Time from Newton to Einstein to Friedman

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doubt go back into the dim past, but for practical purposes one might as well begin with Newton, who systematized and superseded all that went before.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: independent 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 different 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 measured by the one still correspond to a second measured by the other in a hundred 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 hypothesis until forced by observations to adopt a more complicated one. That simpler 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 “experiments”, this ensures the permanent synchrony of all imaginable clocks.

We may digress at this point to recall that physical theories produce mathematical “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 mechanics 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 simultaneous 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 particles 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 inertial 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 invariant 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 station 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 relativity 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 lattice 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 experiments 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

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. Moments in S do not correspond to moments in S’! For example, the mid- day chime of all the lattice clocks in S happens in S’ on a plane that travels at velocity c2/v in the same direction as S, where c is the velocity of light. And also, each lattice clock of S, as it moves through S’, loses time steadily relative to the lattice clocks of S’ as it passes them. This is called “time dilation”. In Special Relativity, in fact, all clocks moving through an inertial frame go slow relative to the lattice clocks of that frame. Of course, the clocks of S’ go slow also as they pass through S. Figure 2 illustrates these facts. It shows three successive moments in an inertial frame S”, through which two other inertial frames, S and S’, move with equal speeds in opposite directions. The “strings of pearls” shown are, in fact, five equally spaced clocks in S and five similar clocks in S’, depicted in S” as they pass each other. Above all, note that though the displayed clocks are synchronized in S and in S’, respectively, they are not synchronous in S”. (We say that simultaneity is relative.) Secondly, note how a single instant, say the instant t’= 0 in S’, races through S”: it is at C’ in the first picture, then at D’ in the second, and at E’ in the third. And lastly, observe how each clock of S and S’ loses steadily relative to the synchronized

Time from Newton to Einstein to Friedman 67 clocks of the other frame: for example, in the first picture A’ reads 4 seconds ahead of A which it just passes, in the second picture it reads only 2 seconds ahead of C, and finally it is not ahead of E at all; by its own reckoning, the journey from A to E as taken 2 seconds, whereas by the clocks of S it has taken 6 seconds! Also note how completely symmetric this effect is between S and S’: each clock of the one loses relative to the synchronized lattice clocks of the other.

We must stress that relativistic time dilation is by now a familiar part of modern physics. One of the most striking examples is the enormously extended lifetime of shortlived subatomic particles when going round and round in so-called magnetic storage rings at speeds close to the speed of light. The theoretical time-dilation factor is given by 1/Ö1-v2/c2, which, for example, is about 10 when v = 0.995c. And mesons in such storage rings and at such speeds do indeed live ten times longer than would otherwise be possible. At the speed of light, clocks would come to a standstill. Of course, no material clock can travel at that speed. Since photons do, they cannot decay like mesons. For if they did, they would be clocks still ticking at the speed of light.

Time dilation is also the stuff of science fiction! Astronauts flying through space at speed 0.995c age only one-tenth as much as their stay-at-home fel- lows. In principle they could return to earth in a hundred years, having them- selves aged only ten years. It may be objected that astronauts and mesons are not clocks, but they could be imagined to carry clocks with them, relative to which they would age “normally”.

We end this brief description of time in Special Relativity by exhibiting the relativistic diagram corresponding to Figure 1, namely Figure 3. It shows the histories or “worldlines” of two observers A and B in relative motion, and their different slicings of world history into moments. Whether the supernova of Figure 1 exploded at the time of the death of Socrates is now a moot question.

Special Relativity is a theory of physics in the absence of gravity - or, at least, in the absence of strong gravity. Einstein had been unhappy with the a priori existence of inertial frames. Science knows of no other entities that act but cannot be acted on. So in his theory of General Relativity the inertial frames can be acted on by the very objects on which they act (which, in fact, they guide along straight lines), namely masses. Masses like the sun curve space-

68 Wolfgang Rindler TIME ® e ® n i l d l r o e

n w i l s ’ d

l B r o nts

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om s m ’ B’s A A’s moments SPACE ®

Figure 3. In Special Relativity the moments, connecting simultaneous events, are observer-dependent. The figure shows the worldlines (the sets of events at the observers) of two observers in uniform relative motion, and the corresponding moments. time, and curved spacetime, in turn, guides free particles like the planets along curved paths: specifically, along the straightest possible paths (“geo- desics”) in curved spacetime. Gravity as a force has disappeared, as has the inverse-square law. On the face of it, it seems a miracle that this alternative model of gravitational orbits can reproduce, within experimental error, essen- tially all the well-established results of Newtonian celestial mechanics. But what is the upshot of all this for the concept of time? Without a doubt, a fur- ther complication away from the simple succession of absolute moments that physicists before Einstein could still envisage.

