Gravity Well - The Rubber-sheet Model

The Rubber-sheet 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, h(x) is the upward displacement of the sheet (assumed to be small), g is the strength of the gravitational field, and ρ(x) 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 Φ(x) is the gravitational potential and ρ(x) 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.