Plane Geometry in Spacetime

Minkowski’s spacetime diagrams are extracted directly from Einstein’s 1905 postulates, using only some very elementary plane geometry. I have spent a significant part of my career looking at familiar results from unfamiliar perspectives. This has been partly because I very much enjoy teaching courses about aspects of physics to nonscientists, for whom the conventional approach is almost always an impenetrable thicket. But it’s also because I have a lot of trouble understanding physics myself. If I don’t think it through in my own terms, more often than not I can’t make much sense of it. The fruits of this self-indulgent approach have elicited a broad spectrum of reactions. The worst was a proposal review from an extremely distinguished senior physicist who anonymously complained that my work was characterized by a surprising lack of originality. (The only encouraging word in the entire report was “surprising”.) Less depressing was “What made you write a paper about that?” from a cosmologist friend and “I still don’t understand what bothers you”, from a beloved postdoctoral advisor. Moving in the positive direction through nervous silence, one passes by “Nice!” in varying degrees of intensity and sincerity until one finally reaches those who have truely appreciated what I was trying to do. Of these two stand out from the others in the warmth of their encouragement, the penetration of their constructive criticisms, and the high regard in which I hold them quite aside from our direct interactions. One was the late Ed Purcell. The other was, and remains to this day, Abner Shimony. So I know that Abner won’t mind my revisiting a subject that has been with us in pretty much its current form for 98 years, and since this meeting is for Abner, the rest of you will just have to put up with it. Even Abner, however, may be tempted to catch up on lost sleep, because much of my talk can be found in Chapter 10 of a new book I sent him last year. Although I can hope that he has not read it, knowing Abner N.D. Mermin Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853-2501, USA e-mail: ndm4@cornell.edu W.C. Myrvold and J. Christian (eds.), Quantum Reality, Relativistic Causality, and Closing 327 the Epistemic Circle, The Western Ontario Series in Philosophy of Science 73, c © Springer Science+Business Media B.V. 2009 328 N.D. Mermin I very much fear that he has. Nevertheless, Abner, pay attention. While I was putting this talk together my muse—curiously enough, also named Abner Shimony—kept raising questions that hadn’t occurred to me, cleverly objecting to obvious points, and otherwise helping me to come up with a better formulation than the one you’ve already read. My subject is how to construct Minkowski’s space-time diagrams (1908) straight from Einstein’s postulates (1905) without the intermediate of the Lorentz transformation or the invariant interval. I also assume no familiarity with graphs or coordinate systems, and in this respect my approach is to the conventional approach as the plane geometry of Euclid is to the analytical geometry of Descartes. As this point of view evolved I found myself led into some entertaining bywaters, which various relativists I consulted seemed not to have visited. In particular, if you do chose to catch up on some lost sleep, keep an ear open for the notion of the light rectangle which I commend to all as a genuinely useful unfamiliar relativistic tool. Here are Einstein’s two postulates. Purcell once complained that I had quoted an execrable translation of them, so I give below my own translation. But because my knowledge of German is limited to Schubert lieder and the operas of Richard Strauss, I also reproduce the original texts. 1. In electrodynamics, as well as in mechanics, no properties of phenomena correspond to the concept of absolute rest. . . . dem Begriffe der absoluten Ruhe nicht nur in der Mechanik, sondern auch in der Elektrodynamik keine Eigenschaften der Erscheinungen entsprechen. 2. Light always propagates in empty space with a definite velocity c, independent of the state of motion of the emitting body. . . . sich das Licht im leeren Raume stets mit einer bestimmten, von Bewegungszustande des emittierenden Korpers unabhangigen Geschwindigkeit V fortpflanze. My Abner-muse kept pointing out that I was slipping in other postulates down the road, so I decided to present up front a third postulate. Having formulated it, I found it sounded familiar, so I looked up Einstein (1905) and sure enough, he states it himself a few pages after the two famous ones: 3. If a clock at A runs synchronously with clocks at B and C, then the clocks at B and C also run synchronously relative to each other. Wenn die Uhr in A sowohl mit der Uhr in B als auch mit der Uhr in C synchron lauft, so laufen auch die Uhren in B und C synchron relativ zueinander. The third postulate comes in an obviously equivalent version: 3′. If event A is simultaneous with event B and event C, then events B and C are also simultaneous. I will combine this with its equivalent under the interchange of space and time: 3′′. If an event A happens in the same place as event B and event C, then the events B and C also happen in the same place. While Postulate 3 can probably be extracted from Postulates 1 and 2 in the case of a single spatial dimension, it has an independent status in two or more spatial dimensions and constructing this talk led me to appreciate why Einstein had made what I had once regarded as an unnecessarily pedantic remark. No pedant he. Plane Geometry in Spacetime 329 Conductor punches Alice’s ticket Front of train crosses highway Bob turns on light Lightning strikes track Cow crosses tracks Fig. 1 Some events of interest to Alice An event, of course, is something that happens at a definite place and time: a point in spacetime. It is, as Einstein (1905) also wisely points out, an idealization and, I would have said, a classical (i.e. pre-quantum) idealization, in that it is an unambiguous phenomenon whose spatial and temporal extent we are willing to treat as inconsequential. Alice wants to make a plane diagram depicting events that happen at various times and places in one spatial dimension—e.g. along a long straight railroad track. Figure 1 gives an example of such a diagram: Alice organizes events in her diagram according to the time at which they take place. Postulate 3 permits her to place all the events to which she assigns the same time on a single straight line, which we shall call an equitemp. (I would have preferred calling it an isochron but an internet search revealed “isochron” to be a standard term in radioactive dating and dendrochronology. “Equitemp” only produced a few furnaces and kettles.) Equitemps associated with different times must be parallel, for if they were not then the event represented by their point of intersection would happen at two different times, contradicting the idea that an event is localized in time. Following the usual conventions of mapmakers (Fig. 2) Alices takes the distance between two equitemps in her diagram to be proportional to the time between the two sets of events they represent. Alice still has the freedom of sliding points along her equitemps and she takes advantage of it to organize events in her diagram according to the location at which they happen, requiring (as Postulate 3 again permits) all events to which she assigns the same place to lie on a single straight line, which we shall call an equiloc. (Here, the more elegant “isotop” is clearly out of the question.) Equilocs associated with different positions must be parallel, or the event represented by their point of intersection would happen in two different places, contradicting the idea that an event is localized in space. Alice again follows mapmakers (Fig. 3) in taking the distance between equilocs in her diagram to be proportional to the spatial separation of the two sets of events they represent. The intersection of an equiloc and equitemp specifies a definite place and time and therefore corresponds to a unique event, so equilocs cannot be parallel to 330 N.D. Mermin = an event Distance between equitemps proportional to time between events (lines of constant time) Equitemps Fig. 2 Some of Alice’s equitemps