A unified approach to the Klein-Gordon equation on Bianchi backgrounds

Abstract

In this paper, we study solutions to the Klein-Gordon equation on Bianchi backgrounds. In particular, we are interested in the asymptotic behaviour of solutions in the direction of silent singularities. The main conclusion is that, for a given solution $u$ to the Klein-Gordon equation, there are smooth functions $u_{i}$, $i=0,1$, on the Lie group under consideration, such that $u_σ(\cdot,σ)-u_{1}$ and $u(\cdot,σ)-u_{1}σ-u_{0}$ asymptotically converge to zero in the direction of the singularity (where $σ$ is a geometrically defined time coordinate such that the singularity corresponds to $σ\rightarrow-\infty$). Here $u_{i}$, $i=0,1$, should be thought of as data on the singularity. Interestingly, it is possible to prove that the asymptotics are of this form for a large class of Bianchi spacetimes. Moreover, the conclusion applies for singularities that are matter dominated; singularities that are vacuum dominated; and even when the asymptotics of the underlying Bianchi spacetime are oscillatory. To summarise, there seems to be a universality as far as the asymptotics in the direction of silent singularities are concerned. In fact, it is tempting to conjecture that as long as the singularity of the underlying Bianchi spacetime is silent, then the asymptotics of solutions are as described above. In order to contrast the above asymptotics with the non-silent setting, we, by appealing to known results, provide a complete asymptotic characterisation of solutions to the Klein-Gordon equation on a flat Kasner background. In that setting, $u_σ$ does, generically, not converge.