I Pseudo-Riemaniann isometries

[cianfa72]

TL;DR Summary

About the isometries of pseudo-riemannian manifolds and the definition of manifold isotropic around a point

I'd ask for clarification about the symmetries of (pseudo) Riemannian manifold $M$ of dimension $n$.

The set of smooth vector fields $\Gamma(TM)$ forms a vector space over $\mathbb R$; the commutator defined as $$[X,Y](f):=X(Y(f)) - Y(X(f))$$ turns it into a (infinite dimensional) Lie algebra.

Consider the set of continuous isometries, i.e. the flows of Killing Vector Fields w.r.t. the tensor metric $g$. KVFs form a linear subspace of $\Gamma (TM)$ that closes w.r.t. the commutator, i.e. it is a Lie algebra on its own. As far as I can tell, this linear subspace/Lie algebra over $\mathbb R$ has always finite dimension $m \geq n$.

That said, consider for instance the case of spherically symmetric Lorentzian metric. It has a specific Lie algebra of rotational KVFs. The latter should be a linear subspace/Lie algebra of the Lie algebra of all KVFs.

Now the question: does the definition of isotropy around a point apply to the entire spacetime as manifold or to its spacelike submanifolds ?

I took as definition of isotropy around a point the one from J. Lee - Introduction to Riemannian manifolds" chapter 3 - Symmetries of Riemannian Manifolds.

 

[jbergman]

Lee's definition of isotropic at a point is in the context of Riemannian Geometry where there isn't a time dimension typically per se so it applies to the entire tangent space at a point. Maybe other authors have different conventions in the context of pseudo-Riemannian geometry.

 

[jbergman]

https://math.stackexchange.com/questions/4613275/are-3-dimensional-einstein-manifolds-isotropic has a definition of isotropic in the semi-riemannian case. Basically non-null tangent vectors at a point of the same length must have an isometry that maps one to the other.

This has some interesting cases. For instance a vector whose spatial components are greater than their time components would have to be map into a pure spatial vector with no time component. Locally these are basically the symmetries of the Lorentz group.

 

[cianfa72]

Maybe other authors have different conventions in the context of pseudo-Riemannian geometry.

S .Carroll in his book "Spacetime and geometry" section 8.1 defines a manifold isotropic around a point $p$ if for any $V,W \in T_pM$ there exists an isometry $\theta_p$ in the subgroup of isometries that leave fixed $p$ and $k \in \mathbb R$ such that $$\theta_p^*(W)=kV$$ where $\theta_p^*$ is the pushforward associated to $\theta_p$ evaluated at $p$.

I believe in this case the relevant manifold $M$ to which the notion of isotropy applies isn't the spacetime as whole but its spacelike hypersurfaces (which are Riemannian manifolds of their own).

https://math.stackexchange.com/questions/4613275/are-3-dimensional-einstein-manifolds-isotropic has a definition of isotropic in the semi-riemannian case. Basically non-null tangent vectors at a point of the same length must have an isometry that maps one to the other.

In that link I believe the relevant condition isn't on the isometry $\theta_p$ fixing the point $p$ by itself, but on its associated pushforward $\theta_p$ at $p$. It applies to the same type of non-null vectors in tangent space (i.e. either spacelike or timelike).

 

[jbergman]

S .Carroll in his book "Spacetime and geometry" section 8.1 defines a manifold isotropic around a point $p$ if for any $V,W \in T_pM$ there exists an isometry $\theta_p$ in the subgroup of isometries that leave fixed $p$ and $k \in \mathbb R$ such that $$\theta_p^*(W)=kV$$ where $\theta_p^*$ is the pushforward associated to $\theta_p$ evaluated at $p$.

I believe in this case the relevant manifold $M$ to which the notion of isotropy applies isn't the spacetime as whole but its spacelike hypersurfaces (which are Riemannian manifolds of their own).

I glanced at Carroll and it appears that he is talking about a specific foliation of spacetime with the requirement that the submanifold spatial leaves are isotropic. However, he notes that a different non-comoving observer, i.e., a different foliation would not find their leaves also isotropic.