General Relativity in the Undergraduate Physics Curriculum James B. Hartle∗ Department of Physics University of California, Santa Barbara, CA 93106-9530 (Dated: February 3, 2008) Einstein’s general relativity is increasingly important in contemporary physics on the frontiers of both the very largest distance scales (astrophysics and cosmology) and the very smallest(elementary particle physics). This paper makes the case for a ‘physics first’ approach to introducing general relativity to undergraduate physics majors. PACS numbers: 01.40.-d,04.20.-q,95.30.Sf I. INTRODUCTION fascination. Students specializing in physics naturally want to know more. A further argument for undergraduate general relativity is accessibility. As I hope to show in this paper, a Einstein’s 1915 relativistic theory of gravity — general rel- number of important phenomena of gravitational physics can ativity — will soon be a century old. It is the classical theory be efficiently introduced with just a basic background in me- of one of the four fundamental forces. It underlies our con- chanics and a minimum of mathematics beyond the usual ad- temporary understandingof the big bang, black holes, pulsars, vanced calculus tool kit. Other subjects of great contempo- quasars, X-ray sources, the final destiny of stars, gravitational rary importance such as high temperature superconductivity waves, and the evolution of the universe itself. It is the intel- or gauge theories of the strong interactions require much more lectual origin of many of the ideas at play in the quest for a prerequisite information. General relativity can be made ac- unified theory of the fundamental forces that includes gravity. cessible to both students and faculty alike at the undergraduate The heart of general relativity is one of the most beautiful and level. revolutionary ideas in modern science — the idea that gravity is the geometry of curved four-dimensional spacetime. Gen- It is probably fruitless to speculate on why a subject as ba- eral relativity and quantum mechanics are usually regarded as sic, accessible, and important as general relativity is not taught the two greatest developments of twentieth-century physics. more widely as part of the undergraduate physics curriculum. Limited time, limited resources, inertia, tradition, and mis- Yet, paradoxically, general relativity — so well established, conceptions may all play a role. Certainly it is not a lack of so important for several branches of physics, and so simple in textbooks. Refs. [2]—[22] are a partial list of texts known its basic conception — is often not represented anywhere in to the author1 that treat general relativity in some way at an the undergraduate physics curriculum. An informal survey by introductory level. William Hiscock [1] of the course offeringsof 32 mid-western Available time is one of the obstacles to introducing general research universities found only a handful that offered an in- relativity at the undergraduate level. The deductive approach termediate (junior/senior) course in general relativity as part to teaching this subject (as for most others) is to assemble of the undergraduate physics curriculum. This has the con- the necessary mathematical tools, motivate the field equations, sequence that many students see gravity first in the context solve the equations in interesting circumstances, and compare arXiv:gr-qc/0506075v2 17 Jun 2005 of planetary orbits in basic mechanics and next, if at all, in the predictions with observation and experiment. This ‘math an advanced graduate course designed in part for prospective first’ order takes time to develop for general relativity which specialists in the subject. There might have been an argu- may not be available to either students or faculty. This arti- ment for such an organization half a century ago. But there cle describes a different, ‘physics first’ approach to introduc- is none today in an era when gravitational physics is increas- ing general relativity at the junior/senior level. Briefly, the ingly important, increasingly topical, increasingly integrated simplest physically relevant solutions to the Einstein equa- with other areas of physics, and increasingly connected with tion are introduced first, without derivation, as curved space- experiment and observation. In the author’s opinion, every times whose properties and observable consequences can be undergraduate physics major should have an opportunity to explored by a study of the motion of test particles and light be introduced to general relativity. rays. This brings the student to interesting physical phenom- Its importance in contemporary physics is not the only ena as quickly as possible. It is the part of the subject most reason for introducing undergraduates to general relativity. directly connected to classical mechanics and the part that re- There are others: First, the subject excites interest in students. quires a minimum of ‘new’ mathematical ideas. Later the Ein- Warped spacetime, black holes, and the big bang are the fo- cus both of contemporary research and of popular scientific 1 This list consists of texts known to the author, published after 1975, and judged to be introductory. It does not pretend to be either complete or ∗Electronic address: [email protected] selective, nor is it a representation that the texts are readily available. 