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I sometimes mistake space-time curvature for gravitational field lines. Do geodesics in some ways represent $g$-field lines? Why is not it traditional to show $g$-field lines around a massive object in general relativity the same as we do for $E$ or $B$ field lines around an electrical charge or a magnet in electromagnetics?

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I would encourage anyone to draw any lines that help either them or others to get a good understanding. But the reason field lines are so much used in electromagnetism, and much less used in gravity, is because they nicely capture a mathematical property of a field whose divergence is zero (on a flat spacetime background). The zero divergence of the field translates to the field lines being continuous, and their spacing then expresses the field strength.

There are aspects of gravity that are a bit like this. In the weak field limit and for simple cases (e.g. a static case or one with steady rotation) one can express gravitational influences using a pair of fields analogous to electric and magnetic fields. But mostly in GR we are interested in stronger effects where the equations are more complicated and non-linear. Then there is nothing quite so convenient as the field lines of electromagnetism. But there are plenty of things one can do. Drawing sets of light cones can help in getting an impression of a region of spacetime. Or one could draw a selection of null geodesics. I like to add to this a set of timelike lines marked off by proper time along them. But such diagrams can be stretched and squeezed, twisted and distorted in all sorts of ways merely by changing the coordinates being used to plot them, so they have to be interpreted with care. If one merely changes coordinates then the diagram changes but the spacetime does not.

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Probably not just as you ask, but there are interesting ideas for visulazations in Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes. II. Stationary black holes by David A. Nichols, Robert Owen, Fan Zhang, Aaron Zimmerman, Jeandrew Brink, Yanbei Chen, Jeffrey D. Kaplan, Geoffrey Lovelace, Keith D. Matthews, Mark A. Scheel, Kip S. Thorne.

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