The discovery [1] [2] that distant supernovae are dimmer than they would be in an Einstein-de Sitter universe which has forced a fresh contemplation of issues almost as old as general relativity itself. If accelerated Hubble expansion driven by a cosmological constant, Λ is the explanation, one is left the difficult task of explaining why Λ is infinitesimally small in natural units of the Planck mass, and just such as to reveal itself in the present epoch [3].

A much discussed alternative is to assume that the cosmological constant vanishes and the acceleration is due to the nonlinearity of the Einstein equations: since the present universe is only homogeneous and isotropic on average, averaging of the Einstein equations will yield the usual FRW model equations plus corrections from “back-reaction” [4]. It is easy to realise this by constructing spherical LTB metrics that can account for the supernova data [5], however, such models assume a co-moving co-ordinate system and rely upon significant shear. Recalling that the inhomogeneity in the solar system is far larger than the cosmological one, yet is readily treated by post-Newtonian approximation, suggests that corrections to the usual linearly perturbed FRW model will be similarly small, and indeed this proves to be the case [6].

Still, the large-scale homogeneity of the universe is much less observationally secure than its isotropy [7], so suggesting the simplest inhomogeneous but isotropic and shear-free generalization of the FRW metric [7]:

d s 2 = N ( t , r ) 2 d t 2 − R ( t , r ) 2 d x → 2 (1)

Assuming a perfect fluid source, T μ ν = ( ρ + p ) u μ u ν − g μ ν p , the resulting Einstein equations¹ were first solved by Wyman [8]; the vanishing of the off-diagonal components of the source in co-moving co-ordinates provides:

R , t N R = R ˙ R = H ( t ) ,   N = R , t H ( t ) R (2)

Herein, e.g. R ˙ = N − 1 R , t indicates the proper time derivative, and H ( t ) is a function of integration. Further, as T i j = − p δ i j implies G r r = G θ θ = G ϕ ϕ , one is led to the “pressure isotropy equation”:

R ″ − 2 R ′ 2 R − R ′ r = 1 2 f ( r ) (3)

The “primes” here denote partial derivatives with respect to r and f ( r ) is another integration function. The remaining Einstein equations can be expressed in the Friedman-like form:

3 H 2 = ρ + 1 R 2 [ 2 R ″ R − ( R ′ R ) 2 + 4 r R ′ R ] ρ ˙ + 3 R ˙ R [ ρ + p ] = 0 (4)

Wyman’s objective was to obtain solutions for a barotropic equation of state p = p ( ρ ) , so that he excluded a solution that would later be rediscovered by Stephani [9]

  f = 0   ,   H = a , t a ,   R = a ( t ) 1 + k a ( t ) r 2 / 4 = a N (5)

The corresponding energy density and pressure follow as:

ρ = 3 [ H 2 + k a 2 ] ,   p = − ρ − 1 3 1 + k a ( t ) r 2 / 4 H ρ , t (6)

That is to say, the energy density is homogeneous while the pressure is inhomogeneous. Thus, while the Stephani model has been considered as an alternative to the ΛCDM model [10], its viability is obscured by the question: what is the nature of the perfect fluid source having these unusual properties?

In this paper we will provide an answer to the aforementioned question: the source is a particular case of “k-essence” [11], having a Lagrangian density that is linear in the “velocity”. Moreover, just as the Stephani metric is exceptional in that it can be embedded into 5-dimensional Minkowski space [9], so too can the k-essence source be lifted to a 5-dimensional nonlinear “electrodynamics”.

The remainder of this paper is organised as follows: in Section 2 we briefly review and reformulate k-essence in a way that makes the choice of Lagrangian density yielding (6) self-evident. Then in Section 3 we show how general k-essence models can be obtained by dimensional reduction from 5-dimensional nonlinear electrodynamics. Finally, our conclusions are presented in Section 4.

K-essence [11] is simply the most general model Lagrangian density for a scalar field φ involving its first covariant derivative φ , μ . We take the derivative to be time-like so

L = L ( φ , Y ≡ g μ ν φ , μ φ , ν ) (7)

The use of the “velocity” Y instead of the usual X = g μ ν φ , μ φ , ν as the kinematic variable considerably simplifies and clarifies the subsequent treatment, e.g. the stress-energy tensor takes the perfect fluid form T μ ν = ( ρ + p ) u μ u ν − g μ ν p with the identifications

u μ = φ , μ / Y ,   p = L   ,   ρ = Y L , Y − L (8)

Indeed, in co-moving coordinates Y = φ ˙ and u μ = δ μ 0 / N , while the energy density is evidently just the Hamiltonian. The φ field equation here reads

( L , Y g μ ν φ , ν / Y ) : μ = L , φ (9)

Imposing the nominal requirements of stability and causality, the adiabatic speed of sound squared is given by²

0 ≤ c s 2 = p , Y ρ , Y = L , Y Y L , Y Y ≤ 1 (10)

That is to say, the Lagrangian density must satisfy the inequalities: L , Y ≥ 0 & L , Y Y ≥ 0 .

