سکوت

If only Newton had set his absolute space as the space of the ‘‘fixed’’ stars, there would have been no Leibniz-Clarke controversy. (Erlichson 1967, 95)

Classical physics made several important contributions to the theory of time, each of which had significant philosophical impact. Galileo worked with the notion of physical time, by which he basically meant clock time, as used in his mechanical experiments. Newton introduced the notion of mathematical time, which he used to define his laws of motion. And Boltzmann arrived at the notion of the arrow of time. The notion of time in classical and modern physics is invariably the notion of measurable time.

 

 

2.8.1  Galileo’s  Physical Time

 

In Galileo’s famous experiments with inclined planes a new notion of time emerges in physics. It is the notion of physical time, which presupposes that time is mea- surable. Although Galileo was a supporter of the Copernican heliocentric model of the universe, his notion of physical time is not derived from cosmological consid- erations. Galileo needed a concrete measure of time for his fall experiments. Galileo’s notion of physical time is an early version of clock time. Clock time indicates a scale of continuous, linear, measurable and abstract units by which the duration of physical events can be measured (Elias 1992, Sects. 21–23; Wendorff 1985, 205–206; Burtt 1980, 91ff). As Galileo’s experiments demonstrate, clock time does not necessarily mean the availability of mechanical clocks. Galileo in fact employed water clocks; and Einstein later used light clocks. Einstein showed that the classic assumption that clocks behave invariantly in all inertial reference frames was mistaken. Galileo was not concerned with the Aristotelian question of ‘why’ bodies moved the way they did on the surface of the Earth. His objective was to describe ‘how’ bodies moved and how this motion could be described mathematically. Although the mechanical clock had been invented at the end of the 13th century, Galileo could not use mechanical clocks to measure the motion of falling bodies, because the time pieces at the beginning of the 17th century were not precise enough to measure the short intervals of falling bodies. What is essential is a regular and invariant process by which the duration of an event can be measured. How did Galileo measure the elapsed times in his experiments? He reports that he used a form of water clock—a bucket with a hole in its bottom, from which a thin steady stream of water flowed. The water was collected, in each experiment, in a cup under the bucket. Galileo weighed the amount of water, which was collected when a ball rolled down the incline from A to B, from B to C, and from C to D (Fig. 2.10). The intervals A–B, B–C, C–D were equally long but the amount of water collected did not remain

the same. In fact, he collected less water for the interval C–D than for the interval A–B, because of the acceleration of the balls down the plane. From the differences in the weights Galileo obtained the proportions of the weights and calculated the proportions of the times. In order to obtain measurements for the fall times of bodies, Galileo experimented with balls rolling down inclined planes.

As a good scientist he carried out the experiments repeatedly on different planes with different inclinations. But in the many repetitions of the experiments, he found no time differences, ‘not even 1=10 of a pulse beat’ in the measured fall times

of the balls in the respective planes. Galileo’s result was that the lengths of the

planes were related to the squares of the times: L * t2. Galileo concludes that

in such experiments, repeated a full hundred times, we always found that the spaces traversed were to each other as the squares of the times, and this was true for all inclinations of the plane, i.e., of the channel, along which we rolled the ball. (Galilei 1954, 179; cf. Galilei 1953)

 

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In modern terms, Galileo found that L v0t 1=2at2; where the parameter t indicates clock time. Note that t can take on any numerical value, even if a mechanical clock cannot measure it. Imagine that t has the value of the square root

 

of p—pp; this value can be plugged into the equation to obtain a hypothetical

length for L, even though such an experiment may not be realizable in a laboratory, certainly not in Galileo’s lifetime. Galileo’s notion of clock time, as realized in his water clocks, satisfies both the criteria of regularity and invariance. First, the water flows regularly, if the same amount of water is taken each time and the same type of bucket is used. Secondly, as Galileo reports himself, no time differences were found in the many repetitions of the experiments. Galileo’s fall experiments were thus at least invariant with respect to changes in time; but they would also have been invariant with respect to changes in locality. The regularity of the water flow is insufficient by modern standards and it is of a linear rather than a periodic type. Nevertheless the water flowed sufficiently regularly from the bottom of the bucket to constitute a clock for the purpose of the experiment.

 

2.8.2 Newton’s Mathematical Time

 

While in Galileo’s experiments the symbol t expresses clock time—or the time interval, which elapses between two events, like the beginning and the end of an inclined-plane experiment—it remains one of the physical parameters, which enter into relations like L ~ t2. In Galileo’s work time has no further philosophical signif- icance. But Newton incorporates his notion of absolute time in his theory of motion in a systematic way. In Newton’s theory the notion of time therefore takes on philo- sophical significance. Newton defines his notion of time, to be used in his theory of motion, before his develops his equations of motion. For Newton mathematical time included two aspects: (a) At first Newton considers, like the Ancients, that time makes a reference to observable physical events, like the orbit of the moons around Jupiter. Although he is a child of the Copernican age he does not believe that the planets orbit the sun with perfect regularity. In fact Newton goes further and conjectures, correctly, that even though the motions of such bodies around their respective centres satisfied the requirement of periodic regularity, this periodicity suffered nevertheless from too many irregularities for the formulation of the laws of motions. For instance, the speed of the Earth’s rotation varies due to tidal frictions, due to seasonal changes and due to a small difference between the geometric and the physical (rotational) axis of the Earth;

(b) Newton therefore postulates an absolute space and an absolute time, which in theory make no reference to material events in the universe. Newton defines ‘immovable’ space as a space with no relation to any external events, objects or processes; equally for time. These definitions stand in stark contrast to, say, the views of Saint Augustine, which make time dependent on physical events.

By a suitable analogy, absolute space can be envisaged as a cosmic container, which exists irrespective of whether it contains material objects, like planets and galaxies. And absolute time can be imagined as a constantly flowing metaphorical river whose regularity constitutes the basis of a clock. All motions in the material world are measured against the constant tick of the clock. One can consider Newtonian space as an empty container, on whose walls are etched a spatial and a temporal axis (Fig. 2.11). If objects are placed inside the container, their locations and motions can then be traced against the fixed spatial and temporal axes. To illustrate, image that this container is populated with two ‘intelligent’ molecules,

which spend their days, like fish in a tank, swimming around aimlessly. To make their  lives  more  interesting,  the  molecules  decide,  on  their  next  encounter,   to determine their respective positions inside the container. Molecule1 asks molecule2, ‘where are you?’ but the answer ‘near you’ is hardly enlightening. Such relative determinations are useless because the molecules change their positions constantly and supply no absolute determination of their location. But the two molecules can determine their positions inside the container if they can refer their respective locations to the same temporal and spatial axes. (The time axis may not consist of mechanical clocks, simple ticking devices would suffice.) The laws of motion are formulated with respect to these absolute notions. As absolute space and time cannot be observed, these notions provoked considerable debate. What function these notions played in Newton’s mechanics will be considered later.

