Apologies for the confusing title, it is late here. I'm wondering exactly what meaning the "time coordinate" has in General Relativity. We always write the line element as $$\tag{1} ds^2=g_{00}(dx^{0})^2+2g_{0i}dx^0dx^i+g_{ij}dx^idx^j,$$ with the assumption that $x^0$ is some kind of time coordinate, and $x^i$ are spatial coordinates. However, as light cone coordinates show, if we pick a random coordinate system, the metric will not be in the form (1). Is the existence of coordinate systems as in (1) an axiom or can it be derived somehow? By a time coordinate I mean that the time coordinate lines should have timelike tangent vectors, and similarly with spacelike coordinates.
Answer
Let $(M^{n+1},g)$ be a Lorentzian manifold. Given $p\in M$, we will show that there is a coordinate system $(x^\mu)$ defined on an open set $p\in U\subset M$ such that $\partial_0$ is a timelike vector field, and $\partial_i$ are spacelike vector fields for $i=1,\dotsc,n$.
Let $(x^\mu)$ be an arbitrary chart defined on $U\ni p$. It is known that $T_pM$ is the span of $\{\partial_0,\partial_1,\dotsc,\partial_n\}$. As $g_p$ has signature $(-,+,\dotsc,+)$, we may find linearly independent vectors $v_\mu$, $\mu=0,1,\dotsc,n$, such that $g_p(v_0,v_0)=-1,$ $g_p(v_i,v_i)=+1$. These vectors are linear combinations of $\{\partial_0,\partial_1,\dotsc,\partial_n\}$. By a linear change of coordinates, we can find a coordinate system $(y^\mu)$ such that $\partial/\partial y^\mu=\partial_\mu'=v_\mu$ at $p$. By continuity, there is a neighborhood $V_0\subset U$ of $p$ such that $g(\partial_0',\partial_0')<0$, i.e., $\partial_0'$ is timelike on $V_0$. Similarly, there exist neighborhoods $V_i$ such that $g(\partial_i',\partial_i')>0$ on $V_i$. We take $V=V_0\cap\cdots \cap V_n$, which is a neighborhood of $p$. By changing each coordinate value by a constant, we can adjust the origin without changing the aforementioned vector fields. Then $(y^\mu)$ is the desired coordinate system on $V$.
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