1
$\begingroup$

Let $X=x\partial_{t}+t\partial_{x}$ and $Y=y\partial_{t}+t\partial_{y}$ be Killing vectors on Minkowski $(-,+,+,+)$. It can be shown that $[X,Y]$ is also Killing. I get the following: \begin{equation} [X,Y]=(x-y)\partial_{t}+x\partial_{y}. \end{equation}

What isometry is this Killing associated with?

$\endgroup$
2
  • 3
    $\begingroup$ You know your last term is wrong, as it is not antisymmetric in x,y. Have you learned about boosts and rotations? $\endgroup$ Nov 19, 2020 at 18:45
  • $\begingroup$ I think it should be $(x-y)$ too. I know it has to do with rotational symmetry. The problem is, I have confusion about the Killing $U=\partial_{t}+\frac{1}{r}\partial_{\phi}$ In Schwarzschild metric. I thought if I understand the Minkowski case, The Schwarzschild would follow from the first case using the same reasoning $\endgroup$
    – M91
    Nov 19, 2020 at 18:50

1 Answer 1

1
$\begingroup$

You did not evaluate you commutator correctly. Your two boosts commute to a rotation on the x,y plane, $$[X,Y]= [x\partial_{t}+t\partial_{x}, y\partial_{t}+t\partial_{y}]=x\partial_y-y\partial_x,$$ and all three annihilate $x^2+y^2+z^2-t^2$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.