In complicated general-relativistic situations, say in the case of globular clusters of stars, in which there is random motion of thousands or even millions of gravitationally bound stars in a relatively small volume, the spacetime is very irregularly curved. No longer are there preferred sets of freely moving observers like the “inertial observers” of Special Relativity, which give rise to the regularly spaced moments shown in Figure 3. Time has become so ambiguous that any continuous slicing of the spacetime into curved “moments” is as good as any other - provided the slices satisfy a certain local condition of being “moment-like”.

For this, we again digress briefly in order to describe Einstein’s so-called Equivalence Principle. Noting Newton’s result that in a freely falling and

Time from Newton to Einstein to Friedman 69 non-rotating cabin, even in a strong gravitational field, ”gravity is eliminated” (recall the televised pictures of astronauts and their utensils floating freely in their capsules!), Einstein in his Equivalence Principle asserted that Special Relativity should be valid in all such “local inertial frames”. In particular, therefore, at all events in a gravitational field, there will still be local sets of “moments”. And the general-relativistic global curved moments must everywhere coincide locally with possible moments in some capsule. That is what is meant by being moment-like. However, outside Special Relativity it will in general be impossible to find a global time slicing such that any freely falling standard clock passes successive moments at equal proper time intervals.

Nevertheless, even in General Relativity there are situations where the concept of time retains some of its regular Newtonian properties. These situations are the so-called stationary fields, caused by mass distributions whose aspect never changes (though they may be in motion). Examples are: any fixed chunk of matter in otherwise empty space; a uniformly rotating spher- ical mass; a rigid tube of arbitrary shape, closed back upon itself, with a heavy fluid uniformly flowing round it. In such fields at least the rates of stationary clocks placed throughout the space are determined up to an arbitrary scale. But here another relativistic phenomenon makes itself felt: gravitational time dilation. It is a consequence of the axioms of General Relativity that standard clocks near heavy masses go slow. But one can compensate for that by suitably speeding them up, where needed, over their natural rates, so that eventually all clocks tick in unison as seen by each other. What is in general not possible is to find a preferred slicing into moments. But having chosen one slice arbitrarily, the others are then fully determined by equal advance- ments of the clocks. Only in the case of static fields (where the sources are not moving), do the moments become fully determinate. Here there is no arbitrariness of the slant of the moments, as there is in Special Relativity, where the spacetime is empty. Now the spacetime is filled with matter - all mutually at rest and thus having parallel worldlines - and the preferred moments are orthogonal to these worldlines (in a certain technical sense).

The only way in which general-relativistic time differs from Newtonian absolute time in a static field is that time runs more slowly near heavy masses. Hence the time interval between successive moments is not the same every-

70 Wolfgang Rindler where when measured with standard clocks. “Youth” can be bought by going to live on a dense planet!

Sometimes there are holes (“black holes”) in a stationary or static field, although they can also occur in arbitrary spacetimes. Each such hole is bounded by a closed surface called its “horizon”, on which the field strengh becomes infinite, and near which the clock rates tend to zero. A clock at the horizon, if such a thing were possible, would stand still. The only particle that can resist an infinite field is a photon, which can remain at rest in it. So the horizon can be thought of as a light front that gets nowhere!

But what goes on inside the horizon is even more surprising. In fact, for the student of time, that is one of the most interesting phenomena of all of General Relativity. Let us distinguish two kinds of surfaces in spacetime. Those that are moment-like, as already defined, and those that are “history-like”, namely generated by worldlines of particles. Now outside a black hole a closed surface surrounding the horizon is history-like: particles can be imagined to sit on it forever, and so their worldlines will be the generators of the spacetime version of this surface. But it turns out that according to Einstein’s field equa- tions a critical change takes place at the horizon: a closed surface inside of it is moment-like. And since moments cannot stand still, this can lead to unstop- pable gravitational collapse.