Whatever the philosophical merits of these notions, Newton emphasizes both the importance of regularity and invariance in his absolute notions. Absolute, immaterial space and the ‘equable flow’ of time are regular with respect to all inertial and accelerated observers (like the ‘intelligent’ molecules). Absolute space and time are also invariant, since the units of absolute space and time do not change as the molecules change their positions or under a switch of perspectives from one molecule to the other. Newton took the need for regularity and invariance to an extreme when he postulated absolute space and time to ground his laws of motion. If absolute space and time could be observed, no observer would have difficulties in measuring all relative motions against its invariant structure. By analogy, the speed of an aeroplane is measured with reference to the surface of the Earth rather than drifting clouds in the sky. Newton even invented some thought experiments, as we shall see, in an effort to demonstrate that the notions of absolute space and time could be inferred from observations of relative motions. For present purposes it is important to note, especially in relation to Leibniz, that Newton introduces approximations to ‘materialize’ his absolute notions. His absolute notions are idealizations from observable physical systems. Thus Newton suggests that ‘Jupiter’s satellites’ may serve as an approximation to absolute time, no doubt because of the perceived regularity of their orbits. If Newton lived today he would certainly change his approximations of absolute time to the recent dis- covery of neutron stars (1967), which rotate very fast and emit very regular pulses of radiation. They have a much greater regularity than planetary motions. As far as absolute space is concerned, Newton regarded the ‘fixed’ stars as a good approximation to the image of ‘immovable’ space because the constellations were regarded as ‘fixed’ in the sky; their apparent rotation through the night sky was due

to the daily rotation of the Earth.

But because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body, considered as immovable, we define all places; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And so, instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs; but in philo- sophical  disquisitions,  we  ought  to  abstract  from  our  senses,  and  consider  things

 

 

Newton conceives of space and time as existing independently of each other. This separation of the temporal and spatial axes would finally be abandoned in Minkowski space–time. Newton’s view that the spatial and temporal axes are absolute reference points for all observers means that these axes remain perpen- dicular to each other, irrespective of the motions involved (Fig. 2.12).

In order to appreciate this point, consider again the measurement of the cir- cumference of the Earth, which today is known to be approximately 40,000 km. Depending on the definition of the units used by Eratosthenes he measured the circumference of the Earth to range between 35,000 and 45,000 km. Now consider an astronaut who flies past the Earth at a speed close to the speed of light. Does the astronaut measure the same value as Eratosthenes? The commonsensical answer is to say ‘yes’, since the measurement of a distance does not seem to depend on the velocity of the frame from which it is measured. And yet, as Einstein’s Special theory of relativity was to show in 1905, this commonsensical answer is mistaken. But the commonsensical answer illustrates well one sense of Newton’s absolute notions: the measurement of temporal and spatial distances depends neither on the location of the measuring device,  nor on the velocity with which it is moving.    On this commonsensical view, if an inhabitant of a distant galaxy could take measurements of the Earth, this galactic creature would measure the same dimensions  as Earthlings.

The absoluteness of time and space can also be understood, less metaphorically, in terms of spatio-temporal structure. (Later it will be shown that Newtonian mechanics can be re-interpreted in the language of space–time.) In this sense,

 

These relations can be illustrated in terms of Newton’s techniques. In the approximation he uses for the abstract notion of absolute space, he considers     that all reference frames, which are in constant uniform motion with respect to each other, may use the ‘fixed stars’ as spatial markers to position themselves   with respect to others. It is as if local observers shunned local landmarks and   used celestial landmarks to position their respective locations. In terms of ‘absolute space’ or the markings on the cosmic container, all observers refer to   the same place. It follows from  the notion of  absolute  time that simultaneity  must also be an absolute notion: the relation of ‘being-at-the-same-temporal- position’ means that all reference frames use the ‘astronomical clock’, which in approximation is provided by the orbits of the Jupiter moons (or  today  the  emitted radiation from pulsars). It follows from the use of the same  astro-  nomical clock, or the temporal markings on the cosmic container walls  that  events, which are simultaneous for some observers, must be simultaneous for all observers  (Fig. 2.13).

It may not come as a great surprise that Newton’s notions of absolute time and space became the subject of an intense debate and generated philosophical con- sequences. Leibniz rejected Newton’s notions and defended a relational view. Later Einstein abandoned Newton’s separation of time and space altogether.

 

 

2.8.3  Newton  and  Leibniz, Compared

 

It has already been stressed that Leibniz (1646–1716) accepted the notion of physical time and that he saw the order of events as subject to a universal principle of causality. Leibniz rejected Newton’s postulates of absolute space and time as merely

metaphysical speculations. He ignored Newton’s need for approximations. Leibniz defined space as the ‘order of coexisting events’ and time as the ‘order of succession of coexisting events’ (Leibniz 1715–1716, Third Paper, 210–215). He thus made an essential reference to the ‘before–after’-relationship in physical events. Leibniz does not specify what he means by ‘events’. Even if his notion of ‘events’ refers to events in the physical world does he mean actual or also possible events? Leibniz had a predecessor to his relational view in the person of Saint Augustine. Saint Augustine, too, insisted on the ‘before–after’ relation between events but he also failed to specify, which events in the physical world he had in mind. It is often claimed in the literature that Leibniz must restrict the admissible events to ‘the set of actual physical events’ (Friedman 1983, 63; Earman 1989, Chap. I). And then Leibniz would face the difficulties, which Newton avoided by resorting to the idealization of absolute space and time. For the ‘set of actual physical events’ may not offer suf- ficient regularity to ground a stable order in the succession of events. What is not often realized, as we shall discuss further in Chap. 3, is that Leibniz admits of ‘possible’ events. Thus Leibniz moves in exactly the opposite direction to Newton. Whereas Newton introduces his idealization of absolute space and time and then suggests approximations, Leibniz begins with ‘actual’ events but then moves to the idealization of possible events and ‘fixed’ existents.

