Does anyone know whether this metric has been studied before or if it has a proper name?

$$ds^2 = -dt^2 + e^{2At} dx^2 + e^{2Bt} dy^2 + e^{2Ct} dz^2$$

i.e. a de Sitter metric which has a different expansion rate in different directions. Or even the more general

$$ds^2 = -dt^2 + a^2(t) dx^2 + b^2(t) dy^2 +c^2(t)dz^2$$

I cannot find any papers/textbooks that mention either of these metrics.

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    $\begingroup$ I think they're reasonably well known: these are Bianchi I metrics, unless I'm very wrong. They don't seem to be very popular since one of the basic postulates (at least in cosmology) is that we live in a homogeneous and isotropic universe, but there have been some studies here, and here, for example. $\endgroup$ – Philip Cherian May 26 '17 at 13:40
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    $\begingroup$ @Philip-Cherian ; Sounds like an answer (or the germ of one). Also this page on Bianchi Universes may be relevant. $\endgroup$ – StephenG May 26 '17 at 15:37

Such metrics describe what are known as Bianchi I universes after Luigi Bianchi who classified 3 dimensional Lie Algebras into eleven classes and showed that every such algebra is isomorphic to one and only one Lie Algebra on this list.

The Wikipedia page gives a brief summary of the classification, while - as @StephenG suggested - the Scholarpedia page is a little more thorough.

As can be clearly seen from the metric, Bianchi I spaces are in general anisotropic and thus not very good candidates to describe the universe (at least in cosmology), since the universe has been observed to be isotropic to great precision. Nevertheless, they have been studied - in particular, the Kasner metric, which describes for example anisotropic inflation.

Other mentions of such metrics can be found here and here.


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