Most general Ansatz for cylindrically symmetric metric in GR?

Question

How would the most general Ansatz for a cylindrically symmetric metric in GR look like?

To make this question more substantial, here is an example of what I have in mind. I ask this question in the spirit of how the Scharzschild solution can be derived from an Ansatz. For Schwarzschild, as for instance described in Carrolls book, we first start with flat Minkowski space

$$ds_{\text{Minkowski}}^2=-dt^2+dr^2+r^2d\Omega$$

and generalize it by modifying components. First we assume time independence as well as time reversal symmetry, which means that any term should be independent of $t$ and any cross components $dtdx_i$ must vanish. Then, perfect spherical symmetry demands that the $d\Omega^2$ part of the metric remains unchanged. Finally, we would define the $r$ coordinate such that the most general Ansatz becomes:

$$ds^2=-e^{2\alpha(r)}dt^2+e^{2\beta(r)}dr^2+r^2d\Omega^2$$

Now, in the case of cylindrical symmetry I am interested in an Ansatz that does not assume time independence or time reversal symmetry. I am tempted to write

$$ds_{\text{cylinder}}^2=-e^{2\alpha(t,r,z)}dt^2+e^{2\beta(t,r,z)}dr^2+r^2d\phi^2+e^{2\gamma(t,r,z)}dz^2$$

But I feel that this expression neglects some cross components between different variables which also would have to appear. What do you guys think?

PS:

Also, please note that I am including a $z$-dependence in the factors. A perfect cylinder would have a translation invariance in the $z$-direction. But what I am interested in is a situation where there is only a killing vector $\partial_\phi$, but no killing vector $\partial_z$.

Answer 1

According to "Exact Solutions of the Einstein Field Equations", the most general cylindrically symmetric metric is

\begin{equation} ds^2 = e^{-2U} (\gamma_{MN} dx^M dx^N + W^2 d\phi^2) + e^{2U} (dz + A d\phi)^2 \end{equation}

With Killing vectors $\eta = \partial_\phi$ and $\zeta = \partial_z$, and all functions independant of $z$ and $\phi$. The other two coordinates can be chose such that

\begin{equation} \gamma_{MN} dx^M dx^N = e^{2k}(d\rho^2 - dt^2) \end{equation}

Additionally, if you also have the reflection symmetry $\phi \rightarrow -\phi$ and $z \rightarrow -z$, $A$ can be made to be 0.

Edit :

If you only have a Killing vector for $\phi$, it is not cylindrically symmetric, it is an axisymmetric spacetime. For just one Killing vector, what you can generally do is just remove the dependance of the coordinate from the metric and eliminate the cross terms with that coordinates.