Understanding Killing vectors of FLRW metric

Question

I am trying to understand how to calculate the Killing vectors of FLRW metric \begin{equation} ds^2 = dt^2 - R(t)^2\left( \frac{dr^2}{1 - k r^2} + r^2 d\theta^2 + r^2 \sin\theta d\phi^2\right). \end{equation} I'm following this article, where they explicitly use Killing equations \begin{equation} \xi_{\mu; \nu} + \xi_{\nu; \mu} = 0. \end{equation}

However, when I get lost when they state the results of solving Killing's equation for the $t=constant$ submanifold \begin{aligned} \xi^{t} &=0 \\ \xi^{r} &=\sqrt{1-k r^{2}}\left(\sin \theta\left(\cos \phi \delta a_{x}+\sin \phi \delta a_{y}\right)+\cos \theta \delta a_{z}\right) \\ \xi^{\theta} &=\frac{\sqrt{1-k r^{2}}}{r}\left[\cos \theta\left(\cos \phi \delta a_{x}+\sin \phi \delta a_{y}\right)-\sin \theta \delta a_{z}\right]+\left(\sin \phi \delta b_{x}-\cos \phi \delta b_{y}\right) \\ \xi^{\phi} &=\frac{\sqrt{1-k r^{2}}}{r}\left[\frac{1}{\sin \theta}\left(\cos \phi \delta a_{y}-\sin \phi \delta a_{x}\right)\right]+\cot \theta\left(\cos \phi \delta b_{x}+\sin \phi \delta b_{y}\right)-\delta b_{z} \end{aligned}

Where they state that there are 6 Killing vector fields, as required by the maximal symmetry of the 3-manifold.

Firstly, Killing equation can only be run for indices $\mu = 0,1,2,3$. Therefore, I suppose that in reality, we have something like $\{\xi^{(i)}_\mu\}$, where the index $i$ refers to different the Killing vectors, and, in the previous case, it would run $i= 1,...6$ and $\mu$ is the component of the $i$-th Killing vector.

Secondly, I don't understand the result given in the mentioned article. I suppose $\delta a_x, \delta a_y, ...$ are the Killing vectors because there are 6 of them, but I don't get why they are arranged in this manner, nor what the $\delta$ means.

Therefore I conclude that I don't understand properly Killing vectors and how to calculate them.

Could someone help me see what I am missing, or point me to any book where it is adequately explained? (I have already looked at MTW and Carroll)

Thanks.

Answer 1

I agree that the notation there is not a model of clarity, but I believe what they are doing is parametrizing the six independent Killing vector fields in the one set of equations (5.1). In other words, the components of Killing's equation (4.11–20) determine the components of $\xi^\mu$ up to a set of six arbitrary constants, and the authors have lumped these six constants into two three-component vectors $\delta \vec{a}$ and $\delta \vec{b}$.

Different choices of these constants lead to different Killing vector fields (KVFs); $\delta a_x \neq 0$ (and all others zero) gives you one KVF, $\delta a_y \neq 0$ (and all others zero) gives you another one, and so on. Note that a linear combination of any two KVFs is itself a KVF, so this notation emphasizes that this manifold has a six-dimensional "space" of Killing vector fields.

The distinction between $\delta \vec{a}$ and $\delta \vec{b}$ is that KVFs with $\delta \vec{a} = 0$ leave the constant-$r$ spheres invariant as well. (All of these KVFs leave the constant-$t$ hypersurfaces invariant, by construction.)

I have no idea why they use the $\delta$'s in the notation for these vectors either. It might be explained later in the paper but I did not see it at a cursory glance.