absolute and relational theories of space and motion

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Absolute and Relational Theories of Space and Motion

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Absolute and Relational Theories of Space andMotionFirst published Fri Aug 11, 2006

Since antiquity, natural philosophers have struggled to comprehend thenature of three tightly interconnected concepts: space, time, and motion.A proper understanding of motion, in particular, has been seen to becrucial for deciding questions about the natures of space and time, andtheir interconnections. Since the time of Newton and Leibniz,philosophers’ struggles to comprehend these concepts have oftenappeared to take the form of a dispute between absolute conceptions ofspace, time and motion, and relational conceptions. This article guides thereader through some of the history of these philosophical struggles.Rather than taking sides in the (alleged) ongoing debates, or reproducingthe standard dialectic recounted in most introductory texts, we havechosen to scrutinize carefully the history of the thinking of the canonicalparticipants in these debates — principally Descartes, Newton, Leibniz,Mach and Einstein. Readers interested in following up either the historicalquestions or current debates about the natures of space, time and motionwill find ample links and references scattered through the discussion andin the Other Internet Resources section below.

1. Introduction2. Aristotle3. Descartes

3.1 The Nature of Motion3.2 Motion and Dynamics

4. Newton4.1 Newton Against the Cartesian Account of Motion — TheBucket4.2 Absolute Space and Motion

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4.2.1 Absolute Space vs Galilean Relativity4.2.2 The Ontology of Absolute Space

5. Absolute Space in the Twentieth Century5.1 The Spacetime Approach5.2 Substantivalism

6. Leibniz6.1 The Ideality of Space6.2 Force and the Nature of Motion

6.2.1 Vis Viva and True Motion6.3 Motion and Dynamics

6.3.1 Leibniz's Mechanics6.3.2 The Equivalence of Hypotheses

6.4 Where Did the Folk Go Wrong?6.5 Leibniz's Response to Newton's Scholium

7. ‘Not-Newton’ versus ‘Be-Leibniz’7.1 Non Sequiturs Mistakenly Attributed to Newton7.2 The Best Explanation Argument Mistakenly Attributed toNewton7.3 Substantivalism and The Best Explanation Argument

7.3.1 The Rotating Spheres7.3.2 Relationist Responses

8. Mach and Later Machians8.1 Two Interpretations of Mach on Inertia8.2 Implementing Mach-heavy8.3 Mach-lite vs Mach-heavy

9. Relativity and Motion9.1 Relations determine state of motion?9.2 The relationist roots of STR and GTR9.3 From Special Relativity to General Relativity9.4 General Relativity and Relativity of Motion

10. Conclusion


2 Stanford Encyclopedia of Philosophy

BibliographyOther Internet ResourcesRelated Entries

1. Introduction

Things change. A platitude perhaps, but still a crucial feature of theworld, and one which causes many philosophical perplexities — see forinstance the entry on Zeno's Paradoxes. For Aristotle, motion (he wouldhave called it ‘locomotion’) was just one kind of change, like generation,growth, decay, fabrication and so on. The atomists held on the contrarythat all change was in reality the motion of atoms into newconfigurations, an idea that was not to begin to realize its full potentialuntil the Seventeenth Century, particularly in the work of Descartes. (Ofcourse, modern physics seems to show that the physical state of a systemgoes well beyond the geometrical configuration of bodies. Fields, whiledetermined by the states of bodies, are not themselves configurations ofbodies if interpreted literally, and in quantum mechanics bodies have‘internal states' such as particle spin.)

While not all changes seem to be merely the (loco)motions of bodies inphysical space. Yet since antiquity, in the western tradition, this kind ofmotion has been absolutely central to the understanding of change. Andsince motion is a crucial concept in physical theories, one is forced toaddress the question of what exactly it is. The question might seem trivial,for surely what is usually meant by saying that something is moving is tosay that it is moving relative to something, often tacitly understoodbetween speakers. For instance: the car is moving at 60mph (relative tothe road and things along it), the plane is flying (relative) to London, therocket is lifting off (the ground), or the passenger is moving (to the frontof the speeding train). Typically the relative reference body is either thesurroundings of the speakers, or the Earth, but this is not always the case.


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surroundings of the speakers, or the Earth, but this is not always the case.For instance, it seems to make sense to ask whether the Earth rotatesabout its axis West-East diurnally or whether it is instead the heavens thatrotate East-West; but if all motions are to be reckoned relative to theEarth, then its rotation seems impossible. But if the Earth does not offer aunique frame of reference for the description of motion, then we maywonder whether any arbitrary object can be used for the definition ofmotions: are all such motions on a par, none privileged over any other? Itis unclear whether anyone has really, consistently espoused this view:Aristotle, perhaps, in the Metaphysics; Descartes and Leibniz are oftenthought to have but, as we'll see, those claims are suspect; possiblyHuygens, though his remarks remain cryptic; Mach at some momentsperhaps. If this view were correct, then the question of whether the Earthor heavens rotate would be meaningless, merely different but equivalentexpressions of the facts.

But suppose, like Aristotle, you take ordinary language accurately toreflect the structure of the world, then you could recognize systematiceveryday uses of ‘up’ and ‘down’ that require some privileged standards— uses that treat things closer to a point at the center of the Earth as more‘down’ and motions towards that point as ‘downwards'. Of course wewould likely explain this usage in terms of the fact that we and ourlanguage evolved in a very noticeable gravitational field directed towardsthe center of the Earth, but for Aristotle, as we shall see, this usage helpedidentify an important structural feature of the universe, which itself wasrequired for the explanation of weight. Now a further question arises: howshould a structure, such as a preferred point in the universe, whichprivileges certain motions, be understood? What makes that pointprivileged? One might expect that Aristotle simply identified it with thecenter of the Earth, and so relative to that particular body; but in fact hedid not adopt that tacit convention as fundamental, for he thought itpossible for the Earth to move from the ‘down’ point. Thus the questionarises (although Aristotle does not address it explicitly) of whether the



arises (although Aristotle does not address it explicitly) of whether thepreferred point is somewhere picked out in some other way by the bodiesin the universe —the center of the heavens perhaps? Or is it picked outquite independently of the arrangements of matter?

The issues that arise in this simple theory help frame the debates betweenlater physicists and philosophers concerning the nature of motion; inparticular, we will focus on the theories of Descartes, Newton, Leibniz,Mach and Einstein, and their interpretations. But similar issues circulatethrough the different contexts: is there any kind of privileged sense ofmotion, a sense in which things can be said to move or not, not justrelative to this or that reference body, but ‘truly’? If so, can this truemotion be analyzed in terms of motions relative to other bodies — tosome special body, or to the entire universe perhaps? (And in relativity, inwhich distances, times and measures of relative motion are frame-dependent, what relations are relevant?) If not, then how is the privilegedkind of motion to be understood, as relative to space itself — somethingphysical but non-material — perhaps? Or can some kinds of motion bebest understood as not being spatial changes — changes of relativelocation or of place — at all?

2. Aristotle

To see that the problem of the interpretation of spatiotemporal quantitiesas absolute or relative is endemic to almost any kind of mechanics onecan imagine, we can look to one of the simplest theories — Aristotle'saccount of natural motion (e.g., On the Heavens I.2). According to thistheory it is because of their natures, and not because of ‘unnatural’ forces,that that heavy bodies move down, and ‘light’ things (air and fire) moveup; it is their natures, or ‘forms’, that constitute the gravity or weight ofthe former and the levity of the latter. This account only makes sense if‘up’ and ‘down’ can be unequivocally determined for each body.According to Aristotle, up and down are fixed by the position of the body


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According to Aristotle, up and down are fixed by the position of the bodyin question relative to the center of the universe, a point coincident withthe center of the Earth. That is, the theory holds that heavy bodiesnaturally move towards the center, while light bodies naturally moveaway.

Does this theory involve absolute or merely relative quantities? It dependson how the center is conceived. If the center were identified with thecenter of the Earth, then the theory could be taken to eschew absolutequantities: it would simply hold that the natural motions of any bodydepend on its position relative to another, namely the Earth. But Aristotleis explicit that the center of the universe is not identical with, but merelycoincident with the center of the Earth (e.g., On the Heavens II.14): sincethe Earth itself is heavy, if it were not at the center it would move there!So the center is not identified with any body, and so perhaps direction-to-center is an absolute quantity in the theory, not understood fundamentallyas direction to some body (merely contingently as such if some bodyhappens to occupy the center). But this conclusion is not clear either. InOn the Heavens II.13, admittedly in response to a different issue, Aristotlesuggests that the center itself is ‘determined’ by the outer spherical shellof the universe (the aetherial region of the fixed stars). If this is what heintends, then the natural law prescribes motion relative to another bodyafter all — namely up or down with respect to the mathematical center ofthe stars.

It would be to push Aristotle's writings too hard to suggest that he wasconsciously wrestling with the issue of whether mechanics requiredabsolute or relative quantities of motion, but what is clear is that thesequestions arise in his physics and his remarks impinge on them. Histheory also gives a simple model of how these questions arise: a physicaltheory of motion will say that ‘under such-and-such circumstances,motion of so-and-so a kind will occur’ — and the question of whetherthat kind of motion makes sense in terms of the relations between bodies



that kind of motion makes sense in terms of the relations between bodiesalone arises automatically. Aristotle may not have recognized the questionexplicitly, but we see it as one issue in the background of his discussionof the center.

3. Descartes

The issues are, however, far more explicit in Descartes' physics; and sincethe form of his theory is different the ‘kinds of motion’ in question arequite different — as they change with all the different theories that wediscuss. For Descartes argued in his 1644 Principles of Philosophy (seeBook II) that the essence of matter was extension (i.e., size and shape)because any other attribute of bodies could be imagined away withoutimagining away matter itself. But he also held that extension constitutesthe nature of space, hence he concluded that space and matter were oneand the same thing. An immediate consequence of the identification is theimpossibility of the vacuum; if every region of space is a region ofmatter, then there can be no space without matter. Thus Descartes'universe is ‘hydrodynamical’ — completely full of mobile matter of indifferent sized pieces in motion, rather like a bucket full of water andlumps of ice of different sizes, which has been stirred around. Sincefundamentally the pieces of matter are nothing but extension, the universeis in fact nothing but a system of geometric bodies in motion without anygaps. (Descartes held that all other properties arise from theconfigurations and motions of such bodies — from geometric complexes.See Garber 1992 for a comprehensive study.)

3.1 The Nature of Motion

The identification of space and matter poses a puzzle about motion: if thespace that a body occupies literally is the matter of the body, then whenthe body — i.e., the matter — moves, so does the space that it occupies.Thus it doesn't change place, which is should be to say that it doesn't


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Thus it doesn't change place, which is should be to say that it doesn'tmove after all! Descartes resolved this difficulty by taking all motion tobe the motion of bodies relative to one another, not a literal change ofspace.

Now, a body has as many relative motions as there are bodies but it doesnot follow that all are equally significant. Indeed, Descartes uses severaldifferent concepts of relational motion. First there is ‘change of place’,which is nothing but motion relative to this or that arbitrary referencebody (II.13). In this sense no motion of a body is privileged, since thespeed, direction, and even curve of a trajectory depends on the referencebody, and none is singled out. Next, he discusses motion in ‘the ordinarysense’ (II.24). This is often conflated with mere change of arbitrary place,but it in fact differs because according to the rules of ordinary speech oneproperly attributes motion only to bodies whose motion is caused bysome action, not to any relative motion. (For instance, a person sitting ona speeding boat is ordinarily said to be at rest, since ‘he feels no action inhimself’.) Finally, he defined motion ‘properly speaking’ (II.25) to be abody's motion relative to the matter contiguously surrounding it, whichthe impossibility of a vacuum guarantees to exist. (Descartes’ definition iscomplicated by the fact that he modifies this technical concept to make itconform more closely to the pre-theoretical sense of ‘motion’; however,in our discussion transference is all that matters, so we will ignore thosecomplications.) Since a body can only be touching one set ofsurroundings, Descartes (dubiously) argued that this standard of motionwas unique.

What we see here is that Descartes, despite holding motion to be themotion of bodies relative to one another, also held there to be a privilegedsense of motion; in a terminology sometimes employed by writers of theperiod, he held there to be a sense of ‘true motion’, over and above themerely relative motions. Equivalently, we can say that Descartes tookmotion (‘properly speaking’) to be a complete predicate: that is, moves-



motion (‘properly speaking’) to be a complete predicate: that is, moves-properly-speaking is a one-place predicate. (In contrast, moves-relative-tois a two-place predicate.) And note that the predicate is complete despitethe fact that it is analyzed in terms of relative motion. (Formally, letcontiguous-surroundings be a function from bodies to their contiguoussurroundings, then x moves-properly-speaking is analyzed as x moves-relative-to contiguous-surroundings(x).)

This example illustrates why it is crucial to keep two questions distinct:on the one hand, is motion to be understood in terms of relations betweenbodies or by invoking something additional, something absolute; on theother hand, are all relative motions equally significant, or is there some‘true’, privileged notion of motion? Descartes' views show that eschewingabsolute motion is logically compatible with accepting true motion; whichis of course not to say that his definitions of motion are themselvestenable.

