Teaching general relativity to undergraduatesNelson Christensen, and Thomas Moore

Citation: Physics Today 65, 6, 41 (2012); doi: 10.1063/PT.3.1605View online: https://doi.org/10.1063/PT.3.1605View Table of Contents: https://physicstoday.scitation.org/toc/pto/65/6Published by the American Institute of Physics

ARTICLES YOU MAY BE INTERESTED IN

What every physicist should know about string theoryPhysics Today 68, 38 (2015); https://doi.org/10.1063/PT.3.2980

A General Relativity WorkbookPhysics Today 67, 54 (2014); https://doi.org/10.1063/PT.3.2387

Gravity: An Introduction to Einstein’s General RelativityPhysics Today 58, 52 (2005); https://doi.org/10.1063/1.2405550

Has elegance betrayed physics?Physics Today 71, 57 (2018); https://doi.org/10.1063/PT.3.4022

Side trips on the road to fusionPhysics Today 72, 28 (2019); https://doi.org/10.1063/PT.3.4131

Introduction to General Relativity, Black Holes, and CosmologyPhysics Today 68, 51 (2015); https://doi.org/10.1063/PT.3.2983

General relativity lies at the heart of awide variety of exciting astrophysicaland cosmological discoveries madeduring the past decade or so.Whereas in 1919 Arthur Eddington

was allegedly at a loss to name a third person whounderstood general relativity, presently its knowl-edge and use are widespread. We have come a longway in a century.

Perhaps the most compelling of the new results isthe 1998 discovery that not only is the universe ex-panding, but the expansion is accelerating. The 2011Nobel Prize in Physics was awarded for observationsof high-redshift supernovae whose distance andbrightness revealed the accelerated expansion andtherefore showed that the cosmos is filled with “darkenergy” (see PHYSICS TODAY, December 2011, page 14).Observations of the cosmic microwave backgroundhave also provided evidence for dark energy—and fornonluminous dark matter. Because the revolution inthe scientific community’s understanding of the uni-verse has been widely reported, undergraduate stu-dents are interested in learning more about cosmology.

New astrophysical observations are also piquingstudents’ curiosity. Astronomers observe supermas-sive black holes in the centers of galaxies, includingour own Milky Way. We now recognize that neutronstars are abundant in our galaxy and that they providethe basis for extreme and intriguing astrophysical phe-

nomena. The Laser Interfe-rometer Gravitational-WaveObservatory and the Virgo in-terferometer are expected tosee gravitational waves withina few years. That discoverywill be an important confirma-tion of general relativity in itsown right; moreover, the grav-itational waves will provide anew means to view the uni-verse and will reveal relativistic events in regionsshielded from electromagnetic observation. Generalrelativity is also important close to home. GravityProbe B, which had been orbiting Earth, recently ob-served frame-dragging and geodetic effects predictedby general relativity. The global positioning system,which students regularly use, requires general rela-tivity to ensure its meter-scale accuracy (see the articleby Neil Ashby in PHYSICS TODAY, May 2002, page 41).In a way that was not true even two decades ago, gen-eral relativity has become an issue of practical concernto mainstream physicists and even engineers.1

www.physicstoday.org June 2012 Physics Today 41

Albert Einstein’s last blackboard, at the Institutefor Advanced Study, Princeton, New Jersey, 1955.(Photograph by Alan Richards, courtesy of the AIPEmilio Segrè Visual Archives.)

Nelson Christensen and Thomas Moore Inspired by new resultsin cosmology and astro-physics, undergraduatesare increasingly eagerto learn about generalrelativity. A number ofinnovative textbooksmake it easier than everbefore to satisfy thatdemand.

Teaching general relativity

to undergraduates

Nelson Christensen (nchriste@carleton.edu) is a professor of physics atCarleton College in Northfield, Minnesota. Thomas Moore

(tmoore@pomona.edu) is a professor of physics at Pomona College inClaremont, California.

The popular press has reported on all of theabove. Not surprisingly, undergraduate studentsare taking notice and wanting to better understandthe physics. They want to learn general relativity toengage the science both in the classroom andthrough research projects.

