Wolfgang Rindler Time from Newton to Einstein to Friedman ABSTRACT This article traces in as untechnical a way as possible the meta- morphoses that Newton’s absolute time has undergone at the hands of Einstein, only to come round almost full-circle in pre- sent-day Friedmannian cosmology. The recurring theme is that of moments being slices through the spacetime of history, and how these slices are affected by gravity. I am keenly aware that scientists, philosophers, his- torians and artists understand the essence of time in profoundly different ways. I shall not presume to speak for the latter three categories, though hope- fully to them, of what I am most familiar with, namely the modern scientific point of view. Its origins no doubt go back into the dim past, but for practical purposes one might as well begin with Newton, who systematized and superseded all that went before. To Newton, time is absolute, which means: inde- pendent of the observer. Its “moments” are world- wide. These moments uniquely connect sets of simultaneous events, such as, for example, the death of Socrates and some supernova explosion in outer space. If we picture the history of the universe four- dimensionally, as in Figure 1, the successive moments slice through spacetime (the set of all events) as does a bread cutter through a loaf. Time progresses as a KronoScope 1:1-2 (2001) © Koninklijke Brill NV, Leiden, 2001 ® TIME (one dimension) SPACE (three dimensions) ® Death of Socrates Supernova ´ ´ Figure 1. The Newtonian view of time as a unique succession of moments, here rep- resented by the horizontal lines. (In reality, they are three-dimensional spaces). regular succession of moments, which both physics and human experience allow to be spaced “equally”, for example by seconds, or years, or millenia. But even this simple scheme presupposes an assumption. For it is not a fore- gone conclusion that there is universal agreement on what is a one-second time interval in different centuries. Suppose we build two different highly accurate clocks and synchronize them now. If they are based on totally dif- ferent physical principles - one, for example, on the oscillations of a balance wheel and the other on the oscillations of a cesium atom - will a second mea- sured by the one still correspond to a second measured by the other in a hun- dred years? The British physicist E.A. Milne (1896-1950) thought not. If he is right, this would enormously complicate physics. Luckily there are no strong reasons to believe him. So, as usual in physics, one adopts the simpler hypoth- esis until forced by observations to adopt a more complicated one. That sim- pler hypothesis is the temporal homogeneity of physics: any two physical experiments which are local, that is, isolated from the rest of the universe, and which are of equal duration now, can be repeated at all future times with equal outcome. And since all clocks are in principle based on repetitive “exper- iments”, this ensures the permanent synchrony of all imaginable clocks. We may digress at this point to recall that physical theories produce mathe- matical “models” of Nature. These models contain ad hoc concepts and axioms which have mathematical consequences that can be tested by experiment. No 64 Wolfgang Rindler theory can ever be “proved”, since, on the one hand, no experiment is ever 100% accurate (unless it involves counting only), and, on the other hand, there might always be tests that have not been thought of or that are beyond one’s capabilities. But theories can be disproved! Indeed, this was the fate of Newton’s theory, including his theory of time, in the 20th century. But we are getting a little ahead of ourselves. The absoluteness of time is needed to make Newton’s basic laws of mechan- ics meaningful. Take Newton’s law of gravity: as the earth goes round the sun along its elliptical orbit, what force does it experience? The force is pro- portional to the inverse square of the instantaneous distance between earth and sun. Thus it is necessary to know unambiguously which are simultane- ous events here and there. Or take Newton’s so-called third law, the equal- ity of action and reaction. If I swirl a stone around on a rope, the stone pulls me as much as I pull it! Formally, Newton’s third law asserts that if particle A exerts a force on particle B (be it by contact during a collision, or via a rope, or via gravity as between the sun and the earth), then B exerts an equal and opposite force on A. But evidently the force that A exerts on B today is not necessarily equal and opposite to that which B exerts on A tomorrow! The simultaneity of the measurements is implied. If simultaneity were observer- dependent, then whose simultaneity are we to take? A’s? B’s? Ours? Luckily, according to Newton, they are all the same. Not according to Einstein! In Einstein’s Special Relativity simultaneity is relative: A’s, B’s, our’s - all are different. No wonder there is no analog of Newton’s third law or of the inverse square law in Einstein’s theory. Both Special Relativity and Newton’s theory in its modern formulation take as their “zeroth axiom” the existence of an infinite set of preferred infinitely extended Euclidean reference frames, all moving uniformly and without rota- tion relative to each other, the so-called inertial frames, in which free parti- cles remain at rest or move uniformly. One of these is identified with the frame of the “fixed stars”, Newton’s “absolute space”. In Newton’s theory all inertial frames share absolute time. And Newton’s laws of mechanics are equally valid with reference to any one of them: this is called “Newtonian Relativity”. In Special Relativity, all inertial frames are equivalent for the performance of all physical experiments, not just mechanical experiments. This is Einstein’s Time from Newton to Einstein to Friedman 65 “Relativity Principle”, or his first axiom. One of its immediate consequences can be shown to be* the existence of a unique invariant velocity among iner- tial frames: a point moving at that velocity in any one inertial frame is judged to move at the same velocity in all other inertial frames. Moreover, the invari- ant velocity is also the maximal velocity at which any particle or signal can travel. In Newton’s theory that invariant velocity is “infinity”: any effect that propagates instantaneously in one inertial frame, so propagates in all others. An example is provided by gravity. According to Newton, if the sun were to explode into two oppositely moving masses, its changed gravitational field would be felt at the earth’s location without time delay, or, in other words, at infinite speed, no matter in what inertial frame the observation is made. In Special Relativity the invariant speed is the speed of light. This is, in fact, Einstein’s “second axiom”. It is hardly avoidable if anything like classical electomagnetism is to hold in all inertial frames. But it is the big stumbling block to newcomers to the theory. If I am in a fast train and I observe a light signal race down its middle, how can I as well as an observer at rest in a sta- tion we are just passing both ascribe the same speed to that signal? Clearly something in our concepts of space and time has to give. This is not the place to retrace the steps that Einstein took to arrive at his results. We shall con- tent ourselves with reporting the outcome. The main outcome is the relativ- ity of time! A useful way to visualize any inertial frame is to think of it as an infinite rigid rectangular lattice, each little lattice cube being, say, one centimeter on each edge, and we imagine ideal little standard clocks affixed to all the lat- tice points. These clocks are synchronized, in each inertial frame, by any method that would synchronize them also in Newton’s absolute space. For example, each clock could exchange light signals with one master clock, and so adjust its zero point that signals in both directions take the same time. Equal read- ings on all these clocks determine “moments” in the frame in question. In every inertial frame, physics is still temporally homogeneous: experiments can be repeated arbitrarily often with always the same outcome. In every inertial frame physics is also spatially homogeneous and isotropic: all exper- iments can be repeated at different locations and in different orientations, with always the same outcome. So far, no surprises. The surprise only comes when we introduce a second inertial frame. Imagine two such frames, S and 66 Wolfgang Rindler (a) 2 1 0 –1 –2 A´ B´ C´ D´ E´ –2 –1 0 1 2 A B C D E (b) 3 2 1 0 –1 A´ B´ C´ D´ E´ –1 0 1 2 3 A B C D E (c) 4 3 2 1 0 A´ B´ C´ D´ E´ 0 1 2 3 4 A B C D E Figure 2. Two sets of clocks, A, B, . and A’, B’, . synchronized in their respective inertial frames are out of synchrony when observed in a third frame. Here they move in opposite directions with the same speed through that third frame. S’, flying through each other at constant speed v. (The two lattices passing through each other can obviously not be made of real sticks, they must be purely in our imagination!) Two amazing things will now become apparent.