## 1. Mach



Between the time of Newton and Leibniz and the 20th century,
Newton's mechanics and gravitation theory reigned essentially
unchallenged, and with that long period of dominance, Newton's
absolute space came to be widely accepted. At least, no natural
philosopher or physicist offered a serious challenge to Newton's
absolute space, in the sense of offering a rival theory that dispenses
with it. But like the action at a distance in Newtonian gravity,
absolute space continued to provoke philosophical unease. Seeking a
replacement for the unobservable Newtonian space, Neumann (1870) and
Lange (1885) developed more concrete definitions of the reference
frames in which Newton's laws
 hold.[1]
 In these and a few other works, the concept of the *set of
inertial frames* (those in which material bodies obey
Newton's three laws of motion) was first clearly expressed,
though it was implicit in both remarks and procedures found in
Newton's *Principia*. (See the entries on
 space and time: inertial frames
 and
 Newton's views on space, time, and motion)
 The most sustained, comprehensive, and influential attack on absolute
space was made by Ernst Mach in his *Science of Mechanics*
(1883).



In a lengthy discussion of Newton's *Scholium* on
absolute space, Mach accuses Newton of violating his own
methodological precepts by going well beyond what the observational
facts teach us concerning motion and acceleration. Mach at least
partly misinterpreted Newton's aims in the *Scholium*,
and inaugurated a reading of Newton's bucket argument (and by
extension the globes argument) that has largely persisted in the
literature since. (See
 absolute and relational space and motion: classical theories,
 section 4, for discussion of Newton's bucket argument.) Mach
viewed the argument as directed against a 'strict' or
'general-relativity' form of relationism, and as an
attempt to establish the existence of absolute space. (Strict
relationism denies that there is any such thing as an absolute motion;
all motion is merely relative, i.e., is nothing more than changes of
location relative to some arbitrarily chosen reference frame.) Mach
points out the obvious gap in the argument when so construed: the
experiment only establishes that acceleration (rotation) of the water
*with respect to the Earth, or the frame of the fixed stars*,
produces the tendency to recede from the center; it does not prove
that a strict relationist theory cannot account for the bucket
phenomena, much less the existence of absolute space.



The reader of the entry on
 absolute and relational space and motion: classical theories
 will recall that Newton's actual aim was simply to show that
Descartes' two kinds of motion are not adequate to account for
rotational phenomena. Newton's bucket argument showed that the
effects of rotational motion could not be accounted for by means of
the motion of the water relative to its immediate surroundings (the
bucket walls); Newton's thought experiment with two globes
connected by a cord was meant to show that one can determine whether
they are rotating about their common center (and if so, in which
direction) without needing any reference to anything external. By
pushing on opposite faces of the two globes and checking for an
increase or decrease in the tension in the cord, one can determine in
which sense the spheres are in rotation, if they are rotating at
all.



Although Mach does not mention the globes thought experiment
specifically, it is easy to read an implicit response to it in the
things he does say: nobody is competent to say what would happen, or
what would be possible, in a universe devoid of matter other than two
globes. In other words, Mach would question Newton's starting
premise that the cord connecting the two globes in an otherwise empty
universe might be under tension, and indeed under a wide range of
different quantities of tension. So, for Mach, neither the bucket nor
the globes can establish the existence of absolute space.


### 1.1 Two Interpretations of Mach on Inertia



Both in Mach's interpretations of Newton's arguments and
in his replies, one can already see two anti-absolute space viewpoints
emerge, though Mach himself never fully kept them apart. The first
strain, which we may call 'Mach-lite', criticizes
Newton's postulation of absolute space as a metaphysical leap
that is neither justified by actual experiments, nor methodologically
sound. The remedy offered by Mach-lite is simple: we should retain
Newton's mechanics and use it just as we already do, but
eliminate the unnecessary posit of absolute space. In its place we
need only substitute the reference frame of the fixed stars, as is the
practice in astronomy in any case. If we find the incorporation of a
reference to contingent circumstances (the existence of a single
reference frame in which the stars are more or less stationary) in the
fundamental laws of nature problematic (which Mach need not, given his
official positivist account of scientific laws), then Mach suggests
that we replace the 1st law with an empirically equivalent
mathematical rival, such as this one:



> 
> Mach's Equation (1960, 287)
>   
> 
> 
> \[ \frac{d^2 (\frac{\Sigma mr}{\Sigma m})}{dt^2} = 0 \]
> 
> 



In this equation the sums are to be taken over all massive bodies in
the universe. Since the top sum is weighted by distance, distant
masses count much more than near ones. In a world with a (reasonably)
static distribution of heavy distant bodies, such as we appear to live
in, the equation entails that the velocity of a free body will be
constant (to an extremely good approximation) in precisely those
frames that we already consider to be 'inertial' frames.
The upshot of this equation is that the frame of the fixed stars plays
the role of absolute space in the statement of the 1st law. This
proposal does not, by itself, offer an alternative to Newtonian
mechanics, and as Mach himself pointed out, the law is not
well-behaved in an infinite universe filled with stars; but the same
can perhaps be said of Newton's law of gravitation (see Malament
1995, and Norton 1993). But Mach did not offer this equation as a
proposed law valid in any circumstances; he avers, "it is
impossible to say whether the new expression would still represent the
true condition of things if the stars were to perform rapid movements
among one another." (p. 289)



It is not clear whether Mach offered this revised first law as a first
step toward a theory that would replace Newton's mechanics,
deriving inertial effects from only relative motions, as Leibniz
desired. But many other remarks made by Mach in his chapter
criticizing absolute space point in this direction, and they have
given birth to the Mach-heavy view, later to be christened
"Mach's Principle" by Albert
 Einstein.[2]
 The Mach-heavy viewpoint calls for a new mechanics that invokes only
relative distances and (perhaps) their 1st and 2nd time derivatives,
and thus is 'generally relativistic' in the sense
sometimes read into Leibniz's remarks about motion (see
 absolute and relational space and motion: classical theories,
 section 6). Mach wished to eliminate absolute time from physics too,
so he would have wanted a proper relationist reduction of these
derivatives also. The Barbour-Bertotti theories, discussed below,
provide this.



