Objective direction of time in general relativity In general relativity, the coordinates $x^\mu$ we put on the manifold are arbitrary and need not have any physical interpretation. For this reason, it is said there is no objective notion of time in GR. However, at the same time there exists the coordinate-independent notion of "timelike", "null", and "spacelike" vectors. Doesn't this necessarily imply the existence of an objective time in the direction of the "most timelike" vector?
To make this more precise, consider an arbitrary manifold $(M,g_{\mu\nu})$ and on it choose a Riemannian metric $h_{\mu\nu}$. Consider all vectors $v^\mu$ satisfying $h_{\mu\nu}v^\mu v^\nu=1$ (or any constant). One of these vectors, $u^\mu$, will minimize $g_{\mu\nu}v^\mu v^\nu$ (using the convention $g_{\mu\nu}t^\mu t^\nu<0$ for timelike vector $t^\mu$) and we take this "most timelike" vector to be the direction of time. The vector field $u^\mu(p)$ constructed this way will generate a family of integral curves which we parameterize using their path length. Since the magnitude of a vector is independent of our choice of coordinates, this construction is coordinate-independent and so represents a unique, objective flow of time.
Where is the error in this construction? Is it possible that this merely corresponds to the proper time of an observer at rest in a given tangent space, equivalently to how the time axis is unique to a given reference frame in special relativity?
I feel like I'm missing something fundamental about how coordinates work in general relativity.
EDIT: My question is not a duplicate of the one asked here. I'm asking specifically about defining a unique time direction, not simply finding vectors that are timelike (which would be satisfied by any $v^\mu$ for which $g_{\mu\nu}v^\mu v^\nu<0$).
 A: The choice of Riemannian metric $h_{\mu \nu}$ is itself arbitrary, since there are multiple inequivalent rank-2 non-degenerate tensors on an arbitrary space;  and different choices of $h_{\mu \nu}$ will lead to different "preferred time directions."
For example, consider the following two rank-2 tensors on Minkowski space with coordinates $t$, $x$, $y$, $z$:
$$
h^{(1)}_{tt} = h^{(1)}_{xx} = h^{(1)}_{yy}=h^{(1)}_{zz}=1 \quad \text{(all other components vanish)}
$$
$$
h^{(2)}_{tt} = h^{(2)}_{xx} = \frac{5}{4} \qquad h^{(2)}_{xt} = h^{(2)}_{tx} = -1 \\ h^{(2)}_{yy}=h^{(1)}_{zz}=1 \qquad \text{(all other components vanish)}
$$
In these coordinates, the vector which minimizes $\eta_{\mu \nu} v^{\mu} v^{\nu}$ subject to the constraint $h^{(1)}_{\mu \nu} v^{\mu} v^{\nu} = 1$ is $v^{\mu} = (1,0,0,0)$, while the vector which minimizes $\eta_{\mu \nu} v^{\mu} v^{\nu}$ subject to the constraint $h^{(2)}_{\mu \nu} v^{\mu} v^{\nu} = 1$ is $v^{\mu} = (\frac43, \frac23, 0, 0)$.
So different choices of the Riemannian metric will lead to different time directions.  In effect, the Riemannian metric introduces a preferred geometric structure on the spacetime, which indirectly encodes the preferred time direction.
