# Kaluza-Klein Cosmology: the bulk metric

## TL;DR

This paper derives a unique five-dimensional vacuum metric under the Cosmological Principle, from which various four-dimensional FRW cosmologies can be projected, including models with signature change and initial singularities.

## Contribution

It provides an explicit, general solution for a 5D vacuum metric that unifies different FRW models and explores signature change and initial singularities within Kaluza-Klein cosmology.

## Key findings

- A unique 5D vacuum metric parametrized by curvature and signature.
- Demonstration of signature change from Euclidean to Lorentzian metrics.
- Identification of a 'Big unfreeze' initial singularity in the model.

## Abstract

The Cosmological Principle is applied to a five-dimensional vacuum manifold. The general (non-trivial) solution is explicitly given. The result is a unique metric, parametrized with the sign of the space curvature ($k=0,\pm 1$) and the signature of the fifth coordinate. Friedmann-Robertson-Walker (FRW) metrics can be obtained from this single 'mother' metric (M-metric), by projecting onto different space-homogeneous four-dimensional hypersurfaces. The expansion factor $R$ is used as time coordinate in order to get full control of the equation of state of the resulting projection. The embedding of a generic (equilibrium) mixture of matter, radiation and cosmological constant is given, modulo a quadrature, although some signature-dependent restrictions must be accounted for. In the 4+1 case, where the extra coordinate is spacelike, the condition ensuring that the projected hypersurface is of Lorentzian type is explicitly given. An example showing a smooth transition from an Euclidian to a Lorentzian 4D metric is provided. This dynamical signature change can be considered a classical counterpart of the Hartle-Hawking 'no-boundary' proposal. The resulting FRW model shows an initial singularity at a finite value of the expansion factor $R$. It can be termed as a 'Big unfreeze', as it is produced just by the beginning of time, without affecting space geometry. The model can be extended in order to fit the present value of the density parameters.