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



> 
> Newton's experiment with the rotating vessel of water simply
> informs us that the relative rotation of the water with respect to the
> sides of the vessel produces no noticeable centrifugal forces, but
> that such forces are produced by its relative rotation with respect to
> the mass of the earth and the other celestial bodies. No one is
> competent to say how the experiment would turn out if the sides of the
> vessel [were] increased until they were ultimately several leagues
> thick. (1883, 284)
> 



The suggestion here seems to be that the relative rotation in stage
(i) of the experiment might immediately generate an outward force
(before any rotation is communicated to the water), if the sides of
the bucket were massive enough. (Note that this response could not
have been made by Leibniz - even if he had wanted to defend
Machian relationism - because it involves action at a distance
between the water and the parts of the bucket, something he regarded
as a metaphysical absurdity.)



![a bucket with walls that are several leagues thick; it has water in it and it is rotating counterclockwise](machs-bucket.jpg)

Mach's thick-sided bucket




More generally, Mach-heavy involves the view that all inertial effects
should be derived from the motions of the body in question relative to
all other massive bodies in the universe. The water in Newton's
bucket feels an outward pull due (mainly) to the relative rotation of
all the fixed stars around it. Mach-heavy is a speculation that an
effect something like electromagnetic induction should be built into
gravity theory. (Such an effect does exist according to the General
Theory of Relativity, and is called 'gravitomagnetic
induction'. The Gravity Probe B mission was designed to measure
a gravitomagnetic induction effect on orbiting gyroscopes due to the
Earth's rotation.) Its specific form must fall off with distance
much more slowly than \(1/r^2\), if the theory is to be empirically
similar to Newtonian physics; but it will certainly predict
experimentally testable novel behaviors. A theory that satisfies all
the goals of Mach-heavy would appear to be ideal for the vindication
of strict relationism and the elimination of absolute quantities of
motion from mechanics.


### 1.2 Implementing Mach-heavy



Direct assault on the problem of satisfying Mach-heavy in a classical
framework proved unsuccessful for a long time, despite the efforts of
others besides Mach - for example, Friedlander (1896),
Fopl (1904), and Reissner (1914, 1915). (Between the late 19th
century and the 1970s, there was of course one extremely important
attempt to satisfy Mach-heavy: the work of Einstein that led to the
General Theory of Relativity. Since Einstein's efforts took
place in a non-classical (Lorentz/Einstein/Minkowski) spacetime
setting, we discuss them in the next section.) One very influential
approach to implementing Mach-heavy was promulgated in the work of
Barbour and Bertotti (1977); this has since developed into the
research programme of 'shape dynamics', and will be
discussed in more detail in section 3 below.


### 1.3 Mach-lite versus Mach-heavy



Mach-lite, like the relational interpretations of Newtonian physics
reviewed in the entry on
 absolute and relational space and motion: classical theories,
 section 5, offers us a way of understanding Newtonian physics without
accepting absolute position, velocity or acceleration. But it does so
in a way that lacks theoretical clarity and elegance, since it does
not delimit a clear set of cosmological models. We know that Mach-lite
makes the same predictions as Newtonian physics for worlds in which
there is a static frame associated with the stars and galaxies; but if
asked about how things will behave in a world with no frame of fixed
stars, or in which the stars are far from 'fixed', it
shrugs and refuses to answer. (Recall that Mach-lite simply says:
"Newton's laws hold in the frame of reference of the fixed
stars.") This is perfectly acceptable according to Mach's
philosophy of science, since the job of mechanics is simply to
summarize observable facts in an economical way. But it is
unsatisfying to those with stronger realist intuitions about laws of
nature.



If there is, in fact, a distinguishable privileged frame of reference
in which the laws of mechanics take on a specially simple form,
without that frame being determined in any way by relation to the
matter distribution, a realist will find it hard to resist the
temptation to view motions described in that frame as the
'true' or 'absolute' motions. If there is a
family of such frames, disagreeing about velocity but all agreeing
about acceleration, then the realist will feel a temptation to think
of at least acceleration as 'true' or
'absolute'. If such a realist believes motion to be by
nature a relation rather than a property (and not all philosophers
accept this; see the entry on
 absolute and relational space and motion: classical theories,
 section 1) then they will feel obliged to accord some sort of
existence or reality to the structure - e.g., the structure of
Galilean spacetime - in relation to which these motions are
defined. For philosophers with such realist inclinations, the ideal
relational account of motion would therefore be some version of
Mach-heavy.


## 2. Einstein



Einstein's Special Theory of Relativity (STR) is notionally
based on a principle of relativity of motion; but that principle is
'special' - meaning, restricted. The relativity
principle built into STR is in fact nothing other than the Galilean
principle of relativity, which is built into Newtonian
 physics.[3]
 In other words, while there is no privileged standard of velocity,
there is nevertheless a determinate fact of the matter about whether a
body has accelerated or non-accelerated (i.e., inertial) motion. In
this regard, the spacetime of STR is exactly like Galilean spacetime
(discussed in the entry on
 absolute and relational space and motion: classical theories,
 section 5). In terms of the question of whether all motion can be
considered purely relative, one could argue that there is nothing new
brought to the table by the introduction of Einstein's STR
- at least, as far as mechanics is concerned. (See the entry on
 space and time: inertial frames
 for a more detailed discussion.)


### 2.1 Relations Determine State of Motion?



In this subsection we will discuss an interesting sense in which, in
STR, the letter (if not the spirit) of classical relationism can be
considered vindicated: the spatio-temporal relations between material
things are, on their own, sufficient to fully determine the state of
motion of a body. The discussion here presupposes acquaintance with
STR and its basic mathematics, and will be hard to follow for readers
lacking that background; such readers should feel free to skip this
subsection, which is not necessary for following the material in the
rest of section 2.



