Homogeneous spacetime - Lie groups

[befj0001]

All Bianchi type spacetimes have metrics that admits a 3-dimensional killing algebra. They are in general not isotropic. Bianchi type IX have a killing algebra that is isomorphic to SO(3), i.e. the rotation group. But what does it mean? If the fourdimensional spacetime is invariant under the rotation group SO(3) doesn't that imply it is isotropic?

My guess is that SO(3) acts as the translation group on our three spatial coordinates, and so the orbits lives in three dimensional space spanned by the spatial coordinates. Therefore, since spacetime includes one extra time coordinate, our three dimensional space is a 3-sphere in four dimensions and SO(3) will be the rotation group on this sphere in four dimensions. But that doesn't seems right, how can SO(3) be the rotation group in both three- and four-dimensional space?

 

[Bill_K]

All Bianchi type spacetimes have metrics that admits a 3-dimensional killing algebra. They are in general not isotropic. Bianchi type IX have a killing algebra that is isomorphic to SO(3), i.e. the rotation group. But what does it mean? If the fourdimensional spacetime is invariant under the rotation group SO(3) doesn't that imply it is isotropic?

The group symmetry operations in Bianchi cosmologies act transitively on the space sections. In Bianchi Type IX, for example, the space sections are 3-spheres and the group operations are the rotations, SO(3).

But Bianchi cosmologies are not generally isotropic, because the expansion may not be isotropic. (I know it is difficult to imagine a 3-sphere expanding nonisotropically, and remaining a sphere, but that's what can happen!)

 

[befj0001]

The group symmetry operations in Bianchi cosmologies act transitively on the space sections. In Bianchi Type IX, for example, the space sections are 3-spheres and the group operations are the rotations, SO(3).

But Bianchi cosmologies are not generally isotropic, because the expansion may not be isotropic. (I know it is difficult to imagine a 3-sphere expanding nonisotropically, and remaining a sphere, but that's what can happen!) Also they may be rotating.

But how can SO(3) describe rotations on a 3-sphere? I've learned that SO(3) describes rotations on a 2-sphere! Does it have to do with the number of generators of the group?

 

[Bill_K]

But how can SO(3) describe rotations on a 3-sphere? I've learned that SO(3) describes rotations on a 2-sphere! Does it have to do with the number of generators of the group?

SO(3) is the rotation group on a 2-sphere, but it also can act on a 3-sphere.

Its action on a 2-sphere is the usual rotation group. It's multiply transitive, that is, there are many different elements of the group that take one point P on the sphere into another point Q. Also, for each group element there are points that remain invariant, namely the poles of the rotation.

By contrast, the action of SO(3) on the 3-sphere is singly transitive, and there are no fixed points. Each group operation moves every point of the sphere. It's more like a group of translations.

The easiest way to understand the action is to consider the group space itself. Each element of SO(3) can be labeled by three parameters, for example the three Euler angles, φ, θ, χ, and the entire group forms a 3-space. And the group multiplication of SO(3) acts on this 3-space in just the way described.

 

[befj0001]

SO(3) is the rotation group on a 2-sphere, but it also can act on a 3-sphere.

Its action on a 2-sphere is the usual rotation group. It's multiply transitive, that is, there are many different elements of the group that take one point P on the sphere into another point Q. Also, for each group element there are points that remain invariant, namely the poles of the rotation.

By contrast, the action of SO(3) on the 3-sphere is singly transitive, and there are no fixed points. Each group operation moves every point of the sphere. It's more like a group of translations.

The easiest way to understand the action is to consider the group space itself. Each element of SO(3) can be labeled by three parameters, for example the three Euler angles, φ, θ, χ, and the entire group forms a 3-space. And the group multiplication of SO(3) acts on this 3-space in just the way described.

I'm still a bit confused about how SO(3) acts on a 3-sphere. When SO(3) acts on a 2-sphere we have a reference point: the origin. When SO(3) acts on a vector in R^3 it just rotates that vector within the 2-sphere defined by the length of that vector. But I don't get how SO(3) can act as the translation group in R^3? When relating SO(3) to the Lie algebra, how can one see that this corresponds to translations?

 

[Bill_K]

When SO(3) acts on a 2-sphere we have a reference point: the origin. When SO(3) acts on a vector in R^3 it just rotates that vector within the 2-sphere defined by the length of that vector.

But note that this action has fixed points, namely the points along the rotation axis, so it's not the same thing.

It may be easier to visualize the group action as primarily a translation in R³, with a bit of a rotation superimposed. That is, the group not only carries point P into point Q, it also brings with it an orientation. Put a triad of basis vectors at P. Then when the group carries P to Q it also carries the basis vectors along with it. Thus every point of the 3-space gets a unique set of basis vectors, and the collection of basis vectors taken together is invariant under the group action.

The metric can then be written in terms of these basis vectors, and that's a convenient way to build the Bianchi symmetry into the solution. For example, this paper uses this approach in the first few equations.

 

[befj0001]

thanks for the answer.

 

[befj0001]

Is there somewhere where I can read about this? Mathematically, how can SO(3) that describes rotations on a 2-sphere in 3-dimensions, describe translations on the spatial part in 4-dimensions?

Another question: What is the difference between SO(3) and SO(2,1) in general relativity?

It's just this sentence:

By contrast, the action of SO(3) on the 3-sphere is singly transitive, and there are no fixed points. Each group operation moves every point of the sphere. It's more like a group of translations.

that I do not fully understand.