I Spherically symmetric manifold

cianfa72
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TL;DR Summary
About the properties of Killing Vector Fields for spherically symmetric manifold
A spherically symmetric manifold has, by definition, a set of 3 independent Killing Vector Fields (KVFs) satisfying: $$\begin{align}[R,S] &=T \nonumber \\ [S,T] &=R \nonumber \\ [T,R] &=S \nonumber \end{align}$$
These 3 KVFs define a linear subspace of the (infinite dimensional) vector space over ##\mathbb R## of the vector fields defined on the manifold ##M##. Indeed any linear combination with constant real coefficients of the 3 KVFs is a KVF and, from the properties of the commutator ##[~.,.]## it follows that the span of ##\{ T,R,S \}## closes w.r.t. it as well.

My question: taken 3 independent linear combinations of ##\{ T,R,S \}##, say ##\{X,Y,Z\}##, do they satisfy the three conditions above as well ?
 
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They will not generally satisfy [X,Y] = Z, [Y,Z] = X, [Z,X] = Y. This is trivial to see, just let them be a multiple of 2 times the original set.
 
Orodruin said:
They will not generally satisfy [X,Y] = Z, [Y,Z] = X, [Z,X] = Y. This is trivial to see, just let them be a multiple of 2 times the original set.
Any linear combination of ##\{ T,R,S \}##, say ##X = aT + bR + cS##, is a KVF that is a generator of rotations about a different axis.

span##\{ T,R,S \}## is a sub Lie algebra of dimension 3 of the Lie algebra of the set of all KVFs. So one can pick a different basis ##\{X,Y,Z\}## within the span##\{ T,R,S \}##, however the conditions in post#1 are no longer true for them.

That said, does the definition of spherically symmetric manifold (e.g. spacetime) actually demand for the existence of 3 linear independent KVFs such that the conditions in post#1 hold true ?
 
Yes.

If you had XYZ you could always create RST as the appropriately normalised basis.
 
##\left\{R,S,T\right\}## represent the cross product in ##\mathbb{R}^3.## The corresponding Lie algebra is of type ##A_1,## the Lie algebra ##\mathfrak{sl}(2)## with a standard basis ##\{H,X,Y\}## where
$$
[H,X]=2X\, , \,[H,Y]=-2Y\, , \,[X,Y]=H.
$$
However, here you need a complex transformation to transform ##\{R,S,T\}## into ##\{H,X,Y\}.## You also need a factor ##-2i## if you use Pauli-matrices. Here are some remarks on it:
https://www.physicsforums.com/insights/journey-manifold-su2-part-ii/
 
Orodruin said:
If you had XYZ you could always create RST as the appropriately normalised basis.
As far as I know, a Lie algebra is characterized by the structure constants. Fixed a basis ##\{ X_i \}## in the underlying vector space they are given by $$[X_i,X_j] = C_{ij}^k X_k$$
Do the structure constants ##C_{ij}^k## change when picking a different set of basis elements ?
 
cianfa72 said:
As far as I know, a Lie algebra is characterized by the structure constants. Fixed a basis ##\{ X_i \}## in the underlying vector space they are given by $$[X_i,X_j] = C_{ij}^k X_k$$
Do the structure constants ##C_{ij}^k## change when picking a different set of basis elements ?
He answered that in post #2. Double all basis elements. It gives you a new basis, with different structure constants.
 
cianfa72 said:
As far as I know, a Lie algebra is characterized by the structure constants. Fixed a basis ##\{ X_i \}## in the underlying vector space they are given by $$[X_i,X_j] = C_{ij}^k X_k$$
Do the structure constants ##C_{ij}^k## change when picking a different set of basis elements ?
##X\longmapsto cX## produces a factor ##c^2## on the product side of the equation but you can compensate only one factor ##c## on the other side of the equation. Now choose ##c## as any number ##\neq \pm 1.##

##\mathbb{F}_2## is a strange case when it comes to Lie algebras and is usually ruled out.
 
martinbn said:
He answered that in post #2. Double all basis elements. It gives you a new basis, with different structure constants.
Ok, so which is the fundamental/distinctive property of the sub Lie algebra of the rotational KVFs of a spherically symmetric manifold ?
 
  • #10
cianfa72 said:
Ok, so which is the fundamental/distinctive property of the sub Lie algebra of the rotational KVFs of a spherically symmetric manifold ?
Being able to pick a basis with those commutation relations doesn't mean that any basis has to have them.
 
  • #11
martinbn said:
Being able to pick a basis with those commutation relations doesn't mean that any basis has to have them.
Ah ok, so the fundamental property/feature is that such a sub Lie algebra admits as basis a set of 3 elements (KVFs) with those commutation relations.

In other words: if one gets a set of 3 independent KVFs with given commutation relations, in order to ascertain that their span generates the sub Lie algebra associated to a spherically symmetric manifold, one must check that there exist independent linear combinations of them having the relevant commutation relations as in post#1.
 
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