As Dorling (1978) first pointed out, there is a sense in which the
standard absolutist arguments against 'strict' relationism
using rotating objects (buckets or globes) fail in the context of STR.
Maudlin (1993) used the same considerations to show that there is a
way of recasting relationism in STR that appears to be successful. STR
incorporates certain novelties concerning the nature of time and
space, and how they mesh together; perhaps the best-known examples are
the phenomena of 'length contraction', 'time
dilation', and the 'relativity of
 simultaneity.'[4]
 In STR both spatial distances and time intervals between events
- when measured in the standard ways - are frame-relative
(observers in different states of motion, i.e. at rest in different
reference frames, will 'disagree' about their sizes). The
standard classical relationist starting point - the
configuration of relative distances between the existing bodies *at
a moment of time* - does not exist, at least not as an
objective, observer- or frame-independent set of facts. Because of
this, when considering what spatial or temporal *relations* a
relationist should postulate as fundamental, it is arguably most
natural to restrict oneself to the frame-invariant spatiotemporal
'distance' between events in spacetime. This is given by
the *interval* between two points: \([\Delta x^2 + \Delta y^2 +
\Delta z^2 - \Delta t^2]\) - the four-dimensional analog of the
Pythagorean theorem, for spacetime distances. If one regards the
spacetime interval relations between point-masses-at-times as
one's basis, on which spacetime is built up as an ideal entity
(analogously to how Leibniz thought of 3-d space as an ideal entity
abstracted from spatial distance relations), then with only mild
caveats relationism works: the spacetime interval relations suffice to
uniquely fix how the material systems can be embedded (up to
isomorphism) in the 'Minkowski' spacetime of STR. The
modern variants of Newton's bucket and globes arguments no
longer stymie the relationist because (for example) the spacetime
interval relations among bits of matter in Newton's bucket at
rest are quite different from the spacetime interval relations found
among those same bits of matter after the bucket is rotating. For
example, the spacetime *interval* relation between a bit of
water near the side of the bucket, at one time, and itself (say) a
second later is *smaller* than the interval relation between a
center-bucket bit of water and itself one second later (times referred
to inertial-frame clocks). The upshot is that, unlike the situation in
classical physics, a non-rotating body cannot have all the same
spatiotemporal relations among its parts as a similar body in
rotation. We cannot put a body or system into a state of rotation (or
other acceleration) without thereby changing the spacetime interval
relations between the various bits of matter at different moments of
time, compared to what they would have been if the body had remained
non-accelerated or non-rotated. The facts about rotation and
acceleration, thus, supervene on spacetime interval
 relations.[5]



It is worth pausing to consider to what extent this victory for (some
form of) relationism satisfies the classical 'strict'
relationism traditionally ascribed to Mach and Leibniz. The
spatiotemporal relations that save the day against the bucket and
globes are, so to speak, mixed spatial and temporal distances. They
are thus quite different from the spatial-distances-at-a-time
presupposed by classical relationists; moreover they do not correspond
to relative velocities (-at-a-time) either. Their oddity is forcefully
captured by noticing that if we choose appropriate bits of matter at
'times' eight minutes apart, I-now am at *zero*
distance from the surface of the sun (of eight minutes
'past', since it took 8 minutes for light from the sun to
reach me-now). So we are by no means dealing here with an innocuous,
'natural' translation of classical relationist quantities
into the STR setting. On the other hand, in light of the relativity of
simultaneity (see
 footnote 5),
 it can be argued that the absolute simultaneity presupposed by
classical relationists and absolutists alike was, in fact, something
that relationists should always have regarded with misgivings. From
this perspective, instantaneous relational configurations -
precisely what one starts with in the theories of Barbour and Bertotti
discussed below - would be the things that should be treated
with suspicion.



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



Maudlin does not claim this version of relationism to be victorious
over an absolutist or substantivalist conception of Minkowski
spacetime, when it comes time to make judgments about the
theory's ontology. There may be more to vindicating relationism
than merely establishing a 1-1 correlation between absolute
motions and patterns of spatiotemporal relations.


### 2.2 The Relationist Roots of STR and GTR



The simple comparison made above between STR and Newtonian physics in
Galilean spacetime is somewhat deceptive. For one thing, Galilean
space*time* is a mathematical innovation posterior to
Einstein's 1905 theory; before then, Galilean spacetime had not
been conceived, and full acceptance of Newtonian mechanics implied
accepting absolute velocities and, arguably, absolute positions, just
as laid down in the *Scholium*. So Einstein's elimination
of absolute velocity was a genuine conceptual advance. Moreover, the
*Scholium* was not the only reason for supposing that there
existed a privileged reference frame of 'rest': the
working assumption of almost all physicists in the latter half of the
19th century was that, in order to understand the wave theory of
light, one had to postulate an aetherial medium filling all space,
wave-like disturbances in which constituted electromagnetic radiation.
It was assumed that the aether rest frame would be an inertial
reference frame; and physicists felt some temptation to equate its
frame with the absolute rest frame, though this was not necessary.
Regardless of this equation of the aether with absolute space, it was
assumed by all 19th century physicists that the equations of
electrodynamic theory would have to look different in a reference
frame moving with respect to the aether than they did in the
aether's rest frame (where they presumably take their canonical
form, i.e., Maxwell's equations and the Lorentz force law). So
while theoreticians labored to find plausible transformation rules for
the electrodynamics of moving bodies, experimentalists tried to detect
the Earth's motion in the aether. Experiment and theory played
collaborative roles, with experimental results ruling out certain
theoretical moves and suggesting new ones, while theoretical advances
called for new experimental tests for their confirmation or - as
it happened - disconfirmation.



As is well known, attempts to detect the Earth's velocity in the
aether were unsuccessful. On the theory side, attempts to formulate
the transformation laws for electrodynamics in moving frames -
in such a way as to be compatible with experimental results -
were complicated and
 inelegant.[6]
 A simplified way of seeing how Einstein swept away a host of problems
at a stroke is this: he proposed that the Galilean principle of
relativity holds for Maxwell's theory, not just for mechanics.
The canonical ('rest-frame') form of Maxwell's
equations should be their form in *any* inertial reference
frame. Since the Maxwell equations dictate the velocity *c* of
electromagnetic radiation (light), this entails that any inertial
observer, no matter how fast she is moving, will measure the velocity
of a light ray as *c* - no matter what the relative
velocity of its emitter may be. Einstein worked out logically the
consequences of this application of the special relativity principle,
and discovered that space and time must be rather different from how
Newton described them. STR undermined Newton's absolute time
just as decisively as it undermined his absolute space.


### 2.3 From Special Relativity to General Relativity



Einstein's STR was the first clear and empirically successful
physical theory to overtly eliminate the concepts of absolute rest and
absolute velocity while recovering *most of* the successes of
classical mechanics and 19th century electrodynamics. It therefore
deserves to be considered the first highly successful theory to
explicitly relativize motion, albeit only partially. But STR only
recovered most of the successes of classical physics: crucially, it
left out gravity. And there was certainly reason to be concerned that
Newtonian gravity and STR would prove incompatible: classical gravity
acted instantaneously at a distance, while STR eliminated the
privileged absolute simultaneity that this instantaneous action
presupposes.



