I was trying to solve a problem in GR with the following metric:

$$ds^2 = -du dv + dx^2 + dy^2 + F(u,x,y) du^2 $$

The coefficients of the metric don't depend on $v$, so $\partial_v$ defines a Killing vector field. Later on the problem you have to consider $F=m(x^2-y^2)$. Then $\partial_u$ also defines a Killing vector field.

I understand the physical meaning of this two Killing vectors but in the last section of the exercise it asks to consider $F=F(x^2+y^2)$ and find a third Killing vector field.

In this case I don't see any candidate vector at first. What I thought is to consider $x^2 + y^2$ as a radius and then after a change of coordinates the Killing vector should be $\partial_\theta$ because the coefficients of the metric shouldn't depend on the angle.

Is this the right approach? Is there any other possible physical interpretation of a symmetry just looking at the coefficients of the metric?


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