extended to a 5-D spacetime. The resulting equations are then separated into three sets, one of which is equivalent to the Einstein 4-D field equations for gravitation, another to Maxwell's equations for the electromagnetic (EM) field, and a third one which describes a scalar field. Independently, Klein3 elaborated a 5-D extension of GR, but with significant differences. Here these two theories are compared and contrasted, and a variant of Kaluza's original formulation, in which the cylinder condition is partially relaxed, is elaborated. Reviews of the theoretical bases of Kaluza-Klein theory, in which both the Kaluza and Klein formulations are examined, include those of Chyba4, Applequist et al.5, Bailin and Love6, Pasini7, Coquereaux and Esposito-Farèse8, Schwarz and Doughty9, Duff10, Overduin and Wesson11, O'Raifeartaigh and Straumann12, van Dongen13, Goenner14, and Ivanov and Prodanov15. Reviews explicitly devoted to Kaluza's theory include those of Fabbri16, Atondo-Rubio et al.17, and Kumar and Suresh18. Here and below, a "brane/bulk" approach to Kaluza-Klein theory is taken, wherein the 4-D sector of the KK metric is considered to reside on a 4-D brane, while the 5-D sector resides in the bulk. II. KALUZA'S 5-D FORMULATION In Kaluza's theory the five dimensional line element is written as 2 = µ 5 µˆ νˆ dsˆ gˆ µˆνˆ (x , x )dxˆ dxˆ (1) where µˆ,νˆ = 1, 2, 3, 4, 5, and x5 is the additional spatial coordinate. The 5-D metric corresponding to this line element is given by 2 g g = µν µ5 gˆ µˆνˆ (2) g 5ν g55 Here g µν is the ordinary 4-D GR metric tensor, µ =1, 2, 3, 4, and the fourth coordinate is the temporal one. To account for the fact that physical qualities depend on the usual 4-D spacetime manifold, Kaluza required that 5-D metric tensor satisfy the "cylinder condition": = gµˆνˆ,5 0 (3) (i.e., derivatives of all the components of the metric with respect to the fifth dimension vanish). The 5-D Christoffel symbols of the first kind are assumed to have the same form as their 4-D counterparts, and are given by Γ = + − (4) 2 λˆµˆνˆ gλˆνˆ,µˆ gλˆµˆ ,νˆ gµˆνˆ,λˆ Symbols involving the fifth coordinate are calculated employing the cylinder condition, expressed in equation (3): Γ = Γ = − 2 λµ5 2 λ5µ gλ5,µ gµ5,λ (5) Γ = + 2 5µν g5ν ,µ g5µ,ν (6) Γ = 2 55µ g55,µ (7) Γ = 2 5µ5 g55,µ (8) Γ = − 2 µ55 2gµ5,5 g55,µ (9) Γ = 2 555 0 (10) At this point, the components g5µ and g55 are set as = α g5µ 2 Aµ (11) = ψ g55 2 (12) where α is a multiplicative constant, and ψ is a scalar field. In Kaluza's formulation, the terms Aµ are identified as the vector and scalar electromagnetic potentials φ [Ax , Ay , Az , e / c]. Equations (5) , (6) and (7) now become 3 Γ = Γ = α − =α λµ5 λ5µ (Aλ,µ A µ ,λ) Fλµ (13) Γ = α + = αΣ 5µν (Aµ,ν Aν ,µ ) µν (14) Γ = Γ =ψ 55µ 5µ5 ,µ (15) The Fνσ terms are related through the equation + + = Fµν ,λ Fνλ,µ Fλµ,ν 0 (16) It is apparent from the above relationships that the terms Fµν can be identified as the components of the covariant electromagnetic field tensor. The Christoffel symbols of the second kind, which are necessary in order to construct the Riemann-Christoffel and Ricci tensors, cannot be obtained directly from those of the first kind, since by definition α αλ Γ µν = g Γλµν (17) αλ and the elements g can only be determined by inversion of the matrix [gαλ ], which is unknown. Thus, as in 4-D GR theory, a general solution to Kaluza's 5-D formulation cannot be obtained, and one must either employ the linearized (weak-field) approximation, or attempt to find an exact solution by specifying a particular geometry, as is employed in obtaining the Schwarzschild solution in GR. Kaluza chose the former option, whereby the 5-D metric gµˆνˆ is expressed in terms of linear deviations hµˆνˆ from the 