Several ways of modifying Newtonian gravity to make it compatible with
the spacetime structure of STR suggested themselves to physicists in
the years 1905-1912, and a number of interesting
Lorentz-covariant theories were proposed (i.e., theories compatible
with the spacetime of STR, which is called 'Minkowski
spacetime' because Hermann Minkowski first revealed the
spacetime structure that Einstein's postulates in STR entail).
Einstein rejected these proposed theories one and all, for violating
either empirical facts or theoretical desiderata. But Einstein's
chief reason for not pursuing the reconciliation of gravitation with
STR's spacetime appears to have been his desire, beginning in
1907, to replace STR with a theory in which not only velocity could be
considered merely relative, but also acceleration. That is to say,
Einstein wanted if possible to completely eliminate all absolute
quantities of motion from physics, thus realizing a theory that
satisfies at least one kind of 'strict' relationism.
(Regarding Einstein's rejection of Lorentz-covariant gravity
theories, see Norton 1992; regarding Einstein's quest to fully
relativize motion, see Hoefer 1994.)



Einstein began to see this complete relativization as possible in
1907, thanks to his discovery of the Equivalence Principle (cf.
Lehmkuhl *forthcoming*). Imagine we are far out in space, in a
rocket ship accelerating at a constant rate \(g = 9.81 m/s^2.\) Things
will feel just like they do on the surface of the Earth; we will feel
a clear up-down direction, bodies will fall to the floor when
released, etc. Indeed, due to the well-known empirical fact that
gravity affects all bodies by imparting a force proportional to their
matter (and energy) content, independent of their internal
constitution, we know that any experiment performed on this rocket
will give the same results that the same experiment would give if
performed on the Earth. Now, Newtonian theory teaches us to consider
the apparent downward, gravity-like forces in the rocket ship as
'pseudo-forces' or 'inertial forces', and
insists that they are to be explained by the fact that the ship is
accelerating in absolute space. But Einstein asked whether there is
any way for the person in the rocket to regard him/herself as being
'at rest' rather than in absolute (accelerated) motion?
And the answer he gave is: Yes. The rocket traveler may regard
him/herself as being 'at rest' in a homogeneous and
uniform gravitational field. Such a field would entail an accelerative
force "downward" on every body that is equal in magnitude
and direction everywhere in space. This is unlike the Earth's
gravitational field, which varies depending on distance from the
Earth's center and points in different directions at different
locations. Positing the existence of such a field will explain all the
observational facts just as well as the supposition that he/she is
accelerating relative to absolute space (or, absolutely accelerating
in Minkowski spacetime). But is it not clear that the latter is the
truth, while the former is a fiction? By no means; if there
*were* a uniform gravitational field filling all space, then it
would affect all the other bodies in the world - the Earth, the
stars, etc, - imparting to them a downward acceleration away
from the rocket; and that is exactly what the traveler observes.



In 1907 Einstein published his first gravitation theory (Einstein
1907), treating the gravitational field as a scalar field that also
represented the (now variable and frame-dependent) speed of light.
Einstein viewed the theory as only a first step on the road to
eliminating absolute motion. In the 1907 theory, the theory's
equations take the same form in any inertial *or uniformly
accelerating* frame of reference. One might say that this theory
reduces the class of absolute motions, leaving only rotation and other
non-uniform accelerations as absolute. But, Einstein reasoned, if
uniform acceleration can be regarded as equivalent to being at rest in
a constant gravitational field, why should it not be possible also to
regard inertial effects from these other, non-uniform motions as
similarly equivalent to "being at rest in a (variable)
gravitational field"? Thus Einstein set himself the goal of
expanding the principle of equivalence to embrace all forms of
'accelerated' motion.



Einstein thought that the key to achieving this aim lay in further
expanding the range of reference frames in which the laws of physics
take their canonical form, to include frames adapted to any arbitrary
motions. More specifically, since the class of all continuous and
differentiable coordinate systems includes as a proper subclass the
coordinate systems adapted to any such frame of reference, if he could
achieve a theory of gravitation, electromagnetism and mechanics that
was *generally covariant* - its equations taking the same
form in any coordinate system from this general class - then the
complete relativity of motion would be achieved. If there are no
special frames of reference in which the laws take on a simpler
canonical form, there is no physical reason to consider any particular
state or states of motion as privileged, nor deviations from those as
representing 'absolute motion'. (Here we are just laying
out Einstein's train of thought; later we will see reasons to
question the last step.) And in 1915, Einstein achieved his aim in the
General Theory of Relativity (GTR).


### 2.4 General Relativity and Relativity of Motion



There is one key element left out of this success story, however, and
it is crucial to understanding why most physicists reject
Einstein's claim to have eliminated absolute states of motion in
GTR. Going back to our accelerating rocket, we accepted
Einstein's claim that we could regard the ship as hovering at
rest in a universe-filling gravitational field. But a gravitational
field, we usually suppose, is generated by matter. How is this
universe-filling field linked to generating matter? The answer may be
supplied by Mach-heavy. Regarding the 'accelerating'
rocket which we decide to regard as 'at rest' in a
gravitational field, the Machian says: all those stars and galaxies,
etc., jointly accelerating downward (relative to the rocket),
'produce' that gravitational field. The mathematical
specifics of how this field is generated will have to be different
from Newton's law of gravity, of course; but it should give
essentially the same results when applied to low-mass, slow-moving
problems such as the orbits of the planets, so as to capture the
empirical successes of Newtonian gravity. Einstein thought, in 1916 at
least, that the field equations of GTR are precisely this mathematical
replacement for Newton's law of gravity, and that they fully
satisfied the desiderata of Mach-heavy relationism. But it was not so.
(See the entry on
 early philosophical interpretations of general relativity.)



In GTR, spacetime is locally very much like STR's flat Minkowski
spacetime. There is no absolute velocity locally, but there are clear
local standards of accelerated vs non-accelerated motion, i.e., local
inertial frames. In these 'freely falling' frames bodies
obey the usual rules for non-gravitational physics familiar from STR,
albeit only approximately (this is sometimes called the 'strong
equivalence principle', and is discussed further in section 4
below). But overall spacetime is curved, and local inertial frames may
tip, bend and twist as we move from one region to another. The
structure of curved spacetime is encoded in the metric field tensor
**gab**, with the curvature encoding gravity
at the same time: gravitational forces are so to speak 'built
into' the metric field, geometrized away. Since the spacetime
structure encodes gravity and inertia, and in a Mach-heavy theory
these phenomena should be completely determined by the relational
distribution of matter (and relative motions), Einstein wished to see
the metric as entirely determined by the distribution of matter and
energy. But what the GTR field equations entail is, in general, only a
partial-determination relation.



