A “frame of reference” is a standard relative to whichmotion and rest may be measured; any set of points or objects that areat rest relative to one another enables us, in principle, to describethe relative motions of bodies. A frame of reference is therefore apurely kinematical device, for the geometrical description of motionwithout regard to the masses or forces involved. A dynamical accountof motion leads to the idea of an “inertial frame,” or areference frame relative to which motions have distinguished dynamicalproperties. For that reason an inertial frame has to be understood asa spatial reference frame together with some means of measuring time,so that uniform motions can be distinguished from accelerated motions.The laws of Newtonian dynamics provide a simple definition: aninertial frame is a reference-frame with a time-scale, relative towhich the motion of a body not subject to forces is always rectilinearand uniform, accelerations are always proportional to and in thedirection of applied forces, and applied forces are always met withequal and opposite reactions. It follows that, in an inertial frame,the center of mass of a closed system of interacting bodies is alwaysat rest or in uniform motion. It also follows that any other frame ofreference moving uniformly relative to an inertial frame is also aninertial frame. For example, in Newtonian celestial mechanics, takingthe “fixed stars” as a frame of reference, we can, inprinciple, determine an (approximately) inertial frame whose center isthe center of mass of the solar system; relative to this frame, everyacceleration of every planet can be accounted for (approximately) as agravitational interaction with some other planet in accord withNewton’s laws of motion.

This appears to be a simple and straightforward concept. By inquiringmore narrowly into its origins and meaning, however, we begin tounderstand why it has been an ongoing subject of philosophicalconcern. It originated in a profound philosophical consideration ofthe principles of relativity and invariance in the context ofNewtonian mechanics. Further reflections on it, in differenttheoretical contexts, had extraordinary consequences for20^th-century theories of space and time.

• 1. Relativity and reference frames in classical mechanics
  • 1.1 The origins of Galilean relativity
  • 1.2 Philosophical controversy over absolute and relative motion
  • 1.3 Galilean relativity in Newtonian physics
  • 1.4 The lingering problem of absolute space
  • 1.5 19th-century analyses of the law of inertia
  • 1.6 The emergence of the concept of inertial frame
  • 1.7 “Quasi-inertial” frames: Newton’s Corollary VI
• 2. Inertial frames in the 20^th century: special and general relativity
  • 2.1 Inertial frames in Newtonian spacetime
  • 2.2 The conflict between Galilean relativity and modern electrodynamics
  • 2.3 Special relativity and Lorentz invariance
  • 2.4 Simultaneity and reference-frames
  • 2.5 From special relativity and Lorentz invariance to general relativity and general covariance
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Relativity and reference frames in classical mechanics

1.1 The origins of Galilean relativity

The term “reference frame” was coined in the19^th century, but it has a long prehistory, beginning,perhaps, with the emergence of the Copernican theory. The significantpoint was not the replacement of the earth by the sun as the center ofall motion in the universe, but the recognition of both the earth andthe sun as merely possible points of view from which the motions ofthe celestial bodies may be described. This implied that the basictask of Ptolemaic astronomy—to represent the planetary motionsby combinations of circular motions—could take any point to befixed, without sacrificing predictive power. Therefore, as Copernicussuggested in the opening arguments ofOn the Revolutions of theHeavenly Spheres, the choice of any particular point requiredsome justification on grounds other than mere successful astronomicalprediction. The most persuasive grounds, seemingly, were physical: wedon’t perceive the physical effects that we would expect theearth’s motion to produce. Copernicus himself noted, however, inreply, that we can indeed undergo motions that are physicallyimperceptible, as on a smoothly moving ship (1543, p.6). At least insome circumstances, we can easily treat our moving point of view as ifit were at rest.

As the basic programme of Ptolemy and Copernicus gave way to that ofearly classical mechanics as developed by Galileo, this equivalence ofpoints of view was made more precise and explicit. Galileo was unableto present a decisive argument for the motion of the earth around thesun. He demonstrated, however, that the Copernican view does notcontradict our experience of a seemingly stable earth. He accomplishedthis through a principle that, in the precise form that it takes inNewtonian mechanics, has become known as the “principle ofGalilean relativity”: mechanical experiments will have the sameresults in a system in uniform motion that they have in a system atrest. Arguments against the motion of the earth had typically appealedto experimental evidence—e.g., that a stone dropped from a towerfalls to the base of the tower, instead of being left behind as theearth rotates during its fall. But Galileo argued persuasively thatsuch experiments would happen just as they do whether the earth weremoving or not, provided that the motion is sufficiently uniform. (SeeFigure 1.) Galileo’s account of this was not precisely theprinciple that we call “Galilean relativity”; he seems tohave thought that a system in uniform circular motion, such as a frameat rest on the rotating earth, would be indistinguishable from a frametruly at rest. The principle was named in his honor because he hadgrasped the underlying idea of dynamical equivalence: he understoodthe composition of motion, and understood how individual motions ofbodies within a system—such as the fall of a stone from atower—are composed with the motion of the system as a whole.This principle of composition, combined with the idea that bodiesmaintain their uniform motion, formed the basis for the idea ofdynamically indistinguishable frames of reference.

Figure 1: Galileo’s Argument Ifthe earth is rotating sufficiently uniformly, a stone dropped from thetower will fall straight to the base, just as a stone dropped from themast of a uniformly moving ship will fall to the foot of the mast. Inboth cases the stone’s vertical motion will be smoothly composedwith its horizontal motion. Hence a sufficiently uniform motion willbe indistinguishable from rest.

1.2 Philosophical controversy over absolute and relative motion

Leibniz, later, articulated a more general “equipollence ofhypotheses”: in any system of interacting bodies, any hypothesisthat any particular body is at rest is equivalent to any other.Therefore neither Copernicus’ nor Ptolemy’s view can betrue—though one may be judged simpler than theother—because both are merely possible hypotheticalinterpretations of the same relative motions. This principle clearlydefines (what we would call) a set of reference frames. They differ intheir arbitrary choices of a resting point or origin, but agree on therelative positions of bodies at any moment and their changing relativedistances through time.

For Leibniz and many others, this general equivalence was a matter ofphilosophical principle, founded in the metaphysical conviction thatspace itself is nothing more than an abstraction from the geometricalrelations among bodies. In some form or other it was a widely sharedtenet of the 17^th-century “mechanicalphilosophy”. Yet it was flatly incompatible with physics asLeibniz himself, and the other “mechanists,” actuallyconceived it. For the basic program of mechanical explanation dependedessentially on the concept of a privileged state of motion, asexpressed in the common assumption that was the forerunner of the“principle of inertia”: bodies maintain a state ofrectilinear motion until acted upon by an external cause. Thus theirfundamental conception of force, as the power of a body to change thestate of another, likewise depended on this notion of a privilegedstate. This dependence was clearly exhibited in the vortex theory ofplanetary motion, which presupposed that any planet would move in astraight line unless impeded. Its actual orbit was therefore explainedby the balance between the planet’s inherent centrifugaltendency (its tendency to follow the tangent to the orbit) and thepressure of the surrounding medium.

For this reason, the notion of a dispute between“relativists” or “relationists” and“absolutists” or “substantivalists”, in the17^th century, is a drastic oversimplification. Newton, inhis controversial Scholium on space, time, and motion, was not merelyasserting that motion is absolute in the face of the mechanists’relativist view. He was arguing that a conception of absolute motionwas already implicit in the views of his opponents—that it wasimplicit in their conception, which he largely shared, of physicalcause and effect. The general equivalence of reference-frames wasimplicitly denied by a physics that characterized forces as powers tochange the states of motion of bodies.

This development placed the subject of reference frames in a newtheoretical context. Having set aside the reference frame of commonsense—the frame in which the earth is at rest in the center,with the heavens revolving around it—the mechanical physics ofthis time naturally tied this subject with novel theoreticalconceptions of motion, and its physical causes and effects. Copernicusargued for a heliocentric system, not from a physical theory ofmotion, but from the comparative simplicity and reasonableness that itintroduced into astronomy; he worked within the established theory ofthe causes of celestial motions, namely, the revolutions of theheavenly spheres. After Copernicus, however—more precisely,after the model of revolving spheres was largelyabandoned—determining the right frame of reference was connectedwith discovering the true physical causes of the planets’motions. Philosophers such as Kepler, Descartes, Huygens, Leibniz, andNewton held vastly differing views of physical causation, motion. andthe relativity of motion. They agreed, however, that the heliocentricpicture was uniquely suited for giving a causal account of planetarymotions, as the effects of physical actions originating in the sun. ToKepler and Descartes, for example, the rotation of the sun on itsaxis, in the same sense in which the planets revolved, identified itas the cause of those revolutions.

