Killing tensor and Conserved quantities

Question

The definition of the killing tensor is written above, as taken from Wikipedia. My question here is two-fold:

1. Can all Killing tensors be build from the Killing vectors of that spacetime?

2. Do Killing tensors also correspond to conserved quantities? If yes, are they independent of whatever are the conserved quantities found from Killing vectors of that spacetime?

Answer 1

1. No; the Killing tensors constructed as tensor products of Killing vectors, such as $\boldsymbol\xi\otimes\boldsymbol\chi$, are considered "trivial". The Killing tensors of physical interest are not formed this way.

2. Absolutely; Killing tensors are also associated with conserved quantities.

For example in Kerr's rotating black hole spacetime, Carter's constant (Carter 1968) is conserved along geodesics, and represents something like a total angular momentum (the interpretation is still being discussed, to my knowledge). Walker & Penrose (1970) showed it originated from a rank 2 Killing tensor, and they comment it is independent of the Killing vectors.

In Friedmann-Lemaitre-Robertson-Walker spacetime, $a^2(g_{\mu\nu}+u_\mu u_\nu)$ is a rank 2 Killing tensor. Here $a$ is the dimensionless scale factor, $\mathbf g$ is the metric, and $\mathbf u$ is the 4-velocity of observers comoving with the Hubble flow. Note the term in parentheses is the spatial metric for these observers. Now this Killing tensor lets you easily derive redshift for geodesic photons, see my description on another site. Carroll (2004, $\S8.5$) is the only GR textbook I know of which describes it, but there are earlier sources even if they don't use the term "Killing tensor". At a guess, Robertson could have done early work on this, see Robertson & Noonan (1968, Appendix B) for a summary of his papers.