I am introducing myself to the topic of killing vectors and therefore, after doing some reading, I try to solve some easy problems. For simplicity, I do my first steps in 2D.

First, I chose the 2D-Minkowski metric, so $ds^2 = -dt^2+dx^2$ holds. The Killing-equation $\nabla_a \xi_b+ \nabla_b \xi_a = 0$ simplifies to $\partial_a\xi_b+\partial_b\xi_a =0$. By differentiating this equation, permuting the indices and adding/subtracting the permuted equations I get a quite easy differential-equation. It's solution reads $\xi_a = A_{ab}X^b+B v_a$ where $A \in \mathbb{R}^{2\times2}$ and $B \in \mathbb{R}^{2\times1}$. Since the Killing-equation has to hold for this solution $C$ has to be anti-symmetric and thus we get the three well known Killing-vector fields $\xi_0 = \partial_t$, $\xi_1 = \partial_x, \xi_2 = -\partial_t+\partial_x$ for the Minkowski-Metric.

Although it wasn't too difficult for me to find the Killing-vectors for Minkowski-metric I struggle when it comes to more sophisticated metrics because I don't see a similar way to find a solution.

Therefore, I chose for my second toy-example the metric associated with $ds^2 = -\frac{1}{x^2}dt^2+\frac{1}{x^2}dx^2$. Since it is time-independent one Killing-vector, $\xi_o = \partial_t$ , can be read out easily. The metric is still diagonal but it depends on $x$ so the Killing-equation doesn't simplify as in the first example. So, things turn out to be much more difficult than in the previous toy-example.

Although I've already spent some time, I haven't figured out a way how to derive some useful equation that helps me finding Killing-vectors.

In the first example (Minkowski-metric) I started from the Killing-equation and by using some information about the given metric I derived all possible Killing-vectors. So the maximum number of linearly independent Killing-vectors was "naturally included" in the solution I derived. In the second problem, I even don't know anything about the number of Killing-vectors that I should expect (I've read in some textbook that $\frac{n(n-1)}{2}$ gives the number of Killing-vectors, but this was stated regarding to a specific class of metrics, so I'm not sure whether this is applicable here).

I tried to find the Killing-vectors as similar as possible to my first approach (maybe that's the problem?). Is there any general way how one can find Killing-vectors? Does anyone have an idea/a hint how I can find the Killing-vectors at least for this specific metric?

Thank's a lot for your help!

PS: My background is solid physics and maths knowledge but only basic knowledge in general-relativity (that's what I want to improve by dealing with this topic).

PPS: I already took a look at this post but since this question is even more general than mine here it didn't help me to proceed.


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