Properties of the magnetic universe with positive cosmological constant

The properties of the Melvin-type spacetime with a positive cosmological constant $\mathrm{\Lambda }$ in $d$-dimensional Einstein–Maxwell gravity is studied. The solution is parametrised in terms of the `de Sitter radius’ $\mathrm{\ell }\propto {\mathrm{\Lambda }}^{-1\mathrm{/}2}$ and the magnetic field parameter $\beta$, and they are warped products of the form ${\mathbb{R}}^{1,d-3}\times S^2$, where ${\mathbb{R}}^{1,d-3}$ is the $(d-2)$-dimensional Minkowski spacetime and $S^2$ is topologically a two-sphere which contains a conical singularity, whose nature depends on the product $\beta \mathrm{\ell }$. In the limit $\mathrm{\ell }\to \mathrm{\infty }$, the $S^2$ decompactifies and the $d$-dimensional Melvin universe is recovered. The Freund–Rubin-type flux compactification model is shown to be another particular limit of this solution. We also calculate the flux and geodesics in this spacetime.