The link between the causal account of motion and the more generalconceptual account of “true” motion was never obvious orstraightforward. Descartes’s heliocentric causal account, inwhich the planets moved in vortices arising from the sun’srotation, was disconnected from his abstract account of motion“according to the truth of the matter,” or “in theproper sense” (Descartes 1642, part II section XXV). Since thereare innumerable objects to which one might refer the motion of anygiven body, the reference may appear to be a matter of arbitrarychoice. The unambiguous reference for the motion of any body, heargued, is provided by the bodies immediately touching it. Newton, inresponse, argued that Descartes’s philosophical account ofmotion is flatly incompatible with his causal account. To Newton, itwas incoherent to appeal to a causal account of motion when explainingthe centrifugal tendency of a body in its orbit around the sun, whileidentifying its “proper” motion only by its relations tothe bodies immediately contiguous to it (Newton 1684a, Stein 1967,Rynasiewicz 2014). Hence Newton’s argument that the onlyunambiguous standard of motion is a body’s change of positionwith respect to space itself. To space understood in this sense, asthe universal frame of reference with respect to which thedisplacements of bodies constitute their true motions, Newton gave thename “absolute space” (1687b, p. 5ff).

Evidently Newton was aware that “absolute space” was not areference frame in any practical sense. He emphasized that “theparts of space cannot be seen,” and that no observable bodiescan be known to be at rest. Hence there is no way to determine motionwith respect to space by direct observation; it must be known by its“properties, causes, and effects” (1687b, p. 7–8).The question arises, then, what properties, causes, or effectsindicate a body’s change of position in absolute space? It isconceivable, for example, that there might be some physical correlatefor velocity, in the sense that a body might have some observablephysical state that depends on its velocity. It follows, in that case,that a body would be in a distinct physical state when it is at restin space. If a body could be known to be in that state, therefore, itwould (in principle) provide a physical marker for a truly restingframe of reference. On Leibniz’s conception of force, forinstance, a given force is required to generate or to maintain a givenvelocity. For objects “passively” resist motion, butmaintain their states of motion only by “active”force—so that, on dynamical grounds, “every body trulydoes have a certain amount of motion, or, if you will, force”(Leibniz 1694, p. 184; see also 1716, p. 404). It would follow fromthis that there must be, in principle, a distinguished frame ofreference in which the velocities of bodies correspond to their truevelocities, i.e. to the amounts of moving force that they trulypossess. It would also follow that, with respect to any frame that isin motion relative to this one, bodies will not have their truevelocities. In short, such a conception of force, if it could beapplied physically, would give a precise physical application ofNewton’s conception of absolute space, by providing a physicalcorrelate for change of absolute place.

1.3 Galilean relativity in Newtonian physics

The difficulty with Newton’s view of absolute space comes, notfrom the epistemological arguments of relationalism, but fromNewton’s own conception of force. If force is defined andmeasured solely by the power to accelerate a body, then obviously theeffects of forces—in short, the causal interactions within asystem of bodies—will be independent of the velocity of thesystem in which they are measured. So the existence of a set ofequivalent “inertial frames” is imposed from the start byNewton’s laws. Suppose that we determine for the bodies in agiven frame of reference—say, the rest frame of the fixedstars—that all observable accelerations are proportional toforces impressed by bodies within the system, by equal and oppositeactions and reactions among those bodies. Then we know that thesephysical interactions will be the same in any frame of reference thatis in uniform rectilinear motion relative to the first one. Thereforeno Newtonian experiment will be able to determine the velocity of abody, or system of bodies, relative to absolute space. In other words,there is no way to distinguish absolute space itself from any frame ofreference that is in uniform motion relative to it. Newton thoughtthat a coherent account of force and motion requires a backgroundspace consisting of “places” that “from infinity toinfinity maintain given positions with respect to one another”(1687b, p. 8–9). But the laws of motion enable us to determinean infinity of such spaces, all in uniform rectilinear motion relativeto each other. The laws furnish no way of singling out any one as“immovable space.”

Oddly enough, no one in the 17^th century, or even beforethe late 19^th century, expressed this equivalence ofreference-frames any more clearly than Newton himself. However, thecredit for articulating this equivalence precisely for the first time,belongs to Christiaan Huygens, who introduced it as one of theHypotheses of his first work on the rules of impact (1656).“Hypothesis I” was the first clear statement of theprinciple of inertia: “Any body, once in motion, if nothingopposes it, continues to move always with the same velocity and alongthe same straight line” (1656, pp. 30–31). The firstprecise statement of the relativity principle followed as HypothesisIII:

The motion of bodies, and their speeds equal or unequal, are to beunderstood respectively, in relation to other bodies which areconsidered as at rest, even though perhaps both the former and thelatter are subject to another motion that is common to them. Inconsequence, when two bodies collide with one another, even if bothtogether undergo another equable motion, they will move each other nodifferently, with respect to a body that is carried by the same commonmotion, than if this extraneous motion were absent from all of them.(1656, p. 32).

Huygens illustrated this principle by the example of an impact thattakes place on a uniformly moving boat, asserting its equivalence tothe same impact taking place at rest. Thus he made precise theargument of Galileo, in light of his more precise understanding of theprinciple of inertia and the dynamical difference between inertial andcircular motion.

Newton’s first statement of the same principle appears in one ofthe series of papers that culminated in thePrincipia,“De motu sphæricorum corporum in fluidis” (1684b).Like Huygens, Newton presents the relativity principle as afundamental principle, “Law 3”:

The motions of bodies included in a given space are the same amongthemselves whether that space is at rest or moves uniformly in astraight line without circular motion. (1684b, p. 40r)

Newton’s first statement of the Galilean relativity principleevidently recapitulates Huygens’ version, which was probablyknown to Newton. The same may be said of “Law 4” in thismanuscript, the principle of conservation of the center of gravity:

By the mutual actions between bodies their common centre of gravitydoes not change its state of motion or rest. (ibid., p. 40r)

But, uniquely, Newton immediately went on to consider the deepertheoretical significance of these principles: they radicallyreconceptualize the problem of “true motion” in theplanetary system. First, they implied that the entire system must beseen as included in a space that may, itself, be either at rest or inuniform motion. Second, they implied that the only truly fixed pointin such a system is the center of gravity of the relevant bodies.This, too, may therefore be in uniform motion or at rest:

Moreover the whole space of the planetary heavens either rests (as iscommonly believed) or moves uniformly in a straight line, and hencethe communal centre of gravity of the planets (by Law 4) either restsor moves along with it. In both cases (by Law 3) the relative motionsof the planets are the same, and their common centre of gravity restsin relation to the whole space, and so can certainly be taken for thestill centre of the whole planetary system. (ibid., p. 47r)

Newton now realized, in short, that the dispute between theheliocentric and geocentric views of the universe had been mistakenlyframed. The proper question about “the system of theworld” was not “which body is at rest in thecenter?” but “where is the center of gravity of thesystem, and which body is closest to it?” For in a system oforbiting bodies, only their common center of gravity will beunaccelerated, and by “Law 3”, the motions of the bodiesin the system will be the same, whether its center of gravity is atrest or in uniform rectilinear motion. By explicitly asserting thedynamical equivalence of “whole spaces” that may movinguniformly or at rest, Newton made it clear that the solution to theproblem of “the system of the world” is the same withrespect to any such moving space as it is with respect to immobilespace. Thus he came as close to articulating the concept of theinertial frame as anyone before the late 19th century.

In the successive drafts of hisPrincipia, Newton graduallyclarified its conceptual structure, and in particular theframe-independent character of its concepts of motion, force, andinteraction. He arrived at the new axiomatic structure whose only lawsare the familiar “Newton’s Laws of Motion”; theprinciple of the conservation of the center of gravity, and therelativity principle, were no longer presupposed, but derived from theLaws as Corollaries IV and V:

Corollary IV: The common center of gravity of bodies does not changeits state, whether of motion or rest, by the actions of those bodiesamong themselves; therefore the common centre of gravity of all bodies(external impediments excluded) rests or moves uniformly in a straightline (1687b, p. 17).

Corollary V: When bodies are enclosed in a given space, their motionsamong themselves are the same whether the space is at rest, or whetherit is moving uniformly straight forward without circular motion(1687b, p. 19).

These principles illuminate the relationship between the theory ofabsolute space, as articulated in Newton’s Scholium to theDefinitions, and the overarching scientific problem of thePrincipia. According to Newton, “the aim for which Icomposed” the book was to show “how to gather the truemotions from their causes, effects, and apparent differences, andconversely, from the motions, true or apparent, to gather their causesand effects” (1687b, p. 11); the more specific aim of Book IIIwas “to exhibit the constitution of the system of theworld” (1687b, p. 401).