Increasingly, college and university teachers areworking to create appropriate courses in general rel-ativity for undergraduate physics majors, aided by anumber of textbooks that offer new strategies for suc-cessfully introducing the subject at a reasonable paceand level. Indeed, our experience is that such a courseneed not be limited to the most gifted students; un-dergraduates at the level of junior or senior physicsmajors are generally quite capable of learning generalrelativity in satisfying depth. In fact, we encourageall institutions offering undergraduate physics de-grees to seriously consider providing a semester-length general relativity course for their students.And to help make that happen, we will share severalpedagogical strategies and describe our and others’experiences of student success with those strategies.

Teaching approachesHistorically, the reason general relativity has notbeen taught to undergraduates is that the subject

has been considered prohibitively difficult. The fulltheory of general relativity is based on the conceptsof differential geometry, most often expressed inthe language of tensor calculus; it thus involvesmathematics beyond what most undergraduatesencounter.

We have identified four distinct approachesthat textbook authors have used to address that dif-ficulty. We call them

‣ The adjusted math-first approach.‣ The calculus-only approach.‣ The physics-first approach.‣ The intertwined + active-learning approach.The adjusted math-first approach was the first

to be developed. It adopts the same basic outlineas a graduate-level course, including an early-onfull treatment of tensor calculus and the mathe-matics of curved spacetime. But it adjusts the pres-entation of that mathematics to be more appropri-ate for mature undergraduates. Excellent texts ofthat type include A First Course in General Relativ-ity by Bernard Schutz and Gravitation and Space-time by Hans Ohanian and Remo Ruffini. (For bib-liographic information for these and all generalrelativity textbooks cited in this article, see the boxon page 44.)

Almost all undergraduate general relativitytexts published in the 21st century turn the focus de-cisively away from the math and toward the physics.But even within that common philosophy, the math-ematical challenge remains, and it is handled differ-ently in each of the remaining three approaches.

The calculus-only approach follows an inno-vative trajectory developed in the final decades ofthe 20th century and described by Edwin Taylorand John Wheeler in their book Exploring BlackHoles. In the calculus-only approach, spacetimemetrics are given, not derived, and the focus is onextracting the implications of the geodesic equa-tion for those metrics. Students who have com-pleted an introductory calculus-based physicscourse can explore many of the most interestingphysical consequences of general relativity with-out tensors or even multivariable calculus.

The physics-first approach is perhaps best ex-emplified by James Hartle’s innovative 2003 text,Gravity. Like the calculus-only approach, it fo-cuses on working through the implications ofgiven metrics, but at the mathematical level ofmost junior and senior undergraduates. (It doesinclude some gently developed tensor calculus.)Hartle’s extraordinary breadth of knowledge con-cerning applications of general relativity helps tomake the physical examples in his text extraordi-narily rich and varied. Although Hartle’s bookstrongly emphasizes the physics, its final chaptersprovide the full mathematics required to under-stand the Einstein equation; however, Hartle hasdesigned his book so that those mathematical sec-tions can be omitted.

The last of the approaches, intertwined + ac-tive-learning, is based on the experimentally sup-ported hypothesis that junior and senior under-graduates can indeed learn the tensor mathematicsneeded to fully understand general relativity—if the

42 June 2012 Physics Today www.physicstoday.org

Teaching relativity

PHYSICS-FIRST(Hartle, 2003)

Geometry as(6%)physics

Newtonian gravityand special relativity

(14%)

Gravity asgeometry (5%)

Metrics andgeodesics (10%)

Schwarzschild metricapplications

(20%)

Spinning objects(6%)

Gravitational waves (3%)

Cosmology (13%)

(optional)More math (10%)

(optional)Applications of the

Einstein equation(13%)

INTERTWINED +ACTIVE-LEARNING

(Moore, 2012)

Overview and specialrelativity (9%)

Index notation, metrics,basic tensors (withsome applications),

geodesics (13%)

Schwarzschild metricapplications (alsobuilds math skills)

(21%)

