## 1. Relativity and reference frames in classical mechanics


### 1.1 The origins of Galilean relativity



The term "reference frame" was coined in the
19th century, but it has a long prehistory, beginning,
perhaps, with the emergence of the Copernican theory. The significant
point was not the replacement of the earth by the sun as the center of
all motion in the universe, but the recognition of both the earth and
the sun as merely possible points of view from which the motions of
the celestial bodies may be described. This implied that the basic
task of Ptolemaic astronomy--to represent the planetary motions
by combinations of circular motions--could take any point to be
fixed, without sacrificing predictive power. Therefore, as Copernicus
suggested in the opening arguments of *On the Revolutions of the
Heavenly Spheres*, the choice of any particular point required
some justification on grounds other than mere successful astronomical
prediction. The most persuasive grounds, seemingly, were physical: we
don't perceive the physical effects that we would expect the
earth's motion to produce. Copernicus himself noted, however, in
reply, that we can indeed undergo motions that are physically
imperceptible, as on a smoothly moving ship (1543, p.6). At least in
some circumstances, we can easily treat our moving point of view as if
it were at rest. 



As the basic programme of Ptolemy and Copernicus gave way to that of
early classical mechanics as developed by Galileo, this equivalence of
points of view was made more precise and explicit. Galileo was unable
to present a decisive argument for the motion of the earth around the
sun. He demonstrated, however, that the Copernican view does not
contradict our experience of a seemingly stable earth. He accomplished
this through a principle that, in the precise form that it takes in
Newtonian mechanics, has become known as the "principle of
Galilean relativity": mechanical experiments will have the same
results in a system in uniform motion that they have in a system at
rest. Arguments against the motion of the earth had typically appealed
to experimental evidence--e.g., that a stone dropped from a tower
falls to the base of the tower, instead of being left behind as the
earth rotates during its fall. But Galileo argued persuasively that
such experiments would happen just as they do whether the earth were
moving or not, provided that the motion is sufficiently uniform. (See
Figure 1.) Galileo's account of this was not precisely the
principle that we call "Galilean relativity"; he seems to
have thought that a system in uniform circular motion, such as a frame
at rest on the rotating earth, would be indistinguishable from a frame
truly at rest. The principle was named in his honor because he had
grasped the underlying idea of dynamical equivalence: he understood
the composition of motion, and understood how individual motions of
bodies within a system--such as the fall of a stone from a
tower--are composed with the motion of the system as a whole.
This principle of composition, combined with the idea that bodies
maintain their uniform motion, formed the basis for the idea of
dynamically indistinguishable frames of reference. 



![figure 1](Fig1.png)

Figure 1: Galileo's Argument If
the earth is rotating sufficiently uniformly, a stone dropped from the
tower will fall straight to the base, just as a stone dropped from the
mast of a uniformly moving ship will fall to the foot of the mast. In
both cases the stone's vertical motion will be smoothly composed
with its horizontal motion. Hence a sufficiently uniform motion will
be indistinguishable from rest.



### 1.2 Philosophical controversy over absolute and relative motion



Leibniz, later, articulated a more general "equipollence of
hypotheses": in any system of interacting bodies, any hypothesis
that any particular body is at rest is equivalent to any other.
Therefore neither Copernicus' nor Ptolemy's view can be
true--though one may be judged simpler than the
other--because both are merely possible hypothetical
interpretations of the same relative motions. This principle clearly
defines (what we would call) a set of reference frames. They differ in
their arbitrary choices of a resting point or origin, but agree on the
relative positions of bodies at any moment and their changing relative
distances through time.



For Leibniz and many others, this general equivalence was a matter of
philosophical principle, founded in the metaphysical conviction that
space itself is nothing more than an abstraction from the geometrical
relations among bodies. In some form or other it was a widely shared
tenet of the 17th-century "mechanical
philosophy". Yet it was flatly incompatible with physics as
Leibniz himself, and the other "mechanists," actually
conceived it. For the basic program of mechanical explanation depended
essentially on the concept of a privileged state of motion, as
expressed in the common assumption that was the forerunner of the
"principle of inertia": bodies maintain a state of
rectilinear motion until acted upon by an external cause. Thus their
fundamental conception of force, as the power of a body to change the
state of another, likewise depended on this notion of a privileged
state. This dependence was clearly exhibited in the vortex theory of
planetary motion, which presupposed that any planet would move in a
straight line unless impeded. Its actual orbit was therefore explained
by the balance between the planet's inherent centrifugal
tendency (its tendency to follow the tangent to the orbit) and the
pressure of the surrounding medium.



For this reason, the notion of a dispute between
"relativists" or "relationists" and
"absolutists" or "substantivalists", in the
17th century, is a drastic oversimplification. Newton, in
his controversial Scholium on space, time, and motion, was not merely
asserting that motion is absolute in the face of the mechanists'
relativist view. He was arguing that a conception of absolute motion
was already implicit in the views of his opponents--that it was
implicit in their conception, which he largely shared, of physical
cause and effect. The general equivalence of reference-frames was
implicitly denied by a physics that characterized forces as powers to
change the states of motion of bodies.



This development placed the subject of reference frames in a new
theoretical context. Having set aside the reference frame of common
sense--the frame in which the earth is at rest in the center,
with the heavens revolving around it--the mechanical physics of
this time naturally tied this subject with novel theoretical
conceptions of motion, and its physical causes and effects. Copernicus
argued for a heliocentric system, not from a physical theory of
motion, but from the comparative simplicity and reasonableness that it
introduced into astronomy; he worked within the established theory of
the causes of celestial motions, namely, the revolutions of the
heavenly spheres. After Copernicus, however--more precisely,
after the model of revolving spheres was largely
abandoned--determining the right frame of reference was connected
with discovering the true physical causes of the planets'
motions. Philosophers such as Kepler, Descartes, Huygens, Leibniz, and
Newton held vastly differing views of physical causation, motion. and
the relativity of motion. They agreed, however, that the heliocentric
picture was uniquely suited for giving a causal account of planetary
motions, as the effects of physical actions originating in the sun. To
Kepler and Descartes, for example, the rotation of the sun on its
axis, in the same sense in which the planets revolved, identified it
as the cause of those revolutions.



The link between the causal account of motion and the more general
conceptual account of "true" motion was never obvious or
straightforward. Descartes's heliocentric causal account, in
which the planets moved in vortices arising from the sun's
rotation, was disconnected from his abstract account of motion
"according to the truth of the matter," or "in the
proper sense" (Descartes 1642, part II section XXV). Since there
are innumerable objects to which one might refer the motion of any
given body, the reference may appear to be a matter of arbitrary
choice. The unambiguous reference for the motion of any body, he
argued, is provided by the bodies immediately touching it. Newton, in
response, argued that Descartes's philosophical account of
motion is flatly incompatible with his causal account. To Newton, it
was incoherent to appeal to a causal account of motion when explaining
the centrifugal tendency of a body in its orbit around the sun, while
identifying its "proper" motion only by its relations to
the bodies immediately contiguous to it (Newton 1684a, Stein 1967,
Rynasiewicz 2014). Hence Newton's argument that the only
unambiguous standard of motion is a body's change of position
with respect to space itself. To space understood in this sense, as
the universal frame of reference with respect to which the
displacements of bodies constitute their true motions, Newton gave the
name "absolute space" (1687b, p. 5ff). 



