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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:

1+1D flat spacetime.

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):

The world line of the mass.

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:

ASD

What I did next was just to join these marks as follows:

Constant time lines

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

This may be wrong...

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.

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You take the 'black axis' in some sense as 'the' reference. Taking this time as reference, though, you construct the so called constant time-lines: they are not constant at all, but I think you want us to interpret this line of events as events that (straight line to the left, curved line to the right) will be simultaneous for observers following the black world-line. So, taking a particular $t_1$ in $t$, to the left - more or less - the simultaneous events occur (following this $t$ world-line) as a straight line, to the right as a curved line.

But still I can not realize the question with relation to the coordinate system. x = constant remains the same (as long as t = constant).

What can be said about the example is that a 'space interval' (thin road, f. e.) that approaches the red world line will look 'smashed' nearby it.

Still the influence of a small mass can be taken via Schwarzschild solution in $(t,x,y,z)$-coordinate system (found in wikipedia). Im not sure you can simply ignore $y$ and $z$, though (as long as they are not Killing vectors).

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