# Is the de Sitter Universe static?

In the book Tolman R.C. Relativity, thermodynamics, and cosmology (3pr., Oxford, 1949) I read that the de Sitter Universe is static as well as the Einstein Universe.

But the de Sitter Universe is an expandable Universe.

I thought "static" meant "not expandable", but perhaps, it is wrong.

What does static universe mean?

## 2 Answers

A static spacetime normally means there is an irrotational global timelike Killing vector, and this isn't the case for the de Sitter geometry so the de Sitter geometry wouldn't normally be described as static.

However the de Sitter metric can be written using static coordinates:

$$ds^2 = -\left(1 - \frac{\Lambda}{3}r^2\right)dt^2 + \frac{dr^2}{\left(1 - \frac{\Lambda}{3}r^2\right)} + r^2d\Omega^2$$

I don't have Tolman's book, but I wonder if that's what he means.

The static chart written in John Rennie's answer covers only part of de Sitter spacetime (and this fact is related to the presence of a cosmological horizon where $g_{00}$ vanishes giving rise to an apparent singularity). There is no static global chart, or equivalently, there is no global timelike Killing vector with an orthogonal spacelike 3-surface. Therefore de Sitter spacetime is only locally static. This fact makes expansion compatible with staticity... Instead, Einstein universe is globally static and no expansion is permitted there since the metric is globally constant with respect to a global notion of time (the parameter of a global irrotational timelike Killing vector).