From a programming perspective, I've always thought of gravitational influence as a kind of vector field,

vector field (crudely drawn)

which seems to attribute to the motions of bodies through the field accurately enough. However, gravity curves the geometry of space (making it non-Euclidean?),

curved space (expertly drawn)

which intuitively doesn't seem the same to me. In the image, the unit squares nearest the influential body look like their angles add up to less than 360 degrees, meaning their areas (and perimeters?) are less. I think these lines represent geodesics, which describe the shortest path between two points on a curved surface. We are viewing those geodesics presumably at infinity and so we see them as being curved, but an observer on one of the geodesics would see it as being perfectly straight (lightlike paths?) in their own frame. (Though I think I've made an error here, I'll continue.)

If we overlay flat spatial curvature seen at infinity onto that curvature seen near the object, we get,


and now it does look as if the unit squares nearest the object have lesser area and the geodesics nearest the object have a bit more length. So, what does this mean in the real world? Does this mean that a path through a gravitational field can be measurably longer (at infinity) than the same path on flat space? And, as consequence, does this mean that a gravitational object has less area/volume than what its circumference ought to purchase?

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    $\begingroup$ If the warped grid lines were geodesics, your diagram would have light (and other objects) repelled by gravity. $\endgroup$ – S. McGrew Feb 24 '19 at 18:56
  • $\begingroup$ @S.McGrew Is the imagery right but I'm using the wrong term, or do I have the imagery backwards? $\endgroup$ – B.fox Feb 24 '19 at 19:06
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    $\begingroup$ Actually, coordinate lines would not generally correspond to geodesics. Your first question needs a bit more thought. "Yes" is the answer to your second question. $\endgroup$ – S. McGrew Feb 24 '19 at 21:19
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    $\begingroup$ This kind of visualization is all over the internet, but it doesn't actually explain very much and isn't actually very accurate. The image that we keep seeing is this one by Christopher Vitale: forbes.com/sites/startswithabang/2019/02/16/… $\endgroup$ – user4552 Feb 24 '19 at 21:44
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    $\begingroup$ To qualify my last comment a bit: in a curved space, the ratio of a sphere's area to its volume is different from the ratio in flat space. In negatively curved space the ratio is "too small", and in positively curved space the ratio is "too large". $\endgroup$ – S. McGrew Feb 24 '19 at 21:48

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