Galaxy recession and Universe expansion

cianfa72 said:

Yes, the identity element of the vector field vector space is the null vector field ##0##. If we "insert" it into post#68 definition we get $$\nabla_Y0 =0, \nabla_Z0=0,\ \forall Y,Z\in\Gamma(TM)$$ hence $$g(0,Z)=0, g(Y,0), \ \forall Y,Z \in\Gamma(TM)$$

Yes. What is the physical meaning of the null vector field, considered as a Killing vector field?

PeterDonis said:

What is the physical meaning of the null vector field, considered as a Killing vector field?

I don't know exactly Maybe is this: if we do not move in any direction from a point the metric can't change.

cianfa72 said:

if we do not move in any direction from a point the metric can't change.

Yes.

Coming back to post#61 now the situation is: on each 3D spacelike hypersurface maximally symmetric there are 6 linearly independent KVFs and 7 KVFs are always linearly dependent. That means there exists a basis of 6 KVFs.

I think the above result implies that a rotation KVFs about a given point on the spacelike hypersurface can always be written as linear combination of 6 linearly independent KVFs.

cianfa72 said:

on each 3D spacelike hypersurface maximally symmetric there are 6 linearly independent KVFs and 7 KVFs are always linearly dependent.

Yes.

cianfa72 said:

That means there exists a basis of 6 KVFs.

Yes, but you can still split them up further into subspaces. See below.

cianfa72 said:

I think the above result implies that a rotation KVFs about a given point on the spacelike hypersurface can always be written as linear combination of 6 linearly independent KVFs.

In the absence of additional structure, that would be the best you could do, yes. But there is additional structure. The rotations centered on a given point form their own closed subspace. So do the translations. So if we pick three linearly independent rotation KVFs, they form a basis by themselves of the subspace of rotations. Similarly, if we pick three linearly independent translation KVFs, they form a basis by themselves of the subspace of translations. Only KVFs that are neither pure rotations nor pure translations would need the full basis of 6 linearly independent KVFs to express them.

PeterDonis said:

The rotations centered on a given point form their own closed subspace. So do the translations. So if we pick three linearly independent rotation KVFs, they form a basis by themselves of the subspace of rotations. Similarly, if we pick three linearly independent translation KVFs, they form a basis by themselves of the subspace of translations. Only KVFs that are neither pure rotations nor pure translations would need the full basis of 6 linearly independent KVFs to express them.

Ok so, fixed a point on the spacelike manifold, the linear subspace of rotation KVFs around it is spanned by 3 linearly independent rotation KVFs around it.

Furthermore the subspace of rotation KVFs around a point can be always given as linear combination of any 3 linearly independent rotation KVFs (e.g. as the linear combination of 3 linearly independent KVFs around a different point).

cianfa72 said:

fixed a point on the spacelike manifold, the linear subspace of rotation KVFs around it is spanned by 3 linearly independent rotation KVFs around it.

Yes.

cianfa72 said:

the subspace of rotation KVFs around a point can be always given as linear combination of any 3 linearly independent rotation KVFs (e.g. as the linear combination of 3 linearly independent KVFs around a different point).

No.

PeterDonis said:

No.

This is the point unclear to me. In post#75 you said

So if we pick three linearly independent rotation KVFs, they form a basis by themselves of the subspace of rotations

Three linearly independent rotation KVFs around a point are not 3 linearly independent rotation KVFs for the manifold ?

cianfa72 said:

Three linearly independent rotation KVFs around a point are not 3 linearly independent rotation KVFs for the manifold ?

What do you mean by "for the manifold"? The set of rotations about a given point induces a set of isometries on the entire manifold. It's just a different set of isometries from the set induced by the set of rotations about a different point.

PeterDonis said:

The set of rotations about a given point induces a set of isometries on the entire manifold. It's just a different set of isometries from the set induced by the set of rotations about a different point.

Yes, but if the set of KVFs on a maximally symmetric 3D spacelike manifold is a vector space of dimension 6 then the rotation KVFs about a given point must be writable as linear combination of any 6 linear independent KVFs.