Evidently Newton was aware that "absolute space" was not a
reference frame in any practical sense. He emphasized that "the
parts of space cannot be seen," and that no observable bodies
can be known to be at rest. Hence there is no way to determine motion
with 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 effects
indicate a body's change of position in absolute space? It is
conceivable, for example, that there might be some physical correlate
for velocity, in the sense that a body might have some observable
physical 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 rest
in space. If a body could be known to be in that state, therefore, it
would (in principle) provide a physical marker for a truly resting
frame of reference. On Leibniz's conception of force, for
instance, a given force is required to generate or to maintain a given
velocity. For objects "passively" resist motion, but
maintain their states of motion only by "active"
force--so that, on dynamical grounds, "every body truly
does have a certain amount of motion, or, if you will, force"
(Leibniz 1694, p. 184; see also 1716, p. 404). It would follow from
this that there must be, in principle, a distinguished frame of
reference in which the velocities of bodies correspond to their true
velocities, i.e. to the amounts of moving force that they truly
possess. It would also follow that, with respect to any frame that is
in motion relative to this one, bodies will not have their true
velocities. In short, such a conception of force, if it could be
applied physically, would give a precise physical application of
Newton's conception of absolute space, by providing a physical
correlate for change of absolute place. 


### 1.3 Galilean relativity in Newtonian physics



The difficulty with Newton's view of absolute space comes, not
from the epistemological arguments of relationalism, but from
Newton's own conception of force. If force is defined and
measured solely by the power to accelerate a body, then obviously the
effects of forces--in short, the causal interactions within a
system of bodies--will be independent of the velocity of the
system in which they are measured. So the existence of a set of
equivalent "inertial frames" is imposed from the start by
Newton's laws. Suppose that we determine for the bodies in a
given frame of reference--say, the rest frame of the fixed
stars--that all observable accelerations are proportional to
forces impressed by bodies within the system, by equal and opposite
actions and reactions among those bodies. Then we know that these
physical interactions will be the same in any frame of reference that
is in uniform rectilinear motion relative to the first one. Therefore
no Newtonian experiment will be able to determine the velocity of a
body, or system of bodies, relative to absolute space. In other words,
there is no way to distinguish absolute space itself from any frame of
reference that is in uniform motion relative to it. Newton thought
that a coherent account of force and motion requires a background
space consisting of "places" that "from infinity to
infinity maintain given positions with respect to one another"
(1687b, p. 8-9). But the laws of motion enable us to determine
an infinity of such spaces, all in uniform rectilinear motion relative
to each other. The laws furnish no way of singling out any one as
"immovable space."



Oddly enough, no one in the 17th century, or even before
the late 19th century, expressed this equivalence of
reference-frames any more clearly than Newton himself. However, the
credit for articulating this equivalence precisely for the first time,
belongs to Christiaan Huygens, who introduced it as one of the
Hypotheses of his first work on the rules of impact (1656).
"Hypothesis I" was the first clear statement of the
principle of inertia: "Any body, once in motion, if nothing
opposes it, continues to move always with the same velocity and along
the same straight line" (1656, pp. 30-31). The first
precise statement of the relativity principle followed as Hypothesis
III: 



> 
> The motion of bodies, and their speeds equal or unequal, are to be
> understood respectively, in relation to other bodies which are
> considered as at rest, even though perhaps both the former and the
> latter are subject to another motion that is common to them. In
> consequence, when two bodies collide with one another, even if both
> together undergo another equable motion, they will move each other no
> differently, with respect to a body that is carried by the same common
> motion, than if this extraneous motion were absent from all of them.
> (1656, p. 32).
> 



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



Newton's first statement of the same principle appears in one of
the series of papers that culminated in the *Principia*,
"De motu sphaericorum corporum in fluidis" (1684b).
Like Huygens, Newton presents the relativity principle as a
fundamental principle, "Law 3": 



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



Newton's first statement of the Galilean relativity principle
evidently recapitulates Huygens' version, which was probably
known to Newton. The same may be said of "Law 4" in this
manuscript, the principle of conservation of the center of gravity:




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



But, uniquely, Newton immediately went on to consider the deeper
theoretical significance of these principles: they radically
reconceptualize the problem of "true motion" in the
planetary system. First, they implied that the entire system must be
seen as included in a space that may, itself, be either at rest or in
uniform motion. Second, they implied that the only truly fixed point
in 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 is
> commonly believed) or moves uniformly in a straight line, and hence
> the communal centre of gravity of the planets (by Law 4) either rests
> or moves along with it. In both cases (by Law 3) the relative motions
> of the planets are the same, and their common centre of gravity rests
> in relation to the whole space, and so can certainly be taken for the
> still centre of the whole planetary system. (ibid., p. 47r)
> 



Newton now realized, in short, that the dispute between the
heliocentric and geocentric views of the universe had been mistakenly
framed. The proper question about "the system of the
world" was not "which body is at rest in the
center?" but "where is the center of gravity of the
system, and which body is closest to it?" For in a system of
orbiting bodies, only their common center of gravity will be
unaccelerated, and by "Law 3", the motions of the bodies
in the system will be the same, whether its center of gravity is at
rest or in uniform rectilinear motion. By explicitly asserting the
dynamical equivalence of "whole spaces" that may moving
uniformly or at rest, Newton made it clear that the solution to the
problem of "the system of the world" is the same with
respect to any such moving space as it is with respect to immobile
space. Thus he came as close to articulating the concept of the
inertial frame as anyone before the late 19th century.



In the successive drafts of his *Principia*, Newton gradually
clarified its conceptual structure, and in particular the
frame-independent character of its concepts of motion, force, and
interaction. He arrived at the new axiomatic structure whose only laws
are the familiar "Newton's Laws of Motion"; the
principle of the conservation of the center of gravity, and the
relativity principle, were no longer presupposed, but derived from the
Laws as Corollaries IV and V:



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



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



These principles illuminate the relationship between the theory of
absolute space, as articulated in Newton's Scholium to the
Definitions, and the overarching scientific problem of the
*Principia*. According to Newton, "the aim for which I
composed" the book was to show "how to gather the true
motions from their causes, effects, and apparent differences, and
conversely, from the motions, true or apparent, to gather their causes
and effects" (1687b, p. 11); the more specific aim of Book III
was "to exhibit the constitution of the system of the
world" (1687b, p. 401). 



On the one hand, Corollary V, like "Law 3" in De Motu,
precisely restricts what Newton's procedure can determine about
the structure of the system of the world. It cannot determine anything
about the velocity of the system as a whole; it can only determine the
position of the center of gravity of the bodies that comprise it, and
the configuration of those bodies with respect to that center. In this
sense it can, in principle, decide between a Keplerian and a Tychonic
interpretation of the motions of these bodies. The system is indeed
approximately Keplerian: the sun has by far the greatest mass and is
therefore little disturbed from the center of gravity by its
interactions with the planets. The sun therefore remains very close to
the common focus of the nearly Keplerian ellipses in which the planets
orbit the sun. But by Corollary V, the actions of the bodies among
themselves would not reveal whether their center was moving uniformly
or at rest. On the other hand, Newton recognized that motion with
respect to absolute space is unknowable. This restriction, therefore,
meant that the solution to the system of the world is secure in spite
of our ignorance. The nearly-Keplerian structure of the system is
known completely independently of the system's state of motion
in absolute space.