Calculus ofcurvature (8%)

Einstein equation(10%)

Cosmology(13%)

Gravitational waves(13%)

Gravitomagnetismand Kerr solution

(13%)

ADJUSTEDMATH-FIRST(Schutz, 1985)

Review ofspecial relativity

(11%)

Vectors andtensors(16%)

Fluids and fluxes(8%)

Curved spaces(19%)

Gravity as geometry andEinstein equation (9%)

Gravitational waves

Spherical stars(20%)

Cosmology(6%)

(11%)

Figure 1. The relative emphasis of physics and mathematics and theirpositioning in exemplary texts for the adjusted math-first, physics-first,and intertwined + active-learning approaches. All three texts are designedfor a one-semester, upper-level course. Blue sections are primarily physicsoriented, yellow sections are primarily math oriented, and green sectionsblend some of each. Cited percentages are based on page counts.

instructor develops the math slowly, on an as-needed basis thoroughly intertwined with thephysics; presents that math at a level really suitedto undergraduates; and uses active-learning tech-niques to ensure that the students get the individualpractice needed to own both the math and thephysics. As in the calculus-only and physics-firstapproaches, the physics is strongly emphasizedover the math. The approach is exemplified by thesoon to be published A General Relativity Workbookby one of us (Moore).

Our four categories are admittedly a bit fuzzyand may not always capture the features of individ-ual textbooks. Still, we think that they highlight im-portant differences worth understanding. Figures 1and 2 illustrate some of those differences. The firstsummarizes and contrasts the exemplary texts forthe three approaches that target upper-level under-graduates. Because the calculus-only approach isqualitatively different, we have not included Taylorand Wheeler’s text. The second figure displays anumber of books, including some calculus-only andgraduate texts, in terms of their degree of mathe-matical sophistication and willingness to delay pre-senting the physics.

When we listed the four approaches, we pre-sented them in approximate chronological order ofdevelopment. However, even though later booksreact to what has gone before, one should not take“later” to be the same as “better.” Rather, the ap-proaches each widen the potential audience; to-gether they support courses for undergraduateswith a broad range of interests and abilities. Manyof our colleagues have reported satisfaction andsuccess with whichever approach they chose.

Adjusted math-firstTraditionally, graduate instruction in general rela-tivity has followed a math-first approach. In the1970s and 1980s, Ohanian and Schutz offered thefirst serious attempts to follow that traditional ap-proach to target undergraduates. The material intheir books mostly parallels the classic graduate-level texts of the era, but the authors treat that ma-terial with substantially less rigor without, inSchutz’s words, “watering down the subject mat-ter.” With the mathematics established, the fieldequations can be derived and then solved for casesof astrophysical interest. The application of thetheory to physical systems comes at the end. Ohan-ian and Ruffini have developed an interesting vari-ant on the adjusted math-first approach: They ini-tially concentrate on a linearized approximation togeneral relativity, study some applications in low-curvature situations, and then move on to the fullgeneral-relativistic theory. Their variant allows foran early discussion of gravitational waves andlight deflection.

We found relatively few undergraduatecourses in the US that currently use the adjustedmath-first approach, and a substantial fraction ofthose are taught in mathematics departments, inwhich emphasizing the underlying mathematicalconcepts before turning to physics makes goodsense. The approach is also used in physics courses

designed for both senior undergraduates and begin-ning graduate students.

Calculus-onlyThe textbook that first developed, and most vividlyexemplifies, the calculus-only approach is Taylorand Wheeler’s Exploring Black Holes. Figure 3 givesan overview of its basic approach and topics; Tay-lor’s own description of its goal is more succinct:“Undergraduate general relativity developed nearthe black hole using calculus, no tensors.” The bookmakes no attempt to develop the full mathematicalmachinery of general relativity or the Einstein equa-tion. Rather, it explores given metrics, teaching stu-dents how to extract meaning from them and to pre-dict the motion of particles in their spacetimes. Thefocus, instead of being on equations, is very much ondeveloping a student’s conceptual understanding.

