# 'Easy way' of finding out the Killing vector fields?

Is there a way for calculating the Killing vector fields of a given metric in a quick way?

Sure I can guess looking at the metric at the symmetries, and then guess some of them, but, for instance, in 4D cases calculating the Killing vector fields can be pretty cumbersome, especially if the metric isn't diagonal.

I made a way to find explicitly Killing equations with Mathematica, but any further solving by it is kinda pointless :\

Or do I need to do it by hand? :\

## 1 Answer

Symmetries may not be manifest when a metric is written in a specific set of coordinates, so in general, there's not much to say other than the fact that you can write down the Killing equation, which is a differential equation, and then look for solutions. So "how do I find Killing vectors?" breaks down to "how do I find linearly independent solutions of a system of differential equations?"

For insight, it may be helpful to calculate a curvature scalar. See examples at http://www.lightandmatter.com/html_books/genrel/ch07/ch07.html#Section7.1 .

• Not only curvature scalar, but other curvature invariants (in particular, for Ricci flat cases). The reason: Killing vector is always orthogonal to gradient of scalar invariant. – user23660 Sep 24 '13 at 3:18
• Thanks for the hints about curvature scalar. But, what if the expression for scalar is really complicated? All in all, I need to solve PDE's on my own :\ I thought that there will be some quicker way. I mean I can always take what others found, plug it in the Killing equation and see if the expression is 0, but I kinda feel like I'm cheating that way :D – dingo_d Sep 24 '13 at 6:22
• If it's just that the calculations are complicated, use a computer algebra system. Maxima is free, and its ctensor package is pretty good. – user4552 Sep 24 '13 at 15:26
• I tried with Maple and Mathematica (I have access to them), and even for 2-sphere I get integral equations :\ I'll see about Maxima, thanks :) – dingo_d Sep 24 '13 at 15:38