Exploring Black Holes

More than half a century ago, Edwin F. Taylor and John Archibald Wheeler wrote the indispensable Spacetime Physics, an unconventional, brilliant, and unique—what else would you expect from John Wheeler?—introduction to special relativity. It has been in print ever since, and now, through Taylor's generosity, the second edition is available for free.¹ But before getting to the sequel, reviewed here, some background seems in order, for a famous book and its legendary authors.

In 1962, Taylor was a young physics professor at Wesleyan University, starting a sabbatical at Princeton, and somewhat prophetically taking up residence in the office of Eric M. Rogers,² on leave. He was about to complete his first book,³ when, a few days into his visit, he fell into conversation with Wheeler. The conversation changed his life, and, if you have studied physics in the past 50 years, probably yours as well; it certainly changed mine. Wheeler was about to engage in one of those wildly audacious enterprises he was famous for: beginning his honors first year class with relativity. (Feynman is quoted in the New York Times' obituary of his old teacher: “Some people think Wheeler's gotten crazy in his later years, but he's always been crazy.”⁴) Would Taylor be willing to attend Wheeler's lectures, transcribe them, and then with him turn these notes into a book? He accepted, and the rest is history.

Taylor described his collaboration in Wheeler's 60th birthday festschrift.⁵ Most of the papers therein, including contributions by Wheeler's two (so far!) Nobel-winning students Feynman and Kip S. Thorne, describe new results by friends and colleagues. The last contribution, by Taylor, stands apart. The narrative details the work Taylor did with Wheeler and its consequence. As important as Niels Bohr was to John Wheeler, so was John Wheeler to Edwin Taylor. Apprentice grew into collaborator, and a younger man was set alight by the inner fire of the older master, and finally in Taylor's own words, “graduated from him to himself.” The experience redirected the trajectory of his life, away from pure physics research to “a path less traveled by,” the art and science of teaching physics. And for nearly 60 years that is what Edwin Taylor has done, with uncommon skill and dedication.

After the sabbatical, he left Wesleyan to join the Science Teaching Center (now the Educational Research Center) at MIT. There he collaborated with Anthony French in writing the last of a series of four undergraduate texts (available in paperback, to keep their price low).⁶ Forty years ago he became one of the first to introduce computers into physics instruction, particularly in relativity and quantum mechanics, where these machines could provide a virtual workshop for phenomena not easily accessible otherwise. In addition to his books and articles, he's served as the editor of this Journal, and was recognized “for notable contributions to the teaching of physics” by the Oersted Medal, in 1998.

So much of Taylor's work has been directed towards making physics, and particularly the physics that takes your breath away, accessible—this is the leitmotif of his professional career. For decades, he has tried to encourage and validate women in physics (in his request for applications to edit this Journal, he pointedly remarked that there had been no appearance of gendered personal pronouns;⁷ in his later books, he made sure that both male and female individuals appeared in examples and exercises). Aware that not everyone can afford them, he has made available for download software, articles, previous books, and now, this most recent publication. Mirroring Wheeler's audacious introduction of relativity to beginning students, in later years, Taylor has championed the early application of the principle of least action as a skeleton key for opening wide the secret chambers of classical mechanics, quantum mechanics, and general relativity,⁸ in courses often taught well in advance of the usual sequence. In this work, he has had the assistance of several colleagues. The calculus of variations may not be appropriate for beginners, but applying it cleverly requires at most ordinary derivatives, well within the reach of neophytes. To get this powerful tool into their hands, you have to be inventive. Taylor and Wheeler have been masterly in showing the rest of us how to do it.⁹ 

I had the good fortune to serve as Taylor's teaching assistant for a couple of weeks in July of 1988, when I was one of 50 high school physics teachers at a Woodrow Wilson institute held at Princeton. A couple of days in, the director asked for help from someone who was comfortable with relativity and computers. I volunteered, was given Taylor's email address, and got in touch with him. He sent me (via the standard delivery system of 1988, the US Postal Service) software he and his MIT students had developed to show some of the peculiar aspects of relativity: “Spacetime,” for length contraction, time dilation, and the failure of simultaneity (allowing the user to frame various simulations, including the twin paradox); “Collision” for calculating final velocities in relativistic collisions, and “Visual,” representing the appearance (in color) of geometrical solids as seen out the windshield of a spaceship traveling past them at a user-adjustable relativistic speed.¹⁰ He was to give a handful of hourly lectures at the summer institute, and introduce us in a laboratory setting to the use of the software (which he then encouraged us to give away to our students). It was my job to get the computer lab ready. I met him at Princeton's “Dinky” shuttle station (he was wearing a jaunty seersucker suit, and tieless; perfect for the hot, sticky summer days in New Jersey¹¹), and drove him to the Nassau Inn. We talked now and again during the week he was there. He reminded me a little of Wheeler (whose sophomore class I'd taken in 1971–1972); expansive, warm, considerate, and enormously enthusiastic. He was a hell of a lot of fun to talk with. Since then I have had only sporadic communication with him. One of these conversations led to an early sighting of the sequel to Spacetime Physics, then called Scouting Black Holes, circa 1990, which I think he invited me to obtain electronically (ftp) from MIT.