3.2 Motion and Dynamics

There is an interpretational tradition which holds that Descartes only tookthe first, ‘ordinary’ sense of motion seriously, and introduced the secondnotion to avoid conflict with the Catholic Church. Such conflict was areal concern, since the censure of Galileo's Copernicanism took placeonly 11 years before publication of the Principles, and had in factdissuaded Descartes from publishing an earlier work, The World. Indeed,in the Principles (III.28) he is at pains to explain how ‘properly speaking’the Earth does not move, because it is swept around the Sun in a giantvortex of matter — the Earth does not move relative to its surroundings inthe vortex.

The difficulty with the reading, aside from the imputation of cowardice tothe old soldier, is that it makes nonsense of Descartes' mechanics, atheory of collisions. For instance, according to his laws of collision if two


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theory of collisions. For instance, according to his laws of collision if twoequal bodies strike each other at equal and opposite velocities then theywill bounce off at equal and opposite velocities (Rule I). On the otherhand, if the very same bodies approach each other with the very samerelative speed, but at different speeds then they will move off together inthe direction of the faster one (Rule III). But if the operative meaning ofmotion in the Rules is the ordinary sense, then these two situations arejust the same situation, differing only in the choice of reference frame,and so could not have different outcomes — bouncing apart versusmoving off together. It seems inconceivable that Descartes could havebeen confused in such a trivial way. (Additionally, as Pooley 2002 pointsout, just after he claims that the Earth is at rest ‘properly speaking’,Descartes argues that the Earth is stationary in the ordinary sense, becausecommon practice is to determine the positions of the stars relative to theEarth. Descartes simply didn't need motion properly speaking to avoidreligious conflict, which again suggests that it has some other significancein his system of thought.)

Thus Garber (1992, Chapter 6-8) proposes that Descartes actually tookthe unequivocal notion of motion properly speaking to be the correctsense of motion in mechanics. Then Rule I covers the case in which thetwo bodies have equal and opposite motions relative to their contiguoussurroundings, while Rule VI covers the case in which the bodies havedifferent motions relative to those surroundings — one is perhaps at restin its surroundings. That is, exactly what is needed to make the rulesconsistent is the kind of privileged, true, sense of motion provided byDescartes' second definition. Insurmountable problems with the rulesremain, but rejecting the traditional interpretation and taking motionproperly speaking seriously in Descartes' philosophy clearly gives a morecharitable reading.

4. Newton



4.1 Newton Against the Cartesian Account of Motion — TheBucket

In an unpublished essay — De Gravitatione (Newton, 2004) — and in aScholium to the definitions given in his 1687 Mathematical Principles ofNatural Philosophy (see Newton, 1999 for an up-to-date translation),Newton attacked both of Descartes' notions of motion as candidates forthe operative notion in mechanics. (see Stein 1967and Rynasiewicz 1995for important, and differing, views on the issue.) (This critique is studiedin more detail in the entry Newton's views on space, time, and motion.)

The most famous argument invokes the so-called ‘Newton's bucket’experiment. Stripped to its basic elements one compares:

i. a bucket of water hanging from a cord as the bucket is set spinningabout the cord's axis, with

ii. the same bucket and water when they are rotating at the same rateabout the cord's axis.

As is familiar from any rotating system, there will be a tendency for thewater to recede from the axis of rotation in the latter case: in (i) thesurface of the water will be flat (because of the Earth's gravitational field)while in (ii) it will be concave. The analysis of such ‘inertial effects' dueto rotation was a major topic of enquiry of ‘natural philosophers' of thetime, including Descartes and his followers, and they would certainlyhave agreed with Newton that the concave surface of the water in thesecond case demonstrated that the water was moving in a mechanicallysignificant sense. There is thus an immediate problem for the claim thatproper motion is the correct mechanical sense of motion: in (i) and (ii)proper motion is anti-correlated with the mechanically significant motionrevealed by the surface of the water. That is, the water is flat in (i) whenit is in motion relative to its immediate surroundings — the inner sides ofthe bucket — but curved in (ii) when it is at rest relative to its immediate


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the bucket — but curved in (ii) when it is at rest relative to its immediatesurroundings. Thus the mechanically relevant meaning of rotation is notthat of proper motion. (You may have noticed a small lacuna in Newton'sargument: in (i) the water is at rest and in (ii) in motion relative to thatpart of its surroundings constituted by the air above it. It's not hard toimagine small modifications to the example to fill this gap.)

Newton also points out that the height that the water climbs up the insideof the bucket provides a measure of the rate of rotation of bucket andwater: the higher the water rises up the sides, the greater the tendency torecede must be, and so the faster the water must be rotating in themechanically significant sense. But supposing, very plausibly, that themeasure is unique, that any particular height indicates a particular rate ofrotation. Then the unique height that the water reaches at any momentimplies a unique rate of rotation in a mechanically significant sense. Andthus motion in the sense of motion relative to an arbitrary reference body,is not the mechanical sense, since that kind of rotation is not unique at all,but depends on the motion of the reference body. And so Descartes’change of place (and for similar reasons, motion in the ordinary sense) isnot the mechanically significant sense of motion.

4.2 Absolute Space and Motion

In our discussion of Descartes we called the sense of motion operative inthe science of mechanics ‘true motion’, and the phrase is used in this wayby Newton in the Scholium. Thus Newton's bucket shows that true(rotational) motion is anti-correlated with, and so not identical with,proper motion (as Descartes proposed according to the Garber reading);and Newton further argues that the rate of true (rotational) motion isunique, and so not identical with change of place, which is multiple.Newton proposed instead that true motion is motion relative to atemporally enduring, rigid, 3-dimensional Euclidean space, which hedubbed ‘absolute space’. Of course, Descartes also defined motion as



dubbed ‘absolute space’. Of course, Descartes also defined motion asrelative to an enduring 3-dimensional Euclidean space; the difference isthat Descartes space was divided into parts (his space was identical with aplenum of corpuscles) in motion, not a rigid structure in which (mobile)material bodies are embedded. So according to Newton, the rate of truerotation of the bucket (and water) is the rate at which it rotates relative toabsolute space. Or put another way, Newton effectively defines thecomplete predicate x moves-absolutely as x moves-relative-to absolutespace; both Newton and Descartes offer the competing completepredicates as analyses of x moves-truly.

4.2.1 Absolute Space vs. Galilean Relativity

Newton's proposal for understanding motion solves the problems that heposed for Descartes, and provides an interpretation of the concepts ofconstant motion and acceleration that appear in his laws of motion.However, it suffers from two notable interpretational problems, both ofwhich were pressed forcefully by Leibniz (in the Leibniz-ClarkeCorrespondence, 1715–1716) — which is not to say that Leibniz himselfoffered a superior account of motion (see below). (Of course, there areother features of Newton's proposal that turned out to be empiricallyinadequate, and are rejected by relativity: Newton's account violates therelativity of simultaneity and postulates a non-dynamical spacetimestructure.) First, according to this account, absolute velocity is a well-defined quantity: more simply, the absolute speed of a body is the rate ofchange of its position relative to an arbitrary point of absolute space. Butthe Galilean relativity of Newton's laws mean that the evolution of aclosed system is unaffected by constant changes in velocity; Galileo'sexperimenter cannot determine from observations inside his cabinwhether the boat is at rest in harbor or sailing smoothly. Put another way,according to Newtonian mechanics, in principle Newton's absolutevelocity cannot be experimentally determined. So in this regard absolutevelocity is quite unlike acceleration (including rotation); Newtonian



velocity is quite unlike acceleration (including rotation); Newtonianacceleration is understood in absolute space as the rate of change ofabsolute velocity, and is, according to Newtonian mechanics, in generalmeasurable, for instance by measuring the height that the water ascendsthe sides of the bucket. (It is worth noting that Newton was well-aware ofthese facts; the Galilean relativity of his theory is demonstrated inCorollary V of the laws of the Principia, while Corollary VI shows thatacceleration is unobservable if all parts of the system accelerate in parallelat the same rate, as they do in a homogeneous gravitational field.) Leibnizargued (rather inconsistently, as we shall see) that since differences inabsolute velocity were unobservable, they could not be genuinedifferences at all; and hence that Newton's absolute space, whoseexistence would entail the reality of such differences, must also be afiction. Few contemporary philosophers would immediately reject aquantity as meaningless simply because it was not experimentallydeterminable, but this fact does justify genuine doubts about the reality ofabsolute velocity, and hence of absolute space.

4.2.2 The Ontology of Absolute Space

The second problem concerns the nature of absolute space. Newton quiteclearly distinguished his account from Descartes' — in particular withregards to absolute space's rigidity versus Descartes' ‘hydrodynamical’space, and the possibility of the vacuum in absolute space. Thus absolutespace is definitely not material. On the other hand, presumably it issupposed to be part of the physical, not mental, realm. In DeGravitatione, Newton rejected both the standard philosophical categoriesof substance and attribute as suitable characterizations. Absolute space isnot a substance for it lacks causal powers and does not have a fullyindependent existence, and yet not an attribute since it would exist even ina vacuum, which by definition is a place where there are no bodies inwhich it might inhere. Newton proposes that space is what we might calla ‘pseudo-substance’, more like a substance than property, yet not quite a



a ‘pseudo-substance’, more like a substance than property, yet not quite asubstance. (Note that Samuel Clarke, in his Correspondence with Leibniz,which Newton had some role in composing, advocates the property view,and note further that when Leibniz objects because of the vacuumproblem, Clarke suggests that there might be non-material beings in thevacuum in which space might inhere.) In fact, Newton accepted theprinciple that everything that exists, exists somewhere — i.e., in absolutespace. Thus he viewed absolute space as a necessary consequence of theexistence of anything, and of God's existence in particular — hencespace's ontological dependence. Leibniz was presumably unaware of theunpublished De Gravitatione in which these particular ideas weredeveloped, but as we shall see, his later works are characterized by arobust rejection of any notion of space as a real thing rather than an ideal,purely mental entity. This is a view that attracts even fewer contemporaryadherents, but there is something deeply peculiar about a non-material butphysical entity, a worry that has influenced many philosophical opponentsof absolute space.

5. Absolute Space in the Twentieth Century

5.1 The Spacetime Approach

After the development of relativity (which we will take up below), and itsinterpretation as a spacetime theory, it was realized that the notion ofspacetime had applicability to a range of theories of mechanics, classicalas well as relativistic. In particular, there is a spacetime geometry —‘Galilean’ or ‘neo-Newtonian’ spacetime — for Newtonian mechanicsthat solves the problem of absolute velocity; an idea exploited by anumber of philosophers from the late 1960s (e.g., Earman 1970, Friedman1983, Sklar 1974 and Stein 1968). For details the reader is referred to theentry on spacetime: inertial frames, but the general idea is that although aspatial distance is well-defined between any two simultaneous points ofthis spacetime, only the temporal interval is well-defined between non-



this spacetime, only the temporal interval is well-defined between non-simultaneous points. Thus things are rather unlike Newton's absolutespace, whose points persist through time and maintain their distances; inabsolute space the distance between p-now and q-then (where p and q arepoints) is just the distance between p-now and q-now. However, Galileanspacetime has an ‘affine connection’ which effectively specifies for everypoint of every continuous curve, the rate at which the curve is changingfrom straightness at that point; for instance, the straight lines are pickedout as those curves whose rate of change from straightness is zero atevery point. (Another way of thinking about this space is as possessing —in addition to a distance between any two simultaneous points and atemporal interval between any points — a three-place relation ofcolinearity, satisfied by three points just in case they lie on a straight line.)

Since the trajectories of bodies are curves in spacetime the affineconnection determines the rate of change from straightness at every pointof every possible trajectory. The straight trajectories thus defined can beinterpreted as the trajectories of bodies moving inertially, and the rate ofchange from straightness of any trajectory can be interpreted as theacceleration of a body following that trajectory. That is, Newton's SecondLaw can be given a geometric formulation as ‘the rate of change fromstraightness of a body's trajectory is equal to the forces acting on the bodydivided by its mass’. The significance of this geometry is that whileacceleration is well-defined, velocity is not — in accord with empiricallydeterminability of acceleration but not velocity according to Newtonianmechanics. (A simple analogy helps see how such a thing is possible:betweenness but not ‘up’ is a well-defined concept in Euclidean space.)Thus Galilean spacetime gives a very nice interpretation of the choice thatnature makes when it decides that the laws of mechanics should beformulated in terms of accelerations not velocities (as Aristotle andDescartes proposed).



