Dirac equation in a de Sitter expansion for massive neutrinos from modern Kaluza-Klein theory

Using the modern Kaluza-Klein theory of gravity (or the Induced Matter theory), we study the Dirac equation for massive neutrinos on a de Sitter background metric from a 5D Riemann-flat (and hence Ricci-flat) extended de Sitter metric, on which is defined the vacuum for test massless 1/2-neutral fields minimally coupled to gravity and free of any other interactions. We obtain that the effective 4D masses of the neutrinos can only take three possible values, which are related to the (static) foliation of the fifth and noncompact extra dimension.


I. INTRODUCTION
In models based on supergravity [1], it has been pointed out the existence of some light particles whose interactions are suppressed at scales close to M = M P /(8π) ≃ 2.4×10 18 GeV.
Such particles have nothing to do with the collider experiments, but may affect the standard scenario of big-bang cosmology [2][3][4]. The gravitino, which is the gauge field associated with local supersymmetry (SUSY), is one of the weakly interacting particles in supergravity models, and we expect the mass of the gravitino to be of the order of the typical SUSYbreaking scale. When the gravitino decays into a neutrino and a sneutrino, the emitted high energy neutrinos scatter off the background neutrinos and produce charged leptons (mainly electrons and positrons), which cause an electro-magnetic cascades and produce many soft photons. Hence, the propagation of neutrinos should be very important during inflation. Inflationary cosmology can be recovered from a 5D vacuum [5][6][7]. The inflationary theory is very consistent with current observations of the temperature anisotropy of the Cosmic Microwave Background (CMB) [8]. The most popular model of supercooled inflation is chaotic inflation [9]. In this model the expansion of the universe is driven by a single scalar field called inflaton. At some initial epoch, presumably the Planck scale, the scalar field is roughly homogeneous and dominates the energy density, which remains almost constant during all the inflationary epoch.
On the other hand inflation can be recovered from the Campbell-Magaard theorem [10][11][12][13][14], which serves as a ladder to go between manifolds whose dimensionality differs by one.
This theorem, which is valid in any number of dimensions, implies that every solution of the 4D Einstein equations with arbitrary energy momentum tensor can be embedded, at least locally, in a solution of the 5D Einstein field equations in vacuum. Because of this, the stress-energy may be a 4D manifestation of the embedding geometry. Physically, the background metric there employed describes a 5D extension of an usual de Sitter spacetime, which is the 4D spacetime that describes an inflationary expansion.
In this Letter we study the Dirac equation for 4D massive neutrinos in a de Sitter expansion using Modern Kaluza-Klein theory of gravity (or Induced Matter theory) [15,16].
In this theory the 5D massless 1/2-spin test fields are considered free from interactions and minimally coupled to gravity on a 5D Ricci-flat metric in which we define the physical vacuum. Our approach is something different (but complementary) than the studied by Wesson in [17], because we make a detailed study of the geometrical spinor properties of 5D vector fields that, once we make a static foliation on the fifth coordinate, can be considered as 4D massive neutrinos.

II. THE 5D CLIFFORD ALGEBRA AND SPINORS IN 5D
We consider the Ponce de León metric [18] The resulting 4D hypersurface after making ψ = ψ 0 describes a de Sitter spacetime. From the relativistic point of view an observer moving with the penta-velocity U ψ = 0, will be moving on a spacetime that describes a de Sitter expansion which has a scalar curvature (4) R = 12/ψ 2 0 = 12 H 2 0 , such that the Hubble parameter is defined by the foliation H 0 = ψ −1 0 . If we foliate ψ = ψ 0 , we get the effective 4D metric which describes a 3D spatially flat, isotropic and homogeneous de Sitter expanding universe with a constant Hubble parameter H 0 .
To define a 5D vacuum we shall consider a Lagrangian for a massless 5D spinor field minimally coupled to gravity (we shall consider = c = 1) whereK = 8πG and γ A are the Dirac matrices which satisfy such that the covariant derivative of the spinor Ψ on (1) is defined in the following form: and the spin connection is given by being ∇ A [e cB ] = ∂ A e cB − Γ D AB e cD the covariant derivative of the five-bein e c B (the symbol ∂ A denotes the partial derivative with respect to x A and η ab = g AB e A a e B b denotes the 5D Minkowski spacetime in Cartesian coordinates), which relates the extended 5D de Sitter metric (1) with the 5D Minkowski spacetime written in Cartesian coordinates: The Dirac matrices γ a are represented in an Euclidean space instead of a Lorentzian space, and are described by the algebra [19,20]: such that γ 4 = γ 0 γ 1 γ 2 γ 3 , and the σ i are the Pauli matrices.