Are the rotation KVFs around a point in the same KVFs subspace as the rotation KVFs around a different point ?

cianfa72 said:

if the set of KVFs on a maximally symmetric 3D spacelike manifold is a vector space of dimension 6 then the rotation KVFs about a given point must be writable as linear combination of any 6 linear independent KVFs.

Yes, but not all of them will be rotation KVFs if you pick a basis that includes rotation KVFs about a different point.

PeterDonis said:

Yes, but not all of them will be rotation KVFs if you pick a basis that includes rotation KVFs about a different point.

You mean not all the KVFs in the picked basis (that includes rotation KVFs about a different point) will be rotation KVFs from the point of view of the given point.

cianfa72 said:

You mean not all the KVFs in the picked basis (that includes rotation KVFs about a different point) will be rotation KVFs from the point of view of the given point.

None of the KVFs in the picked basis will be pure rotation KVFs about the given point if the basis is picked using rotation KVFs centered on a different point.

Note that three of the KVFs in any basis will not be pure rotation KVFs; in the simplest case they will be pure translation KVFs.

PeterDonis said:

Note that three of the KVFs in any basis will not be pure rotation KVFs; in the simplest case they will be pure translation KVFs.

Correct me whether I am wrong. A pure rotation KFV about a point evaluated at that point is the null vector (the 3 pure translation KVFs evaluated at that point form a basis for the tangent space there).

Btw what I quoted from you applies to any other basis of KVFs different from the basis in which the 3 KVFs "act" as pure rotation KVFs about the given point.

cianfa72 said:

A pure rotation KFV about a point evaluated at that point is the null vector.

"Centered" on a particular point is not the same thing as evaluated at that point. You need to take a step back and think more carefully.

Take a simpler example: the 2-dimensional Euclidean plane. This manifold has a three-parameter group of KVFs, or at least that's the way it is usually stated. This can be broken up into a two-parameter group of translations and, as it's usually stated, one rotation, corresponding to an SO(2) isometry on the manifold. But let's unpack that further.

Choose standard Cartesian coordinates on the plane, which requires picking one particular point as the origin. That point then becomes the "center" of the SO(2) rotation isometry.

Now write down a basis for the three-parameter group of KVFs consisting of the translations and the rotations centered on the chosen origin. It looks like this:

$$
\partial_x
$$
$$
\partial_y
$$
$$
- y \partial_x + x \partial_y
$$

It should be easy for you to verify that all three of these are KVFs and that they are linearly independent.

Now, looking at the third KVF above, the "rotation" KVF, you can see that it vanishes at the origin, ##(x, y) = (0, 0)##. In other words, if you evaluate it at the center point of rotation, it vanishes. Or, to put it the way you correctly put it in a previous post, it doesn't "go anywhere" at the center point--it maps the center point to itself. But it doesn't vanish at any other point, and it obviously induces an isometry on the entire manifold that does not map every point on the manifold to itself.

Now suppose we decide to look at a rotation about a different center point, say the point ##(x, y) = (1, 1)##. The KVF that corresponds to this rotation, in the coordinates we used above, would be:

$$
- (y - 1) \partial_x + (x - 1) \partial_y
$$

This is, as should be evident, a linear combination of all three of the KVFs given above. In other words, it is not a pure rotation about the center point we chose above, the point ##(x, y) = (0, 0)##. It's a linear combination of a rotation about that point, and two translations (in the ##x## and ##y## directions).

Thanks @PeterDonis it makes sense. From the point of view of isotropy at a point ##p##, as discussed from Sean Carroll in his book - section 8.1, given two vectors ##V## and ##W## in the tangent space at ##p## must exist a one-parameter diffeomorphism ##\phi_t## that is an isometry such that its pushforward ##\phi_*## (i.e. the differential map ##d\phi##) evaluated at ##p## maps ##V## into a vector parallel to ##W##. This one-parameter diffeomorphism (isometry) maps ##p## into itself and defines a rotation KFV about the point ##p##.