Obtain Lorentzian manifolds from Riemannian ones by Wick rotation

This is a somewhat different take on Igor's answer, and I offer it just in case you are interested.

First, one doesn't need to have any continuous symmetries in order to have this kind of 'Wick rotation' exist. For example, if $(M,g)$ is a real-analytic Riemannian manifold that admits a nontrivial isometric involution $\iota:M\to M$ that fixes a hypersurface $H\subset M$, then, near $H$, one can write the metric in the form $g = dt^2 + h(t^2)$ where $t$ is the distance from the hypersurface and $h(a)$ is the induced metric on the level sets $t^2 = a$. Then the Wick rotation is just $g' = -d\tau^2 + h(-\tau^2)$ in the sense that this is the Lorentian metric induced on the slice $t = i\tau$ in the complexification $(M^\mathbb{C},g^\mathbb{C})$.

Second, one could generalize things a bit and say that two real-analytic (pseudo-)Riemannian metrics are 'Wick-related' if they are '$\mathbb{R}$-slices' of a common connected holomorphic Riemannian complex $n$-manifold $(M^\mathbb{C},g^\mathbb{C})$. By an '$\mathbb{R}$-slice', I mean a real $n$-manifold $N\subset M^\mathbb{C}$ such that the pullback of $g^\mathbb{C}$ to $N$ is real-valued and nondegenerate. In this terminology, I think that two $\mathbb{R}$-slices $N_1,N_2\subset M^\mathbb{C}$ should be said to be related by a 'Wick rotation' if $N_1\cap N_2$ is a submanifold of dimension $n{-}1$. This is certainly what happens in the case above generated by an isometric involution fixing a hypersurface. (Added note: In fact, I don't know an example of a Wick-rotation in this sense that isn't generated by such an isometric involution. It might be interesting to try to prove that this does give them all or else find a counterexample. I note that, to second order, it is true: The hypersurface $N_1\cap N_2$ is always totally geodesic in each of $N_1$ and $N_2$ (with their induced metrics), so reflection in the hypersurface is an isometry at least up to second order.)

Then the problem becomes how to tell when a given connected holomorphic Riemannian complex $n$-manifold $(M^\mathbb{C},g^\mathbb{C})$ has an $\mathbb{R}$-slice (and, of course, to determine them when they exist). Obviously, the complexification of a real-analytic pseudo-Riemannian $n$-manifold has at least one $\mathbb{R}$-slice, but generically, when $n>1$, this is the only one, so 'most' real-analytic pseudo-Riemannian manifolds are not Wick-related to any other, let alone possess a Wick-rotation.

The reason is that $\mathbb{R}$-slices are the integral manifolds of a very restrictive system of PDE for real submanifolds of $(M^\mathbb{C},g^\mathbb{C})$: If one lets $R\subset \mathrm{Gr}^\mathbb{R}_n(TM^\mathbb{C})$ denote the set of real $n$-planes $E\subset T_pM^\mathbb{C}$ to which $g^\mathbb{C}$ restricts to be real-valued (and, of course, nondegenerate), then $R$ is a smooth manifold of dimension $2n+\tfrac12n(n{-}1)$ with $n{+}1$ components (one for each possible index of $g$ when restricted to the $n$-plane $E$), and the basepoint projection $\pi:R\to M^\mathbb{C}$ given by $\pi(E) = p$ is a smooth submersion. It is easy to show that, for any given $E\in R$, there is at most one $\mathbb{R}$-slice that has $E$ as a tangent space. This is because there is a canonical $n$-plane field $H$ on $R$ with the property that the set of tangent spaces of any $\mathbb{R}$-slice is an $n$-manifold in $R$ that is tangent to $H$ everywhere.

The only time $H$ is Frobenius is when $(M^\mathbb{C},g^\mathbb{C})$ is the complexification of a (real) space form, i.e., a Riemannian manifold of constant sectional curvature. Generically, $H$ has no integral manifolds at all, and, generically, when it does have one, it has only one connected component.

For example, in the case $n=2$, if the Gauss curvature $K$ of $(M^\mathbb{C},g^\mathbb{C})$ is not constant, then it has at most a $1$-parameter family of positive definite $\mathbb{R}$-slices, and this happens only when the $\mathbb{R}$-slices are all isometric and have a symmetry vector field (so they are locally surfaces of revolution).