Even so, Taylor and Wheeler, through cleverand judicious simplifications, manage to calculategeodesics with only simple calculus and the princi-ple of maximal aging—the postulate that a free par-ticle follows the world line of maximum propertime. Although Taylor and Wheeler give metricswithout derivation, they work out the physical im-plications of those metrics carefully and completely.Their approach is similar to and roughly at the samelevel as the treatment of the Schrödinger equationin a typical modern physics course in which stu-dents do not see the equation’s deep roots as a po-sition-basis expression of the system’s Hamiltonianoperator but do get some motivation as to why itmight make sense and then work out implicationsfirst in simple one-dimensional situations and thenin successively more complicated cases.

Taylor and Wheeler’s book, like their special-relativity classic Spacetime Physics,2 provides

www.physicstoday.org June 2012 Physics Today 43

MATH LEVEL

GRADUATE

UPPER-LEVEL

UNDERGRAD

LOWER-LEVEL

UNDERGRAD

CALCULUS-ONLY PHYSICS-FIRST

INTERTWINED+ACTIVE-LEARNING

ADJUSTED MATH-FIRST

GRADUATE

Wald, 1984Misner, Thorne, & Wheeler, 1973

Weinberg, 1972

Ryder, 2009Carroll, 2004Schutz, 1985

Ohanian& Ruffini, 1994

Cheng, 2005Moore, 2012

Hartle, 2003

Henriksen, 2011Taylor & Wheeler, 2000

Schutz, 2003DELAY IN THE PHYSICS

Lambourne, 2010

Figure 2. The main sequence. General relativity textbooks differ withregard to their level of mathematical sophistication and how long ittakes for them to get to physical applications beyond special relativity.This is a limited sampling, and locations are somewhat subjective; still,they may provide for useful comparisons.

amazing physical insights at an introductory level.It is especially good about making a sharp distinctionbetween the arbitrary nature of global coordinatesand things that a local observer can actually meas-ure. Most other books do not draw that distinctionnearly so well.

We corresponded with a number of facultymembers who used Exploring Black Holes in coursesfor undergraduates who had only a basic back-ground in introductory physics and calculus. Someof those courses used Spacetime Physics as well, toprovide students with a solid introduction to bothspecial and general relativity. Several instructorstook advantage of the projects that the book sug-gests to give students a chance to explore some-thing on their own. All reported how exhilaratedstudents were to be seriously exploring the impli-cations of relativity. In various ways, our corre-spondents stated that the book provides an excel-lent opportunity for students with a minimalphysics and math background to discover a lot ofinteresting physics and that it gives students

strong motivation for further study.Several later books have, in quite different

ways, targeted audiences with calculus or even pre-calculus math skills. One of the most ambitious andinsightful is Schutz’s Gravity from the Ground Up,which is suitable for students interested in selfstudy or for a course geared toward students notmajoring in physics or math.

Physics-firstIt was Hartle who coined the term “physics first,”to describe his own approach, summarized in fig-ure 1, to teaching general relativity to undergradu-ates.3 Thanks in large part to his work, the physicseducation community has devoted much attentionto that approach. A physics-first class would usu-ally be at the junior- or senior-year level and wouldrequire higher math skills than needed for a calcu-lus-only course. As in a typical upper-division elec-tricity and magnetism course, students in aphysics-first course use the additional math todelve deeper into the subject.

Nonetheless, the course emphasizes concepts,and a goal is to not go too deep into the tensor cal-culus—at least, not right away—but rather to get tothe interesting physics quickly. A geometric presen-tation of special relativity given early in the coursehighlights the metric and invariant quantities andmotivates the general relativistic physics. The textintroduces important metrics, rather than derivingthem, and then uses those metrics for an in-depthexploration of such important physical phenomenaas particle orbits and the deflection of light rays.The results from those calculations are comparedwith experimental results and astrophysical obser-vations; indeed, the connection to current experi-ments and observations provides strong encour-agement for the students. The final part of thecourse can be dedicated to motivating the Einsteinequation and solving it for the spacetime geome-tries previously discussed.