5.2 Substantivalism

Put another way, we can define the complete predicate x accelerates astrajectory(x) has-non-zero-rate-of-change-from-straightness, wheretrajectory maps bodies onto their trajectories in Galilean spacetime. Andthis predicate, defined this way, applies to the water in the bucket if andonly if it is rotating, according to Newtonian mechanics formulated interms of the geometry of Galilean spacetime; it is the mechanicallyrelevant sense of the word in this theory. But all of this formulation anddefinition has been given in terms of the geometry of spacetime, notrelations between bodies; acceleration is ‘absolute’ in the sense that thereis a preferred (true) sense of acceleration in mechanics and which is notdefined in terms of the motions of bodies relative to one another. (Notethat this sense of ‘absolute’ is broader than that of motion relative toabsolute space, which we defined earlier. In the remainder of this articlewe will use it in the broader sense. The reader should be aware that theterm is used in many ways in the literature, and such equivocation oftenleads to massive misunderstandings.) Thus if any of this analysis ofmotion is taken literally then one arrives at a position regarding theontology of spacetime rather like that of Newton's regarding space: it issome kind of ‘substantial’ (or maybe pseudo-substantial) thing with thegeometry of Galilean spacetime, just as absolute space possessedEuclidean geometry. This view regarding the ontology of spacetime isusually called ‘substantivalism’ (Sklar, 1974). The Galilean substantivalistusually sees himself as adopting a more sophisticated geometry thanNewton but sharing his substantivalism (though there is room for debateon Newton's exact ontological views, see DiSalle, 2002). The advantageof the more sophisticated geometry is that although it allows the absolutesense of acceleration apparently required by Newtonian mechanics to bedefined, it does not allow one to define a similar absolute speed orvelocity — x accelerates can be defined as a complete predicate in termsof the geometry of Galilean spacetime but not x moves in general — andso the first of Leibniz's problem is resolved. Of course we see that the



so the first of Leibniz's problem is resolved. Of course we see that thesolution depends on a crucial shift from speed and velocity to accelerationas the relevant senses of ‘motion’: from the rate of change of position tothe rate of rate of change.

While this proposal solves the first kind of problem posed by Leibniz, itseems just as vulnerable to the second. While it is true that it involves therejection of absolute space as Newton conceived it, and with it the need toexplicate the nature of an enduring space, the postulation of Galileanspacetime poses the parallel question of the nature of spacetime. Again, itis a physical but non-material something, the points of which may becoincident with material bodies. What kind of thing is it? Could we dowithout it? As we shall see below, some contemporary philosophersbelieve so.

6. Leibniz

There is a ‘folk-reading’ of Leibniz that one finds either explicitly orimplicitly in the philosophy of physics literature which takes account ofonly some of his remarks on space and motion. The reading underlies vastswathes of the literature: for instance, the quantities captured by Earman's(1999) ‘Leibnizian spacetime’, do not do justice to Leibniz's view ofmotion (as Earman acknowledges). But it is perhaps most obvious inintroductory texts (e.g., Ray 1991, Huggett 2000 to mention a couple).According to this view, the only quantities of motion are relativequantities, relative velocity, acceleration and so on, and all relativemotions are equal, so there is no true sense of motion. However, Leibnizis explicit that other quantities are also ‘real’, and his mechanicsimplicitly — but obviously — depends on yet others. The length of thissection is a measure, not so much the importance of Leibniz's actualviews, but the importance of showing what the prevalent folk view leavesout regarding Leibniz's views on the metaphysics of motion andinterpretation of mechanics.



interpretation of mechanics.

That said, we shall also see that no one has yet discovered a fullysatisfactory way of reconciling the numerous conflicting things thatLeibniz says about motion. Some of these tensions can be put downsimply to his changing his mind (see Cover and Hartz 1988 for anexplication of how Leibniz's views on space developed). However, wewill concentrate on the fairly short period in the mid 1680-90s duringwhich Leibniz developed his theory of mechanics, and was mostconcerned with their interpretation. We will supplement this discussionwith the important remarks that he made in his Correspondence withSamuel Clarke around 30 years later (1715–1716); this discussion isbroadly in line with the earlier period, and the intervening period is one inwhich he turned to other matters, rather than one in which his views onspace were dramatically evolving.

6.1 The Ideality of Space

Arguably, Leibniz's views concerning space and motion do not have acompletely linear logic, starting from some logically sufficient basicpremises, but instead form a collection of mutually supporting doctrines Ifone starts questioning why Leibniz held certain views — concerning theideality of space, for instance — one is apt to be led in a circle. Still,exposition requires starting somewhere, and Leibniz's argument for theideality of space in the Correspondence with Clarke is a good place tobegin. But bear in mind the caveats made here — this argument wasmade later than a number of other relevant writings, and its logicalrelation to Leibniz's views on motion is complex.

Leibniz (LV.47 — this notation means Leibniz's Fifth letter, section 47,and so on) says that (i) a body comes to have the ‘same place’ as anotheronce did, when it comes to stand in the same relations to bodies we‘suppose’ to be unchanged (more on this later). (ii) That we can define ‘a



‘suppose’ to be unchanged (more on this later). (ii) That we can define ‘aplace’ to be that which any such two bodies have in common (here heclaims an analogy with the Euclidean/Eudoxan definition of a rationalnumber in terms of an identity relation between ratios). And finally that(iii) space is all such places taken together. However, he also holds thatproperties are particular, incapable of being instantiated by more than oneindividual, even at different times; hence it is impossible for the twobodies to be in literally the same relations to the unchanged bodies. Thusthe thing that we take to be the same for the two bodies — the place — issomething added by our minds to the situation, and only ideal. As a result,space, which is after all constructed from these ideal places, is itself ideal:‘a certain order, wherein the mind conceives the application of relations’.

It's worth pausing briefly to contrast this view of space with those ofDescartes and of Newton. Both Descartes and Newton claim that space isa real, mind-independent entity; for Descartes it is matter, and for Newtona ‘pseudo-substance’, distinct from matter. And of course for both, theseviews are intimately tied up with their accounts of motion. Leibniz simplydenies the mind-independent reality of space, and this too is bound upwith his views concerning motion. (Note that fundamentally, in themetaphysics of monads that Leibniz was developing contemporaneouslywith his mechanics, everything is in the mind of the monads; but the pointthat Leibniz is making here is that even within the world that is logicallyconstructed from the contents of the minds of monads, space is ideal.)

6.2 Force and the Nature of Motion

So far (apart from that remark about ‘unchanged’ bodies) we have notseen Leibniz introduce anything more than relations of distance betweenbodies, which is certainly consistent with the folk view of his philosophy.However, Leibniz sought to provide a foundation for theCartesian/mechanical philosophy in terms of the Aristotelian/scholasticmetaphysics of substantial forms (here we discuss the views laid out in



metaphysics of substantial forms (here we discuss the views laid out inSections 17-22 of the 1686 Discourse on Metaphysics and the 1695Specimen of Dynamics, both in Garber and Ariew 1989). In particular, heidentifies primary matter with what he calls its ‘primitive passive force’ ofresistance to changes in motion and to penetration, and the substantialform of a body with its ‘primitive active force’. It is important to realizethat these forces are not mere properties of matter, but actually constituteit in some sense, and further that they are not themselves quantifiable.However because of the collisions of bodies with one another, theseforces ‘suffer limitation’, and ‘derivative’ passive and active forces result.(There's a real puzzle here. Collision presupposes space, but primitiveforces constitute matter prior to any spatial concepts — the primitiveactive and passive forces ground motion and extension respectively. SeeGarber and Rauzy, 2004.) Derivative passive force shows up in thedifferent degrees of resistance to change of different kinds of matter (of‘secondary matter’ in scholastic terms), and apparently is measurable.Derivative active force however, is considerably more problematic forLeibniz. On the one hand, it is fundamental to his account of motion andtheory of mechanics — motion fundamentally is possession of force. Buton the other hand, Leibniz endorses the mechanical philosophy, whichprecisely sought to abolish Aristotelian substantial form, which is whatforce represents. Leibniz's goal was to reconcile the two philosophies, byproviding an Aristotelian metaphysical foundation for modern mechanicalscience; as we shall see, it is ultimately an open question exactly howLeibniz intended to deal with the inherent tensions in such a view.

6.2.1 Vis Viva and True Motion

The texts are sufficiently ambiguous to permit dissent, but arguablyLeibniz intends that one manifestation of derivative active force is whathe calls vis viva — ‘living force’. Leibniz had a famous argument withthe Cartesians over the correct definition of this quantity. Descartesdefined it as size times speed — effectively as the magnitude of the



defined it as size times speed — effectively as the magnitude of themomentum of a body. Leibniz gave a brilliant argument (repeated in anumber of places, for instance Section 17 of the Discourse onMetaphysics) that it was size times speed2 — so (proportional to) kineticenergy. If the proposed identification is correct then kinetic energyquantifies derivative active force according to Leibniz; or looked at theother way, the quantity of virtus (another term used by Leibniz for activeforce) associated with a body determines its kinetic energy and hence itsspeed. As far as the authors know, Leibniz never explicitly says anythingconclusive about the relativity of virtus, but it is certainly consistent toread him (as Roberts 2003 does) to claim that there is a unique quantity ofvirtus and hence ‘true’ (as we have been using the term) speed associatedwith each body. At the very least, Leibniz does say that there is a realdifference between possession and non-possession of vis viva (e.g., inSection 18 of the Discourse) and it is a small step from there to true,privileged speed. Indeed, for Leibniz, mere change of relative position isnot ‘entirely real’ (as we saw for instance in the Correspondence) andonly when it has vis viva as its immediate cause is there some reality to it.(However, just to muddy the waters, Leibniz also claims that as a matterof fact, no body ever has zero force, which on the reading proposedmeans no body is ever at rest, which would be surprising given all thecollisions bodies undergo.) An alternative interpretation to the onesuggested here might say that Leibniz intends that while there is adifference between motion/virtus and no motion/virtus, there is somehowno difference between any strictly positive values of those quantities.

It is important to emphasize two points about the preceding account ofmotion in Leibniz's philosophy. First, motion in the everyday sense —motion relative to something else — is not really real. Fundamentallymotion is possession of virtus, something that is ultimately non-spatial(modulo its interpretation as primitive force limited by collision). If thisreading is right — and something along these lines seems necessary if wearen't simply to ignore important statements by Leibniz on motion — then



aren't simply to ignore important statements by Leibniz on motion — thenLeibniz is offering an interpretation of motion that is radically differentfrom the obvious understanding. One might even say that for Leibnizmotion is not movement at all! (We will leave to one side the question ofwhether his account is ultimately coherent.) The second point is thathowever we should understand Leibniz, the folk reading simply does notand cannot take account of his clearly and repeatedly stated view thatwhat is real in motion is force not relative motion, for the folk readingallows Leibniz only relative motion (and of course additionally, motion inthe sense of force is a variety of true motion, again contrary to the folkreading).

6.3 Motion and Dynamics

However, from what has been said so far it is still possible that the folkreading is accurate when it comes to Leibniz's views on the phenomena ofmotion, the subject of his theory of mechanics. The case for the folkreading is in fact supported by Leibniz's resolution of the tension that wementioned earlier, between the fundamental role of force/virtus (whichwe will now take to mean mass times speed2) and its identification withAristotelian form. Leibniz's way out (e.g., Specimen of Dynamics) is torequire that while considerations of force must somehow determine whatform of the laws of motion, the laws themselves should be such as not toallow one to determine the value of the force (and hence true speed). Onemight conclude that in this case Leibniz held that the only quantitieswhich can be determined are those of relative position and motion, as thefolk reading says. But even in this circumscribed context, it is at bestquestionable whether the interpretation is correct.

6.3.1 Leibniz's Mechanics

Consider first Leibniz's mechanics. Since his laws are what is now(ironically) often called ‘Newtonian’ elastic collision theory, it seems that



(ironically) often called ‘Newtonian’ elastic collision theory, it seems thatthey satisfy both of his requirements. The laws include conservation ofkinetic energy (which we identify with virtus), but they hold in all inertialframes, so the kinetic energy of any arbitrary body can be set to any initialvalue. But they do not permit the kinetic energy of a body to take on anyvalues throughout a process. The laws are only Galilean relativistic, andso are not true in every frame. Furthermore, according to the laws ofcollision, in an inertial frame, if a body does not collide then itsLeibnizian force is conserved while if (except in special cases) it doescollide then its force changes. According to Leibniz's laws one cannotdetermine initial kinetic energies, but one certainly can tell when theychange. At very least, there are quantities of motion implicit in Leibniz'smechanics — change in force and true speed — that are not merelyrelative; the folk reading is committed to Leibniz simply missing thisobvious fact.

6.3.2 The Equivalence of Hypotheses

That said, when Leibniz discusses the relativity of motion — which hecalls the ‘equivalence of hypotheses’ about the states of motion of bodies— some of his statements do suggest that he was confused in this way.For another way of stating the problem for the folk reading is that theclaim that relative motions alone suffice for mechanics and that allrelative motions are equal is a principle of general relativity, and couldLeibniz — a mathematical genius — really have failed to notice that hislaws hold only in special frames? Well, just maybe. On the one hand,when he explicitly articulates the principle of the equivalence ofhypotheses (for instance in Specimen of Dynamics) he tends to say onlythat one cannot assign initial velocities on the basis of the outcome of acollision, which requires only Galilean relativity. However, heconfusingly also claimed (On Copernicanism and the Relativity ofMotion, also in Garber and Ariew 1989) that the Tychonic andCopernican hypotheses were equivalent. But if the Earth orbits the Sun in



Copernican hypotheses were equivalent. But if the Earth orbits the Sun inan inertial frame (Copernicus), then there is no inertial frame according towhich the Sun orbits the Earth (Tycho Brahe), and vice versa: thesehypotheses are simply not Galilean equivalent (something else Leibnizcould hardly have failed to notice). So there is some textual support forLeibniz endorsing general relativity, as the folk reading maintains. Anumber of commentators have suggested solutions to the puzzle of theconflicting pronouncements that Leibniz makes on the subject, butarguably none is completely successful in reconciling all of them (Stein1977 argues for general relativity, while Roberts 2003 argues theopposite; see also Lodge 2003).

