Gravity Well - Gravity Wells and General Relativity

Gravity Wells and General Relativity

Both the rigid gravity well and the rubber-sheet model are frequently misidentified as models of general relativity, due to an accidental resemblance to general relativistic embedding diagrams, and perhaps Einstein's employment of gravitational "curvature" bending the path of light, which he described as a prediction of general relativity. In particular, the embedding diagram most commonly found in textbooks (an isometric embedding of a constant-time equatorial slice of the Schwarzschild metric in Euclidean 3-dimensional space) superficially resembles a gravity well.

Embedding diagrams are, however, fundamentally different from gravity wells in a number of ways. Most importantly, an embedding is merely a shape, while a potential plot has a distinguished "downward" direction; thus turning a gravity well "upside down" (by negating the potential) turns the attractive force into a repulsive force, while turning a Schwarzschild embedding upside down (by rotating it) has no effect, since it leaves its intrinsic geometry unchanged. Geodesics on the Schwarzschild surface do bend toward the central mass like a ball rolling in a gravity well, but for entirely different reasons. There is no analogue of the Schwarzschild embedding for a repulsive field: while such a field can be modeled in general relativity, the spatial geometry cannot be embedded in three dimensions.

The Schwarzschild embedding is commonly drawn with a hyperbolic cross section like the potential well, but in fact it has a parabolic cross section which, unlike the gravity well, does not approach a planar asymptote. See Flamm's paraboloid.