We have found that the majority of undergrad-uate general relativity classes taught in physics de-partments use the physics-first approach. The teach-ers for those classes at a wide variety of institutionsreport very good outcomes and positive feedbackfrom students. Some of the students in a physics-first class will be energized to pursue general rela-tivity further, so teachers using that approachshould be prepared to provide further references.

Intertwined + active-learningThere are good reasons to give students a full intro-duction to tensor calculus. In our experience, stu-dents often feel that they are on somewhat shakyground in a course that focuses on results such asmetrics, gravitational-wave formulas, and gravito-magnetic phenomena but that does not provide thefoundation and context for those results. But de-signing a course that presents the full tensor calcu-lus is no easy task. Making undergraduates com-fortable with the math takes time and effort, butdelaying the gratification of interesting physics formany weeks would sap the students’ and instruc-tor’s motivation. In addition, students need time for

44 June 2012 Physics Today www.physicstoday.org

Teaching relativity

Some recommended relativity textsMany texts can form the basis for an excellent undergraduatecourse in relativity. Here are some that we recommend, alongwith a smattering of classic graduate-level books.

‣ S. Weinberg, Gravitation and Cosmology: Principlesand Applications of the General Theory of Relativity,Wiley, New York (1972).

‣ C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation,W. H. Freeman, San Francisco (1973).

‣ R. M. Wald, General Relativity, U. Chicago Press,Chicago (1984).

‣ B. F. Schutz, A First Course in General Relativity, Cam-bridge U. Press, New York (1985; 2nd ed. 2009).

‣ H. C. Ohanian, R. Ruffini, Gravitation and Spacetime,2nd ed. Norton, New York (1994). The first edition waswritten by Ohanian alone in 1976.

‣ E. F. Taylor, J. A. Wheeler, Exploring Black Holes: Intro-duction to General Relativity, Addison Wesley Long-man, San Francisco (2000).

‣ J. B. Hartle, Gravity: An Introduction to Einstein’s Gen-eral Relativity, Addison-Wesley, San Francisco (2003).

‣ B. F. Schutz, Gravity from the Ground Up, Cambridge U.Press, New York (2003).

‣ S. Carroll, Spacetime and Geometry: An Introduction toGeneral Relativity, Addison-Wesley, San Francisco(2004).

‣ T.-P. Cheng, Relativity, Gravitation and Cosmology: ABasic Introduction, Oxford U. Press, New York (2005,2nd ed. 2010).

‣ L. Ryder, Introduction to General Relativity, CambridgeU. Press, New York (2009).

‣ R. J. A. Lambourne, Relativity, Gravitation and Cos-mology, Cambridge U. Press, New York (2010).

‣ R. N. Henriksen, Practical Relativity: From First Princi-ples to the Theory of Gravity, Wiley, Chichester, UK (2011).

‣ T. A. Moore, A General Relativity Workbook, Univer-sity Science Books, Sausalito, CA (in press). Seehttp://pages.pomona.edu/~tmoore/grw.

the nonintuitive concepts of general relativity andthe dizzying new tensor notation to sink in. Inter-weaving the math and physics throughout thecourse is one way to meet the challenge. That ap-proach is exemplified by Moore’s A General Relativ-ity Workbook, summarized in figure 1.

Even when appropriately spread out, the math-ematics presents real difficulties for undergradu-ates. The intertwined + active-learning approachaddresses that particular challenge by pushing stu-dents to work out the math (and physics) them-selves, so that they come to own the math in a waythat would not otherwise be possible. Each chapterin A General Relativity Workbook is meant to corre-spond to a single class day and typically consists offour pages of text that provide an accessible concep-tual overview of the day’s material without gettingsidetracked by derivations or other detailed argu-ments. The students themselves, guided by cues,work out all derivations and details in a series ofboxes. Teachers can thus devote class time to ad-dressing their students’ specific difficulties.