We cannot go into the mathematical details necessary for a full
discussion of the successes and failures of Mach-heavy in the GTR
context. But one can see why the Machian interpretation Einstein hoped
he could give to the curved spacetimes of his theory fails to be
plausible, by considering a few simple 'worlds' permitted
by GTR. In the first place, for our hovering rocket ship, if we are to
attribute the gravity field it feels to matter, there has got to
*be* all this other matter in the universe. But if we regard
the rocket as a mere 'test body' (not itself substantially
affecting the gravity present or absent in the universe), then we can
note that according to GTR, if we remove all the stars, galaxies,
planets etc. from the world, the gravitational field does not
disappear. On the contrary, it stays basically the same locally, and
globally, in the simplest solution of the field equations, it takes
the form of empty Minkowski spacetime - precisely the
quasi-absolute structure Einstein was hoping to eliminate. Solutions
of the GTR field equations for arbitrary realistic configurations of
matter (e.g., a rocket ship ejecting a stream of particles to push
itself forward) are hard to come by, and in fact a realistic two-body
exact solution has yet to be discovered. But numerical methods can be
applied for many purposes, and physicists do not doubt that something
like our accelerating rocket - in otherwise empty space -
is possible according to the
 theory.[7]
 We see clearly, then, that GTR fails to satisfy Einstein's own
understanding of Mach's Principle, according to which, in the
absence of matter, space itself should not be able to exist.



A second example: GTR allows us to model a single *rotating*
object in an otherwise empty universe (e.g., a neutron star).
Relationism of the Machian variety says that such rotation is
impossible, since it can only be understood as rotation relative to
some sort of absolute space. In the case of GTR, this is indeed the
natural way to understand such a model: the rotation is best
understood as rotation relative to a 'background'
spacetime that is identical to the Minkowski spacetime of STR, only
'curved' by the presence of matter in the region of the
star.



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



GTR does not fulfill all the goals of Mach-heavy, at least as
understood by Einstein, and he recognized this fact by 1918 (Einstein
1918). And yet ... GTR comes tantalizingly close to achieving
those goals, in certain striking ways (cf. Hoefer 2014). For one
thing, GTR does predict Mach-heavy effects, known as
'frame-dragging': if we could model Mach's
thick-walled bucket in GTR, it seems clear that it would pull the
water slightly outward, and give it a slight tendency to begin
rotating in the same sense as the bucket (even if the big
bucket's walls were not actually touching the water). While GTR
does permit us to model a lone rotating object, if we model the object
as a shell of mass (instead of a solid sphere) and let the size of the
shell increase (to model the 'sphere of the fixed stars'
we see around us), then as Brill & Cohen (1966) showed, the
frame-dragging becomes complete inside the shell. In other words: our
original Minkowski background structure effectively disappears, and
inertia becomes wholly determined by the shell of matter, just as Mach
posited was the case. This complete determination of inertia by the
global matter distribution appears to be a feature of other models,
including the Friedman-Lemaitre-Robertson-Walker Big Bang models
that best match observations of our universe.



Finally, it is important to recognize that GTR is generally covariant
in a very special sense: unlike all other prior theories (and unlike
many subsequent quantum theories), it postulates no fixed
'prior' or 'background' spacetime structure.
As mathematicians and physicists realized early on, other theories,
e.g., Newtonian mechanics and STR, can be put into a generally
covariant form. But when this is done, there are inevitably
mathematical objects postulated as part of the formalism, whose role
is to represent absolute elements of spacetime structure (see Friedman
1983, Pooley 2017). What is unique about GTR is that it was the first,
and is still the only 'core' physical theory, to have no
such absolute elements in its covariant equations. (Whether these
claims are exactly correct is a matter of ongoing debate, relating to
the question of the 'background independence' of GTR: for
discussion, see e.g. Belot (2011), Pitts (2006), Read (2016), and
Pooley (2017).) The spacetime structure in GTR, represented by the
metric field, is at least partly 'shaped' by the
distribution of matter and energy. And in certain models of the
theory, such as the Big Bang cosmological models, some authors have
claimed that the local standards of inertial motion - the local
'gravitational field' of Einstein's equivalence
principle - are entirely fixed by the matter distribution
throughout space and time, just as Mach-heavy requires (see, for
example, Wheeler and Cuifollini 1995).



Absolutists and relationists are thus left in a frustrating and
perplexing quandary by GTR. Considering its anti-Machian models, we
are inclined to say that motions such as rotation and acceleration
remain absolute, or nearly-totally-absolute, according to the theory.
On the other hand, considering its most Mach-friendly models, which
include all the models taken to be good candidates for representing
the actual universe, we may be inclined to say: motion *in our
world* is entirely relative; the inertial effects normally used to
argue for absolute motion are all understandable as effects of
rotations and accelerations relative to the cosmic matter, just as
Mach hoped. But even if we agree that motions in our world are in fact
all relative in this sense, this does not automatically settle the
traditional relationist/absolutist debate, much less the
relationist/substantivalist debate. Many philosophers (including, we
suspect, Nerlich 1994 and Earman 1989) would be happy to acknowledge
the Mach-friendly status of our spacetime, and argue nevertheless that
we should understand that spacetime as a real thing, more like a
substance than a mere ideal construct of the mind as Leibniz insisted.
By contrast, other philosophers (e.g., Rynasiewicz 1995) argue that
due to the conceptual and mathematical novelties introduced in GTR,
the traditional absolute vs. relational motion debate simply fails to
make sense any more (on this question, see also Hoefer 1998).


## 3. Shape Dynamics



We turn now to a modern-day attempt to implement Mach-heavy known as
'shape dynamics'. (In fact, shape dynamics is just one
theory within this tradition, as we will see below.) This approach was
initiated - albeit not under that name - by Barbour and
Bertotti (1977, 1982). In tackling the problem of implementing
Mach-heavy, rather than formulating a revised law of gravity/inertia
using relative quantities, Barbour and Bertotti used the framework of
Lagrangian mechanics, replacing elements of the mathematics referring
to absolute quantities of motion with new terms invoking only relative
distances, velocities, etc. In this section, we presuppose a basic
familiarity with the Lagrangian framework. For a non-technical
introduction to shape dynamics, see Barbour (1999); for an up-to-date
review of recent work in the field, see Mercati (2018).



In this section, we survey the results and motivations of the shape
dynamics research program, focussing first on the above-mentioned
theory of Barbour and Bertotti (which recovers a subsection of the
solution space of Newtonian particle theory), before turning to the
Machian alternative to general relativity developed by Barbour and
collaborators: it is this latter theory which is shape dynamics
'proper'. Readers uninterested in the technical details of
this work can skip to section 3.5, in which its conceptual upshots are
discussed.