The Galilean relativity principle thus contained, in Newton's
conception, a broader insight: that different states of uniform
motion, or different uniformly-moving frames of reference, determine
only different points of view on the same physically objective
quantities, namely force, mass, and acceleration. We can see this
insight expressed more explicitly in the evolution of Newton's
concept of inertia. The term had been introduced by Kepler, and played
a central role in his physical conception of planetary motion.
Rejecting the Aristotelian idea that the planets are carried by
rotating crystalline spheres, Kepler held that the planets have a
natural tendency to rest in space--what he called their natural
inertia--and argued that they must be moved by active powers that
overcome their natural inertia. Newton, well before working on
*Principia*, had based his conception of inertia on the idea of
Galileo 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 essentially
pre-relativistic, because it implied a conceptual distinction between
a body's power to resist external forces, and the power of a
moving body to change the motion of another. The manuscript *De
gravitatione et aequipondio fluidorum* (1684a), for example, was
evidently written before Newton fully recognized the importance of the
relativity 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 far
> as it is resisted.
> 



> 
> Definition 7: Impetus is force in so far as it is impressed on
> another.
> 



> 
> Definition 8: Inertia is the internal force of a body, so that its
> state 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, was
determined by velocity. Leibniz therefore distinguished this force as
an active power, fundamentally different from the passive power of a
resting body to resist any change of position. Newton, in contrast, as
he developed the *Principia*, and recognized the existence of a
class of indistinguishable relative spaces, gradually came to
understand the "force of inertia" as what we would call a
Galilei-invariant quantity. Impetus and resistance were therefore
recognized as appearances of that invariant quantity in different
frames of reference:



> 
> A body truly exerts this force only in a change of its state brought
> about by another force impressed upon it, and the exercise of this
> force is, under different aspects, both resistance and impetus:
> resistance in so far as the body, to maintain its state, opposes the
> impressed force; impetus insofar as the same body, yielding only with
> difficulty to the force of a resisting obstacle, strives to change the
> state of that obstacle. Resistance is commonly attributed to resting
> bodies and impetus to moving bodies; but motion and rest, as commonly
> understood, are only relatively distinguished from each other; and
> bodies 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 commonly
regarded as distinct (e.g. in the Leibnizian distinction between
passive and active) as merely frame-dependent representations of the
same fundamental property. That is, they represent the same invariant
quantity seen from different points of view. The principle that a body
exerts this force "only in a change of its state"
decisively separates Newton's new view from the older notion of
a specific power that is required to maintain a body in motion. This
change has been noted by modern commentators (see Herivel 1965, p. 26;
see also DiSalle 2013, p. 453; Disalle 2017, in Other Internet
Resources). But it was already noted in Newton's own time by
George Berkeley, who emphasized the contrast between Newton's
conception and that of Leibniz:



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



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



Berkeley thus made clear that the older understanding of inertia,
unlike that expressed in the *Principia*, did not respect the
principle of relativity. Second, Newton's explication implicitly
invokes all three laws of motion (cf. Stein 2002). Newton's
first law alone came to be identified as "the principle of
inertia." Newton himself, however, understood that inertia has
three inseparable aspects: the tendency to persist in motion, the
resistance to change in motion, and the power to react against an
impressed force. All are essential to the explication of inertial mass
as 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 was
Newton's way of drawing attention to the precise role of
inertial mass as an invariant quantity in physical interactions,
underlying the various ways in which its manifestations had been
previously conceived.


### 1.4 The lingering problem of absolute space



Newton understood the Galilean principle of relativity with a degree
of depth and clarity that eluded most of his "relativist"
contemporaries and critics. It may seem bizarre, therefore, that the
notion of inertial frame did not emerge until more than a century and
a half after his death. He had identified a distinguished class of
dynamically equivalent "relative spaces," in any of which
true forces and masses, accelerations and rotations, would have the
same objectively measured values. Yet these spaces, though dynamically
equivalent and empirically indistinguishable, were yet not equivalent
in principle. Evidently, Newton conceived them as moving with various
velocities in absolute space, though those velocities could not be
known. Why should not he, or someone, have recognized the equivalence
of these spaces, and the dispensability of a distinguished resting
space--"absolute" space--immediately?



This is not the place for an adequate answer to this question, if
indeed one is possible. For much of the 20th century, the
accepted answer was that of Ernst Mach: Newton lived in an age
"deficient in epistemological critique." He was therefore
unable to draw the conclusion that these *dynamically*
indistinguishable spaces must be equivalent in *every*
meaningful sense, so that no one of them deserves even in principle to
be designated as "absolute space." Yet even those whom the
20th century credited with more sophisticated
epistemological views, such as Leibniz, evidently had difficulties
understanding force and inertia in a Galilei-invariant way, despite a
philosophical commitment to relativity. We may plausibly suppose that
it was difficult to abandon the intuitive association of force or
motion with velocity in space. It must also have been difficult, in
the mathematical context of Newton's time, to conceive of an
equivalence-class structure as the fundamental spatio-temporal
framework. It required a level of abstraction that became possible
only with the extraordinary development of mathematics, especially of
a more abstract view of geometry, that took place in the
19th century. Newton's arguments established, for the
assumptions of classical dynamics, the need for a dynamical space-time
structure beyond the kinematical structure required to represent
changes of relative position over time. But absolute space, with its
superfluous elements, was the only such structure imagined for the
next two centuries. It was accepted as the only realistic alternative
to theories with no dynamical structure at all. There was as yet no
notion of a structure that expressed *all and only* what was
required by the dynamical laws. Euler, for example, in a penetrating
critique of Leibnizian relationalism (1748), argued that the laws of
motion require a notion of sameness of direction in space, and of
uniform motion with respect to time. The truth of the laws of
motion--which, for Euler, were more securely established than any
principle of metaphysics--could not, therefore, be reconciled
with any account of space and time as merely ideal. But he did not see
the possibility of separating true acceleration and rotation from true
velocity with respect to absolute space. 



In the 17th century, only Huygens came close to expressing
such a view; he held that not velocity, but velocity-difference, was
the fundamental dynamical quantity. He therefore understood, for
example, that the "absoluteness" of rotation had nothing
to do with velocity relative to absolute space. Instead, it arose from
the difference of velocity among different parts of a rotating body.
If a disk is translated through space without rotation, then its parts
move in parallel lines, but if it is rotating, then they move in
different directions, even though they are at rest relative to one
another on account of the bonds holding them together. The
differences, evidently, would be the same irrespective of the velocity
of the body as a whole in absolute space. Unfortunately, Huygens
expressed this view only in manuscripts that remained unpublished for
two centuries. (Cf. Stein, 1977, pp. 9-10 and Appendix III.)
Huygens also reflected on the possibility of replacing absolute space
with (what we would call) empirical frames of reference, again in
unpublished notes that have only been brought to light in recent work
by Stan (2016). But the complete concept of the inertial frame emerged
only in the late 19th century, when it did not seem to be
of any great immediate importance (see below). Indeed, even after the
concept of inertial frame had been widely discussed, the notion
persisted that true rotation could only be understood as rotation with
respect to absolute space. Poincare, for example, convinced of
the essential "relativity of space" as well as the
relativity of motion, considered the concept of absolute space to be
something of a philosophical embarrassment. But it was not clear to
him how the dynamical phenomena of rotation could be understood
without it (cf. DiSalle 2014). So the failure of Newton and Huygens to
formulate the concept of inertial frame, two centuries earlier, seems
less remarkable than the progress that each of them made in
understanding the relativity of motion. As we will see, articulating
this 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 analysis
of the notion of absolute space, for reasons not anticipated by
Newton's contemporary critics. Its starting point was a critical
question about the law of inertia: relative to what is the motion of a
free particle uniform and rectilinear? If the answer is
"absolute space," then the law would appear to be
something other than an empirical claim, for no one can observe the
trajectory of a particle relative to absolute space. Two quite
different answers to the question were offered in 1870, in the form of
revised statements of the law of inertia. Carl Neumann proposed that
when we state the law, we must suppose that there is a body somewhere
in the universe--the "body Alpha"--with respect
to which the motion of a free particle is rectilinear, and that there
is a time-scale somewhere relative to which it is uniform (Neumann
1870). Ernst Mach (1883) claimed that the law of inertia, and
Newton's laws generally, implicitly appeal to the fixed stars as
a spatial reference-frame, and to the rotation of the earth as a
time-scale. At any rate, he maintained, such is the basis for any
genuine empirical content that the laws can claim. The notion of
absolute space, it followed, was only an unwarranted abstraction from
the practice of measuring motions relative to the fixed stars.