Both of us have used the intertwined + active-learning approach, and we usually begin a class ses-sion by asking students which box exercises weremost challenging. We then invite selected studentsto the board to present their work on those challeng-ing exercises. Discussions typically follow, in whichstudents exchange ideas and techniques and weoffer guidance. That format allows us to give stu-dents feedback and help exactly where it is needed.Often we have time to work example homeworkproblems or discuss the physical and mathematicalissues raised in the chapter.

We have found that active learning enables stu-dents to progress much farther and become muchmore confident than do the more passive ap-proaches we have used. In fact, we regard activelearning as a crucial part of the intertwined ap-proach and a technique by which students in thesame target audience as the physics-first approachcan enjoy the benefits of increased mathematical so-phistication. But it is only suitable for instructorscomfortable with active-learning methods.

Tools for successBetween the two of us, we have taught undergrad-uate general relativity courses about 20 times andhave used all four of the approaches discussedabove. We have learned a great deal about whatworks and what does not. Here we share some ideasand resources for you to keep in mind when devel-oping an undergraduate general relativity coursebased on any of the approaches.

Choosing appropriate prerequisites is impor-tant. Most professors using the calculus-only ap-proach require the equivalent of one year of intro-ductory physics and one year of introductorycalculus. The other approaches require more back-ground. In our experience, and in that of most of ourcorrespondents, vector calculus is an absolute min-imum. In addition, to develop the appropriate levelof physics sophistication, students will need a classthat goes beyond the typical sophomore modern-physics course.

Undergraduates gain valuable experience fromlaboratory-based classes in which they learn tomake measurements and tie the results to underly-ing physics. When teaching general relativity, we al-ways have the students ask themselves what theycan measure. Class examples and homework prob-lems should cause students to think constantlyabout taking measurements with meter sticks andclocks in a stationary or freely falling laboratory orin a distant observatory. Like Albert Einstein him-self, your students will have trouble understandingthat the spacetime coordinates involved in globalcoordinate systems have no meaning other thanwhat the metric gives them. Therefore, as empha-sized by Taylor and Wheeler (see figure 3), it is im-portant to help students draw the distinction be-tween global coordinates and real physicalmeasurements performed in a local laboratory. For-tunately, that emphasis is more or less built into thecalculus-only, physics-first, and intertwined + ac-tive-learning approaches.

The emphasis on measurement need not be lim-ited to theory. Your undergraduate general relativitycourse will almost certainly not have an accompany-ing laboratory class. Nonetheless, you can connectsome well-known undergraduate experiments withthe course and provide opportunities for demonstra-tions or projects in which students actually take data.Measuring the speed of light is always instructive.Determining the muon lifetime illustrates how sub-atomic particles can be used as clocks that measureproper time. Many smartphones have apps that dis-play readings from the device’s three-axis ac-celerometer; throwing the smartphone onto a softcushion and then analyzing the logged data can helpclarify the concept of a freely falling reference

www.physicstoday.org June 2012 Physics Today 45

SPECIAL RELATIVITY

EINSTEIN'S FIELD EQUATIONS

THIS BOOK

"Think globally; measure locally."

KEY IDEAS: The metric describes spacetime. The Principle of Maximal Aging describes motion. Make every measurement and observation in an inertial frame --- a local inertial frame when in curved spacetime.

LOCAL METRICis in local "frame" coordinates,which we choose to be inertial.

EVERY MEASUREMENT IS LOCALBut local observer has no global view.

GLOBAL METRICis in global "map" coordinates,which we choose (almost) arbitrarily.

NO SINGLE OBSERVER MEASURES MAP QUANTITIES

THIS BOOK

TOPICSNONSPINNING BLACK HOLE RAIN OBSERVER PANORAMASGLOBAL POSITIONING SYSTEM GRAVITATIONAL MIRAGESINSIDE THE BLACK HOLE SPINNING BLACK HOLE WHY DO THINGS FALL ON EARTH? INSIDE THE SPINNING BLACK HOLEORBITING STONE GRAVITATIONAL WAVESMERCURY'S PERIHELION ADVANCE EXPANDING UNIVERSEGLOBAL LIGHT BEAMS COSMOLOGY

CIRTEM EHT GNIVIRED SAMARONAP RETIBRO ,LLEHS

Principle of Maximal AgingA stone's worldline has maximum wristwatch time between adjacent events. This leads to constants of motion, such as map energy and map angular momentum, which we use to predict global orbits.