On the one hand, Corollary V, like “Law 3” in De Motu,precisely restricts what Newton’s procedure can determine aboutthe structure of the system of the world. It cannot determine anythingabout the velocity of the system as a whole; it can only determine theposition of the center of gravity of the bodies that comprise it, andthe configuration of those bodies with respect to that center. In thissense it can, in principle, decide between a Keplerian and a Tychonicinterpretation of the motions of these bodies. The system is indeedapproximately Keplerian: the sun has by far the greatest mass and istherefore little disturbed from the center of gravity by itsinteractions with the planets. The sun therefore remains very close tothe common focus of the nearly Keplerian ellipses in which the planetsorbit the sun. But by Corollary V, the actions of the bodies amongthemselves would not reveal whether their center was moving uniformlyor at rest. On the other hand, Newton recognized that motion withrespect to absolute space is unknowable. This restriction, therefore,meant that the solution to the system of the world is secure in spiteof our ignorance. The nearly-Keplerian structure of the system isknown completely independently of the system’s state of motionin absolute space.

The Galilean relativity principle thus contained, in Newton’sconception, a broader insight: that different states of uniformmotion, or different uniformly-moving frames of reference, determineonly different points of view on the same physically objectivequantities, namely force, mass, and acceleration. We can see thisinsight expressed more explicitly in the evolution of Newton’sconcept of inertia. The term had been introduced by Kepler, and playeda central role in his physical conception of planetary motion.Rejecting the Aristotelian idea that the planets are carried byrotating crystalline spheres, Kepler held that the planets have anatural tendency to rest in space—what he called their naturalinertia—and argued that they must be moved by active powers thatovercome their natural inertia. Newton, well before working onPrincipia, had based his conception of inertia on the idea ofGalileo and Huygens, that bodies tend to persist in uniform motion:Inertia, on this new conception, was a resistance to change in motion.Even so, Newton’s early understanding of inertia was essentiallypre-relativistic, because it implied a conceptual distinction betweena body’s power to resist external forces, and the power of amoving body to change the motion of another. The manuscriptDegravitatione et aequipondio fluidorum (1684a), for example, wasevidently written before Newton fully recognized the importance of therelativity principle; here Newton’s Definitions distinguish“conatus,” “impetus,” and“inertia” as conceptually separate properties:

Definition 6: Conatus (endeavor) is impeded force, or force in so faras it is resisted.

Definition 7: Impetus is force in so far as it is impressed onanother.

Definition 8: Inertia is the internal force of a body, so that itsstate may not be easily changed by an external force (1684a).

Leibniz (among others), as we saw, made a corresponding distinction:moving force, the power of a body to change the motion of another, wasdetermined by velocity. Leibniz therefore distinguished this force asan active power, fundamentally different from the passive power of aresting body to resist any change of position. Newton, in contrast, ashe developed thePrincipia, and recognized the existence of aclass of indistinguishable relative spaces, gradually came tounderstand the “force of inertia” as what we would call aGalilei-invariant quantity. Impetus and resistance were thereforerecognized as appearances of that invariant quantity in differentframes of reference:

A body truly exerts this force only in a change of its state broughtabout by another force impressed upon it, and the exercise of thisforce is, under different aspects, both resistance and impetus:resistance in so far as the body, to maintain its state, opposes theimpressed force; impetus insofar as the same body, yielding only withdifficulty to the force of a resisting obstacle, strives to change thestate of that obstacle. Resistance is commonly attributed to restingbodies and impetus to moving bodies; but motion and rest, as commonlyunderstood, are only relatively distinguished from each other; andbodies commonly seen as resting are not always truly at rest (1687b,p. 2).

There are two noteworthy points about this explication of inertia.First, it shows that Newton recognized properties that were commonlyregarded as distinct (e.g. in the Leibnizian distinction betweenpassive and active) as merely frame-dependent representations of thesame fundamental property. That is, they represent the same invariantquantity seen from different points of view. The principle that a bodyexerts this force “only in a change of its state”decisively separates Newton’s new view from the older notion ofa specific power that is required to maintain a body in motion. Thischange has been noted by modern commentators (see Herivel 1965, p. 26;see also DiSalle 2013, p. 453; Disalle 2017, in Other InternetResources). But it was already noted in Newton’s own time byGeorge Berkeley, who emphasized the contrast between Newton’sconception and that of Leibniz:

Leibniz confuses impetus with motion. According to Newton, impetus isin truth the same as the force of inertia… (Berkeley 1721[1992], p. 79)

…it is established by experience that it is a primary law ofnature that a body persists “in a state of motion or of rest aslong as nothing happens from elsewhere to change that state,”and therefore it is inferred that the force of inertia is underdifferent aspects either resistance or impetus, in this sense abody can indeed be said to be by nature indifferent to motion andrest. (Berkeley, 1721 [1992], p. 96)

Berkeley thus made clear that the older understanding of inertia,unlike that expressed in thePrincipia, did not respect theprinciple of relativity. Second, Newton’s explication implicitlyinvokes all three laws of motion (cf. Stein 2002). Newton’sfirst law alone came to be identified as “the principle ofinertia.” Newton himself, however, understood that inertia hasthree inseparable aspects: the tendency to persist in motion, theresistance to change in motion, and the power to react against animpressed force. All are essential to the explication of inertial massas a measurable theoretical quantity. To many later commentators,Newton’s use of the phrase “force of inertia”suggested a conceptual confusion. On the contrary, it wasNewton’s way of drawing attention to the precise role ofinertial mass as an invariant quantity in physical interactions,underlying the various ways in which its manifestations had beenpreviously conceived.

1.4 The lingering problem of absolute space

Newton understood the Galilean principle of relativity with a degreeof depth and clarity that eluded most of his “relativist”contemporaries and critics. It may seem bizarre, therefore, that thenotion of inertial frame did not emerge until more than a century anda half after his death. He had identified a distinguished class ofdynamically equivalent “relative spaces,” in any of whichtrue forces and masses, accelerations and rotations, would have thesame objectively measured values. Yet these spaces, though dynamicallyequivalent and empirically indistinguishable, were yet not equivalentin principle. Evidently, Newton conceived them as moving with variousvelocities in absolute space, though those velocities could not beknown. Why should not he, or someone, have recognized the equivalenceof these spaces, and the dispensability of a distinguished restingspace—“absolute” space—immediately?

This is not the place for an adequate answer to this question, ifindeed one is possible. For much of the 20^th century, theaccepted answer was that of Ernst Mach: Newton lived in an age“deficient in epistemological critique.” He was thereforeunable to draw the conclusion that thesedynamicallyindistinguishable spaces must be equivalent ineverymeaningful sense, so that no one of them deserves even in principle tobe designated as “absolute space.” Yet even those whom the20^th century credited with more sophisticatedepistemological views, such as Leibniz, evidently had difficultiesunderstanding force and inertia in a Galilei-invariant way, despite aphilosophical commitment to relativity. We may plausibly suppose thatit was difficult to abandon the intuitive association of force ormotion with velocity in space. It must also have been difficult, inthe mathematical context of Newton’s time, to conceive of anequivalence-class structure as the fundamental spatio-temporalframework. It required a level of abstraction that became possibleonly with the extraordinary development of mathematics, especially ofa more abstract view of geometry, that took place in the19^th century. Newton’s arguments established, for theassumptions of classical dynamics, the need for a dynamical space-timestructure beyond the kinematical structure required to representchanges of relative position over time. But absolute space, with itssuperfluous elements, was the only such structure imagined for thenext two centuries. It was accepted as the only realistic alternativeto theories with no dynamical structure at all. There was as yet nonotion of a structure that expressedall and only what wasrequired by the dynamical laws. Euler, for example, in a penetratingcritique of Leibnizian relationalism (1748), argued that the laws ofmotion require a notion of sameness of direction in space, and ofuniform motion with respect to time. The truth of the laws ofmotion—which, for Euler, were more securely established than anyprinciple of metaphysics—could not, therefore, be reconciledwith any account of space and time as merely ideal. But he did not seethe possibility of separating true acceleration and rotation from truevelocity with respect to absolute space.

In the 17^th century, only Huygens came close to expressingsuch a view; he held that not velocity, but velocity-difference, wasthe fundamental dynamical quantity. He therefore understood, forexample, that the “absoluteness” of rotation had nothingto do with velocity relative to absolute space. Instead, it arose fromthe difference of velocity among different parts of a rotating body.If a disk is translated through space without rotation, then its partsmove in parallel lines, but if it is rotating, then they move indifferent directions, even though they are at rest relative to oneanother on account of the bonds holding them together. Thedifferences, evidently, would be the same irrespective of the velocityof the body as a whole in absolute space. Unfortunately, Huygensexpressed this view only in manuscripts that remained unpublished fortwo centuries. (Cf. Stein, 1977, pp. 9–10 and Appendix III.)Huygens also reflected on the possibility of replacing absolute spacewith (what we would call) empirical frames of reference, again inunpublished notes that have only been brought to light in recent workby Stan (2016). But the complete concept of the inertial frame emergedonly in the late 19^th century, when it did not seem to beof any great immediate importance (see below). Indeed, even after theconcept of inertial frame had been widely discussed, the notionpersisted that true rotation could only be understood as rotation withrespect to absolute space. Poincaré, for example, convinced ofthe essential “relativity of space” as well as therelativity of motion, considered the concept of absolute space to besomething of a philosophical embarrassment. But it was not clear tohim how the dynamical phenomena of rotation could be understoodwithout it (cf. DiSalle 2014). So the failure of Newton and Huygens toformulate the concept of inertial frame, two centuries earlier, seemsless remarkable than the progress that each of them made inunderstanding the relativity of motion. As we will see, articulatingthis concept involved synthesizing (in effect) insights of Newton,Huygens, and Euler.

