THESIS Let there be light?

Could Galileo have worked out the Galileo’s thinking, if carried to its logical extreme importance. Having assumed that principles of the modern theory endpoint, would have led directly the axes of frames I and F are aligned, and of relativity? Could he, even in the to Lorentz invariance, with Galilean so too those of I and F´, it is tempting to mid-seventeenth century, have derived invariance as a sub-case. The argument leap to the conclusion that the axes of F the Lorentz transformations, the existence hinges on what we normally refer to and F´ must also be aligned. of a fundamental limiting velocity, and as the ‘addition’ of velocities, and what But as Feigenbaum argues, there are the equivalence of mass and energy? The one can or cannot say about it from no logical grounds for making such a idea sounds preposterous, especially as the fundamental principles. leap. Rather, although it seems odd, one limitations of the principle of relativity Feigenbaum considers two frames of must allow that the axes of the two frames as Galileo did conceive it only appeared reference, I and F, with their axes aligned. could differ by some rotation R, the nature at the dawn of the twentieth century. Consider yourself situated at rest in I, and of which depends on the two velocities, After all, it was Maxwell’s unification that you see F moving past at velocity V. V and V´. Galileo quite naturally never of electricity and magnetism and his Now suppose you see an object, say a ball, entertained this possibility; he assumed, explanation of the electromagnetic nature moving at velocity v, and wish to calculate in modern language, that the combined of light, along with the Michelson–Morley how an observer at rest in F sees this ball. effect of two subsequent boosts must experiment, that set the stage for Einstein. This is not quite the addition but, more be a third boost at some other velocity. Could special relativity have been That certainly accords with our deepest developed, even in principle, by someone intuitions, and, as Feigenbaum shows, who knew almost nothing of light? leads directly to a function r that Just possibly, the answer is yes. That’s reproduces Galilean invariance. the provocative view, at least, of physicist But if one entertains, as Galileo Mitchell Feigenbaum of The Rockefeller logically might have, that R could be University in New York, who suggests that non-zero — that two non-collinear boosts Galileo, if he’d had access to some modern might lead to some effective rotation — mathematics, might well have followed the results turn out very differently. What his own intuitions about the relativity emerges from the analysis then are the of motion to a theory of relativity in Lorentz transformations, as well as, something akin to today’s form. What ultimately, the other formulae of special makes Feigenbaum’s argument doubly Could special relativity have relativity. Of course, we now know interesting is its emphatic conclusion that these odd rotations as Wigner rotations, the logical foundations of relativity have been developed by someone first derived in 1939 by Eugene Wigner absolutely nothing to do with light, but who knew almost nothing working, of course, from the already follow quite independently from basic developed machinery of the Lorentz logic and symmetry considerations. of light? transformations. These are the rotations It was Galileo’s 1632 treatise A involved in the phenomenon of the Dialogue Concerning the Two Chief properly, the subtraction of velocities, the Thomas precession, and are indeed highly World Systems that got him into such result being some function of V and v, non-intuitive. Hence it is no surprise that trouble with the church. The bulk of the call it r(V,v). Galileo never allowed them as a logical text’s dialogue, between an adherent Determining this function requires possibility. The important point is that the of Aristotelian views, Simplicio, some assumptions. Feigenbaum starts development could have been reversed. and a proponent of the Copernican with the reasonable idea that a uniform Of course, one might object that view, Salviati, argues in favour of the velocity in one frame should correspond the velocity of light appears in the heliocentric world system. During the also to a uniform velocity in the other. Lorentz transformations, suggesting a discussion, Salviati also expresses the Assuming isotropy of space leads primary role for it. Yet in Feigenbaum’s essential insight behind Galileo’s view additionally to the conclusion that r(V,v) arguments, a fundamental limit also of inertia. “Motion which is common to must lie in the plane determined by the appears naturally, some new fundamental many moving things”, he observes, “is vectors v and V. Matters become more constant not in any way linked, a priori, idle and inconsequential to the relation interesting when considering a third to the speed of light. Of course, on of these movables among themselves, frame of reference, F´, which moves at empirical grounds, it turns out to be nothing being changed among them.” velocity V´ as seen in frame I. the speed of light. This development Only relative motion matters, and the Assuming that this frame, like F, suggests, however, that light just happens tendency of an object to remain in motion has its axes aligned with those of our to move at this fundamental speed, the is in all ways equivalent to the tendency of frame I, one can work out some algebraic existence of which has deeper origins. It is an object to remain at rest. relations linking the velocity of the ball fascinating that scientists as long as three In modern terms, we distinguish as seen in frames I and F´. Now we know centuries ago could have worked this out, sharply between the Galilean and Lorentz something about the relationship between and also, perhaps, that we still haven’t invariance of physical laws. But as observations made in I and F, and also found our way completely to the bottom Feigenbaum argues in a paper entitled in I and F´. But what of F and F´? Here of the meaning of relativity. The Theory of Relativity — Galileo’s Child, the development touches on a point of Mark Buchanan nature physics | VOL 4 | AUGUST 2008 | www.nature.com/naturephysics 583 © 2008 Macmillan Publishers Limited. All rights reserved.