I Existence of Killing vectors

1. Dec 15, 2017

Tio Barnabe

By requiring the inner product in two points $x$ and $x'$ having metrics $g$ and $g'$ to be invariant, i.e. $g'(x') = g(x)$, one is lead to the Killing equation. Does general relativity forbiddes spaces where the Killing equation cannot be satisfied?

It seems obvious that we want conserved quantities in our theories. But, is there a way around in which we can consider a space-time having no Killing Vectors at all?

2. Dec 15, 2017

PeroK

A Killing vector represents a symmetry in the metric. If the metric has no symmetries then there are no Killing vectors.

Although the gravity affecting the Earth's solar orbit is nearly symmetric, if you include all the factors, then at a certain level of accuracy it won't be symmetric at all.

3. Dec 15, 2017

Staff: Mentor

How so?

Certainly not.

4. Dec 15, 2017

PAllen

Inner product, so far as I know, is an operation on two vectors producing a scalar. Your usage does not appear to jive with this. Please clarify

5. Dec 15, 2017

Tio Barnabe

I just considered that the two points $x$ and $x'$ were vectors themselves, in which my notation $g(x)$ means the inner product of $x$ with itself (similarly for $g'(x')$). Couldn't I do that?

6. Dec 15, 2017

PAllen

Points in a manifold are not vectors. Vectors live in the tangent space to a manifold at a given point. In flat space, position vectors do happen to form a vector space, but not if there is any curvature. Also, for flat space, you can choose to treat the space as it’s own tangent space, but again you cannot if there is any curvature.

7. Dec 16, 2017

Staff: Mentor

Which, as @PAllen has pointed out, is incorrect. So this entire thread appears to be based on a mistaken premise in your OP.