1.5 19th-century analyses of the law of inertia

The development of this concept began with a renewed critical analysisof the notion of absolute space, for reasons not anticipated byNewton’s contemporary critics. Its starting point was a criticalquestion about the law of inertia: relative to what is the motion of afree particle uniform and rectilinear? If the answer is“absolute space,” then the law would appear to besomething other than an empirical claim, for no one can observe thetrajectory of a particle relative to absolute space. Two quitedifferent answers to the question were offered in 1870, in the form ofrevised statements of the law of inertia. Carl Neumann proposed thatwhen we state the law, we must suppose that there is a body somewherein the universe—the “body Alpha”—with respectto which the motion of a free particle is rectilinear, and that thereis a time-scale somewhere relative to which it is uniform (Neumann1870). Ernst Mach (1883) claimed that the law of inertia, andNewton’s laws generally, implicitly appeal to the fixed stars asa spatial reference-frame, and to the rotation of the earth as atime-scale. At any rate, he maintained, such is the basis for anygenuine empirical content that the laws can claim. The notion ofabsolute space, it followed, was only an unwarranted abstraction fromthe practice of measuring motions relative to the fixed stars.

Mach’s proposal had the advantage of a clear empiricalmotivation; Neumann’s “body Alpha” seemed no lessmysterious than absolute space, and almost sounds comical to themodern reader. But Neumann’s discussion of a time-scale wassomewhat more fruitful, and employed the principle that Euler hadalready expressed (1748): the law of inertia defines a time-scale, bywhich equal intervals of time are those in which a free particletravels equal distances. He also noted, however, that this definitionis quite arbitrary. For, in the absence of a prior definition of equaltimes, any motion whatever can be stipulated to be uniform. It is nohelp to appeal to the requirement of freedom from external forces,since the free particles presumably are known to us only by theiruniform motion. We have a genuine empirical claim only when we stateofat least two free particles that their motions aremutually proportional. Equal intervals of time can then bedefined as those in which two free particles travel mutuallyproportional distances.

1.6 The emergence of the concept of inertial frame

Neumann’s definition of a time-scale directly inspired LudwigLange’s conception of “inertial system” (Lange1885). An inertial coordinate system ought to be one in which freeparticles move in straight lines. But any trajectory may be stipulatedto be rectilinear, and a coordinate system can always be constructedin which it is rectilinear. And so, as in the case of the time-scale,we cannot adequately define an inertial system by the motion of oneparticle. Indeed, for any two particles moving anyhow, a coordinatesystem may be found in which both their trajectories are rectilinear.So far the claim that either particle, or some third particle, ismoving in a straight line may be said to be a matter of convention. Wemust define an inertial system as one in which at least three freeparticles move in straight lines. Then we can state the law of inertiaas the claim that, relative to an inertial system so defined, themotion of any fourth particle, or arbitrarily many particles, will berectilinear. The notions of inertial system and Neumann’stime-scale, which Lange called an “inertial time-scale,”may be combined as follows: relative to a coordinate system in whichthree free particles move in straight linesand travelmutually-proportional distances, the motion of any fourth freeparticle will be rectilinear and uniform. The questionable Newtonianconcepts of absolute rotation and acceleration, Lange proposed, couldnow be replaced by the concepts of “inertial rotation” and“inertial acceleration,” i.e. rotation and accelerationrelative to an inertial system and inertial time-scale. See Figures 2and 3.

(a)	(b)

Figure 2: Neumann’s Time-Scale ByNewton’s first law, a particle not subject to forces travelsequal distances in equal times. But which particles are free offorces? This might appear to be a matter of convention.
(a) Either \(P_1\) or \(P_2\) can be arbitrarily stipulated to be atthe origin of a system of coordinates, and to serve as the measure ofequal times
(b) But one can say of two particles with different velocities: inintervals of time in which one moves a given distance \(d_1\), theother moves a proportional distance \(d_2 = kd_1\) (wherek isa constant; i.e., \(d_1/d_2 = k\)). Or one can compare a particle to afreely rotating planet: in intervals of time through which the planetrotates through equal angles, the particle moves equal distances.

Figure 3: Lange’s Definition of‘inertial system’ (1885) An inertial system is acoordinate system with respect to which three free particles,projected from a single point and moving in non-coplanar directions,move in straight lines and travel mutually-proportional distances. Thelaw of inertia then states that relative to any inertial system, anyfourth free particle will move uniformly.

At about the same time, apparently unaware of the work of Mach,Neumann, and Lange, James Thomson—older brother of WilliamThomson, Lord Kelvin—expressed the content of the law ofinertia, and the appropriate frame of reference and time-scale(“dial-traveller”), in a somewhat simpler manner:

For any set of bodies acted on each by any force, a REFERENCE FRAMEand a REFERENCE DIAL-TRAVELLER are kinematically possible, such thatrelatively to them conjointly, the motion of the mass-centre of eachbody, undergoes change simultaneously with any infinitely shortelement of the dial-traveller progress, or with any element duringwhich the force on the body does not alter in direction nor inmagnitude, which change is proportional to the intensity of the forceacting on that body, and to the simultaneous progress of thedial-traveller, and is made in the direction of the force. (Thomson1884, p. 387)

Thomson did not reject the term “absolute rotation”.Instead, he held that it is properly defined as rotation relative to aframe that satisfies his definition of a reference frame. A body thatis rotating with respect to a reference frame (and dial-traveller) isrotating with respect to any other frame in uniform motion relative tothe first. The definition does not express, as Lange’s does, thedegree of arbitrariness involved in the construction of an inertialsystem by means of free particles. By dispensing with the idealizationof free particles, Thomson’s definition aims to characterize aninertial frame for an actual system of interacting bodies. However, itdoes not quite fulfill its aim. Like Lange’s definition, itleaves out a crucial condition for an inertial system as we understandit: all forces must belong to action-reaction pairs. Otherwise wecould have, as on a rotating sphere, merely apparent (centrifugal)forces that are, by definition, proportional to mass and acceleration,and so the rotating sphere would satisfy Thomson’s definition.Therefore the definition needs to be completed by the stipulation thatto every action there is an equal and opposite reaction. (Thiscompletion was actually proposed by R.F. Muirhead in 1887.)

But, so completed, Thomson’s definition makes the essentialpoint about the relation between Newton’s laws of motion and theinertial frames: that the laws assert the existence of at least oneinertial frame. If one inertial frame is posited, in whichaccelerations properly correspond to Newtonian forces, then any otherinertial frame is in uniform motion with respect to the first; theforces, masses, and accelerations measured in one will have the samemeasures in any other. Any one may be arbitrarily designated as anall-encompassing “immobile space” in which all others aremoving uniformly. Thus the original question, “relative to whatframe of reference do the laws of motion hold?” is revealed tobe wrongly posed. For the laws of motion essentiallydetermine a class of reference frames, and (in principle) aprocedure for constructing them. For the same reason, a skepticalquestion that is still commonly asked about the laws ofmotion—why is it that the laws are true only relative to acertain choice of reference frame?—is also wrongly posed. IfNewton’s laws are true, then we can construct an inertial frame;their truth doesn’t depend on our ability to construct such aframe in advance. Mach expressed the situation particularlyclearly:

It is very much the same whether we refer the laws of motion toabsolute space, or express them abstractly, without express indicationof the system of reference. The latter course is unproblematic andpractical, for in treating particular cases the student of mechanicslooks for a suitable system of reference. But owing to the fact thatthe first way, whenever there was any actual issue at stake, wasnearly always interpreted as having the same meaning as the latter,Newton’s error was much less dangerous than it would otherwisehave been. (1933, p. 269.)

