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

 

[Entropia]

This is a very interesting question and the metric presented certainly raises some intriguing possibilities. To start, let's define what we mean by a "spherically symmetric" metric. In general relativity, a spherically symmetric metric is one that is invariant under rotations around a fixed point. This means that the metric is the same at every point on a sphere centered at the fixed point. In the case of the Minkowski metric in spherical polar coordinates, the coefficient of the angular part, r^2, ensures that the metric is indeed spherically symmetric.

Now, let's consider the modified metric where the coefficient of the angular part is set to a constant, 1. This means that the metric is no longer dependent on the radius, r. This raises an interesting question - can this metric still be considered spherically symmetric? In my opinion, it depends on how we define "spherically symmetric." If we strictly define it as the metric being invariant under rotations around a fixed point, then this modified metric may not pass as spherically symmetric. However, if we broaden our definition to include metrics that exhibit some form of symmetry around a fixed point, then this modified metric may still fit the criteria.

As for what kind of spacetime this modified metric describes, it is difficult to say without further context. It is possible that this could describe a spacetime with some form of anisotropy, where the angles are not influenced by the distance from the center. It could also potentially describe a spacetime with a non-trivial topology, such as a toroidal or cylindrical spacetime. Without more information, it is impossible to say for sure.

In terms of references, I have not come across this specific metric being discussed in the literature. However, there are many alternative metrics that have been proposed and studied, some of which may have similar features to this modified metric. I would suggest looking into alternative metrics and their properties to gain a better understanding of this type of spacetime.

Overall, the modified metric presented is certainly intriguing and raises interesting questions about the nature of spherically symmetric metrics and the possible implications of removing the dependence on the radius. I hope this response has provided some insight and direction for further exploration.