What I recall about Scouting Black Holes (my copy is not to be found; perhaps loaned to a student) is that it began as a continuation of Spacetime Physics; the first chapter was numbered 4. While Spacetime Physics had needed only algebra, Exploring Black Holes assumed a knowledge of derivatives and elementary integrals (nothing beyond the first semester of calculus). Unlike the earlier work, the new book was largely project-driven; a series of explorations were laid out to be gone through more or less sequentially. It was at least as revolutionary as its predecessor. Sometime in the next four years, Scouting Black Holes became Exploring Black Holes, and no longer presupposed a good knowledge of the earlier book (the chapters more conventionally began at 1, and there was some introductory material largely borrowed from Spacetime Physics). The projects were demoted in importance, and the book, now far more expository, was much closer in form to Spacetime Physics. The first edition was published in 2000,¹² and very regrettably is out of print (I've held on to mine). Copies may be found at online secondhand bookstores, but they command at least twice the original list price. Perhaps discouraged by the fate of the first edition, Taylor followed the advice offered by Ruth Chabay to forget about publishing the second edition in hard copy. Although what is online is labeled a draft, it represents a quarter century of writing and thinking about the material. Apart from a very small number of images that need tweaking, and perhaps a section or two to be filled in, it's ready for the printer.

The main idea of the book is explained in the third chapter, “Curving” (Box 4, pp. 3–10). In comparison to the rest of the universe, the black hole is not so forbidding: “its relatively small size allows us to call the black hole our ‘little jugged apocalypse,’ a phrase the writer John Updike uses to describe the view into the portal of a front-loading clothes washing machine.” Its manageable (and mostly simple) size and character “make[s] the black hole a useful example to teach large swaths—but not all—of general relativity.” Not all, but quite a lot! The key to the (non-spinning) black hole is the Schwarzschild metric. Much can be learned from the geodesic trajectories which follow from it by the action principle. In the second edition, unlike in the first, geodesics themselves are not introduced by that name. Instead, they are described by “The Principle of Maximal Aging”: proper time (here called “wristwatch time”) is to be maximized (as it is along a geodesic¹³), and a connection is made between worldlines and Euclid's theorem about the minimum distance being a straight line. In both editions, the Schwarzschild metric is not derived, nor is most of the machinery of general relativity introduced: no Riemann tensor, no Christoffel symbols, certainly no Einstein field equations. Nevertheless, most of the classic tests of general relativity are here: The bending of light and the 1919 eclipse results (Chapter 13), the perihelion shift of Mercury (and other planets) in Chapter 10, and the gravitational red shift (in the treatment of GPS, Chapter 3). A regrettable loss from the first edition is a careful study of the fourth classic test of general relativity, Irwin Shapiro's wonderful experiments of timing radar echoes off Mercury, Venus, and Mars to establish directly the effect of gravitation on the speed of light (Project E).

The most noticeable changes in the second edition are the absence of projects (though they are nearly all laid out as guided examples) and the far greater content of the second edition, reflected in the nearly double number of pages (p. 660 vs. p. 340). There are five chapters and seven projects in the first edition, and twenty-one chapters in the second. A new addition is the considerable time and detail devoted to orbital mechanics around both stationary and spinning black holes, the latter requiring the Kerr metric (in both Boyer–Lindquist and Doran forms).¹⁴ In the first edition, spinning black holes are relegated to a project (F); in the second, they are given the last five chapters. Results which had not been known in 2000 are discussed, in particular LIGO and the discovery of dark energy. (In the first edition, Wheeler states (p. G-11) that he doesn't believe the then recent results found by Saul Perlmutter, Adam Riess, and Brian Schmidt, implying the existence of dark energy. These were soon confirmed and won their discoverers the 2011 Nobel Prize.) Topics covered in the first edition are often given improved and deeper coverage in the second. For example, the familiar “rubber sheet” funnel-shaped diagram of the spacetime near a spherical mass was merely described in the first edition; in the second, the shape of the surface is derived. In the description of the bending of light (Chapter 13, “Gravitational Mirages”), new features include a historical timeline (starting with J. Soldner, and giving the (incorrect) Newtonian calculation as a guided exercise), and much more about gravitational lensing. (Did you know you can visualize an Einstein ring formed by a star using a candle and the stem of a wine glass?) In both editions, the vast majority of calculations are very gently worked out for the benefit of readers who are not so confident in their mathematics.

Because the book is in a form allowing small changes, I suggest a few things that might make it even more attractive: the return of the Shapiro experiment discussion, inclusion of the recent work of Sheperd Doeleman and his team in capturing the first image of a black hole,¹⁵ in particular the result of the photosphere at r = 3GM/c² (a result which is derived in Chapter 11, Exercise 4), and (to steal an innovation from Misner, Thorne, and Wheeler's epochal Gravitation¹⁶) the introduction of the MTW “Track 1” and “Track 2” notations to help readers distinguish the main thread from more intricate results. There was no need for this in the first edition, but perhaps there is in the second. The first edition might serve broadly as the “Track 1” material, and the fine details about orbital mechanics, and nearly all of the content concerning spinning black holes and the Kerr metric, could be “Track 2.”

The second edition of Exploring Black Holes is a remarkable achievement, a lovely capstone to a great and luminous career. As was true of its predecessor, there is really nothing else like it. Professors and students who would like a first look, or perhaps a different look, at the wonders of general relativity would do well to spend time with this wonderful gift.