Spacetime is globally curved.

THIS BOOK

.

Figure 3. The scope, core principles, and topics

covered in Edwin Taylor and John Wheeler’s calculus-only text, Exploring Black Holes. This summary is the anticipated back cover of the forthcoming second edition. (Courtesy of Edwin Taylor.)

frame.4 Students can use a relatively inexpensiveradio telescope to actually measure the galactic rota-tion curve and so discover that our galaxy containsdark matter.5 Although we are unaware of relativityexperiments that use the global positioning system,the many relativistic effects underlying GPS makefor interesting study and discussion.1

Computer advances have made it possible forundergraduates to do calculations and explorephysical problems that would have been muchmore daunting even a decade ago. Hartle hasposted some Mathematica-based programs on thewebpage for his textbook.6 With them a student caneasily generate Christoffel symbols and curvaturetensors for any desired metric. Free computer pro-grams on various sites let students calculate orbitsin the Schwarzschild and Kerr geometries, plot ex-pansion rates for the universe for different cosmo-logical models,7,8 or explore how physicists esti-mate the amount of dark matter in a galaxy.9

Several books, including Schutz’s Gravity fromthe Ground Up and Moore’s Workbook, explicitly dis-cuss how to construct simple numerical models.Figure 4 shows the output for a spreadsheet-basedmodel of cosmic expansion that includes realisticamounts of vacuum energy (the cosmological-constant form of dark energy), matter, and radiation.We have found such assignments extremely valu-able in helping students make predictions about ouruniverse, as distinguished from the toy models usu-ally explored in textbooks. Just as important, suchexercises help students learn about the process ofconstructing numerical models. Moreover, bywrestling with such models, students understand

the physics better: For one thing, they have to learnhow to recognize when an incorrect model is pro-ducing unphysical results.

Don’t spare the drillThe mathematics involved in general relativity isnot that much different from what students have en-countered in their studies of vectors. In a sense, ten-sors are just big vectors, and we have found it valu-able to make as many connections as possible tovector calculus. But the notation truly does look dif-ferent, and in any approach other than calculus-only, students need to become comfortable with thatdifference. Spend some time asking the students toidentify free and summed indices, write out the im-plied sums in an equation, check that the free in-dices on both sides of an equation are consistent, re-name indices in an appropriate way to pull out acommon factor, and identify nonsensical equations.It may seem silly to drill students on such basicthings. Our experience, however, is that spendingtime with such simple drills and addressing stan-dard errors when first teaching the notation reallydo improve student competence and confidence.Few textbooks, in our opinion, provide sufficient at-tention to such drills.

We have also found 2D visualizations to behelpful. Students should practice applying everynew tensor concept to easily visualized 2D flat andcurved spaces before using it in a 4D spacetime.They can work with polar coordinates in flat spaceand longitude–latitude coordinates on a curvedsphere, then move on to stranger coordinate sys-tems for flat space and other types of curved spaces.

46 June 2012 Physics Today www.physicstoday.org

Teaching relativity

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 2 4 6 8 10 12 14 16 18 20 22 24

AGE OF UNIVERSE (gigayears)

a/a

no

w

Define (in Gy , , ,= 0

0

½ [ + ] /Iterate:

= + 2―

= +

H ,

a

t

a ‾

t a a H

a

t t

0-1

m v r

0

0

1 r

1 1 0 0

+ 1

+ 1

Ω Ω Ω Δ

Δ √Ω

Δ √

η

η

η

η

)

===

―――――――

½ [ + ] /

‾

a

a a H

Δ

Δi

i i

i

i + 1 i

− 1

0η

Ω Ω Ωr m v+ +a ai i4

You are here.You are here.

