Kaluza–klein Theory With Torsion Confined to the Extra Dimension

Abstract

Here we consider a variant of the 5 dimensional Kaluza-Klein theory within the framework of Einstein-Cartan formalism that includes torsion. By imposing a set of constraints on torsion and Ricci rotation coefficients, we show that the torsion components are completely expressed in terms of the metric. Moreover, the Ricci tensor in 5D corresponds exactly to what one would obtain from torsion-free general relativity on a 4D hypersurface. The contributions of the scalar and vector fields of the standard K-K theory to the Ricci tensor and the affine connections are completely nullified by the contributions from torsion. As a consequence, geodesic motions do not distinguish the torsion free 4D space-time from a hypersurface of 5D space-time with torsion satisfying the constraints. Since torsion is not an independent dynamical variable in this formalism, the modified Einstein equations are different from those in the general Einstein-Cartan theory. This leads to important cosmological consequences such as the emergence of cosmic acceleration.In any attempt to link fundamental matter fields with intrinsic spin to gravity, it becomes necessary to extend the Riemannian space-time to include torsion, defined to be the antisymmetric part of affine connection. The resulting theory of gravity, known as the Einstein-Cartan theory[1] treats the metric and torsion as two independent geometrical characteristics of space-time [2].Historically, beginning with the Kaluza-Klein (KK) theory, there has been a great interest in introducing extra dimensions of space-time to unify gravity with elementary particle interactions. In the KK theory, the scalar and vector fields, which are the extra dimensional components of the metric tensor contribute to the affine connection and the Ricci tensor and hence modify their values from the corresponding values in 4D space-time [3]. The contribution of these fields to the Einstein tensor are normally interpreted as gravity induced matter.In this work, we incorporate torsion into 5D KK theory [4]. The inclusion of torsion introduces free parameters in the affine connection and the Ricci tensor in addition to the contributions from extra dimensional metric components that occur in the torsion free KK theory. In this paper we impose a minimal set of conditions so as to restrict torsion to purely extra dimensional components and determine all its components in terms of the metric. Interestingly, the imposed conditions lead to a complete cancellation between the modifications induced by the extra dimensional metric components and the contributions from the torsion. Thus the Ricci tensor in 5D space-time in the resulting formalism is exactly the Ricci tensor in a torsion free 4D space-time. In the second part of the paper, we apply the action principle to derive the equations of motion of this formalism. The modified Einstein equations thus derived are finally applied to Robertson-Walker cosmology that leads to a novel expansion history for the universe (see figure 1).To describe the constraints to be i...

Metric theory of gravity with torsion in an extra dimension

“…Here torsion is not an independent degree of freedom coupled to spin, rather it is determined in terms of metric through a set of physically motivated constraints, which serve (i) to confine torsion to the extra dimension, leaving the 4D space-time torsion free, and (ii) to ensure that geodesic motions in 4D remain totally unaffected by the presence of the extra-dimension. These constraints have previously been imposed in terms of veilbeins [17,18], but here it is realized that they impose essentially the requirement that the fifth dimension is hidden at the level of geodesic motion. It turns out that the non-vanishing torsion components are functions of the 5D metric components with the 4D metric g µν obeying the so called cylindrical condition, namely, it is independent of x 5 .…”

“…An alternative formulation of these constraints in terms of vielbeins is worked out in [17]. These constraints are clearly not tensorial in nature because the fifth dimension is singled out.…”

Eternally oscillating zero energy universe

“…This is achieved by imposing the algebraic constraintsΓ µ · ν5 =Γ µ · 55 = 0 andΓ µ · [αβ] = 0. A vierbein formulation of these constraints is elaborated in [25]. By analyzing the geodesic equations, we can note that any motion along the fifth dimension does not affect the 4D components of the geodesic equations.…”

Horizonless, singularity-free, compact shells satisfying NEC

“…It is not dynamically independent and it does not couple to the spin of matter, rather it is completely determined in terms of the metric to ensure that the extra dimension remains hidden. In this section, we briefly summarize the formulation of this theory, but see [8,20] for details.…”