And fixed existents are those in which there has been no cause for a change of the order of coexistence with others or (which is the same thing), in which there has been no motion. (Leibniz 1715–1716 Fifth Paper, Sect. 47, 231; cf. Manders 1982)

These ‘fixed existents’ therefore remain invariant in the order and succession of coexisting events. Despite the often-claimed deep gulf between Newton’s realism and Leibniz’s relationism about time, they share a meeting point in this interaction between idealization and approximation. Leibniz and Newton could therefore agree on the same approximations for the purpose of measuring time and space in the material world. The Leibnizian move to idealizations—‘fixed existents’ and possible events—can be gleaned from a consideration of the order which must  hold between the events. Let us consider the Leibnizian notion of order from the perspectives of universality,  regularity  and  invariance.  Order  may  just  mean  ‘a before–after’-relationship between events, governed by some universal principle of causality. But in his Correspondence with Clarke Leibniz goes one step further when he considers that the ‘before–after’ relationship between events must, in some sense, be law-like. That is, it must be governed by the laws of classical physics. It is the recognition that the cosmic machine is driven by law-like reg- ularities, which makes Leibniz introduce the idealizations. For the laws of classical mechanics require inertial reference frames and these are provided by the invariants in the physical  world.

So there are a number of disagreements and agreements between Leibniz and Newton, whatever other differences may separate them (see Chap.  3). Newton  and Leibniz disagreed about the ultimate nature of time. In his philosophical discourse about time, Newton postulated that time was absolute, since it existed irrespectively  of  material  happenings in  the universe. Even  an  empty  universe

has a ‘clock’ for Newton. For Leibniz the universe must be filled with events;     the order of succession of events constitutes a clock. For Leibniz physical time is not absolute but relational since it depends on the succession of material events    in a particular order. Leibniz agreed with Saint Augustine that before creation   time did not exist. Time is coextensive with the creation of events. Despite these disagreements we can recover some common ground. Leibniz agreed with  Newton that the order of coexisting physical events constituted a universal simultaneity relation between such events. That is, all observers throughout the universe agree that two events, which occur simultaneously for one observer, occur simultaneously for all observers whatever their location in the universe. Leibniz also agreed with Newton that events, which occur in succession, display the same order of succession for all observers. Thus, whether the ‘before–after’ relation refers to actual or possible events, there was no doubt in the minds of any observer, from which ever location they may observe the succession of two events—E1 and E2—that E1 would occur before E2. The aspects of regularity and invariance can also be recovered. Both agreed that the succession of events must be regular and invariant, to a certain degree, across all perspectives for measurable time to be possible. For although Newton and Leibniz did not agree on whether material happenings were needed for time to unfold in a philosophical sense, they were in agreement that time was universal. All observers throughout the universe, who are in motion with respect to each other, use the same clock, i.e., they refer to the same temporal axis. Newton secured regularity of temporal succession and spatial coexistence by the stipulation of an ‘immovable space’ and an ‘equable flow of time’. But absolute space and time are beyond the realm of observation so that Newton introduces appropriate approximations, like the regularity of celestial systems (Jupiter moons, fixed stars). Newton secured invariance by making ‘absolute’ space and time irrespective of particular perspectives. Even in approximation, the regularity of celestial motion is the same for all observers, irrespective of their particular point of observation. Newton’s approximations illustrate the regularity of the clockwork universe, since it is based on the law-like succession of physical events. Leibniz guarantees regularity of temporal and spatial intervals by presenting the universe as a cosmic machine, governed by mechanical laws (Leibniz 1715–1716, Second paper, 208; Fourth Paper, Sect. 32, 219). As Leibniz agrees with Newton about the universality of the order of the succession of events—time is universal for all observers, irrespective of their perspective on the world—the invariance of temporal and spatial relations is also guaranteed. As will be discussed later, Newton’s mechanics has more affinities with Leibnizian relationism than  his philosophy.

We have focussed on the agreement between Newton and Leibniz for the sake of highlighting in this chapter the close connection between time, regularity and invariance in classical physics. The same connection exists in modern physics—in relativity and quantum theory—where it comes to even greater prominence. This is essentially due to the fact that the classical notion of order of events, as defined in classical physics, undergoes a conceptual revision. In the Special theory of rela- tivity  both  the  universality  and  invariance  of  temporal  relations,  in  the  way

Newton and Leibniz understood them, are abandoned. We shall need to see how at least invariance survives in modern physics. With his notion of time as the order of the succession of coexisting events, Leibniz, like Saint Augustine before him, defines a direction of time; in our terminological convention, the ‘before–after’ relation of local events specifies the passage of time. Physicists and philosophers of the classical age did not consider the further question whether time displays a global arrow or whether the universe follows a one-directional evolution. In order to consider such questions, a different physical relationship—entropy—had to be discovered.

 

 

2.8.4  The Arrow of Time

 

The discussion of the topology of time showed that a consideration of the anisotropy of time suggests a distinction between the passage of time and the arrow of time (Denbigh 1981; Grünbaum 1967). The passage of time expresses our daily experience of the one-directional ‘flow’ of time, often from order to disorder, from new to old, from warm to cold. But our impression of the passage of time is compatible with different topologies of time. Time may have the topology of an open line or a closed circle but observers would not be able to tell from their limited experience of the ‘flow’ of time. The arrow of time expresses a global, one- way direction of the universe, from initial to final conditions, where these con- ditions are asymmetric. But the arrow of time cannot directly be observed; it needs to be inferred  from physical criteria.

The arrow of time seems to be quite different from other arrows in our expe- rience, which seem to generate order, information and complexity (cf. Savitt 1995, 4–5; Savitt 1996; Hoyle  1982)

• The psychological arrow. As Saint Augustine emphasized, we remember the past but we anticipate the future; we have the feeling that time ‘flows’ relent- lessly forward into the future towards a period when we will no longer exist.
• The historical arrow: We have records of the past but not of the future; there is evidence of the geological evolution of the Earth (discovery of deep time) and the biological evolution of species towards greater diversity and complexity. Darwin represented this arrow by his image of the tree of life.
• The cosmological arrow: As Kant speculated, the universe has expanded from an initial, chaotic beginning, which is today located in the Big Bang, towards the formation  of  solar  systems,  galaxies  and  clusters  of  galaxies.  Observations

suggest that the cosmological arrow will take the universe towards a Big Chill in the distant future rather than a re-contraction to a Big Crunch. It is at present not clear whether our observable universe is only one amongst an infinity of universes (multiverse), which are born and die.

• But the  thermodynamic  arrow of time seems to point in the opposite  direction,

towards disorder and the destruction of information (Layzer 1975, 56; Layzer

1970; Landsberg 1996, 253; Whitrow 1980, Sect. 7.4; Hawking 1988, 145ff; Ladyman et al. 2008). Could such an increase of disorder be used to characterize the arrow of time? And how are the other arrows related to the thermodynamic arrow?

