system
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human
What does it mean for a coordinate to be timelike?

In the wikipedia article about Schwarzschild black holes, it says : For r < rs the Schwarzschild radial coordinate r becomes timelike and the time coordinate t becomes spacelike.

What does it mean for a coordinate to be "timelike" ? Does it just mean that an object must move forwards in that coordinate?
gpt
It's referring to the sign that that coordinate gets in the metric.

For example, in the Minkowski metric in the (-+++) convention, the metric is 

ds^(2) = -c^(2)dt^(2) + dx^(2) + dy^(2) + dz^(2).

The dx, dy, and dz terms are all positive, while the dt term is negative. So t is a timelike coordinate while x, y, and z are spacelike coordinates.

The Schwarzschild metric is

ds^(2) = -(1 - 2M/r)dt^(2) + (1 - 2M/r)^(-1)dr^(2) + r^(2)dΩ^(2),

where I've used Ω as a shorthand for the angular part containing the two spherical angles.

Focus on the t and r terms. For r > 2M (the Schwarzschild radius), (1 - 2M/r) > 0, so the dt term is overall negative and the dr term is overall positive. That's the normal case, just like in Minkowski. The t coordinate is timelike and the r coordinate is spacelike.

But for r < 2M, (1 - 2M/r) < 0, and the signs flip. The Schwarzschild coordinate t becomes spacelike and the Schwarzschild r becomes timelike. Some people say that this means that "space and time are switching places" at the event horizon, but a more correct explanation is just that you chose bad coordinates. t looks like time and r looks like the radial coordinate from your typical spherical coordinates, but that's not quite true. This set of coordinates has a coordinate singularity at r = 2M. There are other sets of coordinates which behave perfectly fine at the horizon, so this bad behavior is just an artifact of the choice of coordinates.