### 3.1. Configuration Space



For a given physical system, define its 'configuration
space' to be the space of possible instantaneous states of that
system. (For example, the space of possible distributions of N
particles in Euclidean space, according to a Cartesian coordinate
system laid down on that space.) As the system evolves, the point in
configuration space representing the system's instantaneous
state traces out a continuous curve. On this picture, metaphysically
possible worlds are represented by (rising) curves in the product
space formed from configuration space and a one-dimensional space
representing time. Nomologically possible worlds are represented by
those curves that are allowed by the dynamics. For example, in the
Lagrangian formalism, the nomologically possible worlds are
represented by those curves which extremize the action: a particular
functional of such curves.



Consider now, for the sake of concreteness, two Newtonian worlds which
differ by either a static or a kinematic Leibniz shift: that is,
constant translations or velocity boosts of the material content of
the universe (see the companion entry on
 absolute and relational space and motion: classical theories,
 for further discussion of such shifts). These two worlds will be
represented by distinct curves in configuration space. However, given
a configuration space, one can construct a 'reduced'
configuration space, in which certain such histories are
mathematically identified, or 'quotiented', such that they
are mapped to the same unique history in reduced configuration space.
Specifically, proponents of this approach define two such reduced
configuration spaces:


1. 'Relative configuration space', which is configuration
space quotiented by rigid translations and rotations. (I.e., curves in
configuration space which represent worlds which differ by rigid
translations or rotations are mathematically identified.)
2. 'Shape space', which its configuration space
quotiented by rigid translations, rotations, and dilatations. (I.e.,
curves in configuration space which represent worlds which differ by
rigid translations, rotations, or dilatations are mathematically
identified.)



(Two points here. First, recall that a 'dilatation' is a
scale transformation. Second, in what follows we will refer to the
group which consists of the union of translations, rotations and
dilatations as the 'similarity group'.) If these Machian
theorists are able to formulate a dynamics on shape space (i.e., a
dynamics which identifies the curves in shape space which represent
nomologically possible worlds), then that dynamics will, in light of
the above reduction, not bring with it a meaningful notion of absolute
position, or absolute velocity, or absolute scales. Barbour and
collaborators take such a dynamics to realize Mach-heavy: the
undetectable spacetime structure associated with such quantities has
been expunged. Below, we will see how this can be done in the concrete
contexts of Newtonian particle dynamics and general relativity.


### 3.2. Emergent Temporality



The Machian ambitions of Barbour and collaborators do not end there,
for these authors also seek to excise primitive temporal structure.
Initially, one might distinguish histories that correspond to a single
curve in configuration space being traced out at different rates with
respect to the primitive temporal parameter. Those working in this
tradition, however, view each curve in configuration space as
corresponding to exactly one possible history. They therefore elect to
dispose of the auxiliary one-dimensional space representing a
primitive absolute time which was introduced above. Instead, they seek
to construct an 'emergent' notion of temporality from
dynamics defined on configuration space alone. By way of a procedure
known as 'Jacobi's principle', the Machian
relationist selects a unique temporal parameter which maximally
simplifies this dynamics defined on configuration space. For the
details of Jacobi's principle, see Pooley (2013).


### 3.3. Best Matching



It is all well and good speaking of a dynamics defined on relative
configuration space, or shape space. However, it remains incumbent on
our Machian theorists to construct explicit dynamics appropriate for
these spaces: i.e., dynamics which do not recognise solutions related
by the action of the similarity group (viz., translations, rotations,
and dilatations) as being distinct. Given a dynamics on configuration
space, one can indeed achieve this task. The procedure which
implements this is known as 'best matching', and was
developed in the seminal work of Barbour and Bertotti (1982), in which
a version of Newtonian particle mechanics with dynamics formulated on
relative configuration space was first constructed. The extension to
shape space was undertaken in (Barbour 2003).



Informally, the goal of best matching is to use the similarity group
to minimize the intrinsic difference between successive points along a
history in configuration space. To take a simple example drawn from
Barbour (1999), consider the history of a particular triangle: the
triangle may, along that history, rotate, dilate, alter its internal
angles, and so on. However, at each point best matching allows one to
act on the triangle with similarity transformations; thereby,
triangles which at successive points along a history differ merely by
a translation, rotation or dilatation will be regarded as being
identical after best matching. In this way, a 'best
matched' history is selected, in which the intrinsic differences
between successive states of the system under consideration (in the
above example, the triangle) are minimised. While a metric on
configuration space will in general assign a different length to
histories differing by the action of the similarity group, the length
of the best matched history, constructed via the above procedure, will
induce a unique length of paths, and therefore metric, on shape
space.



A little more formally, the best matching procedure works as follows.
Consider a class of paths in configuration space, all corresponding to
the same path in shape space (i.e., consider a class of paths in
configuration space related by the action of the similarity group). As
mentioned above, a given metric on configuration space will in general
assign to each path in that space a different length; as a result, the
length of the associated path in shape space will be underdetermined.
However, starting from any given point p in configuration space, one
can use the action of the similarity group on configuration space to
define a unique curve, by shifting the points of any curve through p
along the corresponding orbits of the similarity group (think of these
'orbits' as contour lines in configuration space, relating
points which differ only by the action of the similarity group) so as
to extremize the length assigned to the curve (relative to the metric
under consideration). It is this extremized length which is assigned
to the unique curve in shape space. With each curve in shape space
assigned a unique length, one can then, as usual, specify a principle
which selects some such curves as representing nomologically possible
worlds, based upon their lengths. (Recall again, for example, that in
Lagrangian mechanics it is those curves which extremize an action
which are regarded as being dynamically possible.)


### 3.4. Relativistic Best Matching



The best matching prescription can be applied not only to Newtonian
particle theories, but also to other spacetime theories, including
GTR. (There is no reason why best matching cannot be applied to
Newtonian field theories, or to special relativistic particle
dynamics, but these steps are usually skipped by Machian relationists
following in the tradition of Barbour and Bertotti, who proceed at
this stage straight to GTR.)