Mach's proposal had the advantage of a clear empirical
motivation; Neumann's "body Alpha" seemed no less
mysterious than absolute space, and almost sounds comical to the
modern reader. But Neumann's discussion of a time-scale was
somewhat more fruitful, and employed the principle that Euler had
already expressed (1748): the law of inertia defines a time-scale, by
which equal intervals of time are those in which a free particle
travels equal distances. He also noted, however, that this definition
is quite arbitrary. For, in the absence of a prior definition of equal
times, any motion whatever can be stipulated to be uniform. It is no
help to appeal to the requirement of freedom from external forces,
since the free particles presumably are known to us only by their
uniform motion. We have a genuine empirical claim only when we state
of *at least two* free particles that their motions are
*mutually* proportional. Equal intervals of time can then be
defined as those in which two free particles travel mutually
proportional distances.


### 1.6 The emergence of the concept of inertial frame



Neumann's definition of a time-scale directly inspired Ludwig
Lange's conception of "inertial system" (Lange
1885). An inertial coordinate system ought to be one in which free
particles move in straight lines. But any trajectory may be stipulated
to be rectilinear, and a coordinate system can always be constructed
in 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 one
particle. Indeed, for any two particles moving anyhow, a coordinate
system may be found in which both their trajectories are rectilinear.
So far the claim that either particle, or some third particle, is
moving in a straight line may be said to be a matter of convention. We
must define an inertial system as one in which at least three free
particles move in straight lines. Then we can state the law of inertia
as the claim that, relative to an inertial system so defined, the
motion of any fourth particle, or arbitrarily many particles, will be
rectilinear. The notions of inertial system and Neumann's
time-scale, which Lange called an "inertial time-scale,"
may be combined as follows: relative to a coordinate system in which
three free particles move in straight lines *and* travel
mutually-proportional distances, the motion of any fourth free
particle will be rectilinear and uniform. The questionable Newtonian
concepts of absolute rotation and acceleration, Lange proposed, could
now be replaced by the concepts of "inertial rotation" and
"inertial acceleration," i.e. rotation and acceleration
relative to an inertial system and inertial time-scale. See Figures 2
and 3.





|  |  |
| --- | --- |
| Figure 2a | Figure 2b |
| (a) | (b) |



Figure 2: Neumann's Time-Scale By
Newton's first law, a particle not subject to forces travels
equal distances in equal times. But which particles are free of
forces? This might appear to be a matter of convention.
  

(a) Either \(P\_1\) or \(P\_2\) can be arbitrarily stipulated to be at
the origin of a system of coordinates, and to serve as the measure of
equal times
  

(b) But one can say of two particles with different velocities: in
intervals of time in which one moves a given distance \(d\_1\), the
other moves a proportional distance \(d\_2 = kd\_1\) (where *k* is
a constant; i.e., \(d\_1/d\_2 = k\)). Or one can compare a particle to a
freely rotating planet: in intervals of time through which the planet
rotates through equal angles, the particle moves equal distances. 




![Figure 3](Fig3.png)

Figure 3: Lange's Definition of
'inertial system' (1885) An inertial system is a
coordinate 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. The
law of inertia then states that relative to any inertial system, any
fourth 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 William
Thomson, Lord Kelvin--expressed the content of the law of
inertia, 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 FRAME
> and a REFERENCE DIAL-TRAVELLER are kinematically possible, such that
> relatively to them conjointly, the motion of the mass-centre of each
> body, undergoes change simultaneously with any infinitely short
> element of the dial-traveller progress, or with any element during
> which the force on the body does not alter in direction nor in
> magnitude, which change is proportional to the intensity of the force
> acting on that body, and to the simultaneous progress of the
> dial-traveller, and is made in the direction of the force. (Thomson
> 1884, p. 387)
> 



Thomson did not reject the term "absolute rotation".
Instead, he held that it is properly defined as rotation relative to a
frame that satisfies his definition of a reference frame. A body that
is rotating with respect to a reference frame (and dial-traveller) is
rotating with respect to any other frame in uniform motion relative to
the first. The definition does not express, as Lange's does, the
degree of arbitrariness involved in the construction of an inertial
system by means of free particles. By dispensing with the idealization
of free particles, Thomson's definition aims to characterize an
inertial frame for an actual system of interacting bodies. However, it
does not quite fulfill its aim. Like Lange's definition, it
leaves out a crucial condition for an inertial system as we understand
it: all forces must belong to action-reaction pairs. Otherwise we
could 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 that
to every action there is an equal and opposite reaction. (This
completion was actually proposed by R.F. Muirhead in 1887.) 



But, so completed, Thomson's definition makes the essential
point about the relation between Newton's laws of motion and the
inertial frames: that the laws assert the existence of at least one
inertial frame. If one inertial frame is posited, in which
accelerations properly correspond to Newtonian forces, then any other
inertial frame is in uniform motion with respect to the first; the
forces, masses, and accelerations measured in one will have the same
measures in any other. Any one may be arbitrarily designated as an
all-encompassing "immobile space" in which all others are
moving uniformly. Thus the original question, "relative to what
frame of reference do the laws of motion hold?" is revealed to
be wrongly posed. For the laws of motion essentially
*determine* a class of reference frames, and (in principle) a
procedure for constructing them. For the same reason, a skeptical
question that is still commonly asked about the laws of
motion--why is it that the laws are true only relative to a
certain choice of reference frame?--is also wrongly posed. If
Newton's laws are true, then we can construct an inertial frame;
their truth doesn't depend on our ability to construct such a
frame in advance. Mach expressed the situation particularly
clearly:



> 
> It is very much the same whether we refer the laws of motion to
> absolute space, or express them abstractly, without express indication
> of the system of reference. The latter course is unproblematic and
> practical, for in treating particular cases the student of mechanics
> looks for a suitable system of reference. But owing to the fact that
> the first way, whenever there was any actual issue at stake, was
> nearly always interpreted as having the same meaning as the latter,
> Newton's error was much less dangerous than it would otherwise
> have been. (1933, p. 269.)
> 



Mach's remark roughly corresponds to Newton's actual
procedure. Even though, for Newton, absolute space was the implicit
reference-frame for stating the laws of motion, the frame for their
application was the standard one for most of the history of astronomy:
the fixed stars. This seemingly arbitrary starting-point did not
undermine Newton's procedure as an account of the "true
motions." For the framework of the fixed stars, initially taken
for granted, turns out to be justified in the course of Newton's
dynamical analysis. If all the accelerations relative to the fixed
stars can be analyzed into action-reaction pairs involving bodies
within the system--leaving no "leftover"
accelerations that need to be traced to some yet-unknown
influence--then we can conclude that the stars are a suitable
(sufficiently inertial) frame of reference after all. Newton could
appeal to a particular case to test this general point: the orbits of
the outer planets were stable with respect to the fixed stars, their
perihelia showing no measurable precession (unlike the perihelion of
Mercury, for a famous example). Newton argued, then, that a relative
space in which these apsides are stable is a sufficient approximation
to a space at rest or in uniform motion (cf. Book III, Proposition
XIV, 1687b, p. 420). 