6.4 Where Did the Folk Go Wrong?

So the folk reading simply ignores Leibniz's metaphysics of motion, itcommits Leibniz to a mathematical howler regarding his laws, and it isarguable whether it is the best rendering of his pronouncementsconcerning relativity; it certainly cannot be accepted unquestioningly.However, it is not hard to understand the temptation of the folk reading.In his Correspondence with Clarke, Leibniz says that he believes space tobe “something merely relative, as time is, … an order of coexistences, astime is an order of successions” (LIII.4), which is naturally taken to meanthat space is at base nothing but the distance and temporal relationsbetween bodies. (Though even this passage has its subtleties, because ofthe ideality of space discussed above, and because in Leibniz's conceptionspace determines what sets of relations are possible.) And if relativedistances and times exhaust the spatiotemporal in this way, then shouldn'tall quantities of motion be defined in terms of those relations? We haveseen two ways in which this would be the wrong conclusion to draw:force seems to involve a notion of speed that is not identified with anyrelative speed, and (unless the equivalence of hypotheses is after all aprinciple of general relativity) the laws pick out a standard of constantmotion that need not be any constant relative motion. Of course, it is hard



motion that need not be any constant relative motion. Of course, it is hardto reconcile these quantities with the view of space and time that Leibnizproposes — what is speed in size times speed2 or constant speed if notspeed relative to some body or to absolute space? Given Leibniz's viewthat space is literally ideal (and indeed that even relative motion is not‘entirely real’) perhaps the best answer is that he took force and hencemotion in its real sense not to be determined by motion in a relative senseat all, but to be primitive monadic quantities. That is, he took x moves tobe a complete predicate, but he believed that it could be fully analyzed interms of strictly monadic predicates: x moves iff x possesses-non-zero-derivative-active-force. And this reading explains just what Leibniz tookus to be supposing when we ‘supposed certain bodies to be unchanged’ inthe construction of the idea of space: that they had no force, nothingcausing, or making real any motion.

6.5 Leibniz's Response to Newton's Scholium

It's again helpful to compare Leibniz with Descartes and Newton, thistime regarding motion. Commentators often express frustration atLeibniz's response to Newton's arguments for absolute space: “I findnothing … in the Scholium that proves or can prove the reality of space initself. However, I grant that there is a difference between an absolute truemotion of a body and a mere relative change …” (LV.53). Not only doesLeibniz apparently fail to take the argument seriously, he then goes on toconcede the step in the argument that seems to require absolute space!But with our understanding of Newton and Leibniz, we can see that whathe says makes perfect sense (or at least that it is not as disingenuous as itis often taken to be). Newton argues in the Scholium that true motioncannot be identified with the kinds of motion that Descartes considers; butboth of these are purely relative motions, and Leibniz is in completeagreement that merely relative motions are not true (i.e., ‘entirely real’).Leibniz's ‘concession’ merely registers his agreement with Newtonagainst Descartes on the difference between true and relative motion; he



against Descartes on the difference between true and relative motion; hesurely understood who and what Newton was refuting, and it was aposition that he had himself, in different terms, publicly argued against atlength. But as we have seen, Leibniz had a very different analysis of thedifference to Newton's; true motion was not, for him, a matter of motionrelative to absolute space, but the possession of quantity of force,ontologically prior to any spatiotemporal quantities at all. There is indeednothing in the Scholium explicitly directed against that view, and since itdoes potentially offer an alternative way of understanding true motion, itis not unreasonable for Leibniz to claim that there is no deductiveinference from true motion to absolute space.

7. ‘Not-Newton’ versus ‘Be-Leibniz’

7.1 Non Sequiturs Mistakenly Attributed to Newton

The folk reading which belies Leibniz has it that he sought a theory ofmechanics formulated in terms only of the relations between bodies. Aswe'll see presently, in the Nineteenth Century, Ernst Mach indeedproposed such an approach, but Leibniz clearly did not; though certainsimilarities between Leibniz and Mach — especially the rejection ofabsolute space — surely helps explain the confusion between the two. Butnot only is Leibniz often misunderstood, there are influential misreadingsof Newton's arguments in the Scholium, influenced by the idea that he isaddressing Leibniz in some way. Of course the Principia was written 30years before the Correspondence, and the arguments of the Scholiumwere not written with Leibniz in mind, but Clarke himself suggests(CIV.13) that those arguments — specifically those concerning the bucket— are telling against Leibniz. That argument is indeed devastating to ageneral principle of relativity — the parity of all relative motions — butwe have seen that it is highly questionable whether Leibniz's equivalenceof hypotheses amount to such a view. That said, his statements in the firstfour letters of the Correspondence could understandably mislead Clarke



four letters of the Correspondence could understandably mislead Clarkeon this point — it is in reply to Clarke's challenge that Leibniz explicitlydenies the parity of relative motions. But interestingly, Clarke does notpresent a true version of Newton's argument — despite some involvementof Newton in writing the replies. Instead of the argument from theuniqueness of the rate of rotation, he argues that systems with differentvelocities must be different because the effects observed if they werebrought to rest would be different. This argument is of course utterlyquestion begging against a view that holds that there is no privilegedstandard of rest!

As we discuss in Section 8, Mach attributed to Newton the fallaciousargument that because the surface of the water curved even when it wasnot in motion relative to the bucket, it must be rotating relative to absolutespace. Our discussion of Newton showed how misleading such a readingis. In the first place he also argues that there must be some privilegedsense of rotation, and hence not all relative motions are equal. Second, theargument is ad hominem against Descartes, in which context a disjunctivesyllogism — motion is either proper or ordinary or relative to absolutespace — is argumentatively legitimate. On the other hand, Mach is quitecorrect that Newton's argument in the Scholium leaves open the logicalpossibility that the privileged, true sense of rotation (and accelerationmore generally) is some species of relative motion; if not motion properlyspeaking, then relative to the fixed stars perhaps. (In fact Newton rejectsthis possibility in De Gravitatione (1962) on the grounds that it wouldinvolve an odious action at a distance; an ironic position given his theoryof universal gravity.)

7.2 The Best Explanation Argument Mistakenly Attributed toNewton

However the kind of folk-reading of Newton that underlies much of thecontemporary literature replaces Mach's interpretation with a more



contemporary literature replaces Mach's interpretation with a morecharitable one. According to this reading, Newton's point is that hismechanics — unlike Descartes' — could explain why the surface of therotating water is curved, that his explanation involves a privileged senseof rotation, and that absent an alternative hypothesis about its relativenature, we should accept absolute space. But our discussion of Newton'sargument showed that it simply does not have an ‘abductive’, ‘bestexplanation’ form, but shows deductively, from Cartesian premises, thatrotation is neither proper nor ordinary motion.

That is not to say that Newton had no understanding of how such effectswould be explained in his mechanics. For instance, in Corollaries 5 and 6to the Definitions of the Principles he states in general terms theconditions under which different states of motion are not — and so byimplication are — discernible according to his laws of mechanics. Nor isit to say that Newton's contemporaries weren't seriously concerned withexplaining inertial effects. Leibniz, for instance, analyzed a rotating body(in the Specimen). In short, parts of a rotating system collide with thesurrounding matter and are continuously deflected, into a series of linearmotions that form a curved path. But the system as Leibniz envisions it —comprised of a plenum of elastic particles of matter — is far too complexfor him to offer any quantitative model based on this qualitative picture.(In the context of the proposed ‘abductive’ reading of Newton, note thatthis point is telling against a rejection of intrinsic rigidity or forces actingat a distance, not narrow relationism; it is the complexity of collisions in aplenum that stymies analysis. And since Leibniz's collision theoryrequires a standard of inertial motion, even if he had explained inertialeffects, he would not have thereby shown that all motions are relative,much less that all are equal.)

7.3 Substantivalism and The Best Explanation Argument



7.3.1 The Rotating Spheres

Although the argument is then not Newton's, it is still an importantresponse to the kind of relationism proposed by the folk-Leibniz,especially when it is extended by bringing in a further example fromNewton's Scholium. Newton considered a pair of identical spheres,connected by a cord, too far from any bodies to observe any relativemotions; he pointed out that their rate and direction of rotation could stillbe experimentally determined by measuring the tension in the rod, and bypushing on opposite faces of the two globes to see whether the tensionincreased or decreased. He intended this simple example to demonstratethat the project he intended in the Principia, of determining the absoluteaccelerations and hence gravitational forces on the planets from theirrelative motions, was possible. However, if we further specify that thespheres and cord are rigid and that they are the only things in theiruniverse, then the example can be used to point out that there areinfinitely many different rates of rotation all of which agree on therelations between bodies. Since there are no differences in the relationsbetween bodies in the different situations, it follows that the observabledifferences between the states of rotation cannot be explained in terms ofthe relations between bodies. Therefore, a theory of the kind attributed tothe folk's Leibniz cannot explain all the phenomena of Newtonianmechanics, and again we can argue abductively for absolute space. (Ofcourse, the argument works by showing that, granted the different statesof rotation, there are states of rotation that cannot merely be relativerotations of any kind; for the differences cannot be traced to any relationaldifferences. That is, granted the assumptions of the argument, rotation isnot true relative motion of any kind.)

This argument (neither the premises nor conclusion) is not Newton's, andmust not be taken as a historically accurate reading, However, that is notto say that the argument is fallacious, and indeed many have found itattractive, particularly as a defense not of Newton's absolute space, but of



attractive, particularly as a defense not of Newton's absolute space, but ofGalilean spacetime. That is, Newtonian mechanics with Galileanspacetime can explain the phenomena associated with rotation, whiletheories of the kind proposed by Mach cannot explain the differencesbetween situations allowed by Newtonian mechanics, but theseexplanations rely on the geometric structure of Galilean spacetime —particularly its connection, to interpret acceleration. And thus — theargument goes — those explanations commit us to the reality ofspacetime — a manifold of points — whose properties include theappropriate geometric ones. This final doctrine, of the reality of spacetimewith its component points or regions, distinct from matter, with geometricproperties, is what we earlier identified as ‘substantivalism’.

7.3.2 Relationist Responses

There are two points to make about this line of argument. First, therelationist could reply that he need not explain all situations which arepossible according to Newtonian mechanics, because that theory is to berejected in favor of one which invokes only distance and time relationsbetween bodies, but which approximates to Newton's if matter isdistributed suitably. Such a relationist would be following Mach'sproposal, which we will discuss next. Such a position would besatisfactory only to the extent that a suitable concrete replacement theoryto Newton's theory is developed; Mach never offered such a theory, butrecently more progress has been made.

Second, one must be careful in understanding just how the argumentworks, for it is tempting to gloss it by saying that in Newtonianmechanics the connection is a crucial part of the explanation of thesurface of the water in the bucket, and if the spacetime which carries theconnection is denied, then the explanation fails too. But this gloss tacitlyassumes that Newtonian mechanics can only be understood in asubstantial Galilean spacetime; if an interpretation of Newtonian



substantial Galilean spacetime; if an interpretation of Newtonianmechanics that does not assume substantivalism can be constructed, thenall Newtonian explanations can be given without a literal connection.Both Sklar (1974) and van Fraassen (1985) have made proposals alongthese lines. Sklar proposes interpreting ‘true’ acceleration as a primitivequantity not defined in terms of motion relative to anything, be it absolutespace, a connection or other bodies. (Notice the family resemblancebetween this proposal and Leibniz's view of force and speed.) VanFraassen proposes formulating mechanics as ‘Newton's Laws hold insome frame’, so that the form of the laws and the ways bodies move picksout a standard of inertial motion, not absolute space or a connection, orany instantaneous relations. These proposals aim to keep the fullexplanatory resources of Newtonian mechanics, and hence admit ‘trueacceleration’, but deny any relations between bodies and spacetime itself.Like the actual Leibniz, they allow absolute quantities of motion, butclaim that space and time themselves are nothing but the relationsbetween bodies. Of course, such views raise the question of how a motioncan be not relative to anything at all, and how we are to understand theprivileging of frames; Huggett (2006) contains a proposal for addressingthese problems. (Note that Sklar and van Fraassen are committed to theidea that in some sense Newton's laws are capable of explaining all thephenomena without recourse to spacetime geometry; that the connectionand the metrical properties are explanatorily redundant. A similar view isdefended in the context of relativity in Brown 2005.)