To see how best matching works in the case of GTR, first note that a
certain subclass of solutions of that theory (namely, those which are
globally hyperbolic) can be formulated in terms of the '3+1
formalism', according to which the state of the universe at a
particular time is represented by a determinate 3-manifold with
associated Riemannian metric; dynamical equations then determine how
such 3-geometries evolve in time. (For a summary of the 3+1 formalism,
see e.g. Gourgoulhon (2012).) The Machian relationists working in the
shape dynamics research program take this 3+1 approach to GTR as their
starting point. They thus assume that instantaneous spaces which are
the points in configuration space have the determinate topology of
some closed 3-manifold without boundary. Configuration space is the
space of Riemannian 3-metrics on that 3-manifold. The natural analogue
of relative configuration space is, then, this space of Riemannian
3-metrics quotiented by diffeomorphisms, which are the generalisations
of Leibniz shifts appropriate to GTR (see the entry on
 the hole argument).
 The analogue of shape space in this case is the space of Riemannian
3-metrics, but quotiented in addition by local dilatations (by
'local', we mean here a transformation which can vary from
point to point).



Having constructed shape space in the relativistic case, one may then
best match in order to construct one's relational theory
implementing Mach-heavy (the metric on configuration space is defined
from the 3+1 dynamics of GTR): conceptually, the approach here is the
same as that presented in the previous section. Moreover, one can
again apply Jacobi's principle, in order to eliminate a
commitment to primitive temporal structure. In this case, the
resulting theory is known as 'shape dynamics', which
involves a commitment only to primitive conformal structure (i.e.,
facts about angles between objects) on the 3-geometries: all other
absolute quantities, the claim goes, have been excised. One way to
understand the relationship between GTR and shape dynamics is that one
trades the relativity of simultaneity but absoluteness of scales in
the former theory, for absolute simultaneity but the relativity of
scales in the latter.


### 3.5. Conceptual Matters



There are important differences between the relationship between
'standard' Newtonian particle mechanics and its
best-matched alternative on the one hand, and the relationship between
GTR and shape dynamics on the other. In the former case, the class of
solutions of the best-matched theory is a proper subset of the
solutions of Newtonian mechanics: for example, it includes only the
sector of the solution space of Newtonian mechanics which ascribes
zero angular momentum to the entire universe. Sometimes, this is
marketed as an advantage of the latter theory: the best-matched theory
predicts what was, in the Newtonian case, an unexplained coincidence.
(For discussion, see Pooley & Brown 2002.) In the latter case, by
contrast, it has been discovered that one can 'glue'
solutions of shape dynamics to construct new solutions, which are not
associated with any particular solution of GTR (in the sense that they
are not the best matched equivalents of any solution of GTR): see
(Mercati 2018). Thus, the solution spaces of GTR and shape dynamics
overlap, but the latter is not a proper subset of the former. Given
this, it is no longer clear that shape dynamics can be presented as a
'more predictive' alternative to GTR.



A second conceptual point to make regarding the Machian relationism of
Barbour and collaborators pertains to its motivations. Barbour claims,
as we have already seen above, that only spatial angles - and
not spatial scales, or a temporal timescale, or absolute velocities or
positions - are directly empirically observable. Thus, the
thought goes that an empiricist of good standing should favour (say)
shape dynamics over GTR, for the former theory, unlike the latter,
renders only such 'directly observable' quantities
meaningful; it does not commit to any absolute quantities which are
not 'directly observable'. There are, however, two central
points at which this reasoning could be questioned. First: one could
repudiate Barbour's empiricist motivations. Second: one could
deny that only angles are directly observable, or, indeed, that this
structure is directly observable at all (see Pooley 2013, p. 47). As
Pooley points out, these are not the strongest grounds on which to
motivate Barbour's project. Rather, a better motivation is this:
best-matched theories have the merit of ontological parsimony, as
compared with the theories such as Newtonian particle mechanics or
general relativity, to which the best-matching procedure is applied. A
second motivation has to do with the potential of this research
programme to present new avenues for exploration in the quest for a
quantum theory of gravity.



Our third and final point is this. Although it is possible to couple
shape dynamics to matter (see e.g. (Gomes 2012)), in this theory, just
as in GTR as discussed in the previous section, one also has vacuum
solutions, with primitive conformal structure on the 3-geometries.
Given the existence of these vacuum solutions, as with GTR, it is far
from clear that the theory makes good on the ambitions of Mach and the
early Einstein to construct a theory in which all spatiotemporal
notions are reduced to facts about matter. That said, it is worth
noting that, unlike in GTR, in shape dynamics one *cannot* have
a solution consisting of a single rotating body: the overall angular
momentum of the universe must vanish.


## 4. The Dynamical Approach



Since 2000, a new 'dynamical' approach to spacetime
structure has emerged in the works of Robert DiSalle (2006) and
especially Oliver Pooley and Harvey Brown (2001, 2006). This approach
is to be situated against an opposing, supposedly orthodox
'geometrical' approach to spacetime structure, as
encapsulated in the works of e.g. Janssen (2009) and Maudlin (2012).
(This is not to say that either the dynamical view or the opposing
geometrical view is a unified edifice, as we will see below.) The
dynamical-geometrical debate has many facets, but one can take the
central bone of contention to pertain to the arrow of explanation: is
it the case that the geometrical structures of spacetime explain why
material bodies behave as they do (as the geometrical view would have
it), or is it rather the case that the geometrical structure of
spacetime is explained by facts about the behaviour of material bodies
(as the dynamical view would have it)? Although this debate connects
with historical debates between substantivalists and relationists, it
should be regarded as a distinct dispute, for reasons to which we will
come.



While it is important to keep in mind the above disagreement regarding
the arrow of explanation when one is considering the
dynamical-geometrical debate, it will be helpful in this article to
hone in on two more specific claims of the dynamical approach, as
presented by Brown (2005), consistent with the above claim that it is
facts about the dynamics of material bodies which explain facts about
spatiotemporal structure, rather than vice versa. These two claims are
the following (Read 2020a):


1. Fixed background space-time structures, such as the Minkowski
space-time of STR, or Newton's absolute space, are to be
ontologically reduced to the symmetries of the dynamical equations
governing matter fields.
2. No piece of space-time structure, whether fixed or dynamical (in
the latter case, as in GTR) is necessarily surveyed by physical
bodies; rather, in order to ascertain whether this is so, one must
attend carefully to the details of the dynamics governing the
particular matter fields which constitute physical bodies.



On the first of these two points: proponents of the dynamical approach
maintain that the spacetime structure of our world is what it is
*because of the dynamical laws of nature and their symmetries*.
That is, the dynamical laws are (at least, relative to spacetime)
fundamental, and spacetime structure is derivative; in this sense, the
view is (at least in some cases) a modern-day form of relationism
(Pooley 2013, SS6.3.2) - albeit of a very different kind
from the relationist approaches considered up to this point. (Note,
though, that this relationism is a *corollary* of the above
explanatory contention of the dynamical approach; moreover, it is one
which is applicable only to theories which fixed spacetime structure
such as Newtonian mechanics or STR - and therefore *not*
to theories with dynamical spacetime structure, such as GTR. For this
reason, as already indicated above, proponents of the dynamical view
are not to be identified naively with relationists.)