Mach took particular notice of Newton's use of the relativity
principle in determining an appropriate reference-frame:



> 
> In order to have a generally valid system of reference, Newton
> introduced Corollary V of the *Principia*. He thought
> of... a coordinate system for which the law of inertia holds,
> fixed in space without rotation relative to the fixed stars. He could
> also allow an arbitrary origin and uniform translation of this
> system... without losing its usefulness... One can see that
> the reduction to absolute space was in no way necessary, since the
> reference frame is just as relatively determined as in any other case.
> (1933, p. 227.)
> 



For Mach, this was an important acknowledgement of Newton's
insight into the relativity of motion. Starting from Corollary V, the
concept of the inertial frame solved the problem of absolute rotation
and acceleration, as a problem internal to the system of
Newton's laws. Absolute space could be dispensed with, without
undermining Newton's dynamical distinctions among states of
motion. Of course, this point did not dismiss Mach's skeptical
questions regarding the laws themselves. Instead, it posed them in a
more precise form: Are Newton's laws truly general laws of
nature, that determine a class of privileged frames? Or do they only
describe motions relative to a particular material frame, the fixed
stars? Empirical evidence was insufficient to decide. Eventually
Mach's question helped to motivate Einstein's search for
new laws in which privileged frames would not play an essential role.
First, however, by comparing Newtonian mechanics with Maxwell's
electrodynamics, Einstein placed the notion of inertial frame on an
entirely new footing (see below,
 Section 2.2
*et seq.*). 


### 1.7 "Quasi-inertial" frames: Newton's Corollary VI



A striking aspect of Newton's treatment of indistinguishable
frames of reference was his discovery of approximately
indistinguishable frames: spaces that are accelerating, yet can be
treated, for practical purposes, as if they were at rest or in uniform
motion. Newton made this notion precise in Corollary VI to the laws of
motion: 



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



As Corollaries IV and V implied, for a given system of interacting
bodies, their center of gravity is unmoved by the actions of the
bodies among themselves, and will remain at rest or in uniform motion
as 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 an
inertial frame. Corollary VI shows that, in very special ideal
circumstances--accelerative forces that act equally on all bodies
within a system, and accelerate them all in parallel
directions--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 made
explicitly in Corollary VI. Rather, it was threefold. The second point
was that there was in fact a force acting equally and in parallel
lines, at least to a high approximation, on important systems of
celestial bodies. The system of Jupiter and its satellites, for
example, is obviously accelerating, as its center of gravity completes
an approximately elliptical orbit around the Sun bound by accelerative
forces toward the Sun's center. But because the accelerations of
all the bodies are nearly equal and parallel, their motions among
themselves are nearly the same as if no such forces acted, and the
system may be treated as the sort of system described in Corollary V.
Evidently the accelerations are unequal, since Jupiter and the
satellites are at varying distances from the sun, and they cannot be
parallel since they are all directed at the center of the sun. But
these differences of distance and direction are so small, in
comparison with the distance of the entire system from the sun, that
they may be neglected. And the same applies to the centripetal
acceleration of Saturn's system. 



Newton applied this same reasoning to the entire solar system: even if
the entire system were accelerating toward some unknown gravitational
source, he could treat the solar system itself as if it were an
isolated system. He argued, from the analysis of accelerations within
the system, that outside forces must be acting more or less equally
and in parallel directions on all parts of the system.



> 
> It may be imagined that the sun and planets are impelled by some other
> force 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 of
> the 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 bearing
> on the phenomena of the heavens.... (1687a, article 13.)
> 



Newton raised this point in order to show that the possibility of such
a force acting on the whole solar system would not affect his
calculations of the forces acting within the system. In the
calculation relevant to this passage, Newton was using Corollary VI to
defend the inference that the force responsible for Jupiter's
orbit is directed to the Sun rather than to the Earth: neglecting any
such imaginary force, "then all the remaining force by
which... Jupiter is urged will tend (by prop. 3, corol. 1) toward
the center of the sun" (ibid). This calculation formed an
important step in the argument for a heliocentric system. Such a use
of Corollary VI parallels, therefore, his use of Corollary V (and its
earlier form, "Law 3") to show that the "frame of
the system of the world" could be determined without regard to
the system's uniform motion in absolute space. 



Indeed, the analogy between the two kinds of case helps to explain
Newton's changing the relativity principle from a law to a
Corollary, for this coincides historically with his first use of
Corollary VI. The two Corollaries identify two classes of frames of
reference that may be treated as equivalent, because they involve,
respectively, theoretically and practically indistinguishable states
of motion. The frames corresponding to Corollary VI may be called
"quasi-inertial," because "approximately
inertial" might be misleading: a closed orbit around the
Sun--like that of Jupiter's system--is not a good
approximation to an inertial motion, and the system can hardly be
considered isolated. But over sufficiently limited segments of its
orbit, its motion is sufficiently close to being inertial.
Moreover--and most important--the accelerative forces toward
the sun are close enough to being equal and parallel that the forces
acting within the system can be effectively isolated from the forces
from without. Hence, while "quasi-inertial" is a useful
term 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 masses
bound in orbit around a larger mass is by no means isolated, yet in
the right conditions it may be treated as if it were. The modern term,
"local inertial frame," is not inappropriate (cf. Schutz
1990, p. 124). But it typically designates the local coordinate frame
of a given inertial observer, rather than the sort of "whole
space" that Newton had in mind, as encompassing a celestial
system 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 inertial
frame, with respect to which any such Newtonian system has a definite
acceleration, 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 mathematical
framework for approximative reasoning, to determine the precise degree
of isolation that a group of interacting bodies may be said to
possess. Proposition III of the *Principia* established
Newton's method for treating a body orbiting a second body which
is, itself, subject to a centripetal force:



> 
> Proposition III, Theorem III: Every body, that, by a radius drawn to
> the centre of another body, in any way moved, describes areas about
> that centre proportional to the times, is urged by a force compounded
> out of the centripetal force tending to that other body, and of all
> the 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 added
to the centripetal force that maintains the orbiting body in its
orbit.



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



> 
> Book I, Proposition LXV, Case 2: Suppose that the accelerative
> attractions towards the greater body to be among themselves
> reciprocally as the squares of the distances; and then, by increasing
> the distance of the great body till the differences of the straight
> lines drawn from that to the others in respect of their length, and
> the inclinations of those lines to each other, are less than any
> given, then the motions of the parts of the system, will continue with
> no errors except such as are less than any given. And because, by the
> small distance of those parts from each other, the whole system is
> attracted as if it were only one body, it will therefore be moved by
> this attraction as if it were one body....(1687b, p. 172.)
> 



Thus the situation described by Corollary VI, in Newton's
analysis, emerges as a limiting case of an orbiting system under an
inverse-square force. As the size of the orbit is arbitrarily
increased, the accelerations toward the center become
indistinguishable from equal and parallel accelerations. Evidently,
Newton's proposition provides, characteristically, a general
method for treating a variety of possible configurations. But it
enabled Newton to address the specific physical fact of the variation
in the Sun's gravity, and its consequences for the superposition
upon it of lesser gravitating systems. At the distance of Jupiter or
Saturn, a revolving system can be a very nearly regular Keplerian
system. As the distance to the Sun decreases, however, differences in
magnitude and direction of the accelerations become significant, and
at the distance of the Earth-Moon system the motions become nearly
intractable. The decisive factor is the proportion between the size of
the orbiting system and its distance from the center of attraction.