 

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In the 19th century physics discovered the laws of thermodynamics, which were considered to shed light on the question of the arrow of time. Of particular importance were the Second law of thermodynamics and the discovery of entropy. Entropy is often characterized as an increase in disorder in a closed system.8 Entropy has also been associated with the loss of information, which is available about a system undergoing entropic change. In Chap. 4 it will be shown that these characterizations are imprecise and that the apparently irreversible increase in entropy is best expressed by a spreading function; entropy itself is best defined in terms of the number of micro-states, which correspond to a given macro-state in phase space. For the time being our discussion is confined to the discovery of entropy and how entropy was used to define the arrow of time.9 There are many everyday illustrations of the concept of entropy: food rots, hot coffee cools, if milk and coffee are mixed they cannot be unmixed, and a sugar lump dissolves in hot coffee and cannot be retrieved. But the total amount of energy always remains the same. According to the first law of thermodynamics—Q     DEk     DEp—the sum  of all energies remains constant in a closed system. But while all mechanical energy (work, W) can be transformed into thermal energy (heat, Q) it is not the case that all thermal energy can be transformed into work: dE  dQ  dW: A  machine can be employed to perform useful work but some of its energy is lost in frictional heat that is no longer available to do work. To illustrate the Second law, consider a child’s playroom (Fig. 4.1a, b). The father puts all the toys into a corner so that the child can play in the morning. When the child enters the room there is a great amount of order in the playroom. There is also precise information as to the whereabouts of the toys. But as the child gets to play, toys are beginning to be tossed about all over the room and when the child takes a lunchtime nap the morning order has been reduced to a lunchtime disorder. The neat ordering of the toys has disappeared and has led to a loss of information as to the whereabouts of the toys. Whilst they were stacked neatly in the corner of the room in the morning,

 

they are now scattered randomly across the room. So the total amount of energy remains the same but entropy has increased.

 

 

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In a slightly more technical sense it is customary to illustrate the increase in entropy by modelling a closed system, like a sealed container, separated by a removable partition, such that compartment  contains gas molecules and com- partment    is empty (Fig. 2.14a). At first all the gas molecules are confined to compartment , so that the probability of finding a gas molecule in compartment  is zero. Furthermore, if an atom is placed in, say, compartment , there is a good chance that it will be hit by gas molecules, so that the molecules can do work on the atom  (W    Fd). If the  partition wall  is removed  (Fig. 2.14b), the  molecules  will

 

spread over the larger volume (phase space) but information about their whereabouts

is lost. The ‘disorder’ of the system is increased, and hence the ability of the system to do useful work is reduced, since, for instance, its pressure decreases.

The notion of entropy can be made more precise, in quantitative terms, by relating it to the Second law of thermodynamics. W. Thomson (Lord Kelvin, 1824–1907) expressed it in the form that every viable engine must have a cold sink (see Atkins 2003, Chap. IV, 113; Kuchling 2001, 292; Fig. 2.15). In other words, heat can only be transformed into work when some of the heat transfers from a warmer to a colder  body.

Rudolf Clausius (1822–1888) stated the Second law in the form that ‘heat never flows from a cooler to a hotter body’ (Atkins 2003, 115; Kuchling  2001, 262;  Fig. 2.16a). In other words: Heat can only be transferred from a colder to a hotter body by means of mechanical work (Fig. 2.16b).

The Clausius statement means that all work can be transformed into heat but not all heat into work: Q = W ? DU. Consider a car engine, for illustration. The car engine burns fuel to move the car but not all of this fuel can be used to transform it into kinetic energy. Some of the energy is needed to run the engine, other energy is lost through friction in the engine parts and in the tyres (etc.). Hence whilst we get work out of the fuel, some of the heat from the fuel irretrievably escapes into the environment; it is not available for further work. By contrast all the work the car performs transforms into various kinds of heat (energy); it moves the car forward but also warms up the interior, it creates an air flow, it wears the tyres. While it is always possible to transform kinetic energy and work into heat, it is never possible to transform all heat into work. Clausius then defined a change in entropy for a reversible process as follows:

 

ð    revÞ

             Energy supplied as heat dQ

 

Change in Entropy ðdSÞ ¼ Temperature at which the heat transfer occurs ðTÞ ¼ 0

ð2:2Þ

And for an irreversible process he defined the change in entropy as dS C dQ/ T [ 0. Clausius further proposed that the amount of entropy, in an irreversible process, never decreases. This leads to the Second law of thermodynamics:

DS 2: 0 (3),                                              ð2:3Þ

(whilst the first law states that the total amount of energy remains constant). In all natural processes the amount of entropy (of disorder) increases, as we are well aware from our everyday experience. Clausius expressed the first law of thermodynamics in the pithy statement:

Die Energie der Welt ist constant. The energy of the world is constant.

And then applies the concept of entropy to the whole universe:

Die Entropie der Welt strebt einem Maximum zu. The entropy of the world strives towards a maximum. (Atkins 2003, 119; Torretti 2007, Sect. 6)

The increase in entropy in natural systems means that the energy available to do useful work decreases. A log of wood keeps the fire burning, which makes the water boil; but once the log has turned into ashes, it can no longer do useful work. Ashes don’t keep the fire  burning.

As we shall discuss, a generation later Ludwig Boltzmann (1844–1906) felt compelled to redefine the notion of entropy for statistical mechanics. But he echoed Clausius’s generalization when he declared that the

If it is true that there is a general degradation of energy, then all forms of energy will eventually be converted into ‘heat’ and the universe will reach thermal equilibrium. This tendency towards equilibrium was called the ‘heat death’ at the end of the 19th century. The scenario of the Heat Death enjoyed great popularity. It was meant to signal that the whole universe would eventually consume the amount of useful available energy, making it impossible to sustain temperature differences. At the end of the universe the total amount of energy was still unchanged but the balance between useful and useless energy had shifted towards thermal equilibrium. The sun, for instance, would no longer sustain life on Earth. Thomson (1852) spoke of ‘a universal tendency in nature to the dissipation of mechanical energy’ and concluded his survey with the ominous warning that ‘the Earth was and will again be unfit for human habitation’.

L. Boltzmann, like W. Thomson, lifts the notion of entropy to a cosmological level in an attempt to identify the arrow of time. Whilst it is generally accepted that entropy increases to a maximum in a closed system, like a container of gas molecules, Boltzmann assumes that the Second law can be applied, under certain reservations, to the whole  universe.

