In all the diagrams I have seen, a large astronomical body eg. a planet or star, shows its gravity well as one dimensional in spacetime. My basic question is: Is it truly one dimensional or multi-dimensional? For example, is it “at the bottom” (which most diagrams shows), “at the top”, “to the left”, “to the right” of the body? Thus, I would say, multi-dimensional, when viewed in space-time. All I ever see is a cone below the body as if spacetime is a flat plain.
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The popular way of demonstrating spacetime curvature as an indentation below the body is somewhat misleading. After all, in reality, there is no absolute "down" so it doesn't make sense to say that the dip is "at the bottom, at the top, to the left, to the right", etc. They always rely on the intuitive concept of down coming from the fact that we live on the Earth.
The simplest setup to consider is a spherically symmetric gravitating body like a star or a planet. Due to the spherical symmetry, the metric tensor and the spacetime curvature take the same value at each spherical shell (determined by its particular radius). The "deepest point of the gravitational well" would then be in the centre of the planet/star. The well itself is multidimensional.
Alessandro Roussel came up with a better visualisation in this video. Essentially, geodesics are continuosly falling into the massive body instead of being "below" it.
Daniil Zhitov
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