What is Spacelike vs. Timelike Coordinate?

[GR191511]

Thanks！

 

[PeterDonis]

Any coordinate $x^\mu$ has a corresponding coordinate basis vector $\partial / \partial x^\mu$. The coordinate is called "spacelike", "timelike", or "null" according to the type of its coordinate basis vector. Or, equivalently, according to the type of direction in spacetime that you are moving along a curve in which $x^\mu$ changes but all other coordinates are held constant.

 

[pervect]

One thing that can be confusing. Suppose we have some coordinates $t,x,y,z$. And we wish to know if the coordinate x is spacelike, timelike, or null at some point P. To do so, we have to associate the coordinate x with some vector field, and evaluate the length of the vector field at some particular point P. The length of the vector at the particular point P can be classified as time-like, space-like, or null, depending on the sign of the length and the sign convention that one adopts.

However, the answer depends on whether we look at the contravariant vector filed $\partial_x$ = $\frac{\partial} {\partial x}$, or the covariant vector field dx.

Most PF posters seems to be the convention to look at the vector field $\partial_x$ rather than the covector field dx. But it's clearer to talk about whether a vector field is time-like, space-like, or null at some point P rather than to talk about whether a coordinate is time-like, space-like, or null at some point P, as we need a map from a coordinate to a vector field in order to perform the classification. Of course it also depends on what point P we choose - for instance, in the Schwarzschild coordinates (t,r,$\theta$, $\phi$), $\partial_t$ is well-known to be a time-like vector outside the event horizion, null at the horizon, and space-like inside the horizon.

Because the Schwarzschild metric is diagonal, dt has the same sign as $\partial_t$. But this is not true in general. For metrics that are not diagonal, such as the Painleve metrics, the sign of $\partial_t$ and dt can be different.