8. Mach and Later Machians

Between the time of Newton and Leibniz and the 20th century, Newton'smechanics and gravitation theory reigned essentially unchallenged, andwith that long period of dominance, absolute space came to be widelyaccepted. At least, no natural philosopher or physicist offered a seriouschallenge to Newton's absolute space, in the sense of offering a rivaltheory that dispenses with it. But like the action at a distance in



theory that dispenses with it. But like the action at a distance inNewtonian gravity, absolute space continued to provoke metaphysicalunease. Seeking a replacement for the unobservable Newtonian space,Neumann (1870) and Lange (1885) developed more concrete definitionsof the reference frames in which Newton's laws hold. In these and a fewother works, the concept of the set of inertial frames was first clearlyexpressed, though it was implicit in both remarks and procedures to befound in the Principia. (See the entries on space and time: inertial framesand Newton's views on space, time, and motion) The most sustained,comprehensive, and influential attack on absolute space was made byErnst Mach in his Science of Mechanics (1883).

In a lengthy discussion of Newton's Scholium on absolute space, Machaccuses Newton of violating his own methodological precepts by goingwell beyond what the observational facts teach us concerning motion andacceleration. Mach at least partly misinterpreted Newton's aims in theScholium, and inaugurated a reading of the bucket argument (and byextension the globes argument) that has largely persisted in the literaturesince. Mach viewed the argument as directed against a ‘strict’ or ‘general-relativity’ form of relationism, and as an attempt to establish the existenceof absolute space. Mach points out the obvious gap in the argument whenso construed: the experiment only establishes that acceleration (rotation)of the water with respect to the Earth, or the frame of the fixed stars,produces the tendency to recede from the center; it does not prove that astrict relationist theory cannot account for the bucket phenomena, muchless the existence of absolute space. (The reader will recall that Newton'sactual aim was simply to show that Descartes' two kinds of motion are notadequate to accounting for rotational phenomena.) Although Mach doesnot mention the globes thought experiment specifically, it is easy to readan implicit response to it in the things he does say: nobody is competentto say what would happen, or what would be possible, in a universedevoid of matter other than two globes. So neither the bucket nor theglobes can establish the existence of absolute space.



globes can establish the existence of absolute space.

8.1 Two Interpretations of Mach on Inertia

Both in Mach's interpretations of Newton's arguments and in his replies,one can already see two anti-absolute space viewpoints emerge, thoughMach himself never fully kept them apart. The first strain, which we maycall ‘Mach-lite’, criticizes Newton's postulation of absolute space as ametaphysical leap that is neither justified by actual experiments, normethodologically sound. The remedy offered by Mach-lite is simple: weshould retain Newton's mechanics and use it just as we already do, buteliminate the unnecessary posit of absolute space. In its place we needonly substitute the frame of the fixed stars, as is the practice in astronomyin any case. If we find the incorporation of a reference to contingentcircumstances (the existence of a single reference frame in which the starsare more or less stationary) in the fundamental laws of nature problematic(which Mach need not, given his official positivist account of scientificlaws), then Mach suggests that we replace the 1st law with an empiricallyequivalent mathematical rival:

The sums in this equation are to be taken over all massive bodies in theuniverse. Since the top sum is weighted by distance, distant masses countmuch more than near ones. In a world with a (reasonably) staticdistribution of heavy distant bodies, such as we appear to live in, theequation entails local conservation of linear momentum in ‘inertial’frames. The upshot of this equation is that the frame of the fixed starsplays exactly the role of absolute space in the statement of the 1st law.(Notice that this equation, unlike Newton's first law, is not vectorial.) Thisproposal does not, by itself, offer an alternative to Newtonian mechanics,

Mach's Equation (1960, 287)



proposal does not, by itself, offer an alternative to Newtonian mechanics,and as Mach himself pointed out, the law is not well-behaved in aninfinite universe filled with stars; but the same can perhaps be said ofNewton's law of gravitation (see Malament 1995, and Norton 1993).[1]

But Mach did not offer this equation as a proposed law valid in anycircumstances; he avers, “it is impossible to say whether the newexpression would still represent the true condition of things if the starswere to perform rapid movements among one another.” (p. 289)

It is not clear whether Mach offered this revised first law as a first steptoward a theory that would replace Newton's mechanics, deriving inertialeffects from only relative motions, as Leibniz desired. But many otherremarks made by Mach in his chapter criticizing absolute space point inthis direction, and they have given birth to the Mach-heavy view, later tobe christened “Mach's Principle” by Albert Einstein.[2] The Mach-heavyviewpoint calls for a new mechanics that invokes only relative distancesand (perhaps) their 1st and 2nd time derivatives, and thus ‘generallyrelativistic’ in the sense sometimes read into Leibniz's remarks aboutmotion. Mach wished to eliminate absolute time from physics too, so hewould have wanted a proper relationist reduction of these derivatives also.The Barbour-Bertotti theories, discussed below, provide this.

Mach-heavy apparently involves the prediction of novel effects due to‘merely’ relative accelerations. Mach hints at such effects in his criticismof Newton's bucket:

Newton's experiment with the rotating vessel of water simplyinforms us that the relative rotation of the water with respect to thesides of the vessel produces no noticeable centrifugal forces, butthat such forces are produced by its relative rotation with respectto the mass of the earth and the other celestial bodies. No one iscompetent to say how the experiment would turn out if the sidesof the vessel [were] increased until they were ultimately several



The suggestion here seems to be that the relative rotation in stage (i) ofthe experiment might immediately generate an outward force (before anyrotation is communicated to the water), if the sides of the bucket weremassive enough.

More generally, Mach-heavy involves the view that all inertial effectsshould be derived from the motions of the body in question relative to allother massive bodies in the universe. The water in Newton's bucket feelsan outward pull due (mainly) to the relative rotation of all the fixed stars

of the vessel [were] increased until they were ultimately severalleagues thick. (1883, 284.)



an outward pull due (mainly) to the relative rotation of all the fixed starsaround it. Mach-heavy is a speculation that an effect something likeelectromagnetic induction should be built into gravity theory. (Such aneffect does exist according to the General Theory of Relativity, and iscalled ‘gravitomagnetic induction’. The recently finished Gravity Probe Bmission was designed to measure the gravitomagnetic induction effectdue to the Earth's rotation.) Its specific form must fall off with distancemuch more slowly than 1/r2, if it is to be empirically similar toNewtonian physics; but it will certainly predict experimentally testablenovel behaviors. A theory that satisfies all the goals of Mach-heavywould appear to be ideal for the vindication of strict relationism and theelimination of absolute quantities of motion from mechanics.

8.2 Implementing Mach-heavy

Direct assault on the problem of satisfying Mach-heavy in a classicalframework proved unsuccessful, despite the efforts of others besidesMach (e.g., Friedländer 1896, Föpl 1904, Reissner 1914, 1915), until thework of Barbour and Bertotti in the 1970s and 80s. (Between the late 19thcentury and the 1970s, there was of course one extremely importantattempt to satisfy Mach-heavy: the work of Einstein that led to theGeneral Theory of Relativity. Since Einstein's efforts took place in a non-classical (Lorentz/Einstein/Minkowski) spacetime setting, we discussthem in the next section.) Rather than formulating a revised law ofgravity/inertia using relative quantities, Barbour and Bertotti attacked theproblem using the framework of Lagrangian mechanics, replacing theelements of the action that involve absolute quantities of motion with newterms invoking only relative distances, velocities etc. Their first (1977)theory uses a very simple and elegant action, and satisfies everything onecould wish for from a Mach-heavy theory: it is relationally pure (evenwith respect to time: while simultaneity is absolute, the temporal metric isderived from the field equations); it is nearly empirically equivalent toNewton's theory in a world such as ours (with a large-scale uniform, near-



Newton's theory in a world such as ours (with a large-scale uniform, near-stationary matter distribution); yet it does predict novel effects such as theones Mach posited with his thick bucket. Among these is an ‘anisotropyof inertia’ effect — accelerating a body away from the galactic centerrequires more force than accelerating it perpendicular to the galacticplane — large enough to be ruled out empirically.

Barbour and Bertotti's second attempt (1982) at a relational Lagrangianmechanics was arguably less Machian, but more empirically adequate. Init, solutions are sought beginning with two temporally-nearby,instantaneous relational configurations of the bodies in the universe.Barbour and Bertotti define an ‘intrinsic difference’ parameter thatmeasures how different the two configurations are. In the solutions of thetheory, this intrinsic difference quantity gets minimized, as well as theordinary action, and in this way full solutions are derived despite notstarting from a privileged inertial-frame description. The theory they endup with turns out to be, in effect, a fragment of Newtonian theory: the setof models of Newtonian mechanics and gravitation in which there is zeronet angular momentum. This result makes perfect sense in terms of strictrelationist aims. In a Newtonian world in which there is a nonzero netangular momentum (e.g., a lone rotating island galaxy), this fact revealsitself in the classic “tendency to recede from the center”. Since a strictrelationist demands that bodies obey the same mechanical laws even in‘rotating’ coordinate systems, there cannot be any such tendency torecede from the center (other than in a local subsystem), in any of therelational theory's models. Since cosmological observations, even today,reveal no net angular momentum in our world, the second Barbour &Bertotti theory can lay claim to exactly the same empirical successes (andproblems) that Newtonian physics had. The second theory does notpredict the (empirically falsified) anisotropy of inertia derivable from thefirst; but neither does it allow a derivation of the precession of the orbit ofMercury, which the first theory does (for appropriately chosen cosmicparameters).



parameters).

8.3 Mach-lite versus Mach-heavy

Mach-lite, like the relational interpretations of Newtonian physicsreviewed in section 5, offers us a way of understanding Newtonianphysics without accepting absolute position, velocity or acceleration. Butit does so in a way that lacks theoretical clarity and elegance, since it doesnot delimit a clear set of cosmological models. We know that Mach-litemakes the same predictions as Newton for worlds in which there is astatic frame associated with the stars and galaxies; but if asked about howthings will behave in a world with no frame of fixed stars, or in which thestars are far from ‘fixed’, it shrugs and refuses to answer. (Recall thatMach-lite simply says: “Newton's laws hold in the frame of reference ofthe fixed stars.”) This is perfectly acceptable according to Mach'sphilosophy of science, since the job of mechanics is simply to summarizeobservable facts in an economical way. But it is unsatisfying to those withstronger realist intuitions about laws of nature.

If there is, in fact, a distinguishable privileged frame of reference inwhich the laws of mechanics take on a specially simple form, without thatframe being determined in any way by relation to the matter distribution,a realist will find it hard to resist the temptation to view motionsdescribed in that frame as the ‘true’ or ‘absolute’ motions. If there is afamily of such frames, disagreeing about velocity but all agreeing aboutacceleration, she will feel a temptation to think of at least acceleration as‘true’ or ‘absolute’. If such a realist believes motion to be by nature arelation rather than a property (and as we saw in the introduction, not allphilosophers accept this) then she will feel obliged to accord some sort ofexistence or reality to the structure — e.g., the structure of Galileanspacetime — in relation to which these motions are defined. Forphilosophers with such realist inclinations, the ideal relational account ofmotion would therefore be some version of Mach-heavy.



motion would therefore be some version of Mach-heavy.

9. Relativity and Motion

The Special Theory of Relativity (STR) is notionally based on a principleof relativity of motion; but that principle is ‘special’ — meaning,restricted. The relativity principle built into STR is in fact nothing otherthan the Galilean principle of relativity, which is built into Newtonianphysics.[3] In other words, while there is no privileged standard ofvelocity, there is nevertheless a determinate fact of the matter aboutwhether a body has accelerated or non-accelerated (i.e., inertial) motion.In this regard, the spacetime of STR is exactly like Galilean spacetime(defined in section 5 above). In terms of the question of whether allmotion can be considered purely relative, one could argue that there isnothing new brought to the table by the introduction of Einstein's STR —at least, as far as mechanics is concerned.

9.1 Relations Determine State of Motion?

As Dorling (1978) first pointed out, however, there is a sense in which thestandard absolutist arguments against ‘strict’ relationism using rotatingobjects (buckets or globes) fail in the context of STR. Maudlin (1993)used the same considerations to show that there is a way of recastingrelationism in STR that appears to be very successful.

STR incorporates certain novelties concerning the nature of time andspace, and how they mesh together; perhaps the best-known examples arethe phenomena of ‘length contraction’, ‘time dilation’, and the ‘relativityof simultaneity.’[4] Since in STR both spatial distances and time intervals— when measured in the standard ways — are observer-relative(observers in different states of motion ‘disagreeing’ about their sizes), itis arguably most natural to restrict oneself to the invariant spacetime



separation given by the interval between two points: [dx2 + dy2 + dz2 —dt2] — the four-dimensional analog of the Pythagorean theorem, forspacetime distances. If one regards the spacetime interval relationsbetween masses-at-times as one's basis on which space-time is built up asan ideal entity, then with only mild caveats relationism works: the‘relationally pure’ facts suffice to uniquely fix how the material systemsare embeddable (up to isomorphism) in the ‘Minkowski’ spacetime ofSTR. The modern variants of Newton's bucket and globes arguments nolonger stymie the relationist because (for example) the spacetime intervalrelations among bits of matter in Newton's bucket at rest are quitedifferent from the spacetime interval relations found among those samebits of matter after the bucket is rotating. For example, the spacetimeinterval relation between a bit of water near the side of the bucket, at onetime, and itself (say) a second later is smaller than the interval relationbetween a center-bucket bit of water and itself one second later (timesreferred to inertial-frame clocks). The upshot is that, unlike the situationin classical physics, a body at rest cannot have all the same spatialrelations among its parts as a similar body in rotation. We cannot put abody or system into a state of rotation (or other acceleration) withoutthereby changing the spacetime interval relations between the various bitsof matter at different moments of time. Rotation and accelerationsupervene on spacetime interval relations.

