# Just a question about calculating Killing vectors

1. Dec 10, 2011

### BasharTeg

1. The problem statement, all variables and given/known data
I am new to the whole GR concept so I just would like to know if what I did is correct or utterly ******** (and if so, why).
I have to find all Killing Vectors to a given metric (1) using equation (2). (2) leads to 3 partial differential equations which I used to calculate them.

2. Relevant equations
(1) $$ds^2=-dt^2 +dx^2$$

(2) $$L_{\xi}g_{\mu \nu}=\xi^{\alpha}\partial_{\alpha}g_{\mu \nu}+g_{\mu \alpha}\partial_{\nu}\xi^{\alpha}+g_{\alpha \nu}\partial_{\mu}\xi^{\alpha}=0$$

3. The attempt at a solution

I used equation to solve the three equations $$L _\xi g_{tt}=0$$, $$L _\xi g_{xx}=0$$ & $$L _\xi g_{tx}=0$$.

$$\partial_t \xi^t=0$$
$$\partial_x \xi^x=0$$
$$\partial_t \xi^x=\partial_x \xi^t$$

$$\xi^t=f(x)$$
$$\xi^x=g(t)$$

and inserting that into the last equation shows that this one has to be equal to a constant since you can completely seperate the x and t depention. So with integration I get:

$$\xi^x=g(t)=At+B$$
$$\xi^t=f(x)=Ax+C$$

So I have 3 constants and the solution:

$$\xi=A \binom{t}{x} + B \binom{0}{1}+C\binom{1}{0}$$

and therefore those 3 vectors are the 3 linear independant Killing vectors.
Is that correct?

Last edited: Dec 10, 2011