Mach’s remark roughly corresponds to Newton’s actualprocedure. Even though, for Newton, absolute space was the implicitreference-frame for stating the laws of motion, the frame for theirapplication was the standard one for most of the history of astronomy:the fixed stars. This seemingly arbitrary starting-point did notundermine Newton’s procedure as an account of the “truemotions.” For the framework of the fixed stars, initially takenfor granted, turns out to be justified in the course of Newton’sdynamical analysis. If all the accelerations relative to the fixedstars can be analyzed into action-reaction pairs involving bodieswithin the system—leaving no “leftover”accelerations that need to be traced to some yet-unknowninfluence—then we can conclude that the stars are a suitable(sufficiently inertial) frame of reference after all. Newton couldappeal to a particular case to test this general point: the orbits ofthe outer planets were stable with respect to the fixed stars, theirperihelia showing no measurable precession (unlike the perihelion ofMercury, for a famous example). Newton argued, then, that a relativespace in which these apsides are stable is a sufficient approximationto a space at rest or in uniform motion (cf. Book III, PropositionXIV, 1687b, p. 420).

Mach took particular notice of Newton’s use of the relativityprinciple in determining an appropriate reference-frame:

In order to have a generally valid system of reference, Newtonintroduced Corollary V of thePrincipia. He thoughtof… a coordinate system for which the law of inertia holds,fixed in space without rotation relative to the fixed stars. He couldalso allow an arbitrary origin and uniform translation of thissystem… without losing its usefulness… One can see thatthe reduction to absolute space was in no way necessary, since thereference frame is just as relatively determined as in any other case.(1933, p. 227.)

For Mach, this was an important acknowledgement of Newton’sinsight into the relativity of motion. Starting from Corollary V, theconcept of the inertial frame solved the problem of absolute rotationand acceleration, as a problem internal to the system ofNewton’s laws. Absolute space could be dispensed with, withoutundermining Newton’s dynamical distinctions among states ofmotion. Of course, this point did not dismiss Mach’s skepticalquestions regarding the laws themselves. Instead, it posed them in amore precise form: Are Newton’s laws truly general laws ofnature, that determine a class of privileged frames? Or do they onlydescribe motions relative to a particular material frame, the fixedstars? Empirical evidence was insufficient to decide. EventuallyMach’s question helped to motivate Einstein’s search fornew laws in which privileged frames would not play an essential role.First, however, by comparing Newtonian mechanics with Maxwell’selectrodynamics, Einstein placed the notion of inertial frame on anentirely new footing (see below,Section 2.2et seq.).

1.7 “Quasi-inertial” frames: Newton’s Corollary VI

A striking aspect of Newton’s treatment of indistinguishableframes of reference was his discovery of approximatelyindistinguishable frames: spaces that are accelerating, yet can betreated, for practical purposes, as if they were at rest or in uniformmotion. Newton made this notion precise in Corollary VI to the laws ofmotion:

If bodies are moved in any way among themselves, and are urged byequal accelerative forces along parallel lines, they will all continueto move among themselves in the same way as if they were not acted onby those forces. (1687b, p. 20.)

As Corollaries IV and V implied, for a given system of interactingbodies, their center of gravity is unmoved by the actions of thebodies among themselves, and will remain at rest or in uniform motionas long as the bodies are not disturbed by any external forces. This,as we noted, was as close as Newton could come to the notion of aninertial frame. Corollary VI shows that, in very special idealcircumstances—accelerative forces that act equally on all bodieswithin a system, and accelerate them all in paralleldirections—an accelerating system of bodies will behave,internally, as if there were no external forces acting on it at all.Yet Newton’s discovery was not limited to the point madeexplicitly in Corollary VI. Rather, it was threefold. The second pointwas that there was in fact a force acting equally and in parallellines, at least to a high approximation, on important systems ofcelestial bodies. The system of Jupiter and its satellites, forexample, is obviously accelerating, as its center of gravity completesan approximately elliptical orbit around the Sun bound by accelerativeforces toward the Sun’s center. But because the accelerations ofall the bodies are nearly equal and parallel, their motions amongthemselves are nearly the same as if no such forces acted, and thesystem may be treated as the sort of system described in Corollary V.Evidently the accelerations are unequal, since Jupiter and thesatellites are at varying distances from the sun, and they cannot beparallel since they are all directed at the center of the sun. Butthese differences of distance and direction are so small, incomparison with the distance of the entire system from the sun, thatthey may be neglected. And the same applies to the centripetalacceleration of Saturn’s system.

Newton applied this same reasoning to the entire solar system: even ifthe entire system were accelerating toward some unknown gravitationalsource, he could treat the solar system itself as if it were anisolated system. He argued, from the analysis of accelerations withinthe system, that outside forces must be acting more or less equallyand in parallel directions on all parts of the system.

It may be imagined that the sun and planets are impelled by some otherforce equally and in the direction of parallel lines; but such a force(by Cor. VI of the Laws of Motion) would not change the situation ofthe planets among themselves, nor would produce any sensible effect;but we are concerned with the causes of sensible effects. Let us,therefore, neglect every such force as precarious, and of no bearingon the phenomena of the heavens…. (1687a, article 13.)

Newton raised this point in order to show that the possibility of sucha force acting on the whole solar system would not affect hiscalculations of the forces acting within the system. In thecalculation relevant to this passage, Newton was using Corollary VI todefend the inference that the force responsible for Jupiter’sorbit is directed to the Sun rather than to the Earth: neglecting anysuch imaginary force, “then all the remaining force bywhich… Jupiter is urged will tend (by prop. 3, corol. 1) towardthe center of the sun” (ibid). This calculation formed animportant step in the argument for a heliocentric system. Such a useof Corollary VI parallels, therefore, his use of Corollary V (and itsearlier form, “Law 3”) to show that the “frame ofthe system of the world” could be determined without regard tothe system’s uniform motion in absolute space.

Indeed, the analogy between the two kinds of case helps to explainNewton’s changing the relativity principle from a law to aCorollary, for this coincides historically with his first use ofCorollary VI. The two Corollaries identify two classes of frames ofreference that may be treated as equivalent, because they involve,respectively, theoretically and practically indistinguishable statesof motion. The frames corresponding to Corollary VI may be called“quasi-inertial,” because “approximatelyinertial” might be misleading: a closed orbit around theSun—like that of Jupiter’s system—is not a goodapproximation to an inertial motion, and the system can hardly beconsidered isolated. But over sufficiently limited segments of itsorbit, its motion is sufficiently close to being inertial.Moreover—and most important—the accelerative forces towardthe sun are close enough to being equal and parallel that the forcesacting within the system can be effectively isolated from the forcesfrom without. Hence, while “quasi-inertial” is a usefulterm for the reference-frame corresponding to such a group of bodies,a useful description for the group itself is George Smith’s“quasi-insular system” (Smith 2019). A system of massesbound in orbit around a larger mass is by no means isolated, yet inthe right conditions it may be treated as if it were. The modern term,“local inertial frame,” is not inappropriate (cf. Schutz1990, p. 124). But it typically designates the local coordinate frameof a given inertial observer, rather than the sort of “wholespace” that Newton had in mind, as encompassing a celestialsystem as large as that of Jupiter, or the solar system as a whole.Moreover, it is typically used in a context in which a global inertialframe, with respect to which any such Newtonian system has a definiteacceleration, would not be assumed to exist.

This last point leads to the third point of Newton’s discovery:that the “quasi-inertial” system is part of a mathematicalframework for approximative reasoning, to determine the precise degreeof isolation that a group of interacting bodies may be said topossess. Proposition III of thePrincipia establishedNewton’s method for treating a body orbiting a second body whichis, itself, subject to a centripetal force:

Proposition III, Theorem III: Every body, that, by a radius drawn tothe centre of another body, in any way moved, describes areas aboutthat centre proportional to the times, is urged by a force compoundedout of the centripetal force tending to that other body, and of allthe accelerative force by which that other body is impelled.(1687b, p.39).

In other words, if an orbiting body obeys Kepler’s area law,then any accelerative force acting on the central body is simply addedto the centripetal force that maintains the orbiting body in itsorbit.

This principle of composition formed the mathematical basis forNewton’s treatment of quasi-inertial frames. When a system oflesser bodies is, as a whole, revolving around a greater body, we havea geometrical framework to describe how closely the motions of thelesser system approximate the conditions of Corollary VI:

Book I, Proposition LXV, Case 2: Suppose that the accelerativeattractions towards the greater body to be among themselvesreciprocally as the squares of the distances; and then, by increasingthe distance of the great body till the differences of the straightlines drawn from that to the others in respect of their length, andthe inclinations of those lines to each other, are less than anygiven, then the motions of the parts of the system, will continue withno errors except such as are less than any given. And because, by thesmall distance of those parts from each other, the whole system isattracted as if it were only one body, it will therefore be moved bythis attraction as if it were one body….(1687b, p. 172.)

