Newtonian approximation of the metric tensor

A stationary spacetime $M$ is one which admits a complete timelike Killing vector field $boldsymbol xi = xi^a frac{partial}{partial x^a}$. The integral curves of $boldsymbol xi$ define a flow which preserves the metric. That is, if you define a map $phi_t:Mrightarrow M$ which takes some point $pin M$ and "flows" along the vector field $boldsymbol xi$ for a parameter distance $t$, then the metric doesn't change. This is captured by the fact that the Lie derivative of the metric along $boldsymbol xi$ vanishes:

$$mathcal L_{boldsymbol xi} mathbf g = 0 iff nabla_mu xi_nu + nabla_nu xi_mu = 0$$

The implication of this can be seen by noting that if such a vector field exists, then in any neighborhood where $boldsymbol xi neq 0$, we can choose a coordinate system in which we use the flow parameter $t$ as the temporal coordinate $x^0$. If we do so, then $boldsymbol xi = frac{partial}{partial t}$ and the Lie derivative reduces to $(mathcal L_{boldsymbol xi} mathbf g)_{munu} =partial_t g_{munu} = 0$, i.e. a spacetime is stationary if there exists a coordinate system in which none of the metric components depend on time.

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A static spacetime is one which is stationary, but has the additional requirement that there exist a spacelike hypersurface $Sigma$ which is everywhere orthogonal to the orbits of $phi_t$. In a neighborhood of such a surface, every point $p$ lies on an integral curve of $mathbf xi$ which intersects $Sigma$ at a unique point $p_0$, and can therefore be labeled by coordinates $(t,x^1,x^2,x^3)$ where $t$ is the Killing flow parameter (which we set equal to $0$ at $p_0$) and ${x^a},a=1,2,3$ are arbitrarily chosen coordinates on $Sigma$.

The requirement that $Sigma$ be orthogonal to the orbits of $phi_t$ means that for any vector $mathbf v=v^a frac{partial}{partial x^a},a=1,2,3$ (that is, any vector tangent to $Sigma$), we have that $mathbf g(boldsymbol xi, mathbf v) = v^a g_{ta} = 0$ for any choice of $v^a, a=1,2,3$. This immediately implies that $g_{ta}=0$, and therefore that

$$mathbf g = -alpha(x^1,x^2,x^3) dt^2 + beta_{ab}(x^1,x^2,x^3) dx^a dx^bqquad a,b=1,2,3$$

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From a physical standpoint, one should interpret a stationary spacetime as one which possesses time-translation symmetry and a static spacetime as one which possesses both time-translation and time-reversal symmetries (in our judiciously-chosen coordinate system); note that $trightarrow -t$ leaves a stationary metric unchanged iff there are no $dt dx^a$ cross terms.

Roughly, stationary-but-not-static spacetimes are those for which, given any spacelike hypersurface $Sigma$, the timelike Killing flow $phi_t$ pushes points along $Sigma$ as well as forward in time. In other words, the infinitesimal time-translation map $phi_{delta t}$ induces a change in the spatial coordinates as well as the time coordinate. This is manifested as frame-dragging, and is characteristic of spacetimes around rotating bodies - see e.g. the Kerr spacetime, which is stationary (no $t$-dependence in the metric components) but not static.