Can we always choose a gauge in GR in which time is constant? In General relativity the metric describes the curvature of 4D space-time. But due to diffeomorphism invariance, many metrics describe the same physics.
Can we always choose a metric such that we can foliate space-time into slices which have curvature but time runs constantly between slices. So that $g_{00}=1$ and $g_{0i}=0$.
So that knowing the curvature of the space slice is all we need to know.
In other words can we always write metrics in terms of space curvature without time curvature. So it is a theory in which the curvature of 3-space changes with time.
 A: Locally yes. And by locally, I mean "in an open domain", not just infinitesimally.
To see this consider a spacelike hypersurface $\Sigma$, equipped with local coordinates $\xi^a$. Let $n^\mu$ be the future-directed unit normal field to $\Sigma$. Now, given an initial point and an initial vector, the geodesic equation has unique solutions, but there is no guarantee that they are indefinitely extendible. At a point $x\in\Sigma$ let $\gamma(\lambda)$ be the geodesic with initial velocity $n^\mu(x)$.
If the coordinates of $x\in\Sigma$ are $\xi^1,...,\xi^n$, then we should assign to the point $\gamma(\lambda)$ the coordinates $(\lambda,\xi^1,...,\xi^n)$.
It is not difficult to see that there is at least an open neighborhood of the $n-1$ dimensional coordinate domain on $\Sigma$ in which this is a well-defined coordinate system. It can also be seen that as long as this construct is well defined, the $\lambda$ coordinate will remain orthogonal to the $\xi$ coordinates.
The metric in this coordinate system has components $g_{00}=g_{\mu\nu}n^\mu n^\nu=-1$, $g_{0a}=g_{\mu\nu}n^\mu(\partial_a)^\nu=0$, so it is in required form.
This construction is referred to as Gaussian normal coordinates.