People have been amazed to find as an ultimate consequence of this proposition that the whole world must be hurrying towards an end state in which all occurrences will cease, but this result is obvious if one regards the world as finite and subject to the second law. (Boltzmann 1904, 170; cf. Uffink 2001)

It will emerge later that this assumption is questionable. Boltzmann himself pointed out that the direction of increasing entropy could be identified with the arrow of time only in certain parts of the universe.

There must be in the universe, which is in thermal equilibrium as a whole and therefore dead, here and there relatively small regions of the size of our galaxy (which we call worlds), which during a relatively short time of eons deviate significantly from thermal equilibrium. Among these worlds the state probability increases as often as it decreases. (….) a living being that finds itself in such a world at a certain period of time can define the time direction as going from less probable to more probable states (the former will be the ‘‘past’’ and the latter the ‘‘future’’) and by virtue of this definition he will find that this small region, isolated from the rest of the universe, is ‘‘initially’’ always in an improbable state. (Boltzmann 1897, 242; cf. Boltzmann 1895a, 415; Sklar 1992, Chap. 3; Zwart 1976,

Chap. VI, Sect. 3; Whitrow 1980, Sect. 7.3; van Fraassen 1970, 90ff)

Boltzmann holds that the arrow of time lies in the direction, in which a small region of the universe moves closer to equilibrium. Such small regions house life- supporting planets like the Earth. Boltzmann proposes that the surrounding universe is in thermal equilibrium but there are local, entropy-decreasing fluctuations; our world is in an ordered state, away from equilibrium, because it exists in an entropy- reduced fluctuation state. Although the entire universe languishes in a state of Heat Death,  in which  time  no longer  exists, the inhabitants of low-entropy fluctuation

 

 

2.8.5   Maxwell’s Demon

 

Boltzmann, however, soon encountered problems with the thermodynamic notions of entropy and disorder, which forced him to reinterpret the Second law in a statistical manner. The understanding of the Second law as a probabilistic, rather than a deterministic law, in turn had serious consequences for the identification of the arrow of time with entropy. This shift constituted the beginning of the tran- sition from thermodynamics to statistical mechanics.

In a famous thought experiment, involving ‘a being with superior faculties’, James Maxwell attempted to show that the Second law of thermodynamics only possessed statistical validity (Maxwell 1875, 328–329; cf. Earman and Norton 1998, 1999; Leff and Rex 2002; Hemmo and Shenker 2010). This being, later dubbed ‘Maxwell’s demon’, is able to ‘follow every molecule in its course’. In an appropriate setup such a being would be able, says Maxwell, to sort the molecules according to their respective velocities. The setup is simply a container, divided into two chambers by a partition, in which there is an opening (Fig. 2.17). The demon’s only work involves the opening and closing of the hole ‘so as to allow only the swifter molecules to pass from A to B, and only the slower ones to pass from B to A.’ Maxwell  concludes:

He will thus, without expenditure of work, raise the temperature of B and lower that of A, in contradiction to the second law of thermodynamics.

If it can be broken, the Second law cannot enjoy deterministic validity. It must be a statistical principle, which holds with overwhelming probability for systems composed of many particles. It follows that findings, which apply to the macro- scopic level, may not apply to the microscopic level:

This is only one of the instances, in which conclusions which we have drawn from our experience of bodies consisting of an immense number of molecules may be found not to be applicable to the more delicate observations and experiments which we may suppose made by one who can perceive and handle the individual molecules which we deal with only  in large classes.

Following this insight, Maxwell spells out the classical view of probability, which results in averages over micro-states (of molecules) but leaves us ignorant about their individual properties (position, momentum):

In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus. (Maxwell 1875, 329; cf. Myrvold  2011)

Physicists ever since have attempted to show that Maxwell’s demon cannot achieve his aim; and that the demon himself is subject to the Second law.10 For if the demon succeeded, our energy problems would be solved:

Machines of all kinds could be operated without batteries, fuel tanks or power cords. For example, the demon would enable one to run a steam engine continuously without fuel, by keeping the engine’s boiler perpetually hot and its condenser perpetually cold. (Bennett 1987, 88)

One reason why the demon cannot escape the effects of the Second law is that, in order to sort the molecules, he must be in contact with them and thus is affected by their thermal motions. The demon would absorb more heat from the molecules than he can expand. In other words, he would warm up. He would begin to shake from the Brownian motion of the molecules inside his body, which would make him unfit to perform his  task.

It turns out, if we build a finite-sized demon, that the demon himself gets so warm that he cannot see very well after a while. The simplest demon, as an example, would be a trap door held over the hole by a spring. A fast molecule comes through, because it is able to lift the trap door. The slow molecule cannot get through, and bounces back. But this thing is nothing but our ratchet and pawl in another form, and ultimately the mechanism will heat up. If we assume that the specific heat of the demon is not infinite, it must heat up. It has but a finite number of internal gears and wheels, so it cannot get rid of the extra heat that it gets from observing the molecules. Soon it is shaking from Brownian motion so much that it cannot tell whether it is coming or going, much less whether the molecules are coming or going, so it does not work. (Feyman 1963, Sect. 46.3)

 

Nevertheless, J. C. Maxwell achieved his aim of ‘picking a hole in the second law’. The Second law is a statistical principle, which raises several questions for the notion of  time:

• Reversibility: It is not a violation of the Second law if all the molecules in the container in the course of time return to their initial state. But this return is unlikely to happen in the lifetime of the universe. Yet there is a fundamental split between the reversibility of the micro-states and the overwhelming irre- versibility of the macro-states. What then is more fundamental: the temporal symmetry of the dynamic laws, which govern the micro-states or the temporal asymmetry of the macro-states, with which we are familiar?
• Arrow of Time: If the second law is a statistical law, is it still possible to

associate the arrow of time with an increase in entropy? Do the Boltzmann fluctuations mean that the arrow of time reverses when entropy decreases? And is the universe on a trajectory towards a Big Chill or a Big Crunch, a return to its original state? Is the universe symmetric rather than asymmetric?

As will be shown in Chap. 4 such questions cannot be answered within classical thermodynamics. They require tools as furnished by quantum cosmology. But this Chapter first completes its survey of the contributions of physics to the notion of physical time.

 

 

1. Time in Modern Physics

 

It is important to make a sharp distinction between an asymmetry of the world in time from an intrinsic asymmetry of time itself. (Davies 1994, 120; italics in original)

It has been argued that there is a need for physical time and that physical time, because of the uniformity in the sequence of events, on which it is based, is measurable time. Both the theory of relativity and quantum theory made significant contributions to our understanding of measurable time, because they threw further light on the notions of regularity and invariance.

