Carleton College Carleton Digital Commons Faculty Work Physics and Astronomy 2012 Teaching general relativity to undergraduates Nelson Christensen Carleton College Thomas Moore Pomona College Follow this and additional works at: https://digitalcommons.carleton.edu/phys_faculty Part of the Physics Commons Recommended Citation Christensen, Nelson, and Thomas Moore., "Teaching general relativity to undergraduates". Physics Today, vol. 65, no. 6, 2012. Available at: https://doi.org/10.1063/PT.3.1605. [Online]. Accessed via Faculty Work. Physics and Astronomy. Carleton Digital Commons. https://digitalcommons.carleton.edu/phys_faculty/ 11 The definitive version is available at https://doi.org/10.1063/PT.3.1605 This Article is brought to you for free and open access by the Physics and Astronomy at Carleton Digital Commons. It has been accepted for inclusion in Faculty Work by an authorized administrator of Carleton Digital Commons. For more information, please contact [email protected]. Teaching general relativity to undergraduates Nelson Christensen and Thomas Moore Inspired by new results eneral relativity lies at the heart of a nomena. The Laser Interfe- in cosmology and astro- wide variety of exciting astrophysical rometer Gravitational-Wave physics, undergraduates and cosmological discoveries made Observatory and the Virgo in- are increasingly eager during the past decade or so. terferometer are expected to to learn about general Whereas in 1919 Arthur Eddington see gravitational waves within relativity. A number of Gwas allegedly at a loss to name a third person who a few years. That discovery innovative textbooks understood general relativity, presently its knowl- will be an important confirma- edge and use are widespread. We have come a long tion of general relativity in its make it easier than ever way in a century. own right; moreover, the grav- before to satisfy that Perhaps the most compelling of the new results is itational waves will provide a demand. the 1998 discovery that not only is the universe ex- new means to view the uni- panding, but the expansion is accelerating. The 2011 verse and will reveal relativistic events in regions Nobel Prize in Physics was awarded for observations shielded from electromagnetic observation. General of high-redshift supernovae whose distance and relativity is also important close to home. Gravity brightness revealed the accelerated expansion and Probe B, which had been orbiting Earth, recently ob- therefore showed that the cosmos is filled with “dark served frame-dragging and geodetic effects predicted energy” (see PHYSICS TODAY, December 2011, page 14). by general relativity. The global positioning system, Observations of the cosmic microwave background which students regularly use, requires general rela- have also provided evidence for dark energy—and for tivity to ensure its meter-scale accuracy (see the article nonluminous dark matter. Because the revolution in by Neil Ashby in PHYSICS TODAY, May 2002, page 41). the scientific community’s understanding of the uni- In a way that was not true even two decades ago, gen- verse has been widely reported, undergraduate stu- eral relativity has become an issue of practical concern dents are interested in learning more about cosmology. to mainstream physicists and even engineers.1 New astrophysical observations are also piquing students’ curiosity. Astronomers observe supermas- Nelson Christensen ([email protected]) is a professor of physics at sive black holes in the centers of galaxies, including Carleton College in Northfield, Minnesota. Thomas Moore our own Milky Way. We now recognize that neutron ([email protected]) is a professor of physics at Pomona College in stars are abundant in our galaxy and that they provide Claremont, California. the basis for extreme and intriguing astrophysical phe- Albert Einstein’s last blackboard, at the Institute for Advanced Study, Princeton, New Jersey, 1955. (Photograph by Alan Richards, courtesy of the AIP Emilio Segrè Visual Archives.) [[[This articlewww.physicstoday.org is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. June 2012 Physics Today 41Downloaded to ]]] IP: 137.22.11.134 On: Tue, 19 Nov 2013 21:48:39 Teaching relativity The popular press has reported on all of the has been considered prohibitively difficult. The full above. Not surprisingly, undergraduate students theory of general relativity is based on the concepts are taking notice and wanting to better understand of differential geometry, most often expressed in the physics. They want to learn general relativity to the language of tensor calculus; it thus involves engage the science both in the classroom and mathematics beyond what most undergraduates through research