Particular classes of k-essence are factorizable models, L ( φ , Y ) = − V ( φ ) F ( Y ) (which includes tachyon models [12] [13] for F ( Y ) = 1 − Y 2 and the Chaplygin gas [14] in the subcase of constant potential), and purely kinetic models (such as Scherrer’s [15] model, L ( Y ) ≃ L ( Y 0 ) + L ″ ( Y 0 ) ( Y − Y 0 ) 2 / 2 ). Herein we are interested in models of the type [16]:

L ( φ , Y ) = F ( Y ) − V ( φ ) = p ⇒ ρ = Y F ′ ( Y ) − F ( Y ) + V ( φ ) (11)

This is because in co-moving co-ordinates φ is a function of x 0 = t only, so that the energy density will be homogeneous if Y F ′ ( Y ) − F ( Y ) = 0 , i.e. for some constant K

L ( φ , Y ) = K Y − V ( φ ) (12)

Note that in this linear velocity model the pressure is nonetheless inhomogeneous via the lapse function Y = φ , t / N ( t , x i ) .

For the model (12)

V = 3 [ H 2 + k a 2 ] (13)

Combining (2), (9) and (12) implies the Hubble expansion is directly related to the potential:

3 K H = − ∂ V / ∂ φ (14)

Taking the partial time derivative of (13), and using (14),

φ , t = 2 K [ k a 2 − H , t ] (15)

Hence, given a ( t ) equations (13) and (15) allow one to reconstruct the potential (at least in parametric form). For the power law expansion a ( t ) = ( t / t 0 ) n

V ( t ) = 3 [ n 2 t 2 + k ( t 0 t ) 2 n ] φ ( t ) = − 2 K [ n t + k t 0 2 n − 1 ( t 0 t ) 2 n − 1 ] (16)

In the de Sitter-like case a ( t ) = e H ( t − t 0 )

V ( t ) = 3 [ H 2 + k e − 2 H ( t − t 0 ) ] φ ( t ) = − k K e − 2 H ( t − t 0 ) ∴ V ( φ ) = 3 ( H 2 − K φ ) (17)

Albeit the model (12) has the requisite properties to serve as the source in the Stephani universe, one seems to have traded one mystery for another: how is one to understand the linear dependence on Y? To answer this we recall that long before Kaluza and Klein, Nordstrom [17] proposed to obtain a scalar gravity theory from 5-dimensional “electrodynamics” by applying a “cylinder condition”. More specifically, let the 5-dimensional co-ordinates be denoted by x M = ( x μ , y ) and for the 5-vector potential A M = ( A μ , φ ) ; assuming A M is independent of the fifth co-ordinate, the 5-dimensional field strength F M N ( 5 ) = A N , M − A M , N decomposes as F μ ν ( 5 ) = F μ ν , F μ 5 ( 5 ) = φ , μ . Then taking the 5-dimensional metric g M N ( 5 ) of the form g μ ν ( 5 ) = g μ ν , g μ 5 ( 5 ) = 0 , g 55 ( 5 ) = − 1 , we have:

− 1 2 F ( 5 ) ⋅ F ( 5 ) ≡ − 1 2 F M N ( 5 ) F ( 5 ) M N = − 1 2 F μ ν F μ ν + g μ ν φ , μ φ , ν = − 1 2 F ⋅ F + Y 2 (18)

As [ A • A ] ( 5 ) = A M A M = A μ A μ − φ 2 = A • A − φ 2 , for compact y it follows that any k-essence model L ( φ , Y ) can be obtained from a 5-dimensional model³ L ( 5 ) ( − [ A • A ] ( 5 ) , − 1 2 F ( 5 ) ⋅ F ( 5 ) ) by dimensional reduction provided we also set A μ = 0 .

Taking the range of the fifth co-ordinate as 0 ≤ y ≤ l 5 , for our model source in the Stephani universe

l 5 L ( 5 ) = K − 1 2 F ( 5 ) ⋅ F ( 5 ) − V ( − [ A • A ] ( 5 ) ) (19)

Similar kinetic terms appear in the context of nonlinear Born-Infeld electrodynamics and D-branes in string/M-theory. Of particular note is that Nielson and Oleson [18] proposed a Lagrangian density of the form − 1 2 F ⋅ F as a field theory for closed dual strings identified as magnetic field lines. In our case F 05 ( 5 ) ≠ 0 is electric but its dual is the Kalb-Ramond field strength H i j k that is purely magnetic [19]. We thus suggest that the kinetic part of (19) be understood as originating in string/M-theory space filling magnetic branes.

In this paper, we have considered the issue of the matter source in the Stephani universe as an inhomogeneous alternative to the FRW model with a cosmological constant. We have shown that a form of k-essence has the requisite properties to be that source, and that this k-essence can be obtained by dimensional reduction of a 5-dimensional model truncation of string/M-theory.

This work was supported by a grant from the National Research Foundation.

The authors declare no conflicts of interest.

Tupper, G.B., Marais, M. and Helayël-Neto, J.A. (2021) The Stephani Universe, K-Essence and Strings in the 5-th Dimension. Open Access Library Journal, 8: e3654. https://doi.org/10.4236/oalib.1103654