2 stein equation can be motivated and solved to show where the solutions come from. When time is limited this is a surer and more direct route to getting at the applications of general rel- ativity that are important in contemporary science. Section II expands very briefly on the importance of gen- eral relativity in contemporary physics. Section III outlines the basic structure of the subject. Sections IV and V describe the ‘math first’ and ‘physics first’ approaches to introducing general relativity to undergraduate physics majors. This is not an even-handed comparison. The ‘math first’ approach is de- scribed only to contrast it with the ‘physics first’ approach which is advocated in this paper. Section VI illustrates how ideas from classical mechanics can be used to calculate im- portant effects in general relativity. Section VII reports the personal experiences of the author in using the ‘physics first’ method. II. WHERE IS GENERAL RELATIVITY IMPORTANT? Gravity is the weakest of the four fundamental forces at accessible energy scales. The ratio of the gravitational force to the electric force between two protons separated by a distance r is (in Gaussian electromagnetic units) Gm2 /r2 Gm2 Fgrav p p ∼ −40 = 2 2 = 2 10 . (2.1) Felec e /r e Gravity might thus seem to be negligible. But three other facts FIG. 1: Where relativistic gravity is important. This figure shows explain why it is important and where it is important. First, selected phenomena plotted on a graph of their characteristic mass gravity is a universal force coupling to all forms of mass and M vs. their characteristic size R. The distance scale ranges from energy. Second, gravity is a long-range force in contrast to the the smallest considered in today’s physics (the Planck scale) to the weak and strong forces which are characterized by nucleus- largest (the size of the visible universe). Masses range from those of size ranges and below. Third, and most importantly, gravity is elementary particles to the mass in the visible universe. Phenomena 2 unscreened. There is no negative “gravitational charge”; mass above the line 2GM = c R are inside black holes and inaccessible. is always positive. Phenomena on that line, or close to it and below, are the ones for which general relativity is important [cf. (2.2)] and are indicated by These three facts explain why gravity is the dominant force dots. Phenomena further away, for which relativity is unimportant, governing the structure of the universe on the largest scales of are indicated by squares. For the universe R is the Hubble distance space and time — the scales of astrophysics and cosmology. and M the mass inside it. Our universe roughly evolves along the line The strong and weak forces are short range. The relatively 2GM = c2R from the smallest scales of quantum gravity to the largest much greater strength of electromagnetic forces ensures that characterizing present cosmology. (Reproduced from [11] which was charges will be screened in an electrically neutral universe like adapted from a figure prepared by C. Will for [25]). ours. Only gravity is left to operate on very large scales. Relativistic gravity — general relativity — is important for an object of mass M and size R when general relativistic effects are important for the operation of the Global Positioning System (GPS) [24]. GM Relativistic gravity is also important on the smallest scales q ≡ ∼ 1. (2.2) Rc2 considered in contemporary physics — those of quantum gravity. These are characterized by the Planck length ℓ Neutron stars (q ∼ .1) and black holes (q ∼ .5) are relativistic objects by this rough criterion. So is our universe (q ∼ 1) if 3 1 −33 ℓPl = (Gh¯/c ) 2 ∼ 10 cm. (2.3) we take R to be the present Hubble distance and M to be the mass within it. Figure 1 displays some phenomena for which This is much, much smaller than even the scale of the strong relativity is important and ones for which it is not. interactions ∼ 10−13 cm. Yet, this is the scale which many General relativity can sometimes be important even when contemporary explorers believe will naturally characterize the q is small provided compensating observational precision can final theory unifying the four fundamental forces including be achieved. For the Sun q ∼ 10−6, yet the solar system is the gravity. This is the characteristic scale of string theory. This domain of the precision tests that confirm general relativity to is the scale that will characterizethe union of the two great de- as much as 1 part in 105 [23]. For the Earth q ∼ 10−9, yet velopments of twentieth century physics — general relativity 3 and quantum mechanics.