What follows is a long self-made example to deal with my conceptual issues of visualizing curved spacetime.
Imagine an observer floating somewhere in space. He feels no strain on his body, indicating he is free-falling and is thus moving along a geodesic.
He is a magician and has the power to conjure up glowing sticks within his hands.
Their construction is as follows (the following are valid in his vicinity of course):
1) Each stick is very thin and of 1m in length when he conjures it in his hands.
2) Each stick is rigid and mass-less
3) It glows along it's cylindrical surface uniformly. The glow varies with intensity such that it goes from complete dark to peak brightness to dark in 1 heartbeat of the magician in his local rest frame. It is thus like a clock synchronized with his heartbeat.
The magician starts conjuring up billions of these sticks in his vicinity and starts assembling them into a cubic lattice in his vicinity. As he starts assembling the lattice, it grows in size and moves further out far beyond his vicinity. He effectively creates a crude cartesian co-ordinate system (without any markings).
In curvature free space-time, his eyes will project the glowing grid lines as straight lines intersecting at infinity and the whole system will glow in the same way.
Now if space-time had some arbitrary curvature (not too high), how would the glowing grid 'lines' be projected onto his eyes?
Alternatively he could create a spherical system by making concentric glowing shells of increasing radii (where the latitudes and longitudes glow) made with uniform curvature in his vicinity. He could fix the spatial origin as wherever he wants. In flat space-time, the concentric shells would be projected onto his eyes as concentric circles. What would he see in curved space-time?
Is this how experimenters construct their 'imaginary' co-ordinates systems for work in astro-physics?