The actual existence of quasi-inertial frames, corresponding to the
abstract cases of Proposition LXV, was a crucial part of
Newton's argument for universal gravitation--more
precisely, that the force holding the planets and their satellites in
their respective orbits is, in fact, the same force as gravity. One
crucial ground for the identification was the fact that the
interplanetary force shares the most striking feature of terrestrial
gravity, namely, that it imparts the same acceleration to all
terrestrial bodies. This principle was discovered by Galileo, of
course, but Newton tested it more severely, and with a greater variety
of test bodies. He constructed pendulums of identical wooden boxes
suspended from strings of equal length, which he filled with different
materials; he found that these differences made no difference to the
speed of falling over many oscillations of the pendulums. By this
means he showed that Galileo's principle holds to a much higher
degree of accuracy than Galileo was able to show, and inferred that a
body's weight toward the earth is generally proportional to its
mass. (1687b, Book III, Proposition VI). But Newton extended this
principle beyond terrestrial gravity, to the accelerative forces
acting on the planets and their satellites. Proposition IV, Corollary
VI, from Book I, showed that an orbiting body that obeys
Kepler's third law is urged toward the center by an
inverse-square force. Newton could then show that the centripetal
forces acting on Jupiter's moons depend only on the
inverse-square of the distance towards Jupiter's center: 



> 
> since the satellites of Jupiter perform their revolutions in times
> that observe the sesquiplicate proportion of their distances from
> Jupiter's center, their accelerative gravities towards Jupiter
> will be inversely as the squares of their distances from
> Jupiter's centre; that is, equal, at equal distances....
> And by the same argument, if the circumsolar planets were let fall at
> equal distances from the Sun, they would, in their descent towards the
> Sun, describe equal spaces in equal times. But forces that equally
> accelerate unequal bodies must be as those bodies; that is to say, the
> weights of the planets towards the Sun must be as their quantities of
> matter. (Ibid.)
> 



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



The proportionality of weight to mass was understood in its broader
foundational significance, as the equivalence of gravitational and
inertial mass, through Einstein's "principle of
equivalence" (cf. Einstein 1916; see also Norton 1985). In
Einstein's reasoning, the identity of inertia and gravitation
helped to undermine the special status of inertial motion, and
suggested the extension of the relativity principle from inertial
frames to frames in any state of motion whatever. If an inertial
frame, K, cannot be distinguished from another frame K' that is
uniformly accelerated with respect to K, then K' may equally be
treated as a "privileged" or "stationary"
frame: "they have equal title as systems of reference for the
description of physical phenomena" (Einstein 1916, p. 114). This
circumstance undermines a defining characteristic of inertial frames:
that with respect to a given inertial frame, every other inertial
frame 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 of
> co-ordinates, K and K', we call the "principle of
> equivalence;" this principle is evidently intimately connected
> with the theorem of the equality between the inert and the
> gravitational mass, and signifies an extension of the principle of
> relativity to co-ordinate systems which are in non-uniform motion
> relatively to each other. In fact, through this conception we arrive
> at the unity of the nature of inertia and gravitation. (Einstein
> 1922).
> 



This reasoning, in turn, suggested the connection between the
gravitational field and the curvature of space-time. (See Einstein
1916; see also Related Entries:
 Einstein, Albert: philosophy of science |
 
 general relativity: early philosophical interpretations of).
 



Even independently of Einstein's theory, however, one might see
in hindsight that Newton's application of Corollary VI, and his
identification of what we've been calling quasi-inertial frames,
already undermined the idea of an inertial frame. As we saw, Newton
could treat a quasi-inertial system like that of Jupiter and its
moons, not as essentially different from the system of the Earth and
the Moon, but as a limiting case of such a system: as the whole system
becomes sufficiently far from the central body, the differences among
the accelerations of its parts toward their common center become
negligibly small. In other words, Newton could treat such systems, and
the mathematical description of the differences among them, as
revealing the structure and workings of the sun's gravity,
rather than as questioning the fundamental distinction between uniform
and accelerated motion. In that sense, he was not proposing the
"complete physical equivalence" of uniformly accelerating
systems with uniformly moving systems (inertial frames). But even
Newton recognized, as we also saw, that the solar system as a whole
might have an unknown and practically unknowable acceleration. In
effect, he explained why his analysis of accelerations of the bodies
in the system, among themselves, required no knowledge of any absolute
accelerations. In the 19th century, Maxwell, without questioning the
underlying framework of absolute space and time, pointed out that
Corollary VI implied a kind of relativity of acceleration (1878, pp.
51-52). Viewing the foregoing in hindsight, more recent
literature has suggested that the physics of the Principia did not
really require the notion of inertial frame at all, and therefore that
a weaker geometry than Newtonian spacetime would be a sufficient
background 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 be
unified in a curved space-time (cf. section 9, below), as a Newtonian
version of general relativity (cf. Malament 2012, chapter 4; see also
Knox, 2014 and Weatherall 2018). The relevant conceptual resources and
mathematical techniques for such approaches, evidently, developed only
in the aftermath of general relativity. We return to this theme in
2.5.


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


### 2.1 Inertial frames in Newtonian spacetime



By the early years of the 20th century, the notion of
inertial system seems to have been widely accepted as the basis for
Newtonian mechanics, even if the specific works of Lange and Thomson
were little noticed. In writing "On the electrodynamics of
moving bodies" in 1905, Einstein took it to be obvious to his
readers that classical mechanics does not require a single privileged
frame of reference, but an equivalence-class of frames, all in uniform
motion relative to each other, and in any of which "the
equations of mechanics hold good." Two inertial frames with
coordinates \((x, y, z, t)\) and \((x', y', z', t')\) are related by
the *Galilean transformations*,

 

\[\begin{align\*}
 x' &= x - vt \\
 y' &= y \\
 z' &= z \\
 t' &= t
\end{align\*}\]


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



The lack of a privileged spatial frame, combined with the obvious
existence of privileged states of motion--paths defined as
rectilinear in space and uniform with respect to time--suggests
that the geometrical situation ought to be regarded from a
four-dimensional *spatio-temporal* point of view. The
structure defined by the class of inertial frames can be captured in
the statement that *space-time* is a four-dimensional affine
space, whose straight lines (geodesics) *are the trajectories of
particles in uniform rectilinear motion*. See Figure 4.



![Figure 4](Fig4.png)

Figure 4: Inertial Trajectories as
Straight Lines of Space-time




That is, space-time is a structure whose automorphisms--the
Galilean transformations that relate one inertial frame to
another--are affine transformations: they take straight lines
into straight lines, and parallel lines into parallel lines. The
former condition implies that an inertial motion in one frame will be
an inertial motion in any other frame, and likewise for an
accelerating or rotational motion. The latter implies that
uniformly-moving particles or observers who are relatively at rest in
one frame will also be relatively at rest in another. An inertial
frame can be characterized as a family of parallel straight lines
"filling" space-time, representing the possible
trajectories of a family of free particles that are relatively at
rest. See Figure 5. Therefore, to assert that an inertial frame exists
is to impose a global structure on space-time; it is equivalent to the
assertion that space-time is an affine space that is flat. 



![Figure 5](Fig5.png)

Figure 5: Each of these families of
straight lines, \(F\_1\) and \(F\_2\), represents the trajectories of a
family of free particles that are relatively at rest, and therefore
each defines an inertial frame. Relative to each other, the frames
defined by \(F\_1\) and \(F\_2\) are in uniform motion.
  

