I Killing's vectors

1. Jun 29, 2017

davidge

I have tried to find the three Killing vectors for the metric $$ds^2 = dr^2 + r^2d \theta^2$$ that is, the Euclidean metric of $\mathbb{R}^2$ written in polar coordinates. I found these to be

$$\bigg(\text{first}\bigg) \ \ \xi_r = \text{Cos} \theta \\ \xi_\theta = -\text{rSin} \theta \\ \bigg(\text{second}\bigg) \ \ \xi_r = \text{Sin} \theta \\ {\xi_\theta = \text{rCos} \theta} \\ \bigg(\text{third}\bigg) \ \ \xi_r = 0 \\ \xi_\theta = \text{r²}$$ As I have found solutions only for 3d on web, I would like to know whether these are correct or not.

2. Jun 29, 2017

Orodruin

Staff Emeritus
Why don't you check whether or not they satisfy the Killing equations?

3. Jun 29, 2017

davidge

I did
And they do satisfy the Killing equation.

4. Jun 29, 2017

Orodruin

Staff Emeritus
And thus they are Killing vector fields ...

5. Jun 29, 2017

davidge

6. Jun 29, 2017

davidge

What bothers me is that in 2d we should have only two independent vectors. So I should be able to get one of those three above by a linear combination of the other two, but when I do that, I get non constant coefficients multiplying them.

7. Jun 29, 2017

Orodruin

Staff Emeritus
These are vector fields, not vectors.

8. Jun 29, 2017

davidge

So they are'nt vectors? Can you say a bit more on this please

9. Jun 29, 2017

Orodruin

Staff Emeritus
There is no such thing as a "Killing vector". The Killing equation is a differential equation and as such describes vector fields, ie, assignments of one vector to each point in the space.

10. Jun 29, 2017

davidge

For instance, what could be such one vector?

11. Jun 29, 2017

Orodruin

Staff Emeritus
You wrote down several vector fields (in coordinate basis) in the firs post.

12. Jun 29, 2017

davidge

But you say they aren't vectors. I asked for an example of assigment of a vector by a vector field

13. Jun 29, 2017

Orodruin

Staff Emeritus
Your field is an assignment of a (dual) vector to every point in space!

For example, for $\theta = 0$ (and arbitrary r) your first field takes the value $\xi = dr$, where $dr$ is the coordinate basis dual vector.

14. Jun 29, 2017

davidge

I got it. Thanks.

15. Jun 29, 2017

davidge

Is it correct to say that they are three independent vector fields? In the sense that one cannot be expressed as a multiple of another one.

Also, if we evaluate any of them at a particular point $(r, \theta)$ do they form three linearly dependent vectors?