Now, in fact, each $\mathbb{R}$-slice (of whatever index) lies in the (real) hypersurface in $M^\mathbb{C}$ on which $K$ takes values in $\mathbb{R}$. Set $E = g^\mathbb{C}(\nabla K,\nabla K)$ (everything computed in the holomorphic category). For most metrics $g^\mathbb{C}$, the holomorphic functions $K$ and $E$ will not be functionally dependent, i.e., $\mathrm{d}K\wedge\mathrm{d}E$ will vanish only on a (possibly empty) complex-analytic subvariety $C\subset M^\mathbb{C}$ of complex dimension $1$. Go ahead and define $F = g^\mathbb{C}(\nabla K,\nabla E)$ and $G = g^\mathbb{C}(\nabla E,\nabla E)$.

Obviously, any $\mathbb{R}$-slice $N\subset M^\mathbb{C}$ must lie inside the locus on which the holomorphic functions $K$, $E$, $F$, and $G$ assume real values. Outside of $C$, the set $L\subset M^\mathbb{C}\setminus C$ on which $K$ and $E$ take real values is a (possibly empty) real submanifold of dimension $2$ and its components are the only possible $\mathbb{R}$-slices that lie outside of $C$ (of course, any $\mathbb{R}$-slice can only intersect $C$ in a (real) $1$-dimensional curve at most). A component of $L$ actually is an $\mathbb{R}$-slice if and only if $F$ and $G$ are real-valued on it. Two components are related by a Wick-rotation in the above sense if they intersect (in a real $1$-dimensional curve that necessarily lies inside $C$).

A similar, but slightly more involved analysis can be done for the case in which $K$ and $E$ are dependent everywhere on $M^\mathbb{C}$. Moreover, a similar, but more complicated analysis can be used in higher dimensions, with, say, the symmetric functions of the eigenvalues of the holomorphic Ricci tensor used in the place of $K$.

I'm going to guess that your interest in Wick rotation comes from the role it plays in the formulation of quantum field theories (QFTs) on Miknowski spacetime and some other curved spacetimes, like the examples you mentioned. Of course, it is natural to ask, like you probably have, whether Wick rotation plays a similarly important role in the formulation of QFTs on more general Lorentzian manifolds. And the answer to that question seems to be No. At least, the current state of the art methods in dealing with QFTs in curved spacetimes do not make use of Wick rotation.

Now, on to the question you actually asked. As far as I know, there is no general theory about the conditions under which a Wick rotation can be performed, except the property of being static, where the formal substitution trick $t\mapsto i\tau$ is already sufficient. However, the only way to even attempt at trying to find such general conditions, whatever they are is to make the notion of Wick rotation mathematically precise and coordinate independent. So I'll finish by doing that below.

You need to start with an analytic manifold $M$ with an analytic Lorentzian metric $g$. The manifold $M$ will have a complexification $M_\mathbb{C}$, where it embeds as a real submanifold $M \subset M_\mathbb{C}$. The same can be said about the cotangent bundle $T^*M \to T^*M_\mathbb{C}$ and any tensor powers of it. So, the metric $g$ naturally defines an analytic section of $T^*M_\mathbb{C}$ over the real submanifold $M$, and thus extends uniquely to a section $g_\mathbb{C}$ on all of $M_\mathbb{C}$ by analytic continuation. Of course, there is no reason for $g_\mathbb{C}$ to remain of definite signature or even real off $M$. But let us say that the Lorentzian manifold $(M,g)$ has a Wick rotation if the embedding $M\subset M_\mathbb{C}$ can be smoothly deformed into another embedding $M'\subset M_\mathbb{C}$ such that $g_\mathbb{C}$ restricted to $M'$ is equal to a real Riemannian metric $g'$.

In the case of a static spacetime, the metric can be written as $g = -a\, dt^2 + h$, where $t$ is a special time coordinate so that $a$ and $h$ do not depend on $t$ or $dt$. The deformation $(t,x) \mapsto (e^{i\theta} t,x)$ is a deformation of $M\subset M_\mathbb{C}$ that terminates at a Wick rotation for $\theta=\pi/2$.