Thus the situation described by Corollary VI, in Newton’sanalysis, emerges as a limiting case of an orbiting system under aninverse-square force. As the size of the orbit is arbitrarilyincreased, the accelerations toward the center becomeindistinguishable from equal and parallel accelerations. Evidently,Newton’s proposition provides, characteristically, a generalmethod for treating a variety of possible configurations. But itenabled Newton to address the specific physical fact of the variationin the Sun’s gravity, and its consequences for the superpositionupon it of lesser gravitating systems. At the distance of Jupiter orSaturn, a revolving system can be a very nearly regular Kepleriansystem. As the distance to the Sun decreases, however, differences inmagnitude and direction of the accelerations become significant, andat the distance of the Earth-Moon system the motions become nearlyintractable. The decisive factor is the proportion between the size ofthe orbiting system and its distance from the center of attraction.

The actual existence of quasi-inertial frames, corresponding to theabstract cases of Proposition LXV, was a crucial part ofNewton’s argument for universal gravitation—moreprecisely, that the force holding the planets and their satellites intheir respective orbits is, in fact, the same force as gravity. Onecrucial ground for the identification was the fact that theinterplanetary force shares the most striking feature of terrestrialgravity, namely, that it imparts the same acceleration to allterrestrial bodies. This principle was discovered by Galileo, ofcourse, but Newton tested it more severely, and with a greater varietyof test bodies. He constructed pendulums of identical wooden boxessuspended from strings of equal length, which he filled with differentmaterials; he found that these differences made no difference to thespeed of falling over many oscillations of the pendulums. By thismeans he showed that Galileo’s principle holds to a much higherdegree of accuracy than Galileo was able to show, and inferred that abody’s weight toward the earth is generally proportional to itsmass. (1687b, Book III, Proposition VI). But Newton extended thisprinciple beyond terrestrial gravity, to the accelerative forcesacting on the planets and their satellites. Proposition IV, CorollaryVI, from Book I, showed that an orbiting body that obeysKepler’s third law is urged toward the center by aninverse-square force. Newton could then show that the centripetalforces acting on Jupiter’s moons depend only on theinverse-square of the distance towards Jupiter’s center:

since the satellites of Jupiter perform their revolutions in timesthat observe the sesquiplicate proportion of their distances fromJupiter’s center, their accelerative gravities towards Jupiterwill be inversely as the squares of their distances fromJupiter’s centre; that is, equal, at equal distances….And by the same argument, if the circumsolar planets were let fall atequal distances from the Sun, they would, in their descent towards theSun, describe equal spaces in equal times. But forces that equallyaccelerate unequal bodies must be as those bodies; that is to say, theweights of the planets towards the Sun must be as their quantities ofmatter. (Ibid.)

In each of these cases, that is, Newton found that the centripetalacceleration behaves like gravitational acceleration, and so thebodies’ forces toward their respective centers are, essentially,their weights toward those centers. Moreover, the orbits ofJupiter’s moons provided a completely novel test ofGalileo’s principle, on an extremely large scales of mass anddistance. For he showed that Jupiter and its moons—within thelimits of observational accuracy—undergo the same accelerationstoward the sun (cf. 1687b, Book I, Proposition 65; Book III,Proposition VI). Any non-negligible difference in these accelerationswould produce corresponding irregularities in the satellites’orbits.

The proportionality of weight to mass was understood in its broaderfoundational significance, as the equivalence of gravitational andinertial mass, through Einstein’s “principle ofequivalence” (cf. Einstein 1916; see also Norton 1985). InEinstein’s reasoning, the identity of inertia and gravitationhelped to undermine the special status of inertial motion, andsuggested the extension of the relativity principle from inertialframes to frames in any state of motion whatever. If an inertialframe, K, cannot be distinguished from another frame K′ that isuniformly accelerated with respect to K, then K′ may equally betreated as a “privileged” or “stationary”frame: “they have equal title as systems of reference for thedescription of physical phenomena” (Einstein 1916, p. 114). Thiscircumstance undermines a defining characteristic of inertial frames:that with respect to a given inertial frame, every other inertialframe is in uniform rectilinear motion. Corollary VI points the way,after all, toward an extended relativity principle.

The assumption of the complete physical equivalence of the systems ofco-ordinates, K and K′, we call the “principle ofequivalence;” this principle is evidently intimately connectedwith the theorem of the equality between the inert and thegravitational mass, and signifies an extension of the principle ofrelativity to co-ordinate systems which are in non-uniform motionrelatively to each other. In fact, through this conception we arriveat the unity of the nature of inertia and gravitation. (Einstein1922).

This reasoning, in turn, suggested the connection between thegravitational field and the curvature of space-time. (See Einstein1916; see also Related Entries:Einstein, Albert: philosophy of science |general relativity: early philosophical interpretations of).

Even independently of Einstein’s theory, however, one might seein hindsight that Newton’s application of Corollary VI, and hisidentification of what we’ve been calling quasi-inertial frames,already undermined the idea of an inertial frame. As we saw, Newtoncould treat a quasi-inertial system like that of Jupiter and itsmoons, not as essentially different from the system of the Earth andthe Moon, but as a limiting case of such a system: as the whole systembecomes sufficiently far from the central body, the differences amongthe accelerations of its parts toward their common center becomenegligibly small. In other words, Newton could treat such systems, andthe mathematical description of the differences among them, asrevealing the structure and workings of the sun’s gravity,rather than as questioning the fundamental distinction between uniformand accelerated motion. In that sense, he was not proposing the“complete physical equivalence” of uniformly acceleratingsystems with uniformly moving systems (inertial frames). But evenNewton recognized, as we also saw, that the solar system as a wholemight have an unknown and practically unknowable acceleration. Ineffect, he explained why his analysis of accelerations of the bodiesin the system, among themselves, required no knowledge of any absoluteaccelerations. In the 19th century, Maxwell, without questioning theunderlying framework of absolute space and time, pointed out thatCorollary VI implied a kind of relativity of acceleration (1878, pp.51–52). Viewing the foregoing in hindsight, more recentliterature has suggested that the physics of the Principia did notreally require the notion of inertial frame at all, and therefore thata weaker geometry than Newtonian spacetime would be a sufficientbackground structure for Newton’s dynamical reasoning (cf.Saunders, 2013). It is further suggested, following Cartan (1923,1924), that the space-time structure and the gravitational field beunified in a curved space-time (cf. section 9, below), as a Newtonianversion of general relativity (cf. Malament 2012, chapter 4; see alsoKnox, 2014 and Weatherall 2018). The relevant conceptual resources andmathematical techniques for such approaches, evidently, developed onlyin the aftermath of general relativity. We return to this theme in2.5.

2. Inertial frames in the 20^th century: special and general relativity

2.1 Inertial frames in Newtonian spacetime

By the early years of the 20^th century, the notion ofinertial system seems to have been widely accepted as the basis forNewtonian mechanics, even if the specific works of Lange and Thomsonwere little noticed. In writing “On the electrodynamics ofmoving bodies” in 1905, Einstein took it to be obvious to hisreaders that classical mechanics does not require a single privilegedframe of reference, but an equivalence-class of frames, all in uniformmotion relative to each other, and in any of which “theequations of mechanics hold good.” Two inertial frames withcoordinates \((x, y, z, t)\) and \((x', y', z', t')\) are related bytheGalilean transformations,

(assuming that the x-axis is defined to be the direction of theirrelative motion). These transformations clearly preserve the invariantquantities of Newtonian mechanics, i.e. acceleration, force, and mass(and therefore time, length, and simultaneity). As far as Newtonianmechanics was concerned, then, the problem of absolute motion wascompletely solved; all that remained was to express the equivalence ofinertial frames in a simpler geometrical structure.

The lack of a privileged spatial frame, combined with the obviousexistence of privileged states of motion—paths defined asrectilinear in space and uniform with respect to time—suggeststhat the geometrical situation ought to be regarded from afour-dimensionalspatio-temporal point of view. Thestructure defined by the class of inertial frames can be captured inthe statement thatspace-time is a four-dimensional affinespace, whose straight lines (geodesics)are the trajectories ofparticles in uniform rectilinear motion. See Figure 4.

Figure 4: Inertial Trajectories asStraight Lines of Space-time

That is, space-time is a structure whose automorphisms—theGalilean transformations that relate one inertial frame toanother—are affine transformations: they take straight linesinto straight lines, and parallel lines into parallel lines. Theformer condition implies that an inertial motion in one frame will bean inertial motion in any other frame, and likewise for anaccelerating or rotational motion. The latter implies thatuniformly-moving particles or observers who are relatively at rest inone frame will also be relatively at rest in another. An inertialframe can be characterized as a family of parallel straight lines“filling” space-time, representing the possibletrajectories of a family of free particles that are relatively atrest. See Figure 5. Therefore, to assert that an inertial frame existsis to impose a global structure on space-time; it is equivalent to theassertion that space-time is an affine space that is flat.

