Understanding the Plane Symmetric Metric I don't understand as to what is the point of having a plane symmetric universe / metric at all? I mean shouldn't any physically sensical cosmological model (e.g. FLRW Model) entail a spherically symmetric metric?
What scenarios are modelled by planar symmetry? I am relatively new to GR and cosmology, and would be very glad if some one can explain.
 A: *

*Einstein's equations are, in general, extremely difficult to solve. As such, exact solutions are valuable for a number of reasons - they are pedagogically useful, they provide insight about symmetries and their consequences, they may serve as limiting cases or rough approximations to more realistic models, etc.

*Spherical symmetry is by no means a general feature.  In fact, it's rather hard to come up with a realistic physical system which does have spherical symmetry.  Stars, planets, and (presumably) most black holes possess non-zero angular momentum, which means that at most they are cylindrically symmetric (though if the angular momentum is small then a spherically symmetric model may suffice).

*Planar symmetry is used e.g. to model the presence of domain walls, or to embed (in the non-technical sense) two-dimensional toy models in 4D spacetime.


I don't understand the point of solving the field equations in cylindrically symmetric scenarios. In lieu of homogeneous isotropic universe, why deviate from spherical symmetry at all?



*As stated above, anything with non-zero angular momentum cannot be spherically symmetric. Large structures which do have angular momentum - spiral galaxies, accretion disks, etc - tend to be very obviously non-spherical, but may have approximate cylindrical symmetry.