Let us pause for a moment to consider a profound difference between space and time: At any point of space I can stand still; but there is no way I can stand still at a point in time. Time inexorably marches on. And so it is with moment-like surfaces in spacetime. We know from their local special-relativistic interpretation that the moment t = noon must become the moment t = one o’clock, and then two o’clock, and so on. It turns out that time inside a black hole progresses inwards. Consider a typical heavy star about three times more massive than our sun. Eventually it will burn up all its fuel, the central furnace will no longer resist the inward pressure of gravity, and the star will begin to collapse. Once it collapses within the horizon - a sphere of some 10 km radius - it disappears from our view and no force in the world can now stop its total collapse to zero volume. Why? Its surface has become a “moment in time” and no force can stop the march of time. It is not the case that gravity has become irresistible, but rather that the force of gravity

Time from Newton to Einstein to Friedman 71 has been replaced by time! There is now a black hole with undiminished external gravity where there used to be a star.

Our story now draws to an end. But, perhaps unexpectedly, it is a happy end. After exposing all the pitfalls of relativistic time at the level of “small” gravitating systems like globular clusters, or even galaxies and clusters of galaxies, General Relativity, when applied to the universe as a whole, brings us full-circle almost back to Newton’s absolute time and even Newton’s absolute space. The basic reason for this is the incredible and unexplained regularity of our universe. Admittedly, the universe is full of irregular arrays of matter; but it is so huge that on a cosmic scale those irregularities are negligible - far more negligible, for example, than are the little ups and downs on the surface of an orange! The universe turns out to be isotropic and homogeneous. How do we know? It is impossible to observe its homogeneity directly, that is, to check that all its regions are identical. For light takes billions of years to get to us from the really distant regions, which we therefore see as they were billions of years ago; and we have no accurate knowledge how our region looked that long ago. But we can directly observe that the universe is perfectly isotropic around us: in no direction does it look any different than in any other direction, both for optical and radio observations. Now our sun is but one of billions and billions of similar stars. At least 1022 are known to exist (a “one” followed by 22 zeros) - the number of pinheads that would fill a cube 15 miles on edge! So, surely, our position in the universe is not unique. But then the universe must be isotropic about every point. And that implies homogeneity. For if region A had developed differently from region B, this would show up as a lack of isotropy from any region C equidis- tant from A and B.

The real father of modern cosmology is the Russian physicist Alexander Friedman (1888-1925). He was the first ever to countenance (in 1922) a dynamic universe - a huge (and possibly infinite) system of galaxies all moving under their mutual gravity according to Einstein’s laws of General Relativity. No one before Friedman, not even Einstein, had dared to give up the picture of a static universe! The cosmological model that Friedman constructed is accepted to this day. It expands equally in all directions, having apparently started with a “Big Bang”. Each galaxy can consider itself to be at the center of the expansion. Friedman realized that a homogeneous evolving universe deter-

72 Wolfgang Rindler mines its own unique preferred time, nowadays known a cosmic time. Its moments are defined as global sets of events at which the smoothed-out universe is in the same state (same density of matter, same density of radiation, same recession rate of nearby galaxies, etc.). And the interval between moments is defined as the proper time on any standard clock that partakes of the smoothed-out motion pattern. Cosmic time, so defined, allows us to assign in an observer-independent way - just as in Newton’s theory - a unique date AB (“after the Big Bang”) to events anywhere in the universe.

The smoothed-out motion pattern also defines a local state of rest everywhere in the universe at each instant, just as Newton’s absolute space had done; only now this absolute space is expanding! It has been determined that our solar system moves with a proper velocity of some 600km/sec through it. (The universe of background radiation is far more finely homogeneous than the universe of matter, and it is comparatively easy to measure our velocity relative to that.)

So the picture that finally emerges is as follows: the history of the smoothed- out universe, regarded as a spacetime, can be sliced uniquely and with per- fect regularity into cosmic moments. However, this regular global slicing is deformed locally, by the lumpiness of the actual universe, into the irregular and often arbitrary slicing we have discussed. Thus globally, let us say in the realm of philosophy, time has preserved its Newtonian features. Only locally, where physicists must dirty their hands, has time become a little tricky. Here Einstein reigns.

* The interested reader will find the various results mentioned in this article fully elaborated in the author’s book, “Relativity: Special, General, and Cosmological” published by Oxford University Press in August 2001.