 

 

 

2.9    Time  in Modern Physics                                                 63

 

These absolute reference frames were often regarded as privileged in comparison with the mere relative motions. In Greek cosmology, for instance, the position of the motionless Earth, near or at the centre of the universe, constituted such a privileged position. The natural motion of earthly objects, like falling stones, was to ‘strive’ back towards the Earth, since the ‘centre’ was their ‘natural’ place. The ‘fixed’ stars also provided absolute reference systems, against which the orbits of the planets could be tracked. Although around the time of the Scientific Revolution the Aristotelian theory of motion had long been abandoned, Newton still formulated both space and time as absolute reference frames, against which all other, merely relative motions are to be measured. Such systems are called ‘absolute’ because they furnish the ultimate unchanging standards, against which all other motions can be gauged. Recall the two ‘intelligent’ molecules floating randomly in their con- tainer world. When they meet, the question ‘where are you?’ makes little sense as long as they only have their relative motions and positions to orient themselves. An answer like ‘within your sight’ or ‘at arm’s length’ would not be very informative. But if on the inside of the container two axes are painted: one vertical ‘y’ axis along the side and a horizontal ‘x’ axis along the floor, it is easy for the two intelligent molecules to position themselves inside the container. The ‘x–y’ system on the inside of the container would serve as an absolute reference frame, compared to their relative motions (Fig. 2.11). Such ‘absolute’ reference frames were often regarded as fixed—like the stationary Earth in Greek cosmology—and immaterial, like Newton’s philosophical notion of absolute container space and the meta- phorical river of time (cf. Smart 1956). They were non-dynamic, since they served as a canvass against which the material world evolved (cf. Rynasiewicz 2000; Friedman 1983). And although Leibniz did not agree with Newton that these absolute reference frames were immaterial, he regarded them, in agreement with classical science, as universal in the sense that all observers in the universe agreed with these standards.

As science progressed in the 19th century it failed to find evidence for the existence of absolute reference frames. At first, 19th century physicists believed that an all-pervasive ether filled all of space, which served as an absolute scale, against which all relative motions on Earth could be measured. Such an ether- filling medium would fulfil the same functions as Newton’s absolute space, since it would constitute an absolute reference frame, which is both regular and invariant. However, physicists could not find any experimental evidence for the ether—the famous Michelson-Morley experiment (1887) drew a null result. The experiment showed that the existence of such an ether or absolute reference frame could not be detected. If there was no evidence of absolute space, there was no evidence of absolute time either. Absolute motion could not be detected.

At the beginning of the 20th century, Einstein’s famous Special theory of relativity (1905) showed further that such absolute reference frames were not needed for the formulation of the laws of physics. As we shall see (Chap. 3), the Special theory of relativity reawakened interest in the Parmenidean-Kantian view of time and it requires a reinterpretation of the Leibnizian notion of order. The Special theory of relativity does not apply to cosmology but it laid the foundation

for the General theory of relativity, which applies to large-scale cosmic structures. In both theories the time used is clock time, since the effect of motion on various kinds of clocks is taken into consideration. The Special theory is not, strictly speaking, a revolutionary theory. It is best thought of as an extension of classical mechanics. But it had a revolutionary effect on the classical notion of time in the same way in which the General theory had an effect on the notion of space. Both theories work with the notion of space–time. The Special theory had an immediate impact on the measurement of time. Einstein used light clocks because his theory of relativity shows that mechanical clocks are no longer reliable indicators of universal time. The impact was threefold—it concerned (a) the simultaneity of events; (b) the synchronization of clocks and (c) the universality of time.

1. In one of his famous thought experiments Einstein considers a fast-moving train whose front and rear ends, as it passes an embankment, are hit by bolts of lightning (Fig. 2.18). Two observers witness the event. One sits in the middle of the train, the other observes the events from the embankment. Only the embankment observer will see the two events as simultaneous, whilst the train passenger experiences the events as non-simultaneous. Within their respective frames the events occur regularly but they are not invariant across the two frames moving inertially with respect to each other. The reason for this curious situation is the finite, constant velocity of light. The observer on the train ‘rushes’ towards the ray of light coming from the front but ‘runs away’ from the ray coming from the rear end of the train. The two signals hit the embankment observer’s retina at the same time. Nevertheless, this situation does not give rise to relativism, since the observers can calculate the respective values of the arrival times of the signals. The rules they use are known as the Lorentz transformations. The effect of these rules can be compared to the transformation rules used by Archimedes and Eratosthenes, who—according to a previous thought experiment—used their knowledge of astronomy to calculate the highest point of the sun in their respective locations.

Einstein’s train thought experiment illustrates the so-called ‘relativity of simultaneity’, which is a consequence of the Special theory of relativity. In this illustration, an event, which is simultaneous for the embankment observer is no longer  simultaneous  for  the  train-bound  observer.  Thus  Einstein  abandons the

 

universal simultaneity assumption in classical physics. Relative simultaneity, as Einstein insists, is one of the arguments for the relativity of time.

Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference-body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event. (Einstein 1920, 26; italics in original)

1. Consider some space travellers who experiment with the clocks, which are attached to their space rockets. Their rockets serve as inertial reference frames, which travel near the speed of light. There are four observers, each one occu- pying a corner of the spaceship (1A, 2A and 1B, 2B, respectively) travelling in opposite directions (Fig. 2.19). They have set their clocks internally according to the method of light clocks, discussed below. They now wish to compare their clocks. When the spaceships’ two midpoints coincide, a light signal is set off, which propagates towards the front and rear ends respectively. As light travels with finite constant velocity, and the spaceships travel relative to each other, the observers 2A and 2B will be closer to the light source than observers 1A and 1B.
   • The signal will reach observer 2A earlier than 1B who will receive it after some additional time. According to 2A the clock of 1B lags behind time.
   • The signal will reach 1A later than 2B, so that 1A will conclude that 2B’s clock is ahead of  time.

As their clocks are synchronized within their respective  spaceships, observers in A will claim a delay between the watches of the other observers in B, and vice versa. Both A and B are correct from their respective points of view. It makes no sense to ask whose clock is ‘really’ showing the right time. The notion of absolute

simultaneity has vanished. Clearly the clock readings are perspectival. There are other effects on clocks, to be discussed later, which lead to similar consequences.

