Is a static spacetime always spherically symmetric?

Question

I'm a bit confused. In this question it is suggested that a static spacetime can be spherically asymmetric. A static spacetime is one for which the metric doesn't change in time. It's irrotational too. Doesn't that mean that a static spacetime is automatically spherically symmetric?
By irrotational I mean that the spacetime cannot be rotated (well, it can, but you won't see a difference). But I'm not sure if this is right. But what else can be meant? That there is no torsion in spacetime,  or something like that?

Michael Seifert · Accepted Answer

No.  Imagine the spacetime surrounding a teapot floating in deep space.1  The teapot is not spherically symmetric, and so the gravitational field it creates is not spherically symmetric.  But the situation is static, since it does not change with time.
Now, it is true that many astronomical objects cannot hold a rigid shape, and their own gravity will eventually reshape them into spherical shapes (or spheroidal shapes if they're rotating.)  But this fact is really due to the material strengths of large astronomical bodies relative to their own self-gravity;  there is nothing in general relativity that prohibits static objects without spherical symmetry.
Edit:  I see that the confusion stems from what the word "irrotational" in the Wikipedia article on static spacetimes actually means.  It does not mean that you can't rotate the spacetime without changing it.  Instead, it's a mathematical condition that has to do with the fact that the spacetime doesn't change with time, and a particular property of a special spacetime vector field (the so-called timelike Killing vector field) that must exist if there is to be a notion of "not changing in time."  The mathematical condition itself is very loosely analogous to this vector field not having a curl, which is where the word "irrotational" comes from (remember that the "curl" of a vector field is sometimes called its "rotation".)

1 The existence of such a teapot is left as an exercise to the reader.

Anders Sandberg · Answer

You could make a static 4+1D spacetime by taking the standard 3+1D Schwarzschild (spherically symmetric and static) metric and just adding an extra space coordinate $z$: $$ds^2 = - left(1-frac{2M}{r}right)dt^2 + frac{dr^2}{1-frac{2M}{r}} + r^2(dtheta^2 + sin^2(theta) dphi^2) + dz^2$$
This is spherically symmetric along $z=$constant slices, but overall just axisymmetric.
There are others, like the 3+1D Bonnor Beam (cylindric symmetry). Generally people look for symmetric solutions since the difficulty of finding less symmetric solutions go up a lot, but there is nothing preventing static solutions from being asymmetric in space.

Is a static spacetime always spherically symmetric?

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