Figure 5: Each of these families ofstraight lines, \(F_1\) and \(F_2\), represents the trajectories of afamily of free particles that are relatively at rest, and thereforeeach defines an inertial frame. Relative to each other, the framesdefined by \(F_1\) and \(F_2\) are in uniform motion.
Each of the surfacesS is a “hypersurface of absolutesimultaneity” representing all of space at a given moment;evidently (given the Galilean transformations) two inertial frameswill agree on which events in space-time are simultaneous.

The form of the Galilean transformations shows that, in addition tobeing affine transformations, they also preserve metrical relations ontime and space. Distinct inertial frames will agree on simultaneity,and on (ratios of) time-intervals; they will also agree on the spatialdistance between points at a given moment of time. Therefore, in thefour-dimensional picture, the decomposition of space-time intohypersurfaces of absolute simultaneity is independent of the choice ofinertial frame. Another way of putting this is that Newtonianspace-time is endowed with aprojection of space onto time,i.e. a function that identifies space-time points that have the sametime-coordinate. Similarly, absolute space arises from a projection ofspace-time onto space, i.e. a function that identifies space-timepoints that have the same spatial coordinates. See Figure 6.

Figure 6

But Galilean relativity implies that this latter projection isarbitrary. While it assumes that we can identify the same time atdifferent spatial locations, Newtonian mechanics provides no physicalway of identifying the same spatial point at different times. Thus theequivalence of inertial frames can be thought of as the arbitrarinessof the projection of space-time onto space. Any such projection is,essentially, the arbitrary choice of some particular inertial frame asa rest-frame. In the relativized version of Newton’s theory,then, the class of inertial frames replaces absolute space, whileabsolute time remains. The structure of Newtonian space-time (alsoknown as Galilean space-time, or neo-Newtonian space-time) expressesthis fact in a direct and obvious way. (See Stein 1967 and Ehlers 1973for further explanation.)

(a)	(b)

Figure 7:(a) Here is a space-time diagram of motions relative to the inertialframe in which \(O_1\), \(O_2\), and \(P_1\) are at rest. This can beseen as arising from the projection of each of their inertialtrajectories onto a single point of space. \(O_3\) is in uniformmotion. \(O_4\) is accelerating any old way. \(O_5\) and \(O_6\) arerevolving around their common center of gravity \(P_1\), which (asnoted above) is at rest. \(O_7\) and \(O_8\) are revolving aroundtheir center of gravity \(P_2\), which is in uniform motion.
(b) Here is the same situation viewed from an inertial frame in which\(O_3\) and \(P_2\) are at rest. Now \(O_1,\) \(O_2,\) and \(P_1\) arein uniform motion. \(O_4\) is accelerating any old way. \(O_5\) and\(O_6\) are revolving around their common center of gravity \(P_1\),which is in uniform motion. \(O_7\) and \(O_8\) are revolving aroundtheir center of gravity \(P_2\), which (as noted above) is atrest.

2.2 The conflict between Galilean relativity and modern electrodynamics

By the time that this representation of the Newtonian space-timestructure was developed, however, the Newtonian conception of inertialframe had been essentially overthrown. First, 19^th-centuryelectrodynamics raised again the question of a privileged frame ofreference: the conception of light as an electromagnetic wave in theether implied that the rest-frame of the ether itself should play adistinguished role in electrodynamical phenomena. On the one hand,physicists such as Maxwell and Lorentz were careful to point out thatvelocity relative to the ether was not equivalent to absolutevelocity, because the state of motion of the ether itself wasnecessarily unknown. In other words, this conception of light did notnecessarily violate the classical principle of relativity. On theother hand, the existence of such a preferred frame made theequivalence of inertial frames correspondingly less interesting, evenif it was true in principle. This is why the appearance of the idea ofinertial frame in the 1880s, as suggested earlier, was not of pressingphysical interest to the majority of physicists, and seemed to be amere philosophical sidelight. The attempts to measure the effects ofmotion relative to the ether commanded considerably moreattention.

Second, the abandonment of the ether—following the failure ofattempts to measure velocity relative to the ether and, moregenerally, the apparent independence of all electrodynamical phenomenaof motion relative to the ether—did not vindicate the Newtonianinertial frame. Rather, it required a dramatically revised conception.Special relativity might be said to have applied the relativityprinciple of Newtonian mechanics to Maxwell’s electrodynamics,by eliminating the privileged status of the rest-frame of the etherand admitting that the velocity of light is independent of the motionof the source. As Einstein expressed it, “the same laws ofelectrodynamics and optics will be valid for all frames of referencefor which the equations of mechanics hold good.” (1905, p. 38.)But as Einstein also pointed out, the invariance of the velocity oflight and the principle of relativity, at least in its Galilean form,are incompatible. It simply makes no sense, according to Galileanrelativity, that any velocity should appear to be the same in inertialframes that are in relative motion.

2.3 Special relativity and Lorentz invariance

Einstein solved this difficulty through his analysis of simultaneity:frames in relative motion can agree on the velocity of light only ifthey disagree on simultaneity. Only the relativity of simultaneitymakes possible the invariance of the velocity of light. This meansthat the transformations between inertial frames that preserve thevelocity of light will not preserve simultaneity. These are theLorentz transformations:

Evidently these transformations preserve the velocity of light, butthey do not preserve length and time. So the invariant quantities ofNewtonian mechanics, which presuppose invariant measures of length andtime, must now depend on the choice of inertial frame. In other words,acceleration is an invariant quantity only when it is zero, for themagnitude of a non-zero acceleration will differ for observers inrelative motion. Therefore the notions of force, mass, andacceleration no longer provides an appropriate definition of aninertial frame, except in the ideal case of a frame in which no forcesare acting. Einstein’s definition instead appeals to theinvariant quantities of electrodynamics: an inertial frame is one inwhich light travels equal distances in equal times in arbitrarydirections. What seems impossible, from the point of view of Galileanrelativity, is that a frame that moves uniformly relative to one framethat satisfies Einstein’s definition should also satisfy thedefinition. But that seeming impossibility rests, again, on theassumption that two inertial frames will have a common measure ofsimultaneity. Einstein showed that light-signals provide anempirically and theoretically reasonable definition of simultaneity,in light of the empirical soundness of Maxwell’s equations andthe apparent invariance of the velocity of light. In the absence of anempirically reasonable alternative compatible with Galileaninvariance, there is no sound criterion of simultaneity that will givethe same results in different inertial frames. Of course, it wouldremain true that in a given inertial frame, the motions of freeparticles would satisfy the requirements of Lange’s definition,and particles that move uniformly in one such frame would also moveuniformly in any other. But such a definition is not a substitute forEinstein’s definition, since it must itself presuppose adefinition of simultaneity. Otherwise, its appeal to the measurementand comparison of times and distances is without a sound empiricalbasis. Einstein’s definition places such measurements on anempirical basis. Their results, however, will depend on the choice ofan inertial frame, and will vary systematically according to therelative velocities of different inertial frames.

The space-time structure of special relativity thus differsessentially from Newtonian space-time, and is called “Minkowskispace-time” since Minkowski (1908) first formulatedEinstein’s theory in its four-dimensional form. It is an affinespace, like Newtonian space-time. In both cases, the trajectories offree particles are straight lines of the affine structure, and a setof parallel inertial trajectories (geodesics) corresponds to aninertial frame. As we just saw, however, Newtonian space-timepresupposes the invariant division of space-time intothree-dimensional hypersurfaces of absolute simultaneity, and anobjective measure of distance between points at a given moment ofabsolute time. Minkowski space-time is a four-dimensional vector spacewith an invariant four-dimensional metrical structure, imposed by theinvariance of the speed of light. Instead of an invariant spatialinterval between simultaneous events, there is an invariantspatio-temporal interval between any two points in space-time. Sincethere is no invariant relation of simultaneity, the sets ofsimultaneous events for any inertial frame are the hyperplanesorthogonal to the trajectories that determine that frame. In otherwords, the choice between two inertial frames determines a choicebetween two distinct divisions of space-time into space and time. SeeFigure 8:

Figure 8: The inertial frames \(F\) and\(F'\) are in relative motion, and therefore, as the Lorentztransformations indicate, they disagree on simultaneity. \(F\) and\(F'\) thus determine distinct decompositions of space-time intoinstantaneous spaces, \(S\) and \(S'\), respectively

2.4 Simultaneity and reference-frames

The details of Einstein’s argument and the structure ofMinkowski space-time can be found elsewhere (see, e.g., Einstein 1951and Geroch 1978). Here only one more point is worth making. It couldbe argued that Einstein’s and Lorentz’s view arecompletely equivalent. That is, we could assume that there is indeed aprivileged frame of reference, and that the apparent invariance of thevelocity of light is explained by the effects on bodies of theirmotion through the ether (the Lorentz contraction and time dilation).This purported distinction between empirically indistinguishableframes has often been criticized on straightforward methodologicalgrounds, but it could be (and surely has been) argued that it is moreintuitively plausible than the relativity of simultaneity. After all,knowing that (as Einstein showed) the Lorentz contraction can bederived from the invariance of the velocity of light does not, byitself, entitle us to say which of the two is the more convincingstarting-point.