1. Finally, consider the loss of the universality of temporal measurements, which was inherent in Newtonian physics. Recall that Leibniz agreed with Newton that time ticks at the same rate for every observer in the universe. They agreed that time was universal although they disagreed about whether it was also absolute (independent of material events). Einstein abandoned the universality assumptions of classical physics. When clocks do not seem to tick at the same rate for all observers, it becomes difficult to agree on the temporal lengths of events. By abandoning the classical universality notion, he also abandoned the classical invariance notion. Thus the clocks run regularly for inertial observers within their own reference frames but across inertial reference frames the respective clocks are seen as running either ‘fast’ or ‘slow’, depending on the velocity of the frame, and the viewpoint from which they are observed. Spatial and temporal measurements become perspectival features of inertial reference frames, a phenomenon known as ‘time dilation’.

From the point of view of the measurement of time, the most striking conse- quence of these thought experiments is the loss of universality and invariance, though not regularity. The classical notion of universality of clock readings is lost because every reference frame carries its own clock but if these clocks are in inertial motion with respect to each other the clock readings will not agree. The invariance across different reference frames is lost because clocks, as judged from respective reference frames in inertial motion with respect to each other, are considered to tick at different rates (time-dilation). In other words, clock readings are path-dependent. Clocks record the length of a time-like curve connecting two events. But clock time retains its regularity within a particular reference frame— according to observers in these particular frames, the clocks run ‘normally’. It seems that the only invariant clock time, which still exists in the Special theory of relativity, is the clock time measured along the world line of each particular reference frame (Denbigh 1981; Chap. 3, Sect. 7; cf. Petkov 2005, Sect. 4.9; Hoyle 1982, 94). However, the loss of the universality and invariance of clock time is compensated for by new types of invariance. For Einstein postulated that the velocity of light, c, is invariant in all inertial reference frames. In 1676 the Danish physicist Olaf  Rømer  had determined the velocity of light by  observations of the

Jupiter moons; he found the value of c & 225,000 km/s (or 2.25 9 108 m/s, which is relatively close to the modern value of 2.99 9 108 m/s in vacuum). In his

concern for invariance Einstein shares the same worry as Newton who postulated absolute time and space as a background for the formulation of the laws of motion. These ‘invariants’ are one reason why Einstein’s theory of relativity is not a relativist theory. What Einstein means is that the theory of relativity needs to recover invariant relationships, despite the perspectival nature of spatial and temporal measurements. Absolute yardsticks like the ether, absolute space and time are replaced by a number of invariant relationships, which may serve as the basis for the measurement of physical time.

 

 

This new velocity theorem yields the old theorem in the limit when v and v0 are much smaller than c: But it never renders a velocity greater than c. Let the train move at a velocity near the speed of light, c1, and let the train-bound observer again shine a light, c2, in both directions. On applying the new velocity theorem:

 

the embankment observer will calculate a combined velocity of c for the two situations, in agreement with the postulate. For the train-bound observer w = c.

 

-      ¼

Einstein also used light clocks to compensate for the lack of synchronization of moving clocks (Einstein 1905, 30). How can two clocks, which are separated in space, be synchronized? An observer, C, can be placed at midpoint between the two clocks at A and B. A and B emit light signals towards C. The two clocks are synchronized when the signals arrive at C at the same time (Fig. 2.20). These two clocks will be in synchrony according to this procedure, if the condition tA     tC

 

 

-

tB     tC  is satisfied.

 

The use of light clocks also allows a definition of the simultaneity of two events in a given reference frame. Imagine that observers A and B, after having syn- chronized their clocks, move to positions A and B, where they explode bombs.  This is a version of the train thought experiment. C, who is equidistant between    A and B, will see the light from the two explosions at the same time and hence will conclude that they exploded simultaneously. Thus two events in a given reference frame are simultaneous, according to this procedure, if the light signals from the events reach an observer halfway between the events at the same time, according to his clock.11 But, as has been emphasized, two events, which are simultaneous in one reference frame, S (like the train platform or spaceship A), are not simulta- neous in another reference frame, S0 (like the moving train or spaceship B).

As we shall see in Chap. 3, this train thought experiment may be dubbed the locus classicus for the prevailing view that the passage of time, according to the Special theory of relativity, is a human illusion, since every reference frame records its own clock readings, which depend on the velocity of the frame. There are ‘as many times, as there are reference frames’ (Pauli 1981, 15). There are also as many ‘Nows’ as there are reference frames, with the consequence that there is no universal Now. It seems, then, that time cannot be a feature of the physical universe. For the slicing of four-dimensional space–time into a ‘3 ? 1’ perspec- tive—three spatial and one temporal dimension—depends on the velocity of the reference frame, to which respective observers are attached. If the passage of  

is a human construction, then the physical universe must be a Parmenidean block universe. Heraclitean flux is an illusion. This reasoning drove many scientists and philosophers to accept a Kantian view of time.

Due to the absence of an observer-independent simultaneity relation, the Special theory of relativity does not support the view that ‘the world evolves in time’. Time, in the sense of an all-pervading ‘now’ does not exist. The four-dimensional world simply is, it does not evolve. (Ehlers 1997, 198)

But does this perspectivalism justify the claim, often made in connection with such thought experiments, that time is a mental construct and that the physical world is an atemporal  block universe?

 

2.9.3  Further Invariants in Space–Time

 

Although temporal and spatial measurements are perspectival in the  Special  theory of relativity—there are as many times as there are reference frames in inertial motion—there exists nevertheless an invariant four-dimensional space– time structure—Minkowski space–time—which is the same for all inertial observers.

Minkowski space–time is a geometric construction, which reflects both the perspectival aspects of the theory and its invariant structure. The invariant struc- ture is expressed by the invariance of the speed of light, c and the space–time interval, ds.

Consider, first, how the perspectival features of the geometric representation are captured in space–time diagrams. In Fig. 2.21a, b the simultaneity planes of the

respective observers are inclined at oblique angles, which reflect the different velocities of the reference frames to which each observer is attached. The observers disagree about the simultaneity of events, as well as their duration and the spatial extent between them. Nevertheless, they are able to compute each other’s measurements by the employment of the Lorentz transformations, which are  therefore invariant.

Second, the invariance of the speed of light, c, for all observers invests the geometric construction with a causal structure (Fig. 2.22).

From each event, p, a light cone diverges into the future, and a light cone converges from the past onto p. These light cones can be connected by signals, and hence an earlier event can have a causal effect on a later event, within the light cone, but events lying beyond the boundary of the light cones cannot be reached by either light or other electromagnetic signals. The invariant structure of Minkowski space–time means that the temporal order of events is fixed for all inertial observers, even though they judge the simultaneity of events differently.

Chapter 3 will investigate the philosophical consequences of Minkowski space– time, in particular the claim that the perspectivalism of temporal measurements means that time is a mental construct in the Kantian sense, and that the physical world  is atemporal.