projects. encounter. Increasingly, college and university teachers are We have identified four distinct approaches working to create appropriate courses in general rel- that textbook authors have used to address that dif- ativity for undergraduate physics majors, aided by a ficulty. We call them number of textbooks that offer new strategies for suc- ‣ The adjusted math-first approach. cessfully introducing the subject at a reasonable pace ‣ The calculus-only approach. and level. Indeed, our experience is that such a course ‣ The physics-first approach. need not be limited to the most gifted students; un- ‣ The intertwined + active-learning approach. dergraduates at the level of junior or senior physics majors are generally quite capable of learning general The adjusted math-first approach was the first relativity in satisfying depth. In fact, we encourage to be developed. It adopts the same basic outline all institutions offering undergraduate physics de- as a graduate-level course, including an early-on grees to seriously consider providing a semester- full treatment of tensor calculus and the mathe- matics of curved spacetime. But it adjusts the pres- length general relativity course for their students. entation of that mathematics to be more appropri- And to help make that happen, we will share several ate for mature undergraduates. Excellent texts of pedagogical strategies and describe our and others’ that type include A First Course in General Relativ- experiences of student success with those strategies. ity by Bernard Schutz and Gravitation and Space- Teaching approaches time by Hans Ohanian and Remo Ruffini. (For bib- liographic information for these and all general Historically, the reason general relativity has not relativity textbooks cited in this article, see the box been taught to undergraduates is that the subject on page 44.) Almost all undergraduate general relativity texts published in the 21st century turn the focus de- ADJUSTED INTERTWINED + cisively away from the math and toward the physics. MATH-FIRST PHYSICS-FIRST ACTIVE-LEARNING But even within that common philosophy, the math- (Schutz, 1985) (Hartle, 2003) (Moore, 2012) ematical challenge remains, and it is handled differ- Review of Geometry as Overview and special ently in each of the remaining three approaches. special relativity physics(6%) relativity (9%) The calculus-only approach follows an inno- (11%) Newtonian gravity Index notation, metrics, vative trajectory developed in the final decades of and special relativity basic tensors (with the 20th century and described by Edwin Taylor Vectors andtensors (14%) some applications), (13%) and John Wheeler in their book Exploring Black (16%) Gravity as geodesics geometry (5%) Holes. In the calculus-only approach, spacetime metrics are given, not derived, and the focus is on Metrics and Schwarzschild metric Fluids and fluxes geodesics (10%) applications (also extracting the implications of the geodesic equa- (8%) builds math skills) tion for those metrics. Students who have com- (21%) pleted an introductory calculus-based physics Curved spaces Schwarzschild metric course can explore many of the most interesting (19%) applications Calculus of physical consequences of general relativity with- (20%) curvature (8%) out tensors or even multivariable calculus. The physics-first approach is perhaps best ex- Gravity as geometry and Einstein equation(10%) Spinning objects emplified by James Hartle’s innovative 2003 text, Einstein equation (9%) (6%) Gravity. Like the calculus-only approach, it fo- Gravitational waves (3%) Gravitational waves Cosmology cuses on working through the implications of (11%) (13%) given metrics, but at the mathematical level of Cosmology (13%) most junior and senior undergraduates. (It does include some gently developed tensor calculus.) Gravitational waves Spherical stars (optional) (13%) Hartle’s extraordinary breadth of knowledge con- (20%) More math (10%) cerning applications of general relativity helps to make the physical examples in his text extraordi- (optional) Gravitomagnetism Applications of the and Kerr solution narily rich and varied. Although Hartle’s book Cosmology strongly emphasizes the physics, its final chapters (6%) Einstein equation (13%) (13%) provide the full mathematics required to under- stand the Einstein equation; however, Hartle has Figure 1. The relative emphasis of physics and mathematics and their designed his book so that those mathematical sec- positioning in exemplary texts for the adjusted math-first, physics-first, tions can be omitted. and intertwined + active-learning approaches. All three texts are designed The last of the approaches, intertwined + ac- for a one-semester, upper-level