A. Variables separation for the Dirac equation in 5D
Finally, using the fact that γ A = e A a γ a , we obtain the Dirac equation on the metric (1) where we shall use (5) and (6) and Cartesian coordinates to describe the 3D Euclidean hypersurface. The Dirac equation (13) can be written as The relevant second kind Christoffel symbols are Hence, the Dirac equation on the 5D Riemann-flat metric (1) results to be In order to make separation of variables we shall use the method introduced in [21]. After some algebraic manipulation, Eq. (13) can be rewritten as whereK 04 Φ = kΦ = −K 123 Φ and Φ = γ 0 γ 4 Ψ [By making det |kΦ +K 123 Φ| = 0, we can evaluate the variable separation constant: k = | k|]. The operators of separationK 04 and K 123 are given byK where the condition K 04 ,K 123 = 0 must be fulfilled. Since the metric (1) is 3D spatially isotropic, one obtains that the 3D spatial solutions can be expanded in term of harmonic functions, so that one can write such that k is the wavenumber of propagation on the 3D isotropic and homogeneous Euclidean space. Furthermore, one obtainŝ After some algebra, we obtain that such thatK where we have used that Υ(t, ψ) = (γ 0 ) −1 Φ 0 (t, ψ), and that comply with K 0 ,K 4 = 0. Using the first equation in (20), with the variables separa- we obtain the differential equation for Λ(ψ) which is the same differential equation obtained in [22], but in a different framework. The solution for this equation is where Λ 0 is a constant of integration and M = M 0 /ψ 0 is a separation constant. For M 0 < 2 the function Λ(ψ) tends to 0 for ψ → ±∞, but is divergent for ψ → 0. In order for the function Λ(ψ) to be real, we must ask M 0 to take integer values and M 0 ≤ 2 : M 0 = 2, 1, 0, −1, −2, −3, −4, .... An interesting property is that for even |M 0 | values the function Λ(ψ) is even but for odd |M 0 | values the function is also odd.

SITTER SPACETIME
We assume a static foliation of the 5D spacetime on the 4D hypersurface Σ 0 , on which the 4D energy momentum tensor is described by a perfect fluidT αβ = e A α e B β T AB ψ 0 = (ρ+P )u α u β −P h αβ , where ρ(t, r, ψ 0 ) and P (t, r, ψ 0 ) are the energy density and pressure of the induced matter, respectively. The 4-velocities u α are related to the 5-velocities U A by u a = e A α U A , and h αβ = e A α e B β g AB are the components of the metric tensor in (2). The Campbell-Magaard theorem, which is valid in any number of dimensions, implies that every solution of the 4D Einstein equations with arbitrary energy momentum tensor can be embedded, at least locally, in a solution of the 5D Einstein field equations in vacuum. Because of this, the tensorT µν is induced as a 4D manifestation of the embedding geometry.

IV. FINAL REMARKS
We shows how the mass of the neutrinos on a 4D de Sitter spacetime can be induced from a free massless 5D test spinors on a extended Riemann-flat (and hence Ricci-flat) metric which has non-zero connections Γ a bc . But more strong is the result that the masses of cosmological neutrinos should be dependent on the energy scale of the universe because they are dependent on the Hubble parameter. However, these results must be taken carefully because we have neglected any kind of interactions of the spinor fields with other fields. (These results are only valid for free 4D neutrinos that propagate freely in a 4D de Sitter background.) A more complete treatment will be studied in a future work.