Figure 4. How to construct a universe.Even with a simple spreadsheet, studentscan construct an accurate numericalmodel for the expansion of our universe,as described by the scale parameter a.This graph was generated from spread-sheet data assuming a flat universewhose current Hubble constant is givenby 1/H0 = 13.7 Gy. The current cosmic energy fractions in matter (Ωm), vacuumenergy (Ωv), and radiation (Ωr) are 0.272,0.728, and 0.000084, respectively. The algorithm is based on a pair of simple difference equations representing the differential equations for the scale factorand time t as a function of the parameterη. Note the inflection point at about 9 Gy.That is the time at which vacuum energybegins to dominate matter and the cosmic expansion accelerates.

Practicing the formalism and applying it to con-crete, easily visualizable systems helps students toavoid errors and build an intuition about and con-fidence in the mathematics.

Whatever approach you favor, you should tryto use active-learning methods in your class. One ofthe most robust results of physics education re-search is that students learn the material much moreeffectively when they are actively engaged in thelearning process instead of passively receiving thematerial in lectures or readings.10 Students need totake ownership of ideas by actively processing themand holding them up against their own experience.Most of the work demonstrating the efficacy of ac-tive learning has focused on introductory courses,but we have found that our upper-level students ex-posed to active-learning techniques perform muchbetter than those in more traditional courses. Fortu-nately, most general relativity classes are likely to besmall enough that active learning should be reason-ably easy to implement. Still, to do that, you willneed to make sure that students come to class pre-pared, design appropriate activities for both in andout of class, and give students appropriate feedbackon their work.

Advances in astronomy and cosmology haveopened a whole new world of opportunities that iswaiting for undergraduate physics students with abackground in general relativity. Those who teachthe subject can take advantage of a decade’s worthof pedagogical progress. Various educational tech-niques long used in teaching electrodynamics andquantum mechanics to undergraduates can now besuccessfully applied to general relativity, and newtextbooks are making such approaches broadly ac-cessible. General relativity is therefore rapidly be-coming as important as many topics traditionallycovered for a physics degree. We strongly recom-mend that any major-granting department considera regular course offering in this fascinating subject.

We thank the numerous physics teachers and textbookauthors who have helped us acquire information for thisarticle.

References1. H. F. Fliegel, R. S. DiEsposti, “GPS and relativity: An engi-

neering overview” (1996), http://tycho.usno.navy.mil/ptti/1996/Vol%2028_16.pdf.

2. E. F. Taylor, J. A. Wheeler, Spacetime Physics: Introduc-tion to Special Relativity, 2nd ed., W. H. Freeman, NewYork (1992).

3. J. B. Hartle, Gravity: An Introduction to Einstein’s GeneralRelativity, Addison-Wesley, San Francisco (2003); Am. J.Phys. 74, 14 (2006).

4. P. Wogt, J. Kuhn, Phys. Teach. 50, 182 (2012).5. MIT Haystack Observatory, Undergraduate Research,

SRT, http://www.haystack.mit.edu/edu/undergrad/srt.6. J. B. Hartle, Mathematica Notebooks, http://wps.aw.com/

aw_hartle_gravity_1/0,6533,512496,00.html. See alsoK. Lake, http://arxiv.org/abs/physics/0509108.

7. T. Moore, A General Relativity Workbook, http://pages.pomona.edu/~tmoore/grw/download.html.

8. Open Source Physics, http://www.opensourcephysics.org.

9. Rotation Curves, http://burro.astr.cwru.edu/JavaLab/RotcurveWeb/main.html.

10. R. Hake, Am. J. Phys. 66, 64 (1998). ■

June 2012 Physics Today 47

Lasers | Lenses | Mirrors | Assemblies

Windows | Shutters | Waveplates | Mounts

Americas +1 505 296 9541

Europe +31 (0)316 333041

Asia +81 3 3407 3614

Custom Solutions for any Atmosphere.

Vacuum compatible actuators

Thermal and Vibrational analysis

Mounted optics flatness testing

Special cleaning and packaging

Cleanroom assembled and tested

Size 12 to 450 mm

UNDER PRESSURE TO FIND STABLE MOUNTS?

18 inch mirror mountVibrational stability

For more information go to: cvimellesgriot.com/Vacuum