This is a very basic question about General Relativity. I haven't take any GR course yet.
Suppose a flat spacetime with one space direction and one time direction, as follows:
Now add a mass at rest at $x_0$. The world line of this mass can be drawn in the spacetime diagram as follows (red line):
Now I wonder how the constant time and constant space lines would look like?, i.e. I would like to depict the grid of this spacetime.
My reasoning (may be wrong) was as follows: Over the $t$ axis I marked down the units of time. Then, I know that inside a gravitational field clocks run slower, so I marked down (over the mass' world line) the slower units of time. Like this:
What I did next was just to join these marks as follows:
I don't care about the exact shape of the constant time lines, I just want to know if this reasoning is good.
I also wander how are the constant space lines... If I consider the following
then a very small test mass placed at rest at some $x$ near the big mass will eventually feel an attraction and at $t\to\infty$ will collide with the big mass. This must be wrong because in real life masses do not need infinite time to fall. But I cannot manage to find the constant $x$ lines.