# Time-dependent compactification to de Sitter space: a no-go theorem

## TL;DR

The paper proves that while the Strong Energy Condition alone does not prevent time-dependent compactifications to de Sitter space, the combined SEC and NEC do, leading to singularities in the higher-dimensional metric.

## Contribution

It demonstrates that the null energy condition, together with the strong energy condition, rules out non-singular time-dependent compactifications to de Sitter space.

## Key findings

- SEC alone does not exclude time-dependent compactifications.
- Combined SEC and NEC exclude such compactifications due to singularities.
- Provides example illustrating the impact of energy conditions on compactification solutions.

## Abstract

It is known that the Einstein gravitational field equations in $D>4$ spacetime dimensions have no time-independent non-singular compactification solutions to de Sitter space if the $D$-dimensional stress tensor satisfies the Strong Energy Condition (SEC). Here we show, by example, that the SEC alone does not exclude time-dependent non-singular compactifications to de Sitter space, in Einstein conformal frame. However, this possibility is excluded by the combined SEC and Null Energy Condition (NEC) because the NEC forces a time-evolution towards a singular $D$-metric.