Just a question about calculating Killing vectors

[BasharTeg]

Homework Statement

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.

Homework Equations

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

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

The Attempt at a Solution

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

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

the first 2 lead to

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

and inserting that into the last equation shows that this one has to be equal to a constant since you can completely separate 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?

 

[mark726]

Hi there,

Your approach to finding the Killing vectors using equation (2) looks correct. However, I would suggest double checking your results by plugging in the vectors you found into the equation L_{\xi}g_{\mu \nu}=0 and verifying that it holds for all components of the metric.

Also, remember that the Killing vectors are not unique. So while your solution is correct, there may be other linearly independent Killing vectors as well. It would be helpful to double check your solution with a known solution or to compare it with other solutions found in the literature.

Overall, great job on tackling this concept and using the equations correctly! Keep up the good work.