Exact solutions of embedding the four-dimensional perfect fluid in a five- or higher-dimensional Einstein spacetime and the cosmological interpretations

TL;DR

This paper derives exact solutions for embedding 4D perfect fluids into higher-dimensional Einstein spacetimes, revealing connections among various solutions and implications for cosmology and extra dimension compactification.

Contribution

It provides a general exact solution for embedding 4D perfect fluids in 5D and higher-dimensional Einstein spacetimes, linking different models and exploring extra dimension phenomenology.

Findings

The 4D effective metric matches the Robertson-Walker cosmology.

The solutions connect braneworld and black hole scenarios.

Periodic dependence on extra dimension allows for compactification.

Abstract

We investigate an exact solution that describes the embedding of the four-dimensional (4D) perfect fluid in a five-dimensional (5D) Einstein spacetime. The effective metric of the 4D perfect fluid as a hypersurface with induced matter is equivalent to the Robertson-Walker metric of cosmology. This general solution shows interconnections among many 5D solutions, such as the solution in the braneworld scenario and the topological black hole with cosmological constant. If the 5D cosmological constant is positive, the metric periodically depends on the extra dimension. Thus we can compactify the extra dimension on $S^1$ and study the phenomenological issues. We also generalize the metric ansatz to the higher-dimensional case, in which the 4D part of the Einstein equations can be reduced to a linear equation.