We shall also see that there are two different ways of interpreting c, giving rise to two different views of Minkowski space–time. The geometric view is the standard approach, in which c is a postulate, which defines the geometric limit of the light cones, within which all material particles trace out their world lines. But  c can also be seen as an optical signal and this defines an optical approach. From the  point  of  view  of  an  optical  approach,  which  leads  to  a  light  geometry,  c propagates with a finite velocity between events in space–time and is subject to

thermodynamic considerations. These two approaches lead to very different interpretations of Minkowski space–time and the notion of time (Chap. 3).

A further invariant relationship is the space–time interval, ds, which starts from the well-known fact that separate measurements of spatial and temporal lengths in Minkowski space–time are subject to the phenomena of time dilation and length contraction. This has the result that observers, who are attached to different ref- erence frames, moving inertially with respect to each other, will not agree on

(a) the simultaneity of two events (Fig. 2.18), (b) the temporal duration of events (Fig. 2.23), (c) the spatial lengths of rods. For these observers the clocks run regularly within their respective frames (proper time), but are not judged as ticking invariably across the observers’ inertial frames (co-ordinate time). However, if the observers unite space and time, in Minkowski’s words, then they will agree on the space–time interval ds (Fig. 2.23) such that

Although the observers record different clock times in their respective frames, the space–time separation, ds, of events in space–time is invariant. This means that observers will agree on a basic succession of events in space–time, as long as the possibility of a causal signal connection between them exists. Thus they will disagree on the simultaneity of events but agree on their order and space–time separation. The time order of events is invariant within the respective light cones. Whilst ds and c are the best-known invariant relationships in the Special theory, and often used in philosophical arguments against the block universe, they are not the only invariants. In the geometric approach ds and c appear as geometric constructions, which either set the limit of the light cones or specify a space–time interval between events, which is invariant for all time-like related frames. But from the point of view of an optical approach, which leads to a light geometry, c propagates with a finite velocity and is subject to thermodynamic considerations.

This means that the space–time interval ds, which guarantees invariance, must be grounded in empirical  relations,  still to be specified.

 

 

2.9.4 The Measurement of Time in the General Theory of Relativity

 

As is well-known, soon after the completion of his Special theory of relativity (1905), Einstein realized that this theory had limitations, which needed to be overcome. The first limitation was that the Special theory still gives undue pref- erence to inertial reference frames. It extended the Galilean relativity principle from mechanical to electrodynamic phenomena but it was a ‘principle of the physical relativity of all uniform motion’ (Einstein 1920, Chap. XVIII, 59; italics in original). In order to include non-uniform motion in the principle of relativity, Einstein extended  the special  principle of relativity to a general  principle.

All bodies of reference K, K’, etc. are equivalent for the description of natural  phenomena (formulation of the general laws of nature), whatever their state of motion. (Einstein  1920, 61)

Thus the General theory of relativity deals with uniform and accelerated motion and assumes the equivalence of inertial and gravitational mass. The Special theory of relativity is only valid in the absence of gravitational fields. But gravitational fields have an effect on the acceleration of particles. In a famous thought exper- iment Einstein concluded that the force, which particles experience as gravitational acceleration, could be attributed to the local curvature of space–time. The second limitation of the Special theory was that it treated Minkowski space–time as an undynamic geometric structure. It affected the behaviour of measuring rods (length contraction) and mechanical clocks (time dilation) but the material world had no effect on the space–time structure. In the General theory the space–time structure itself becomes, in Wheeler’s words, fully dynamic in the sense ‘that matter tells space–time  how to curve  and curved space–time  tells  matter how to move’.  For

Einstein space–time is ‘not necessarily something to which one can ascribe a separate existence independently of the actual objects of physical reality’ (Einstein 1920, Note to 15th  edition).

 

þ      /

What effect does gravitation have on the measurement of time? The ticking rate of a clock will be affected by its position in the gravitational field. The stronger the gravitational field, the slower its rate. If two clocks are placed in a gravitational field, which varies in strength, the two clocks will not tick at the same rate. A clock at the bottom of a mountain will run more slowly than a clock at the top of a mountain, because of the gravitational gradient between top and bottom—this phenomenon is  called  gravitational  time  dilation.  Generally,  clocks  at  rest in a strong  gravitation  field  run more slowly by a  factor  of 1      GM c2r  than  in a

 

weaker field (Clemence 1966). If a clock moves with uniform velocity through a

gravitational field both effects—time dilation due to its velocity and its position in the gravitational field—have to be taken into account. In a famous experiment— the Maryland experiment (1975–1976)—atomic clocks were flown around the Earth in commercial airlines at a constant height and their time was compared to atomic clocks stationed on Earth. Compared to the earth-bound clocks, the moving clocks experienced two effects: due to the velocity effect the airborne clocks ran more slowly by 6 ns; and due to the gravitational effect they ran faster than the earth-bound clocks by 53 ns. The net difference is 53 - 6 ns = 47 ns (see Sexl  and Schmidt 1978, 37–39; Thorne 1995, 100–103).

Thus the General theory reaffirms the relativity of temporal measurements. The space–time of General relativity also seems to suggest a fundamental, timeless world. Einstein himself affirmed that the General theory had deprived time of     the ‘last vestiges of physical reality’ (Einstein 1916, Sect. 3). As we shall see, in the General theory the Lorentz transformations of the Special theory are replaced by more general coordinate or manifold transformations (so-called diffeomor- phisms). Yet there are no transformations between coordinate systems without the retention of some invariant structure. In the case of the General theory, this is still the space–time interval, ds, but in the presence of gravitational fields it takes the general form:

ds  ¼ gikdxidxk;

where the gik are functions of the spatial coordinates, x1, x2, x3 and the temporal coordinate  x4  (cf. Wald 1984).

The measurement of time in the General theory then reemphasizes the point, which reverberates throughout this chapter. On the one hand, there is a strong tendency in modern physics to relegate the notion of time to a Kantian ‘pure form of intuition’ and to consider the physical world to be a Parmenidean block universe; on the other hand, there are a number of invariant relationships, whose impact on the notion of time has not yet been properly appreciated. The question, which will be considered in the following chapters, is what effect the criterion of invariance will have on the notion of time in modern physical theories and whether it  allows  an inference  to a  Heraclitean  universe. Although  the  General   theoryapplies to the large-scale structures of the universe, it does not seem to change the basic doubt that some writers may harbour about the objective passage of time. Will this doubt persist if the attention turns to the very small and probes the measurement of time in quantum mechanics?