S15GeneralRelativity

General Relativity, a comparison between theoretical models and balls rolling on a warped spandex fabric

Joseph Svetlik And Daniel Landberg

In 1915 Einstein published the theory of General relativity shaping how we currently understand gravity. Unlike in Newtonian mechanics, the acceleration of objects in General Relativity models isn’t due to the force of one object on another due to their respective masses and directional separation; rather, in General Relativity, acceleration between two or more massive bodies is due to the warping of space-time by all relevant masses and energy distributions.

Purpose

Provide an analogy for students being introduced to the subject of general relativity. The concept of massive objects warping space can be confusing; the simple demonstration using spandex and spheres is a wonderful analogy.

Theory

To model the warped space of general relativity, a piece of spandex is stapled to a wooden ring on stilts, and a massive sphere is placed in the center.

Figure 1: Shows a view of the height, z, of the spandex surface with the radius R, and a top down view, displaying radius and angle phi

In this experiment, M is the center mass, σ is the spandex mass per area, z is the spandex height, and r is the radius.

The shape of the spandex being used must be determined so that Energy equations may be applied to the system. Spandex shape is determined by a Lagrangian of the spandex's and central masses total potential energy. The Energies are

for the elastic P.E. of the spandex, where

for the Center mass

for the spandex mass

Adding these terms up and taking the lagrangian with respect to radius lets us solve for z'(r).

Where either of the mass terms can dominate and cause the other to be neglible. In our case, the center mass far outweighed the spandex mass. Expanding the equation then gives

Using z(r) the elasticity of the fabric can be determined as well as the potential energy of a sphere at any radius. Using potential energy and kinetic energy over time, near circular orbits of any sphere can be modeled.

Data acquisition/analysis

Videos of orbits were taking by a video mounted directly above the spandex. These videos were analyzed using tracker to plot the position of the marble at each time frame of the video. Lag in the video capturing process was accounted for automatically by tracker. This data was transferred to excel for analysis in Mathematica.

A picture of the side view of the spandex warped by the center mass was loaded into imageJ where the slope of x in figure 1 was plotted for z and r coordinates. This data was transferred into excel for analysis in Mathematica.

figure 2: Graph o Kinetic Energy versus time on left, phi versus t on right. The solid lines are from fitted equations and the dots are data points.

X and y co-ordinates were used to calculate the radius and angle of the orbiting sphere. The radial velocity was determined and used to calculate the kinetic energy of the rolling sphere over time, which was then fitted to a decaying exponential function.

Angle over time was also determined and fitted to a growing exponential function.

The radius r of any near circular orbit then is

figure 3: Radius(meters) versus time. The solid lines are from fitted equations and the dots are data points.

Conclusions

Overall, the outcome of this experiment was a failure, as we did not find all the conclusions sought and were unable to do error analysis. However, we did find an equation that can be used as a base for future experiments like this should someone wish to go further.

References

O'Connor, J.J. and E.F. Robertson (1996), "General relativity". Mathematical Physics index, School of Mathematics and Statistics, University of St. Andrews, Scotland, May, 1996.

Comment on “The shape of ‘the Spandex’ and the orbits upon its surface,” by Gary D. White and Michael Walker [Am. J. Phys. 70 (1), 48–52 (2002)]

Circular orbits on a warped spandex fabric” - Chad A. Middleton, Michael Langstrom. [Am. J. Phys. 82, 287 (2014); doi 10.1119/1.4848635