Popular demonstrations commonly use stretched spandex fabric to illustrate the way in which curved spacetime mimics the force of gravity in general relativity. There are significant potential conceptual pitfalls to such an approach. In particular, it obscures the fact that most of what we ordinarily feel as gravity is due to the warping of time rather than space, a concept that is admittedly harder to demonstrate. Nevertheless, with appropriate caveats simulations of this kind can convey some of the wonder of Einstein’s theory to non-specialists.
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One way to try and correct this misinterpretation without abandoning demonstrations altogether is to appeal to the Newtonian limit. The premise of the spandex-sheet demonstration is that a marble follows a circular path around the central mass even though no force acts “through the fabric.” But general relativity, whatever else it may say, must reduce to Newton’s laws for weak fields like those prevailing in the solar system. Newton’s first law states that planets and marbles alike move on straight lines when no force acts. What then causes the motion of a marble (or planet) to deviate so strongly from straightness, if no force is acting? The answer is that its path is very close to straight in spacetime, not in space. The Earth, for instance, travels along a helix whose radius in space is only one astronomical unit, but each of whose spiral turns stretch across a light-year in time (i.e., the duration of one orbit, expressed in units of distance). The straightness of this trajectory can be conveyed to non-physicists by asking them to imagine a Slinky with 100 coils, stretched out until its length is 3 million times greater than its width (i.e., the ratio of 100 light-years to two astronomical units). There is still some curvature here, so the motion does violate Newton’s first law, but only slightly. It is this slight curvature of spacetime, rather than the gross spatial curvature suggested by the spandex, that mimics the force of gravity in Einstein’s theory. Its smallness reflects the weak gravitational field of the Sun.
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Similar demonstrations can be seen online in several places, including those by Steve Gould (https://www.youtube.com/watch?v=dw7U3BYMs4U), LIGO-Caltech (https://www.youtube.com/watch?v=YfSyhcFu_MM), the Arvin Gottlieb Planetarium (https://www.youtube.com/watch?v=wnWmGr_523s), and Benjamin Giblin and Ben Morton at the University of Edinburgh (https://www.youtube.com/watch?v=T6B1U-5oAp4).
19.
Spandex fabric in the size used here can be ordered online, and the other components for our demonstration found in any large hardware store, for a total cost of less than $100.
20.
We note that the sound speed cs found in this way is an average. The actual speed of waves in the fabric may depend on distance from the center. This could perhaps be determined using a camera facing down onto the spandex from above, in combination with the tracking feature of PASCO Capstone. In general, the propagation of two-dimensional elastic waves in a circular membrane is mathematically challenging (more so than the rectangular case, which is a simple extension of the linear string model that most students are familiar with). Another, more indirect way to determine the mean sound speed might be to use a PASCO force probe to measure the tension force per length τ in the fabric (in N/m or kg/s2) and weigh a sample of known area to obtain the surface mass density σ (in kg/m2). Then cs = √(τ/σ).
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).© 2018 American Association of Physics Teachers.
2018
American Association of Physics Teachers
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