In physics, a gravity well is the gravitational potential field around a massive body (a particular kind of potential well). Physical models of gravity wells are sometimes used to illustrate orbital mechanics. Gravity wells are frequently confused with general relativistic embedding diagrams, but the two concepts are unrelated.
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The external gravitational potential of a spherically symmetric body of mass M is given by . A plot of this function in two dimensions is shown in the figure. This plot has been completed with an interior potential proportional to , corresponding to an object of uniform density, but this interior potential is generally irrelevant since the orbit of a test particle cannot intersect the body.
The potential function has a hyperbolic cross section; the sudden dip in the center is the origin of the name "gravity well."
In a uniform gravitational field, the gravitational potential at a point is proportional to the height. Thus if the graph of a gravitational potential Φ(x,y) is constructed as a physical surface and placed in a uniform gravitational field so that the actual field points in the − Φ direction, then each point on the surface will have an actual gravitational potential proportional to the value of Φ at that point. As a result, an object constrained to move on the surface will have roughly the same equation of motion as an object moving in the potential field Φ itself. Gravity wells constructed on this principle can be found in many science museums.
There are several sources of inaccuracy in this model:
Consider an idealized rubber sheet suspended in a uniform gravitational field normal to the sheet. In equilibrium, the elastic tension in each part of the sheet must be equal and opposite to the gravitational pull on that part of the sheet; that is,
where k is the elastic constant of the rubber, is the upward displacement of the sheet (assumed to be small), g is the strength of the gravitational field, and is the mass density of the sheet. The mass density may be viewed as intrinsic to the sheet or as belonging to objects resting on top of the sheet. This equilibrium condition is identical in form to the gravitational Poisson equation
where is the gravitational potential and is the mass density. Thus, to a first approximation, a massive object placed on a rubber sheet will deform the sheet into a correctly shaped gravity well, and (as in the rigid case) a second test object placed near the first will gravitate toward it in an approximation of the correct force law. More generally, a collection of objects placed on the sheet will mutually gravitate in roughly the way predicted by Newton's law of gravitation.
This model is somewhat less suited to classroom demonstration than the rigid gravity well because a physical twodimensional rubber sheet will deform according to the twodimensional analogue of Newtonian gravity, which has a 1/r force law. To obtain the correct 1/r² force law, one needs a threedimensional rubber sheet bending into a fourth spatial dimension.
Both the rigid gravity well and the rubbersheet model are frequently misidentified as models of general relativity, due to an accidental resemblance to general relativistic embedding diagrams. In particular, the embedding diagram most commonly found in textbooks (an isometric embedding of a constanttime equatorial slice of the Schwarzschild metric in Euclidean 3space) 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.
Students of physics often have trouble conceiving of how the model of the gravity well is an acceleration vector field representation as opposed to an actual physical constant.^{[citation needed]}
The rubbersheet universe model both helps and hinders, in that the model posits a gravitational field pulling objects down into a rubber sheet to produce the gravity well effect (or the rubber sheet can be thought of as accelerating upward). The reality is somewhat different, and educators in physics often go to great pains to explain this^{[citation needed]}, although many students, even very advanced ones, struggle to shake off the paradigm of the rubber sheet.