Each of the surfaces *S* is a "hypersurface of absolute
simultaneity" representing all of space at a given moment;
evidently (given the Galilean transformations) two inertial frames
will agree on which events in space-time are simultaneous.




The form of the Galilean transformations shows that, in addition to
being affine transformations, they also preserve metrical relations on
time and space. Distinct inertial frames will agree on simultaneity,
and on (ratios of) time-intervals; they will also agree on the spatial
distance between points at a given moment of time. Therefore, in the
four-dimensional picture, the decomposition of space-time into
hypersurfaces of absolute simultaneity is independent of the choice of
inertial frame. Another way of putting this is that Newtonian
space-time is endowed with a *projection* of space onto time,
i.e. a function that identifies space-time points that have the same
time-coordinate. Similarly, absolute space arises from a projection of
space-time onto space, i.e. a function that identifies space-time
points that have the same spatial coordinates. See Figure 6. 



![Figure 6](Fig6aandb.png)

Figure 6




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





|  |  |
| --- | --- |
| Figure 7a | Figure 7b |
| (a) | (b) |


Figure 7:
(a) Here is a space-time diagram of motions relative to the inertial
frame in which \(O\_1\), \(O\_2\), and \(P\_1\) are at rest. This can be
seen as arising from the projection of each of their inertial
trajectories onto a single point of space. \(O\_3\) is in 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 (as
noted above) is at rest. \(O\_7\) and \(O\_8\) are revolving around
their 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 \(P\_1\) are
in 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 around
their center of gravity \(P\_2\), which (as noted above) is at
rest.



### 2.2 The conflict between Galilean relativity and modern electrodynamics



By the time that this representation of the Newtonian space-time
structure was developed, however, the Newtonian conception of inertial
frame had been essentially overthrown. First, 19th-century
electrodynamics raised again the question of a privileged frame of
reference: the conception of light as an electromagnetic wave in the
ether implied that the rest-frame of the ether itself should play a
distinguished role in electrodynamical phenomena. On the one hand,
physicists such as Maxwell and Lorentz were careful to point out that
velocity relative to the ether was not equivalent to absolute
velocity, because the state of motion of the ether itself was
necessarily unknown. In other words, this conception of light did not
necessarily violate the classical principle of relativity. On the
other hand, the existence of such a preferred frame made the
equivalence of inertial frames correspondingly less interesting, even
if it was true in principle. This is why the appearance of the idea of
inertial frame in the 1880s, as suggested earlier, was not of pressing
physical interest to the majority of physicists, and seemed to be a
mere philosophical sidelight. The attempts to measure the effects of
motion relative to the ether commanded considerably more
attention.



Second, the abandonment of the ether--following the failure of
attempts to measure velocity relative to the ether and, more
generally, the apparent independence of all electrodynamical phenomena
of motion relative to the ether--did not vindicate the Newtonian
inertial frame. Rather, it required a dramatically revised conception.
Special relativity might be said to have applied the relativity
principle of Newtonian mechanics to Maxwell's electrodynamics,
by eliminating the privileged status of the rest-frame of the ether
and admitting that the velocity of light is independent of the motion
of the source. As Einstein expressed it, "the same laws of
electrodynamics and optics will be valid for all frames of reference
for which the equations of mechanics hold good." (1905, p. 38.)
But as Einstein also pointed out, the invariance of the velocity of
light and the principle of relativity, at least in its Galilean form,
are incompatible. It simply makes no sense, according to Galilean
relativity, that any velocity should appear to be the same in inertial
frames 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 if
they disagree on simultaneity. Only the relativity of simultaneity
makes possible the invariance of the velocity of light. This means
that the transformations between inertial frames that preserve the
velocity of light will not preserve simultaneity. These are the
*Lorentz transformations*:

 

\[
x' = \frac{x-vt}{\sqrt{1-v^2/c^2}} \quad y'=y \quad z'=z \quad t'=\frac{t-vx/c^2}{\sqrt{1-v^2/c^2}}

\]


Evidently these transformations preserve the velocity of light, but
they do not preserve length and time. So the invariant quantities of
Newtonian mechanics, which presuppose invariant measures of length and
time, must now depend on the choice of inertial frame. In other words,
acceleration is an invariant quantity only when it is zero, for the
magnitude of a non-zero acceleration will differ for observers in
relative motion. Therefore the notions of force, mass, and
acceleration no longer provides an appropriate definition of an
inertial frame, except in the ideal case of a frame in which no forces
are acting. Einstein's definition instead appeals to the
invariant quantities of electrodynamics: an inertial frame is one in
which light travels equal distances in equal times in arbitrary
directions. What seems impossible, from the point of view of Galilean
relativity, is that a frame that moves uniformly relative to one frame
that satisfies Einstein's definition should also satisfy the
definition. But that seeming impossibility rests, again, on the
assumption that two inertial frames will have a common measure of
simultaneity. Einstein showed that light-signals provide an
empirically and theoretically reasonable definition of simultaneity,
in light of the empirical soundness of Maxwell's equations and
the apparent invariance of the velocity of light. In the absence of an
empirically reasonable alternative compatible with Galilean
invariance, there is no sound criterion of simultaneity that will give
the same results in different inertial frames. Of course, it would
remain true that in a given inertial frame, the motions of free
particles would satisfy the requirements of Lange's definition,
and particles that move uniformly in one such frame would also move
uniformly in any other. But such a definition is not a substitute for
Einstein's definition, since it must itself presuppose a
definition of simultaneity. Otherwise, its appeal to the measurement
and comparison of times and distances is without a sound empirical
basis. Einstein's definition places such measurements on an
empirical basis. Their results, however, will depend on the choice of
an inertial frame, and will vary systematically according to the
relative velocities of different inertial frames.



The space-time structure of special relativity thus differs
essentially from Newtonian space-time, and is called "Minkowski
space-time" since Minkowski (1908) first formulated
Einstein's theory in its four-dimensional form. It is an affine
space, like Newtonian space-time. In both cases, the trajectories of
free particles are straight lines of the affine structure, and a set
of parallel inertial trajectories (geodesics) corresponds to an
inertial frame. As we just saw, however, Newtonian space-time
presupposes the invariant division of space-time into
three-dimensional hypersurfaces of absolute simultaneity, and an
objective measure of distance between points at a given moment of
absolute time. Minkowski space-time is a four-dimensional vector space
with an invariant four-dimensional metrical structure, imposed by the
invariance of the speed of light. Instead of an invariant spatial
interval between simultaneous events, there is an invariant
spatio-temporal interval between any two points in space-time. Since
there is no invariant relation of simultaneity, the sets of
simultaneous events for any inertial frame are the hyperplanes
orthogonal to the trajectories that determine that frame. In other
words, the choice between two inertial frames determines a choice
between two distinct divisions of space-time into space and time. See
Figure 8:



![Figure 8](Fig8.png)

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



### 2.4 Simultaneity and reference-frames



The details of Einstein's argument and the structure of
Minkowski space-time can be found elsewhere (see, e.g., Einstein 1951
and Geroch 1978). Here only one more point is worth making. It could
be argued that Einstein's and Lorentz's view are
completely equivalent. That is, we could assume that there is indeed a
privileged frame of reference, and that the apparent invariance of the
velocity of light is explained by the effects on bodies of their
motion through the ether (the Lorentz contraction and time dilation).
This purported distinction between empirically indistinguishable
frames has often been criticized on straightforward methodological
grounds, but it could be (and surely has been) argued that it is more
intuitively plausible than the relativity of simultaneity. After all,
knowing that (as Einstein showed) the Lorentz contraction can be
derived from the invariance of the velocity of light does not, by
itself, entitle us to say which of the two is the more convincing
starting-point.



