# Killing vectors and isometry

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: $$$$[X,Y]=(x-y)\partial_{t}+x\partial_{y}.$$$$

What isometry is this Killing associated with?

• You know your last term is wrong, as it is not antisymmetric in x,y. Have you learned about boosts and rotations? Nov 19, 2020 at 18:45
• 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
– M91
Nov 19, 2020 at 18:50

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 $$x^2+y^2+z^2-t^2$$.