Bianchi class B spacetimes with electromagnetic fields

We carry out a thorough analysis on a class of cosmological space-times which admit three spacelike Killing vectors of Bianchi class B and contain electromagnetic fields. Using dynamical system analysis, we show that a family of electro-vacuum plane-wave solutions of the Einstein-Maxwell equations is the stable attractor for expanding universes. Phase dynamics are investigated in detail for particular symmetric models. We integrate the system exactly for some special cases to confirm the qualitative features. Some of the obtained solutions have not been presented previously to the best of our knowledge. Finally, based on those analyses, we discuss the relation between those homogeneous models and perturbations of open Friedmann-Lemaitre-Robertson-Walker universes. We argue that the electro-vacuum plane-wave modes correspond to a certain long-wavelength limit of electromagnetic perturbations.

Figure 1

Phase portrait of type V and ${\mathrm{VII}}_h$ LRS models for dust ($\gamma =1$). The diagonal dotted line represents the projection of equilibrium points $\mathcal{P}{M}_h$ onto the LRS invariant set.

Figure 2

Phase portrait of type V and ${\mathrm{VII}}_h$ LRS models for radiation ($\gamma =\frac{4}{3}$). All the orbits achieve their most isotropic state at late time.

Figure 3

Phase portrait of type V and ${\mathrm{VII}}_h$ LRS models for $\gamma =\frac{2}{3}$. The orbits are attracted towards the line of equilibrium points $\mathcal{F}(\tilde{h})$ indicated by the vertical dotted lines.