The (Newtonian) Shell Theorem is fairly sensitive to spherical symmetry. In General Relativity one can write down a metric wherein inside any boundary surface there is flat spacetime. It's easiest to do this for a spherical boundary, but one can work out a metric which is axisymmetric (e.g. oblate and spinning or prolate and tidally deformed) and probably all sorts of other weird shapes following ideas from Gauss's Law for Gravitation. Writing down a metric for that is hard though -- really hard if the idea is to make it time-independent, and really really hard if the idea is to make it time-dependent but static (as in a complex Gaussian surface doesn't relax into a more spherical shell). For example, bumps raised on each other by binary black holes will vanish after merger (or if they fly away on hyperbolic trajectories, having "grazed" each other), leaving you with a spherical horizon (if nonspinnning) or an oblate one (if spinning).
Essentially to break spherical symmetry (or axisymmetry where there's spin) and keep it broken you have to introduce something like a dark energy. One can do that outside (retaining flat space inside) or inside (leading to the equivalent direction-dependent attraction of outside objects).
Interesting! Now, that sounds like a more unintuitive result to me. Can you give examples of symmetries for which it doesn't work?
The (Newtonian) Shell Theorem is fairly sensitive to spherical symmetry. In General Relativity one can write down a metric wherein inside any boundary surface there is flat spacetime. It's easiest to do this for a spherical boundary, but one can work out a metric which is axisymmetric (e.g. oblate and spinning or prolate and tidally deformed) and probably all sorts of other weird shapes following ideas from Gauss's Law for Gravitation. Writing down a metric for that is hard though -- really hard if the idea is to make it time-independent, and really really hard if the idea is to make it time-dependent but static (as in a complex Gaussian surface doesn't relax into a more spherical shell). For example, bumps raised on each other by binary black holes will vanish after merger (or if they fly away on hyperbolic trajectories, having "grazed" each other), leaving you with a spherical horizon (if nonspinnning) or an oblate one (if spinning).
Essentially to break spherical symmetry (or axisymmetry where there's spin) and keep it broken you have to introduce something like a dark energy. One can do that outside (retaining flat space inside) or inside (leading to the equivalent direction-dependent attraction of outside objects).