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First of all, How do you differentiate between timelike coordinates and spacelike coordinates ?

My understanding is that if we are given a metric tensor, a coordinate will be timelike if $dx^{2}$ has a negative term and spacelike if that term is positive. For example, in Minkowski spacetime, $g=-dt^2+dx^2+dy^2+dz^2$ so we say t is timelike and x,y,z are spacelike. (To check causality of vector $v$ we use $g(v,v)$ but here I am talking about coordinate)

Is this enough or there is any other thing that we should also consider ?

Secondly, if $x^\mu$ is a timelike coordinate then How do we calculate its gradient and why its gradient $\nabla x^\mu$ should also be timelike ?

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  • $\begingroup$ Related post by OP: physics.stackexchange.com/q/857311/2451 $\endgroup$ Commented Aug 10 at 15:26
  • $\begingroup$ $t$ is a timelike/null/spacelike coordinate if and only if the vector $\partial_t$ is a timelike/null/spacelike vector. $\endgroup$ Commented Aug 10 at 17:03
  • $\begingroup$ @Prahar we use metric tensor $g(\partial_{t},\partial_{t})$ to check that causality. right ? $\endgroup$ Commented Aug 10 at 22:00
  • $\begingroup$ @Prahar If i write two coordinates $(t,r,\theta,\phi)$ and $(v,r,\theta,\phi)$ and if $\partial_{v}=\partial_{t}$, then both coordinates should have same causal character everywhere in spacetime? $\endgroup$ Commented Aug 10 at 22:20
  • $\begingroup$ Assuming both those things, the answer is yes. $\endgroup$ Commented Aug 10 at 23:34

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