It is worth pausing to consider to what extent this victory for (some formof) relationism satisfies the classical ‘strict’ relationism traditionallyascribed to Mach and Leibniz. The spatiotemporal relations that save theday against the bucket and globes are, so to speak, mixed spatial andtemporal distances. They are thus quite different from the spatial-distances-at-a-time presupposed by classical relationists; moreover theydo not correspond to relative velocities (-at-a-time) either. Their oddity isforcefully captured by noticing that if we choose appropriate bits ofmatter at ‘times’ eight minutes apart, I-now am at zero distance from thesurface of the sun (of eight minutes ‘past’, since it took 8 minutes for



surface of the sun (of eight minutes ‘past’, since it took 8 minutes forlight from the sun to reach me-now). So we are by no means dealing herewith an innocuous, ‘natural’ translation of classical relationist quantitiesinto the STR setting. On the other hand, in light of the relativity ofsimultaneity (see note[4]), it can be argued that the absolute simultaneitypresupposed by classical relationists and absolutists alike was, in fact,something that relationists should always have regarded with misgivings.From this perspective, instantaneous relational configurations — preciselywhat one starts with in the theories of Barbour and Bertotti — would bethe things that should be treated with suspicion.

If we now return to our questions about motions — about the nature ofvelocities and accelerations — we find, as noted above, that matters in theinterval-relational interpretation of STR are much the same as inNewtonian mechanics in Galilean spacetime. There are no well-definedabsolute velocities, but there are indeed well-defined absoluteaccelerations and rotations. In fact, the difference between an acceleratingbody (e.g., a rocket) and an inertially moving body is codified directly inthe cross-temporal interval relations of the body with itself. So we arevery far from being able to conclude that all motion is relative motion of abody with respect to other bodies. It is true that the absolute motions arein 1-1 correlation with patterns of spacetime interval relations, but it isnot at all correct to say that they are, for that reason, eliminable in favorof merely relative motions. Rather we should simply say that no absoluteacceleration can fail to have an effect on the material body or bodiesaccelerated. But this was already true in classical physics if matter ismodeled realistically: the cord connecting the globes does not merelytense, but also stretches; and so does the bucket, even if imperceptibly,i.e., the spatial relations change.

Maudlin does not claim this version of relationism to be victorious overan absolutist or substantivalist conception of Minkowski spacetime, whenit comes time to make judgments about the theory's ontology. There may



it comes time to make judgments about the theory's ontology. There maybe more to vindicating relationism than merely establishing a 1-1correlation between absolute motions and patterns of spatiotemporalrelations.

9.2 The Relationist Roots of STR and GTR

The simple comparison made above between STR and Newtonian physicsin Galilean spacetime is somewhat deceptive. For one thing, Galileanspacetime is a mathematical innovation posterior to Einstein's 1905theory; before then, Galilean spacetime had not been conceived, and fullacceptance of Newtonian mechanics implied accepting absolute velocitiesand, arguably, absolute positions, just as laid down in the Scholium. SoEinstein's elimination of absolute velocity was a genuine conceptualadvance. Moreover, the Scholium was not the only reason for supposingthat there existed a privileged reference frame of ‘rest’: the workingassumption of almost all physicists in the latter half of the 19th centurywas that, in order to understand the wave theory of light, one had topostulate an aetherial medium filling all space, wave-like disturbances inwhich constituted electromagnetic radiation. It was assumed that theaether rest frame would be an inertial reference frame; and physicists feltsome temptation to equate its frame with the absolute rest frame, thoughthis was not necessary. Regardless of this equation of the aether withabsolute space, it was assumed by all 19th century physicists that theequations of electrodynamic theory would have to look different in areference frame moving with respect to the aether than they did in theaether's rest frame (where they presumably take their canonical form, i.e.,Maxwell's equations and the Lorentz force law.) So while theoreticianslabored to find plausible transformation rules for the electrodynamics ofmoving bodies, experimentalists tried to detect the Earth's motion in theaether. Experiment and theory played collaborative roles, withexperimental results ruling out certain theoretical moves and suggestingnew ones, while theoretical advances called for new experimental tests



new ones, while theoretical advances called for new experimental testsfor their confirmation or — as it happened — disconfirmation.

As is well known, attempts to detect the Earth's velocity in the aetherwere unsuccessful. On the theory side, attempts to formulate thetransformation laws for electrodynamics in moving frames — in such away as to be compatible with experimental results — were complicatedand inelegant.[5] A simplified way of seeing how Einstein swept away ahost of problems at a stroke is this: he proposed that the Galileanprinciple of relativity holds for Maxwell's theory, not just for mechanics.The canonical (‘rest-frame’) form of Maxwell's equations should be theirform in any inertial reference frame. Since the Maxwell equations dictatethe velocity c of electromagnetic radiation (light), this entails that anyinertial observer, no matter how fast she is moving, will measure thevelocity of a light ray as c — no matter what the relative velocity of itsemitter. Einstein worked out logically the consequences of this applicationof the special relativity principle, and discovered that space and time mustbe rather different from how Newton described them. STR underminedNewton's absolute time just as decisively as it undermined his absolutespace (see note [4]).

9.3 From Special Relativity to General Relativity

Einstein's STR was the first clear and empirically successful physicaltheory to overtly eliminate the concepts of absolute rest and absolutevelocity while recovering most of the successes of classical mechanicsand 19th century electrodynamics. It therefore deserves to be consideredthe first highly successful theory to explicitly relativize motion, albeitonly partially. But STR only recovered most of the successes of classicalphysics: crucially, it left out gravity. And there was certainly reason to beconcerned that Newtonian gravity and STR would prove incompatible:classical gravity acted instantaneously at a distance, while STR eliminatedthe privileged absolute simultaneity that this instantaneous action



the privileged absolute simultaneity that this instantaneous actionpresupposes.

Several ways of modifying Newtonian gravity to make it compatible withthe spacetime structure of STR suggested themselves to physicists in theyears 1905-1912, and a number of interesting Lorentz-covariant theorieswere proposed (set in the Minkowski spacetime of STR). Einsteinrejected these efforts one and all, for violating either empirical facts ortheoretical desiderata. But Einstein's chief reason for not pursuing thereconciliation of gravitation with STR's spacetime appears to have beenhis desire, beginning in 1907, to replace STR with a theory in which notonly velocity could be considered merely relative, but also acceleration.That is to say, Einstein wanted if possible to completely eliminate allabsolute quantities of motion from physics, thus realizing a theory thatsatisfies at least one kind of ‘strict’ relationism. (Regarding Einstein'srejection of Lorentz-covariant gravity theories, see Norton 1992;regarding Einstein's quest to fully relativize motion, see Hoefer 1994.)

Einstein began to see this complete relativization as possible in 1907,thanks to his discovery of the Equivalence Principle. Imagine we are farout in space, in a rocket ship accelerating at a constant rate g = 9.98 m/s2.Things will feel just like they do on the surface of the Earth; we will feela clear up-down direction, bodies will fall to the floor when released, etc.Indeed, due to the well-known empirical fact that gravity affects allbodies by imparting a force proportional to their matter (and energy)content, independent of their internal constitution, we know that anyexperiment performed on this rocket will give the same results that thesame experiment would give if performed on the Earth. Now, Newtoniantheory teaches us to consider the apparent downward, gravity-like forcesin the rocket ship as ‘pseudo-forces’ or ‘inertial forces’, and insists thatthey are to be explained by the fact that the ship is accelerating inabsolute space. But Einstein asked: “Is there any way for the person in therocket to regard him/herself as being ‘at rest’ rather than in absolute



rocket to regard him/herself as being ‘at rest’ rather than in absolute(accelerated) motion?” And the answer he gave is: Yes. The rockettraveler may regard him/herself as being ‘at rest’ in a homogeneous anduniform gravitational field. This will explain all the observational factsjust as well as the supposition that he/she is accelerating relative toabsolute space (or, absolutely accelerating in Minkowski spacetime). Butis it not clear that the latter is the truth, while the former is a fiction? Byno means; if there were a uniform gravitational field filling all space, thenit would affect all the other bodies in the world — the Earth, the stars,etc, imparting to them a downward acceleration away from the rocket;and that is exactly what the traveler observes.

In 1907, Einstein published his first gravitation theory (Einstein 1907),treating the gravitational field as a scalar field that also represented the(now variable and frame-dependent) speed of light. Einstein viewed thetheory as only a first step on the road to eliminating absolute motion. Inthe 1907 theory, the theory's equations take the same form in any inertialor uniformly accelerating frame of reference. One might say that thistheory reduces the class of absolute motions, leaving only rotation andother non-uniform accelerations as absolute. But, Einstein reasoned, ifuniform acceleration can be regarded as equivalent to being at rest in aconstant gravitational field, why should it not be possible also to regardinertial effects from these other, non-uniform motions as similarlyequivalent to “being at rest in a (variable) gravitational field”? ThusEinstein set himself the goal of expanding the principle of equivalence toembrace all forms of ‘accelerated’ motion.

Einstein thought that the key to achieving this aim lay in furtherexpanding the range of reference frames in which the laws of physics taketheir canonical form, to include frames adapted to any arbitrary motions.More specifically, since the class of all continuous and differentiablecoordinate systems includes as a subclass the coordinate systems adaptedto any such frame of reference, if he could achieve a theory of gravitation,



to any such frame of reference, if he could achieve a theory of gravitation,electromagnetism and mechanics that was generally covariant — itsequations taking the same form in any coordinate system from thisgeneral class — then the complete relativity of motion would be achieved.If there are no special frames of reference in which the laws take on asimpler canonical form, there is no physical reason to consider anyparticular state or states of motion as privileged, nor deviations fromthose as representing ‘absolute motion’. (Here we are just laying outEinstein's train of thought; later we will see reasons to question the laststep.) And in 1915, Einstein achieved his aim in the General Theory ofRelativity (GTR).

9.4 General Relativity and Relativity of Motion

There is one key element left out of this success story, however, and it iscrucial to understanding why most physicists reject Einstein's claim tohave eliminated absolute states of motion in GTR. Going back to ouraccelerating rocket, we accepted Einstein's claim that we could regard theship as hovering at rest in a universe-filling gravitational field. But agravitational field, we usually suppose, is generated by matter. How isthis universe-filling field linked to generating matter? The answer may besupplied by Mach-heavy. Regarding the ‘accelerating’ rocket which wedecide to regard as ‘at rest’ in a gravitational field, the Machian says: allthose stars and galaxies, etc., jointly accelerating downward (relative tothe rocket), ‘produce’ that gravitational field. The mathematical specificsof how this field is generated will have to be different from Newton's lawof gravity, of course; but it should give essentially the same results whenapplied to low-mass, slow-moving problems such as the orbits of theplanets, so as to capture the empirical successes of Newtonian gravity.Einstein thought, in 1916 at least, that the field equations of GTR areprecisely this mathematical replacement for Newton's law of gravity, andthat they fully satisfied the desiderata of Mach-heavy relationism. But itwas not so. (See the entry on early philosophical interpretations of general



was not so. (See the entry on early philosophical interpretations of generalrelativity.)

In GTR, spacetime is locally very much like flat Minkowski spacetime.There is no absolute velocity locally, but there are clear local standards ofaccelerated vs non-accelerated motion, i.e., local inertial frames. In these‘freely falling’ frames bodies obey the usual rules for non-gravitationalphysics familiar from STR, albeit only approximately. But overallspacetime is curved, and local inertial frames may tip, bend and twist aswe move from one region to another. The structure of curved spacetime isencoded in the metric field tensor gab, with the curvature encodinggravity at the same time: gravitational forces are so to speak ‘built into’the metric field, geometrized away. Since the spacetime structure encodesgravity and inertia, and in a Mach-heavy theory these phenomena shouldbe completely determined by the relational distribution of matter (andrelative motions), Einstein wished to see the metric as entirely determinedby the distribution of matter and energy. But what the GTR fieldequations entail is, in general, only a partial-determination relation.

