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A classic example of a stationary, axisymmetric metric in GR is the Kerr metric. In Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ it is obvious that the metric is independent of $t,\phi$ and so is stationary and axisymmetric.

Now, often in GR we want to work in a covariant, coordinate independent way and just deal with 4-vectors, tensors etc. In this case the metric is just represented by $g^{\mu \nu}$.

My question is, is there a way to enforce stationarity and axisymmetry onto this metric tensor $g^{\mu \nu}$, without reference to a coordinate system? For instance, can this be done with Killing vectors?

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A spacetime is said to be stationary if it has an (asimptotically) timelike Killing vector.

Similarly, if one has a Killing vector which has closed spacelike trajectories, then we get an ignorable coordinate, which corresponds to the axisymmetry.

In the example of the Kerr metric, the timelike Killing vector is $\frac{\partial}{\partial t}$ and the "axisymmetric" Killing vector is of course $\frac{\partial}{\partial \phi}$.

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The definition of stationary and axialsymmetric normally comes after one has specified a local coordinate expression, (writing $g_{\mu\nu}$ is still a local coordinates expression, since it has indices).

However as you might know, an abstract symmetry of a metric is associated to a Killing vector as has been pointed out. The fact that you can identify such Killing vector with stationarity or rotations, has to do with extra requirements on these Killing vector fields such as forcing it to be "timelike" or satisfying some algebra of rotations ($U(1)$ plus spacelike for the axial example or $SO(3)$ plus spacelike for the spherical case). Then you could work with vector fields that are Killing vectors and that satisfy some additional property you are interested in, perhaps this extra properties are what you should look into, if you don't want to specify a particular frame.

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