A weird spherically symmetric metric

[smallphi]

a weird "spherically symmetric" metric

Minkowski metric in spherical polar coordinates [t, r, theta, phi] is

$$ds^2 = - dt^2 + dr^2 + r^2\,(d\theta^2 + sin^2(\theta)\, d\phi^2)$$.

The question is what happens when the coefficient of the angular part is set to constant, say 1, instead of r^2:

$$ds^2 = - dt^2 + dr^2 + (d\theta^2 + sin^2(\theta)\, d\phi^2)$$.

Supposedly this is still 'spherically symmetric' spacetime since it contains the spherically symmetric angular part but strangely the angular part doesn't depend on the radius anymore - the 'angles' are not influenced by how far from the 'center' you are anymore.

Can someone explain what kind of spacetime, the above metric describes and can it still pass for 'spherically symmetric'? A reference would be nice, since I've never seen that discussed anywhere.

 

[Mentz114]

Rewriting your metric as -

$$ds^2 = - dt^2 + dr^2 + R^2(d\theta^2 + sin^2(\theta)\, d\phi^2)$$

where R is a constant, the curvature scalar is R^-2 and the non-zero components of the Einstein tensor are G₀₀= R^-2 and G₁₁= -R^-2.

One should work out the Killing vectors, but that's beyond me right now.

So, it's not a vacuum solution, and the 'matter' is pretty weird. I doubt if it represents anything that can exist.

M