We cannot go into the mathematical details necessary for a full discussionof the successes and failures of Mach-heavy in the GTR context. But onecan see why the Machian interpretation Einstein hoped he could give tothe curved spacetimes of his theory fails to be plausible, by considering afew simple ‘worlds’ permitted by GTR. In the first place, for our hoveringrocket ship, if we are to attribute the gravity field it feels to matter, therehas got to be all this other matter in the universe. But if we regard therocket as a mere ‘test body’ (not itself substantially affecting the gravitypresent or absent in the universe), then we can note that according toGTR, if we remove all the stars, galaxies, planets etc. from the world, thegravitational field does not disappear. On the contrary, it stays basicallythe same locally, and globally it takes the form of empty Minkowskispacetime, precisely the quasi-absolute structure Einstein was hoping toeliminate. Solutions of the GTR field equations for arbitrary realistic



eliminate. Solutions of the GTR field equations for arbitrary realisticconfigurations of matter (e.g., a rocket ship ejecting a stream of particlesto push itself forward) are hard to come by, and in fact a realistic two-body exact solution has yet to be discovered. But numerical methods canbe applied for many purposes, and physicists do not doubt that somethinglike our accelerating rocket — in otherwise empty space — is possibleaccording to the theory.[6] We see clearly, then, that GTR fails to satisfyEinstein's own understanding of Mach's Principle, according to which, inthe absence of matter, space itself should not be able to exist. A secondexample: GTR allows us to model a single rotating object in an otherwiseempty universe (e.g., a neutron star). Relationism of the Machian varietysays that such rotation is impossible, since it can only be understood asrotation relative to some sort of absolute space. In the case of GTR, this isbasically right: the rotation is best understood as rotation relative to a‘background’ spacetime that is identical to the Minkowski spacetime ofSTR, only ‘curved’ by the presence of matter in the region of the star.

On the other hand, there is one charge of failure-to-relativize-motionsometimes leveled at GTR that is unfair. It is sometimes asserted that thesimple fact that the metric field (or the connection it determines)distinguishes, at every location, motions that are ‘absolutely’ acceleratedand/or ‘absolutely rotating’ from those that are not, by itself entails thatGTR fails to embody a folk-Leibniz style general relativity of motion(e.g. Earman (1989), ch. 5). We think this is incorrect, and leads tounfairly harsh judgments about confusion on Einstein's part. The localinertial structure encoded in the metric would not be ‘absolute’ in anymeaningful sense, if that structure were in some clear sense fullydetermined by the relationally specified matter-energy distribution.Einstein was not simply confused when he named his gravity theory. (Justwhat is to be understood by “the relationally specified matter-energydistribution” is a further, thorny issue, which we cannot enter into here.)



GTR does not fulfill all the goals of Mach-heavy, at least as understoodby Einstein, and he recognized this fact by 1918 (Einstein 1918). And yet… GTR comes tantalizingly close to achieving those goals, in certainstriking ways. For one thing, GTR does predict Mach-heavy effects,known as ‘frame-dragging’: if we could model Mach's thick-walledbucket in GTR, it seems clear that it would pull the water slightlyoutward, and give it a slight tendency to begin rotating in the same senseas the bucket (even if the big bucket's walls were not actually touchingthe water. While GTR does permit us to model a lone rotating object, ifwe model the object as a shell of mass (instead of a solid sphere) and letthe size of the shell increase (to model the ‘sphere of the fixed stars’ wesee around us), then as Brill & Cohen (1966) showed, the frame-draggingbecomes complete inside the shell. In other words: our originalMinkowski background structure effectively disappears, and inertiabecomes wholly determined by the shell of matter, just as Mach positedwas the case. This complete determination of inertia by the global matterdistribution appears to be a feature of other models, including theFriedman-Robertson-Walker-Lemâitre Big Bang models that best matchobservations of our universe.

Finally, it is important to recognize that GTR is generally covariant in avery special sense: unlike all other prior theories (and unlike manysubsequent quantum theories), it postulates no fixed ‘prior’ or‘background’ spacetime structure. As mathematicians and physicistsrealized early on, other theories, e.g., Newtonian mechanics and STR, canbe put into a generally covariant form. But when this is done, there areinevitably mathematical objects postulated as part of the formalism,whose role is to represent absolute elements of spacetime structure. Whatis unique about GTR is that it was the first, and is still the only ‘core’physical theory, to have no such absolute elements in its covariantequations. The spacetime structure in GTR, represented by the metricfield (which determines the connection), is at least partly ‘shaped’ by thedistribution of matter and energy. And in certain models of the theory,



distribution of matter and energy. And in certain models of the theory,such as the Big Bang cosmological models, some authors have claimedthat the local standards of inertial motion — the local ‘gravitational field’of Einstein's equivalence principle — are entirely fixed by the matterdistribution throughout space and time, just as Mach-heavy requires (see,for example, Wheeler and Cuifollini 1995).

Absolutists and relationists are thus left in a frustrating and perplexingquandary by GTR. Considering its anti-Machian models, we are inclinedto say that motions such as rotation and acceleration remain absolute, ornearly-totally-absolute, according to the theory. On the other hand,considering its most Mach-friendly models, which include all the modelstaken to be good candidates for representing the actual universe, we maybe inclined to say: motion in our world is entirely relative; the inertialeffects normally used to argue for absolute motion are all understandableas effects of rotations and accelerations relative to the cosmic matter, justas Mach hoped. But even if we agree that motions in our world are in factall relative in this sense, this does not automatically settle the traditionalrelationist/absolutist debate, much less the relationist/substantivalistdebate. Many philosophers (including, we suspect, Nerlich 1994 andEarman 1989) would be happy to acknowledge the Mach-friendly statusof our spacetime, and argue nevertheless that we should understand thatspacetime as a real thing, more like a substance than a mere idealconstruct of the mind as Leibniz insisted. (Nerlich (1994) and Earman(1989), we suspect, would take this stance.) Some, though not all,attempts to convert GTR into a quantum theory would accord spacetimethis same sort of substantiality that other quantum fields possess.

10. Conclusion

This article has been concerned with tracing the history and philosophy of‘absolute’ and ‘relative’ theories of space and motion. Along the way wehave been at pains to introduce some clear terminology for various



have been at pains to introduce some clear terminology for variousdifferent concepts (e.g., ‘true’ motion, ‘substantivalism’, ‘absolutespace’), but what we have not really done is say what the differencebetween absolute and relative space and motion is: just what is at stake?Recently Rynasiewicz (2000) has argued that there simply are no constantissues running through the history that we have discussed here; that thereis no stable meaning for either ‘absolute motion’ or ‘relative motion’ (or‘substantival space’ vs ‘relational space’). While we agree to a certainextent, we think that nevertheless there are a series of issues that havemotivated thinkers again and again; indeed, those that we identified in theintroduction. (One quick remark: Rynasiewicz is probably right that theissues cannot be expressed in formally precise terms, but that does notmean that there are no looser philosophical affinities that shed useful lighton the history.)

Our discussion has revealed several different issues, of which we willhighlight three as components of the ‘absolute-relative debate’. (i) Thereis the question of whether all motions and all possible descriptions ofmotions are equal, or whether some are ‘real’ — what we have called, inSeventeenth Century parlance, ‘true’. There is a natural temptation forthose who hold that there is ‘nothing but the relative positions andmotions between bodies' (and more so for their readers) to add ‘and allsuch motions are equal’, thus denying the existence of true motion.However, arguably — perhaps surprisingly — no one we have discussedhas unreservedly held this view (at least not consistently): Descartesconsidered motion ‘properly speaking’ to be privileged, Leibnizintroduced ‘active force’ to ground motion (arguably in his mechanics aswell as metaphysically), and Mach's view seems to be that the distributionof matter in the universe determines a preferred standard of inertialmotion. (Again, in general relativity, there is a distinction between inertialand accelerated motion.)



That is, relationists can allow true motions if they offer an analysis ofthem in terms of the relations between bodies. Given this logical point,and given the historical ways thinkers have understood themselves, itseems unhelpful to characterize the issues in (i) as constituting anabsolute-relative debate, hence our use of the term ‘true’ instead of‘absolute’. So we are led to the second question: (ii) is true motiondefinable in terms of relations or not? (Of course the answer depends onwhat kind of definitions will count, and absent an explicit definition —Descartes' proper motion for example — the issue is often taken to be thatof whether true motions supervene on relations, as Newton's globes areoften supposed to refute.) It seems reasonable to call this issue that ofwhether motion is absolute or relative. Descartes and Mach are relationistsabout motion in this sense, while Newton is an absolutist. Leibniz is alsoan absolutist about motion in his metaphysics, and if our reading iscorrect, also about the interpretation of motion in the laws of collision.This classification of Leibniz's views runs contrary to his customaryidentification as relationist-in-chief, but we will clarify his relationistcredentials below. Finally, we have discussed (ii) in the context ofrelativity, first examining Maudlin's proposal that the embedding of arelationally-specified system in Minkowski spacetime is in general uniqueonce all the spacetime interval-distance relations are given. This proposalmay or may not be held to satisfy the relational-definability question of(ii), but in any case it cannot be carried over to the context of generalrelativity theory. In the case of GTR we linked relational motion to thesatisfaction of Mach's Principle, just as Einstein did in the early years ofthe theory. Despite some promising features displayed by GTR, andcertain of its models, we saw that Mach's Principle is not fully satisfied inGTR as a whole. We also noted that in the absence of absolutesimultaneity, it becomes an open question what relations are to bepermitted in the definition (or supervience base) — spacetime intervalrelations? Instantaneous spatial distances and velocities on a 3-dhypersurface? (In recent works, Barbour has argued that GTR is fully



hypersurface? (In recent works, Barbour has argued that GTR is fullyMachian, using a 3-d relational-configuration approach. See Barbour,Foster and Murchadha 2002.)

The final issue is that of (iii) whether absolute motion is motion withrespect to substantival space or not. Of course this is how Newtonunderstood acceleration — as acceleration relative to absolute space.More recent Newtonians share this view, although motion for them iswith respect to substantival Galilean spacetime (or rather, since theyknow Newtonian mechanics is false, they hold that this is the bestinterpretation of that theory). Leibniz denied that motion was relative tospace itself, since he denied the reality of space; for him true motion wasthe possession of active force. So despite his ‘absolutism’ (our adjectivenot his) about motion he was simultaneously a relationist about space:‘space is merely relative’. Following Leibniz's lead we can call thisdebate the question of whether space is absolute or relative. Thedrawback of this name is that it suggests a separation between motion andspace, which exists in Leibniz's views, but which is otherwiseproblematic; still, no better description presents itself.

Others who are absolutists about motion but relationists about spaceinclude Sklar (1974) and van Fraassen (1985); Sklar introduced aprimitive quantity of acceleration, not supervenient on motions relative toanything at all, while van Fraassen let the laws themselves pick out theinertial frames. It is of course arguable whether any of these threeproposals are successful; (even) stripped of Leibniz's Aristotelianpackaging, can absolute quantities of motion ‘stand on their own feet’?And under what understanding of laws can they ground a standard ofinertial motion? Huggett (2006) defends a similar position of absolutismabout motion, but relationism about space; he argues — in the case ofNewtonian physics — that fundamentally there is nothing to space butrelations between bodies, but that absolute motions supervene — not onthe relations at any one time — but on the entire history of relations.



the relations at any one time — but on the entire history of relations.

Bibliography

Works cited in text

Aristotle, 1984, The Complete Works of Aristotle: The RevisedOxford Translation, J. Barnes (ed.), Princeton: Princeton UniversityPress.Barbour, J. and Bertotti, B., 1982, “Mach's Principle and theStructure of Dynamical Theories,” Proceedings of the Royal Society(London), 382: 295-306.–––, 1977, “Gravity and Inertia in a Machian Framework,” NuovoCimento, 38B: 1-27.Brill, D. R. and Cohen, J., 1966, “Rotating Masses and their effectson inertial frames,” Physical Review 143: 1011-1015.Brown, H. R., 2005, Physical Relativity: Space-Time Structure froma Dynamical Perspective, Oxford: Oxford University Press.Descartes, R., 1983, Principles of Philosophy, R. P. Miller and V. R.Miller (trans.), Dordrecht, London: Reidel.Dorling, J., 1978, “Did Einstein need General Relativity to solve theProblem of Space? Or had the Problem already been solved bySpecial Relativity?,” British Journal for the Philosophy of Science,29: 311-323.Earman, J., 1989, World Enough and Spacetime: Absolute andRelational Theories of Motion. Boston: M.I.T. Press.–––, 1970, “Who's Afraid of Absolute Space?,” Australasian Journalof Philosophy, 48: 287-319.Einstein, A., 1918, “Prinzipielles zur allgemeinenRelativitätstheorie,” Annalen der Physik, 51: 639-642.–––, 1907, “Über das Relativitätsprinzip und die aus demselbengezogenen Folgerungen,” Jahrbuch der Radioaktivität undElectronik 4: 411-462.