5-D Minkowski η metric µˆνˆ as ≈η + << gµˆνˆ µˆνˆ hµˆνˆ with hµˆνˆ 1 (18) The Christoffel symbols of the second kind may now be obtained from those of the first kind using the relations αˆ αˆλˆ Γ µˆνˆ ≈η Γ = −Γ µˆνˆλˆ µˆνˆαˆ (19) The weak-field 5-D Riemann-Christoffel curvature tensor is given by the relations µˆ µˆ µˆ R νˆλˆσˆ = Γ νˆσˆ ,λˆ − Γ νˆλˆ,σˆ (20) in which terms involving products of Christoffel symbols have been neglected. The corresponding 5-D Ricci array is αˆ βˆ = Γ − Γ ˆ Rµˆνˆ µˆνˆ,αˆ µˆβ ,νˆ (21) 4 The 4-D (gravitational) Ricci tensor is then α β Rµν = Γ µν ,α − Γ µβ ,ν (22) The field equations that describe Kaluza's 5-D unified gravity and electromagnetism are then: 1 Gravity: Rµν = −κ (Tµν − ηµν T) (23) 2 = −κ EM: R5µ T5µ (24) = − ψ = −κ − 1η Scalar field: R55 T55 55T (25) 2 2 Here κ = 8π G / c is the gravitational constant, and Tµν is the 4-D energy-momentum array for the mass source. The R5µ (electromagnetic) components can be expressed in terms of the electromagnetic potentials or the covariant electromagnetic field tensor: = −α − = −α R5ν (Aσ ,ν Aν ,σ ),σ Fσν ,σ (26) and the gravitational equation (23) may be written in terms of the Einstein tensor 1 Gµν = Rµν − ηµν R = −κ Tµν (27) 2 The components of the 4-D Ricci tensor are now expressed in terms of the metric deviations in the form 1 β β β Rµν = ()hµ ,νβ + hν ,µβ − hµν β − h µν (28) 2 , , 19 with h being the trace of the metric deviation tensor hµν . Defining the "trace-reversed" metric deviation tensor 1 hµν = hµν − ηµν h (29) 2 the Einstein tensor becomes 1 ()β βγ β β Gµν = − hµν β +ηµν hβγ − hµβ ν − hνβ µ (30) 2 , , , , 5 19 Both hµν and hµν transform as components of a tensor in 4-D flat spacetime . After µν = imposing the harmonic gauge h,ν 0 one obtains the 4-D field equations of linearized gravitation in terms of the trace-reversed deviations: (4) hµν = −2Gµν = 2κ Tµν (31) in which (4) is the 4-D d'Alembertian operator, given by ∂2 1 2 (4) = − ∇ (32) c2 ∂t 2 Similarly, the electromagnetic terms can be expressed in terms of the deviations (field potentials) hµ5 : 1 Gµ = Rµ = − (4) hµ = −κ Tµ (33) 5 5 2 5 5 Employing equations (11), (18) and (29) one obtains = α hµ5 2 Aµ (34) and = ψ h55 2 (35) The potentials for all three fields can now be written in 5-D matrix form as α h11 h12 h13 h14 2 Ax α h12 h22 h23 h24 2 Ay = α hµˆνˆ h13 h23 h33 h34 2 Az (36) αφ h14 h24 h34 h44 2 e / c α α α α φ ψ 2 Ax 2 Ay 2 Az 2 e / c 2 The 4-D gravitational field components are contained in the first four rows and columns of the array (36), the electromagnetic potentials in the fifth row and column, and the scalar field occupies the (5,5) position in the array. This array is not a tensor, and Kaluza did not discuss its transformation properties in his 5-D formulation. Einstein and Bergmann20 examined these properties in some depth, but obtained a formulation that is closer to Klein’s than Kaluza’s. At this point a general expression for all three field potential equations has been obtained in terms of the trace-reversed deviations: 6 = κ (4) hµˆνˆ 2 Tµˆνˆ (37) The EM and scalar field compartments are thus = κ (4) hµ5 2 Tµ5 (38) = κ (4) h55 2 T55 (39) If the gravitational source consists of a distribution of non-interacting moving matter ("dust"), the components of the 4-D source (energy-momentum) tensor are given by = ρ = ρ Tµν muµuν . Kaluza extends this to 5-D through the specification Tµˆνˆ muµˆ uνˆ . The source array then has the form ρ 2 ρ u u ρ u u ρ ρ mux m x y m x z mcux m c5ux 2 ρ ρ u ρ u u ρ cu ρ c u mu yux m y m y z m y m 5 y = ρ ρ ρ 2 ρ ρ Tµˆνˆ muzux muzu y muz mcuz m c5uz (40) 2 ρ cu ρ cu ρ cu ρ c ρ cc m x m y m z m m 5 ρ ρ c u ρ ρ ρ 2 m c5 ux m 5 y m c5 uz m cc5 m c5 As in the case of the field array, this object is not a tensor.