An introductory approach to black holes is presented along with astronomical observational data pertaining to the presence of a supermassive black hole at the center of our galaxy. Concepts of conservation of energy and Kepler's third law are employed so students can apply formulas from their physics class to determine the mass of the black hole that resides in the center of the Milky Way.
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Many years ago (1979) at a UNC-Chapel Hill workshop, Hendrik van Dam derived this formula for a group of visiting introductory teachers. He facetiously stated that we used the “wrong formulas” (i.e., nonrelativistic) for kinetic energy and potential energy, and our two mistakes canceled to get the correct general-relativistic result!
4.
Technically in general relativity, one must interpret the r-coordinate as that computed from the circumference of a sphere surrounding the center. Due to the curvature of space, the directly measured radial distance between two nested spheres is larger than the difference found using the general-relativistic r-coordinate. See E. F. Taylor and J. A. Wheeler, Exploring Black Holes: Introduction to General Relativity (Addison-Wesley Longman, San Francisco, 2000), Chap. 2, pp. 7–11, 28.
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A situation where Kepler's third law needs to be modified is a binary system of black holes orbiting near each other. A nice diagram showing deviations from Kepler's third law in such an extreme case is found in the otherwise advanced paper
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© 2008 American Association of Physics Teachers.
2008
American Association of Physics Teachers
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