Electronik 4: 411-462.Einstein, A., Lorentz, H. A., Minkowski, H. and Weyl, H., 1952, ThePrinciple of Relativity. W. Perrett and G.B. Jeffery, trs. New York:Dover Books.Föppl, A. “Über absolute und relative Bewegung,” Sitzungsberichteder Münchener Akad.. 35:383.Friedländer, B. and J., 1896, Absolute und relative Bewegung,Berlin: Leonhard Simion.Friedman, M., 1983, Foundations of Space-Time Theories:Relativistic Physics and Philosophy of Science, Princeton: PrincetonUniversity Press.Garber, D., 1992, Descartes' Metaphysical Physics, Chicago:University of Chicago Press.Garber, D. and J. B. Rauzy, 2004, “Leibniz on Body, Matter andExtension,” The Aristotelian Society: Supplementary Volume, 78: 23-40.Hartz, G. A. and J. A. Cover, 1988, “Space and Time in theLeibnizian Metaphysic,” Nous, 22: 493-519.Hoefer, C., 1994, “Einstein's Struggle for a Machian GravitationTheory,” Studies in History and Philosophy of Science, 25: 287-336.Huggett, N., 2006, “The Regularity Account of RelationalSpacetime,” Mind, 115: 41-74.–––, 2000, “Space from Zeno to Einstein: Classic Readings with aContemporary Commentary,” International Studies in the Philosophyof Science, 14: 327-329.Lange, L., 1885, “Ueber das Beharrungsgesetz,” Berichte derKöniglichen Sachsischen Gesellschaft der Wissenschaften zu Leipzig,Mathematisch-physische Classe 37 (1885): 333-51.Leibniz, G. W., 1989, Philosophical Essays, R. Ariew and D. Garber(trans.), Indianapolis: Hackett Pub. Co.Leibniz, G. W., and Samuel Clarke, 1715–1716, “Correspondence”,in The Leibniz-Clarke Correspondence, Together with Extracts from



in The Leibniz-Clarke Correspondence, Together with Extracts fromNewton's “Principia” and “Opticks”, H. G. Alexander (ed.),Manchester: Manchester University Press, 1956.Lodge, P., 2003, “Leibniz on Relativity and the Motion of Bodies,”Philosophical Topics, 31: 277-308.Mach, E., 1883, Die Mechanik in ihrer Entwickelung, historisch-kritisch dargestellt. 2nd edition. Leipzig: Brockhaus. Englishtranslation (6th edition, 1960): The Science of Mechanics, La Salle,Illinois: Open Court Press.Malament, D., 1995, “Is Newtonian Cosmology Really Inconsistent?,” Philosophy of Science 62, no. 4.Maudlin, T., 1993, “Buckets of Water and Waves of Space: WhySpace-Time is Probably a Substance,” Philosophy of Science,60:183-203.Minkowski, H. (1908). “Space and time,” In Einstein, et al. (1952),pp. 75-91.Nerlich, Graham, 1994, The Shape of Space (2nd edition),Cambridge: Cambridge University Press.Neumann, C., 1870, Ueber die Principien der Galilei-Newton'schenTheorie. Leipzig: B. G. Teubner, 1870.Newton, I., 2004, Newton: Philosophical Writings, A. Janiak (ed.),Cambridge: Cambridge University Press.Newton, I. and I. B. Cohen, 1999, The Principia: MathematicalPrinciples of Natural Philosophy, I. B. Cohen and A. M. Whitman(trans.), Berkeley ; London: University of California Press.Norton, J., 1995, “Mach's Principle before Einstein,” in J. Barbourand H. Pfister (eds.) Mach's Principle: From Newton's Bucket toQuantum Gravity: Einstein Studies, Vol. 6. Boston: Birkhäuser, pp.9-57.Norton, J., 1993, “A Paradox in Newtonian Cosmology,” in M.Forbes , D. Hull and K. Okruhlik (eds.) PSA 1992: Proceedings ofthe 1992 Biennial Meeting of the Philosophy of Science Association.



the 1992 Biennial Meeting of the Philosophy of Science Association.Vol. 2. East Lansing, MI: Philosophy of Science Association, pp.412-20.–––, 1992, “Einstein, Nordström and the Early Demise of Scalar,Lorentz-Covariant Theories of Gravitation,” Archive for History ofExact Sciences, 45: 17-94.Pooley, O., 2002, the Reality of Spacetime, D.Phil thesis, OxfordUniversity.Ray, C., 1991, Time, Space and Philosophy, New York: Routledge.Roberts, J. T., 2003, “Leibniz on Force and Absolute Motion,”Philosophy of Science, 70: 553-573.Rynasiewicz, R., 1995, “By their Properties, Causes, and Effects:Newton's Scholium on Time, Space, Place, and Motion — I. TheText,” Studies in History and Philosophy of Science, 26: 133-153.Sklar, L., 1974, Space, Time and Spacetime, Berkeley: University ofCalifornia Press.Stein, H., 1977, “Some Philosophical Prehistory of GeneralRelativity,” in Minnesota Studies in the Philosophy of Science8: Foundations of Space-Time Theories: , J. Earman, C. Glymourand J. Stachel (eds.), Minneapolis: University of Minnesota Press.–––, 1967, “Newtonian Space-Time,” Texas Quarterly, 10: 174-200.Wheeler, J.A. and Ciufolini, I., 1995, Gravitation and Inertia,Princeton, N.J.: Princeton U. Press.

Notable Philosophical Discussions of the Absolute-RelativeDebates

Barbour, J. B., 1982, “Relational Concepts of Space and Time,”British Journal for the Philosophy of Science, 33: 251-274.Belot, G., 2000, “Geometry and Motion,” British Journal for thePhilosophy of Science, 51: 561-595.Butterfield, J., 1984, “Relationism and Possible Worlds,” BritishJournal for the Philosophy of Science, 35: 101-112.



Journal for the Philosophy of Science, 35: 101-112.Callender, C., 2002, “Philosophy of Space-Time Physics,” in TheBlackwell Guide to the Philosophy of Science, P. Machamer (ed.),Cambridge: Blackwell. 173-198.Carrier, M., 1992, “Kant's Relational Theory of Absolute Space,”Kant Studien, 83: 399-416.Dieks, D., 2001, “Space-Time Relationism in Newtonian andRelativistic Physics,” International Studies in the Philosophy ofScience, 15: 5-17.Disalle, R., 1995, “Spacetime Theory as Physical Geometry,”Erkenntnis, 42: 317-337.Earman, J., 1986, “Why Space is Not a Substance (at Least Not toFirst Degree),” Pacific Philosophical Quarterly, 67: 225-244.–––, 1970, “Who's Afraid of Absolute Space?,” Australasian Journalof Philosophy, 48: 287-319.Earman, J. and J. Norton, 1987, “What Price SpacetimeSubstantivalism: The Hole Story,” British Journal for the Philosophyof Science, 38: 515-525.Hoefer, C., 2000, “Kant's Hands and Earman's Pions: ChiralityArguments for Substantival Space,” International Studies in thePhilosophy of Science, 14: 237-256.–––, 1998, “Absolute Versus Relational Spacetime: For Better OrWorse, the Debate Goes on,” British Journal for the Philosophy ofScience, 49: 451-467.–––, 1996, “The Metaphysics of Space-Time Substantialism,”Journal of Philosophy, 93: 5-27.Huggett, N., 2000, “Reflections on Parity Nonconservation,”Philosophy of Science, 67: 219-241.Le Poidevin, R., 2004, “Space, Supervenience and Substantivalism,”Analysis, 64: 191-198.Malament, D., 1985, “Discussion: A Modest Remark aboutReichenbach, Rotation, and General Relativity,” Philosophy of



Reichenbach, Rotation, and General Relativity,” Philosophy ofScience, 52: 615-620.Maudlin, T., 1993, “Buckets of Water and Waves of Space: WhySpace-Time is Probably a Substance,” Philosophy of Science, 60:183-203.–––, 1990, “Substances and Space-Time: What Aristotle would haveSaid to Einstein,” Studies in History and Philosophy of Science, 531-561.Mundy, B., 1992, “Space-Time and Isomorphism,” Proceedings ofthe Biennial Meetings of the Philosophy of Science Association, 1:515-527.–––, 1983, “Relational Theories of Euclidean Space and MinkowskiSpace-Time,” Philosophy of Science, 50: 205-226.Nerlich, G., 2003, “Space-Time Substantivalism,” in The OxfordHandbook of Metaphysics, M. J. Loux (ed.), Oxford: Oxford UnivPr. 281-314.–––, 1996, “What Spacetime Explains,” Philosophical Quarterly, 46:127-131.–––, 1994, What Spacetime Explains: Metaphysical Essays on Spaceand Time, New York: Cambridge Univ Pr.–––, 1973, “Hands, Knees, and Absolute Space,” Journal ofPhilosophy, 70: 337-351.Rynasiewicz, R., 2000, “On the Distinction between Absolute andRelative Motion,” Philosophy of Science, 67: 70-93.–––, 1996, “Absolute Versus Relational Space-Time: An OutmodedDebate?,” Journal of Philosophy, 93: 279-306.Teller, P., 1991, “Substance, Relations, and Arguments about theNature of Space-Time,” Philosophical Review, 363-397.Torretti, R., 2000, “Spacetime Models for the World,” Studies inHistory and Philosophy of Modern Physics, 31B: 171-186.



Other Internet Resources

St. Andrews School of Mathematics and Statistics Index ofBiographiesThe Pittsburgh Phil-Sci Archive of pre-publication articles inphilosophy of scienceNed Wright's Special Relativity tutorialAndrew Hamilton's Special Relativity pagesUsenet Relativity FAQ (SR, GTR and Cosmology)

Related Entries

Descartes, René: physics | general relativity: early philosophicalinterpretations of | Newton, Isaac: views on space, time, and motion |space and time: inertial frames | space and time: the hole argument | Zenoof Elea: Zeno's paradoxes

Notes to Absolute and Relational Theories of Spaceand Motion

1. See Malament (1995), and Norton (1993).] But Mach did not offer thisequation as a proposed law valid in any circumstances; he avers, "It isimpossible to say whether the new expression would still represent thetrue condition of things if the stars were to perform rapid movementsamong one another." (p. 289)

2. Norton (1995) offers an extensive discussion of the two interpretationspossible for Mach's discussion of inertia, arguing that on balance it seemsmost likely that Mach advocated only a reformulation of Newtonianmechanics — i.e., what we are calling "Mach-lite".

3. Many authors have differentiated the Galilean relativity principle fromthe Special Principle of Relativity, on the grounds that the meaning of



the Special Principle of Relativity, on the grounds that the meaning of"frame of reference" changes with the introduction of the latter theory.The Special Principle is thus sometimes equated with the assertion of theLorentz covariance of physical laws. But Brown (2005) convincinglyargues that the Special Principle properly speaking — the principleintroduced by Einstein and used in the derivation of the Lorentztransformations — is in fact exactly the same as Galileo's.

4. Special relativity applies the Galilean relativity principle toelectromagnetic theory. Since the speed of light is determined by basicequations of that theory, if the relativity principle is to hold, we canconclude that the speed of light must be the same for observers in anyinertial frame, regardless of the velocity of the light's source. This isprofoundly counter-intuitive, once one explores what it means. Three ofthe immediate consequences of the constancy of light's velocity are therelativity of simultaneity, length contraction (apparent shortening, in thedirection of motion, of rapidly moving objects), and time dilation(apparent slowing down of fast-moving clocks). We can explain aninstance of the first phenomenon here, using an example first crafted byEinstein. Suppose a fast-moving train is passing a signal light on theground. Just as the center of the car you are in passes the lamp, it emits aflash of red light. Since you know the speed of light is a fixed constant cin inertial frames, and you are in an inertial frame, you conclude that theflash of light arrives simultaneously at the back and the front of your traincar. The two arrival events, for you, are simultaneous. But now considerhow an observer standing next to the signal lamp on the ground viewsevents. The light travels at the same speed c toward the back of the trainand toward the front. But the train is moving, fast, forward. So the lightwill reach the rear of your car first, and then the forward-going light rays,having to catch up to the rapidly advancing front, will arrive some timelater. That is: for the observer on the ground, the two events (light arrivesat back of car; light arrives at front) are definitely not simultaneous. Andif the Special Principle of relativity is to be taken seriously, we must



if the Special Principle of relativity is to be taken seriously, we mustadmit that neither observer's perspective is “correct”; judgment that twoevents in different locations are “simultaneous” is simply relative –relative to state of motion, or reference frame. (The other two surprisingeffects just mentioned can be deduced from similarly simple thought-experiments, but we will not do so here.) For a complete introduction tospecial relativity, the reader may wish to look at one of the followingreliable web-based tutorials:

Special Relativity (Andrew Hamilton)Relativity Tutorial (Ned Wright)Usenet Relativity FAQ

5. This is perhaps an unfair description of the later theories of Lorentz,which were exceedingly clever and in which most of the famous "effects"of STR (e.g., length contraction and time dilation) were predicted. SeeBrown 2005 for a sympathetic philosophical exposition of such precursorsto STR.

6. “Something like” must be understood in a fairly loose sense; GTR byitself cannot model the chemical reactions that drive a rocket, forexample, nor the solid material body of the rocket itself. What seems clearfrom studies of both existence theorems and numerical methods is that alarge number of as-yet unexplored solutions exist that display “absoluteaccelerations” (especially rotations) of a kind that Mach's Principle wasintended to rule out. See Living Reviews in Relativity for review articlescovering these issues.

Copyright © 2009 by the authors Nick Huggett and Carl Hoefer



absolute and relational theories of space and motion

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