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I'm having an interpretation problem with two models of magnetic universes in General Relativity.

Firstly, consider the Melvin universe described by the following static metric: \begin{equation}\tag{1} ds^2 = (1 + a \, r^2)^2 (dt^2 - dr^2 - dz^2) - \frac{r^2}{(1 + a \, r^2)^2} \, d\varphi^2, \end{equation} where $a = G B^2/\mu_0$ is a constant. This metric describes a constant magnetic field distributed not exactly uniformly on an infinite plane (covered by the polar coordinates $r$ and $\varphi$), which curves space around it. The outward magnetic pressure in that plane is counter-balanced by its own gravitational field, which may explains the static nature of the metric (1) (however, see my question below).

Now, consider the Bertotti-Robinson metric, which describes another magnetized universe: \begin{equation}\tag{2} ds^2 = dt^2 - R^2 (d\vartheta^2 + \sin^2 {\! \vartheta} \, d\varphi^2) - \sin^2 {\!( t/R)} \: dz^2, \end{equation} where $R = \sqrt{\mu_0 / 4 \pi G B^2}$ is another constant. Metric (2) describes an inflating and collapsing universe (along the $z$ axis) which contains an uniformly distributed magnetic field over a closed compact plane (which has the topology of a 2-sphere of radius $R$).

I interpret the Bertotti-Robinson magnetic field as creating an outward pressure in the compact plane counter-balanced by the inward gravity, which may explain why the spherical "plane" is static (the radius $R$ doesn't depend on time $t$). This physical interpretation is similar to the Melvin magnetic field.

However, magnetic field lines also produce a negative pressure (i.e. a tension) along the lines (i.e. along the $z$ axis), which explains the $\sin^2 {\! (t/R)}$ factor in front of the $dz^2$ part of metric (2). Field lines tension and gravity make that metric evolving in time up to a Big Crunch!

Now, I wonder why the Melvin metric (1) doesn't show a similar contraction/instability feature. Why the negative pressure (i.e. tension) produced by the Melvin magnetic field lines isn't producing any time evolution of metric (1) along its $z$ axis?

Of course, I know that these metrics are exact solutions to the Einstein-Maxwell equations, but I'm looking for a proper physical interpretation/explanation of the Melvin static feature.

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    $\begingroup$ Note, that $t$ – $z$ part of the metric represents an $AdS_2$ space. It is a maximally symmetric 2D Lorentzian manifold and so the time evolution you observe is an artifact of the particular coordinate choice. $t=0$ and $t=\pi R$ are not true but coordinate singularities, and the spacetime around them is the same as anywhere else on the manifold. There are other coordinate choices for $AdS$, for example, one where the metric is static. $\endgroup$ – A.V.S. Aug 12 at 6:51
  • $\begingroup$ @A.V.S., true! I need to find that coordinates transformation to remove the time dependance. In that case, the discrepency in interpretation of the Melvin and BR metrics would be gone. You should write a full answer! $\endgroup$ – Cham Aug 12 at 13:14

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