This is why it is so important that Einstein’s 1905 paper beginswith a critical analysis of the entire notion of a frame of reference.It is tacitly assumed by Lorentz’s theory, and classicalelectrodynamics generally, that we have a reference-frame in which wecan measure the velocity of light, and that there is a distinguishedframe—at rest with respect to the ether—in which its truevelocity would be measured. But how is such a reference-framedetermined? The distances between points in space can only bedetermined if it is possible to determine which events aresimultaneous. In practice this is always done by light-signalling, ifonly in the informal sense that we identify simultaneous events whenwe see them at the same time. But if the spatial frame of reference isdetermined by light-signals, and is then to be used to measure thespeed of light, we would appear to be going in a circle. For thisreason, Poincaré concluded that determining the speed of lightis partly a matter of convention (1898). Before Einstein, however, itwas tacitly assumed that, while light-signalling is useful andpractical, it is not essential to the definition of simultaneity.There would be, therefore, a fact of the matter about which events aresimultaneous that is independent of this method of signalling. Thisassumption was actually made explicit by James Thomson. Herecognized—as few did before Poincaré andEinstein—that the measurement of distance involves

the difficulty as to imperfection of our means of ascertaining orspecifying, or clearly idealizing, simultaneity at distant places. Forthis we do commonly use signals by sound, by light, by electricity, byconnecting wires or bars, and by various other means. The timerequired in the transmission of the signal involves an imperfection inhuman powers of ascertaining simultaneity of occurrences at distantplaces. It seems, however, probably not to involve any difficulty ofidealizing or imagining the existence of simultaneity. Probably it maynot be felt to involve any difficulty comparable to that of attemptingto form a distinct notion of identity of place at successive times inunmarked space. (1884, p. 380).

In other words, Thomson assumed that it was not a difficulty inprinciple, like the difficulty of determining rest in absolute space.But Einstein showed that it was precisely the same kind of difficulty:determinations of simultaneity involve reference to an arbitrarychoice of reference-frame, just as much as determinations of velocity.Einstein’s conclusion is, of course, entirely contingent on theempirical facts of electrodynamics. It could have been avoided ifthere were in nature a useful signal of some kind whose transmissionwould provide a criterion of absolute simultaneity, so that the sameevents would be determined to be simultaneous in all inertial frames.Or, experiments such as those of Michelson and Morley might havesucceeded in exhibiting the dependence of the velocity of light on thestate of motion of the source. Then synchronization by light-signalscould still have been regarded as a mere practical substitute for anotion of absolute simultaneity that stood on independent grounds,empirically as well as conceptually. But as Einstein saw, because ofthe apparent independence of the velocity of light of the motion ofthe source, even “idealizing or imagining the existence ofsimultaneity” involves light-signalling more essentially thananyone could have realized. Unless some other criterion ofsimultaneity is provided, therefore, the establishment of a spatialframe of reference involves light-signalling in an essential way. Inthe absence of such a criterion the speed of light cannot be, asLorentz supposed, empirically measured against the background of aninertial frame. By appealing to the speed of light in definingsimultaneity, Einstein gave an empirically sound construction both forspatio-temporal measurement and for a dynamically distinct class ofreference-frames. (Cf. DiSalle 2006, ch. 4.)

2.5 From special relativity and Lorentz invariance to general relativity and general covariance

It may seem surprising that, after this insightful analysis of theconcept of inertial frame and its role in electrodynamics, Einsteinshould have turned almost immediately to call that concept intoquestion. But he became convinced, largely by his reading of Mach,that the central role of inertial frames was an “epistemologicaldefect” that special relativity shared with Newtonian mechanics.Only relative motions are observable, yet both of these theoriespurport to identify a privileged state of motion and use it to explainobservable effects (such as centrifugal forces). Coordinate systemsare not observable, yet both of these theories assign a fundamentalphysical role to certain kinds of coordinate system, namely, theinertial systems. In either theory, inertial coordinates aredistinguished from all others, and the laws of physics are said tohold only relative to inertial coordinate systems. In anepistemologically sophisticated theory, both of these problems wouldbe solved at once: the new theory would only refer to what isobservable, which is relative motion; it would admit arbitrarycoordinate systems, instead of confining itself to a special class ofsystem. Why, after all, should any genuine physical phenomenon dependon the choice of coordinate system?

Another way of expressing Einstein’s view is to say that, inNewtonian mechanics and special relativity, rotation is“absolute” because the transformations between inertialframes (Galilean or Lorentzian) preserve rotational states. Thus the“absoluteness” of rotation arises precisely from singlingout one type of frame, by one type of transformation, instead ofallowing arbitrary transformations and arbitrary frames. Einstein heldthat this epistemological insight had a natural mathematicalrepresentation in the principle ofgeneral covariance, or theprinciple that the laws of nature are to be invariant underarbitrary coordinate transformations. More precisely, whatthis means is that coordinate transformations are no longer required(as in the affine spaces of Newtonian mechanics and specialrelativity) to take straight lines to straight lines, and to preservefurther metrical structures appropriate to each theory, but only topreserve the smoothness of curves (i.e. their differentiability). Thegeneral theory of relativity was intended to be a generally covarianttheory of space-time, and its general covariance was intended toexpress the general relativity of motion (Cf. Einstein 1916, section3). The extent to which the theory realized Einstein’s originalaims remains a topic of philosophical debate. (Cf. Related Entries:“space and time: the hole argument,”“Einstein’s philosophy of science”.)

The central role of inertial frames in Newtonian and Minkowskispace-time theories, in sum, rested on their shared assumption of theuniformity of space-time. In Newtonian mechanics and specialrelativity, the formal relations between inertial coordinatesystems—the Galilean and Lorentz transformations,respectively—correspond to symmetry transformations of uniformspace-time, that is, a space-time with non-trivial global symmetries.In Einstein’s context, a coordinate transformation from thecoordinates of one “stationary system” to those of anotherwould generally not reflect a global symmetry of space-time, just tothe extent that the two were relatively accelerating.

At the same time, we can say that the concept of inertial frameretains at least one aspect of the relevance that it had for Newton.As we saw, Newton was aware that he could not determine (to put it inour language) an actual inertial frame from his analysis of the solarsystem; Corollary VI implied that all local phenomena were compatiblewith an acceleration of the entire system, by nearly equal and nearlyparallel accelerative forces acting on the sun, the planets, and theirsatellites. By the very same reasoning, however, he established thatsuch a uniformly accelerating frame was sufficient for a successfulanalysis of the relevant causes acting within the system—thosecauses that determine the system’s configuration. From thisanalysis he could determine the configuration of the system, that is,determine that our system is approximately Keplerian, with smalldeviations from Keplerian motion accounted for by perturbative actionsof the members of the system on each other. In other words, though atrue inertial frame cannot be determined, a sufficient approximationto an inertial frame provides a sufficient basis for this causalaccount.

In empirical testing and measurement in general relativity, thetreatment of individual gravitating systems follows an analogouspattern. For the analysis of interactions within a local system, anasymptotically flat solution to Einstein’s equation plays a roleanalogous to the Newtonian system that approximates the conditions ofCorollary VI. In general relativity, there can be non-negligiblepost-Newtonian effects, such as spatial curvature, and the non-linearsuperposition of gravitational fields. When they appear, empiricalconsequences such as anomalous precessions and light-bending providetests of general relativity and other relativistic gravitationtheories. (See Will 2018, ch. 4, especially 4.1–4.3).Abstractly, we might wish to treat such a system as isolated within aregion of space-time that “flattens out” at infinity.Practically, it suffices to consider a region in which, at asufficient remove from a system of mutually gravitating masses,curvature becomes negligible in comparison with the curvature inducedby those masses locally. In other words, analogously to the Newtoniancase, it is not necessary that such a system be isolated from externalinfluences. It suffices that external influences make a negligibledifference to the actions of the masses among themselves, and to theresulting configuration of the system. In short, instead of providingan exact account of the global symmetries of space-time, the idea ofan inertial frame still provides a crucial practical tool for theempirical study of actual physical interactions.

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• DiSalle, R., 2017, “Absolute space and Newton's theory of relativity”, presentation at The Philosophy of Howard Stein (Conference at U. Chicago, June 9-11, 2017).

Related Entries

Einstein, Albert: philosophy of science |general relativity: early philosophical interpretations of |geometry: in the 19th century |space and time: conventionality of simultaneity |space and time: the hole argument

Acknowledgments

I am deeply grateful to Sona Ghosh for herinvaluable assistance with philosophical, mathematical, and technicalmatters. I would also like to thank an anonymous referee for the SEPfor very helpful comments on previous versions.