This is why it is so important that Einstein's 1905 paper begins
with a critical analysis of the entire notion of a frame of reference.
It is tacitly assumed by Lorentz's theory, and classical
electrodynamics generally, that we have a reference-frame in which we
can measure the velocity of light, and that there is a distinguished
frame--at rest with respect to the ether--in which its true
velocity would be measured. But how is such a reference-frame
determined? The distances between points in space can only be
determined if it is possible to determine which events are
simultaneous. In practice this is always done by light-signalling, if
only in the informal sense that we identify simultaneous events when
we see them at the same time. But if the spatial frame of reference is
determined by light-signals, and is then to be used to measure the
speed of light, we would appear to be going in a circle. For this
reason, Poincare concluded that determining the speed of light
is partly a matter of convention (1898). Before Einstein, however, it
was tacitly assumed that, while light-signalling is useful and
practical, it is not essential to the definition of simultaneity.
There would be, therefore, a fact of the matter about which events are
simultaneous that is independent of this method of signalling. This
assumption was actually made explicit by James Thomson. He
recognized--as few did before Poincare and
Einstein--that the measurement of distance involves



> 
> the difficulty as to imperfection of our means of ascertaining or
> specifying, or clearly idealizing, simultaneity at distant places. For
> this we do commonly use signals by sound, by light, by electricity, by
> connecting wires or bars, and by various other means. The time
> required in the transmission of the signal involves an imperfection in
> human powers of ascertaining simultaneity of occurrences at distant
> places. It seems, however, probably not to involve any difficulty of
> idealizing or imagining the existence of simultaneity. Probably it may
> not be felt to involve any difficulty comparable to that of attempting
> to form a distinct notion of identity of place at successive times in
> unmarked space. (1884, p. 380).
> 



In other words, Thomson assumed that it was not a difficulty in
principle, 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 arbitrary
choice of reference-frame, just as much as determinations of velocity.
Einstein's conclusion is, of course, entirely contingent on the
empirical facts of electrodynamics. It could have been avoided if
there were in nature a useful signal of some kind whose transmission
would provide a criterion of absolute simultaneity, so that the same
events would be determined to be simultaneous in all inertial frames.
Or, experiments such as those of Michelson and Morley might have
succeeded in exhibiting the dependence of the velocity of light on the
state of motion of the source. Then synchronization by light-signals
could still have been regarded as a mere practical substitute for a
notion of absolute simultaneity that stood on independent grounds,
empirically as well as conceptually. But as Einstein saw, because of
the apparent independence of the velocity of light of the motion of
the source, even "idealizing or imagining the existence of
simultaneity" involves light-signalling more essentially than
anyone could have realized. Unless some other criterion of
simultaneity is provided, therefore, the establishment of a spatial
frame of reference involves light-signalling in an essential way. In
the absence of such a criterion the speed of light cannot be, as
Lorentz supposed, empirically measured against the background of an
inertial frame. By appealing to the speed of light in defining
simultaneity, Einstein gave an empirically sound construction both for
spatio-temporal measurement and for a dynamically distinct class of
reference-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 the
concept of inertial frame and its role in electrodynamics, Einstein
should have turned almost immediately to call that concept into
question. But he became convinced, largely by his reading of Mach,
that the central role of inertial frames was an "epistemological
defect" that special relativity shared with Newtonian mechanics.
Only relative motions are observable, yet both of these theories
purport to identify a privileged state of motion and use it to explain
observable effects (such as centrifugal forces). Coordinate systems
are not observable, yet both of these theories assign a fundamental
physical role to certain kinds of coordinate system, namely, the
inertial systems. In either theory, inertial coordinates are
distinguished from all others, and the laws of physics are said to
hold only relative to inertial coordinate systems. In an
epistemologically sophisticated theory, both of these problems would
be solved at once: the new theory would only refer to what is
observable, which is relative motion; it would admit arbitrary
coordinate systems, instead of confining itself to a special class of
system. Why, after all, should any genuine physical phenomenon depend
on the choice of coordinate system?



Another way of expressing Einstein's view is to say that, in
Newtonian mechanics and special relativity, rotation is
"absolute" because the transformations between inertial
frames (Galilean or Lorentzian) preserve rotational states. Thus the
"absoluteness" of rotation arises precisely from singling
out one type of frame, by one type of transformation, instead of
allowing arbitrary transformations and arbitrary frames. Einstein held
that this epistemological insight had a natural mathematical
representation in the principle of *general covariance*, or the
principle that the laws of nature are to be invariant under
*arbitrary* coordinate transformations. More precisely, what
this means is that coordinate transformations are no longer required
(as in the affine spaces of Newtonian mechanics and special
relativity) to take straight lines to straight lines, and to preserve
further metrical structures appropriate to each theory, but only to
preserve the smoothness of curves (i.e. their differentiability). The
general theory of relativity was intended to be a generally covariant
theory of space-time, and its general covariance was intended to
express the general relativity of motion (Cf. Einstein 1916, section
3). The extent to which the theory realized Einstein's original
aims 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 Minkowski
space-time theories, in sum, rested on their shared assumption of the
uniformity of space-time. In Newtonian mechanics and special
relativity, the formal relations between inertial coordinate
systems--the Galilean and Lorentz transformations,
respectively--correspond to symmetry transformations of uniform
space-time, that is, a space-time with non-trivial global symmetries.
In Einstein's context, a coordinate transformation from the
coordinates of one "stationary system" to those of another
would generally not reflect a global symmetry of space-time, just to
the extent that the two were relatively accelerating. 



At the same time, we can say that the concept of inertial frame
retains 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 in
our language) an actual inertial frame from his analysis of the solar
system; Corollary VI implied that all local phenomena were compatible
with an acceleration of the entire system, by nearly equal and nearly
parallel accelerative forces acting on the sun, the planets, and their
satellites. By the very same reasoning, however, he established that
such a uniformly accelerating frame was sufficient for a successful
analysis of the relevant causes acting within the system--those
causes that determine the system's configuration. From this
analysis he could determine the configuration of the system, that is,
determine that our system is approximately Keplerian, with small
deviations from Keplerian motion accounted for by perturbative actions
of the members of the system on each other. In other words, though a
true inertial frame cannot be determined, a sufficient approximation
to an inertial frame provides a sufficient basis for this causal
account. 



In empirical testing and measurement in general relativity, the
treatment of individual gravitating systems follows an analogous
pattern. For the analysis of interactions within a local system, an
asymptotically flat solution to Einstein's equation plays a role
analogous to the Newtonian system that approximates the conditions of
Corollary VI. In general relativity, there can be non-negligible
post-Newtonian effects, such as spatial curvature, and the non-linear
superposition of gravitational fields. When they appear, empirical
consequences such as anomalous precessions and light-bending provide
tests of general relativity and other relativistic gravitation
theories. (See Will 2018, ch. 4, especially 4.1-4.3).
Abstractly, we might wish to treat such a system as isolated within a
region of space-time that "flattens out" at infinity.
Practically, it suffices to consider a region in which, at a
sufficient remove from a system of mutually gravitating masses,
curvature becomes negligible in comparison with the curvature induced
by those masses locally. In other words, analogously to the Newtonian
case, it is not necessary that such a system be isolated from external
influences. It suffices that external influences make a negligible
difference to the actions of the masses among themselves, and to the
resulting configuration of the system. In short, instead of providing
an exact account of the global symmetries of space-time, the idea of
an inertial frame still provides a crucial practical tool for the
empirical study of actual physical interactions.