Quantitative rubber sheet models of gravitation wells using Spandex

Long a staple of introductory treatments of general relativity, the rubber sheet model exhibits Wheeler's concise summary---``Matter tells space-time how to curve and space-time tells matter how to move''---very nicely. But what of the quantitative aspects of the rubber sheet model: how far can the analogy be pushed? We show$^{1}$ that when a mass M is suspended from the center of an otherwise unstretched elastic sheet affixed to a circular boundary it exhibits a distortion far from the center given by h = A*(M*r$^{2})^{1/3}$ . Here, as might be expected, h and r are the vertical and axial distances from the center, but this result is not the expected logarithmic form of 2-D solutions to LaPlace's equation (the stretched drumhead). This surprise has a natural explanation and is confirmed experimentally with Spandex as the medium, and its consequences for general rubber sheet models are pursued. $^{1}$``The shape of `the Spandex' and orbits upon its surface'', \textit{American Journal of Physics}, \textbf{70}, 48-52 (2002), G. D. White and M. Walker. See also the comment by Don S. Lemons and T. C. Lipscombe, also in \textit{AJP}, \textbf{70, }1056-1058 (2002).