On the second of these two points: the idea - what Butterfield
(2007) calls 'Brown's moral' - is that one
cannot simply posit a piece of geometrical structure in one's
theory, e.g. a Minkowski metric field in STR, and know *ab
initio* that material bodies (in particular rods and clocks) will
read off intervals of that structure; rather, whether this is the case
or not will depend upon the constitution of, and dynamics governing,
those material bodies. We will see below specific theories in which
any such assumption seems to fail. Note that this second point is
again consistent with the explanatory contention taken above to be
characteristic of the dynamical approach: a given piece of structure
inherits its operational significance as spacetime by dint of the
behaviour of material bodies.



Before addressing the second of these two points, we should consider
the first in greater detail. The claim that fixed spatiotemporal
structure is to be ontologically reduced to facts about material
bodies invites many questions, chief among which is perhaps the
following: to what could this ontological reduction possibly amount?
In the following section, we will see one particular metaphysical
programme which promises to make good on this claim.


### 4.1 The Dynamical Approach and Regularity Relationism



There is arguably a tight relationship between the geometrical
symmetries of a spacetime and the symmetries of a theory that
describes the physics of matter (in a broad sense, including fields)
in it. (Theories such as GTR, in which space-time has its own dynamics
are more complicated, and will be discussed later; for further
discussion of symmetries in physics, see the entry on
 symmetry and symmetry breaking.)
 Each symmetry is a set of transformations, with a rule of
composition: formally a 'group'. For instance, the group
of rotations in the plane has a distinct element for every angle in
the range 0-360 degrees; the composition of two rotations is the
single rotation through the sum of their angles. Spacetime symmetries
are those transformations which leave invariant a piece of spacetime
structure (e.g., the symmetries of Minkowski spacetime are
translations, spatial rotations and Lorentz boosts: together, the
so-called Poincare transformations); dynamical symmetries are
those transformations which leave invariant a set of dynamical
equations (e.g., the symmetries of Maxwell's equations of
electromagnetism are again the Poincare transformations). There
are good reasons to hold that the symmetry groups of theory and
spacetime must agree. First, since the theory describes matter, and
hence (arguably) what is measurable, any theoretical symmetries not
reflected in the postulated spacetime structure indicate unmeasurable
geometry: for instance, if an absolute present were postulated in
relativistic physics. While in the other direction, if there were
extra spacetime symmetries beyond those found in the dynamics, then
*per impossible* one could measure nonexistent geometric
quantities: for instance, a theory that depends on absolute velocities
cannot be formulated in Galilean spacetime (see the entry on
 absolute and relational space and motion: classical theories
 for further discussion of these Newtonian spacetime structures).
Famously, Earman (1989, ch. 3) declares that the matching of
space-time and dynamical symmetries is, thus, an 'adequacy
condition' on a physical theory.



Given this 'adequacy condition', a given geometry for
spacetime formally constrains the allowable theories to those with the
just the right symmetries: not too many, and not too few. It was an
assumption of many substantivalists (the views of whom are discussed
below) that this constraint was not merely formal, but ontological:
that the geometry is more fundamental than the laws, or that geometry
offers a 'real' explanation of the form of the laws
- such authors would, by the above categorization, qualify as
proponents of a geometrical view. However, that the symmetries should
agree does not specify any direction of dependence, and it could be
reversed, so that the geometric symmetries are ontologically
determined by those of the laws of the theory: hence the geometry
itself is an expression of the (symmetry properties of the) dynamics
of matter - transparently, this is consistent with the first of
the two specific commitments of the dynamical view discussed above. In
the words of Brown and Pooley (2006) (making these points about STR):
"... space-time's Minkowskian structure cannot be
taken to explain the Lorentz covariance of the dynamical laws. From
our perspective ... the direction of explanation goes the other
way around. It is the Lorentz covariance of the laws that underwrites
the fact that the geometry of space-time is Minkowskian."



Of the opposing geometrical approach to spacetime, Brown and Pooley
(2006, p. 84) question the *mechanism* by which autonomous
spacetime structure is supposed to explain or constrain the behaviour
of material bodies. Although we will keep our attention focussed on
the dynamical view in this subsection, rather than upon its opponents
(see the following subsection for more on the explanatory capacities
of spacetime), one might, however, ask at this point: does the
dynamical view really do better in this regard? How is it that
dynamical symmetries are supposed to explain, or account for,
spacetime structure? In the context of theories with fixed spacetime
structure, this question is answered by proponents of the dynamical
view via an *ontological reduction* of spatiotemporal structure
to symmetries of dynamical equations governing matter fields, as
indicated in (1) above. (In fact, this 'reduction' is
better described as a form of *elimination*, as we will see.)
But this, in turn, invites yet more questions: how, metaphysically, is
this ontological reduction operating? Can one in fact state dynamical
laws, or understand them as "holding" or
"governing", without *presupposing* facts about
spacetime structure?



Take, for example, Newton's laws of motion. The 1st law asserts
that bodies not acted upon by an external force will move with
constant velocity; similarly for the 2nd law and acceleration. These
laws seem to presuppose that these are meaningful terms, but in
spacetime terms their meaning is given by geometric structures: for
instance, constant velocity in Galilean spacetime means having a
straight spacetime trajectory. And the problem is not restricted to
Newtonian physics; the same point can be made regarding theories that
presuppose the Minkowski background spacetime structure, e.g., the
quantum field theories of the Standard Model.



It is increasingly well-appreciated that one suitable metaphysical
program to which the dynamical approach can appeal here is
Huggett's (2006) regularity relationism: see (Huggett 2009;
Pooley 2013; Stevens 2020). The idea is to consider the dynamical laws
as regularities that systematize and describe the patterns of events
concerning an underlying ontology/ideology that involves or
presupposes only very limited spatiotemporal features. To illustrate
how this approach might go, consider Pooley's (2013, SS6.3)
proposal that the dynamical approach to STR might postulate only
R4 topological spatiotemporal structure, which could be
(for example) attributed to a massive scalar field. Suppose we are
given a full 4D field description of such a field, in terms of some
arbitrary coordinate system. This would describe a simple
'Humean mosaic', to use David Lewis' term for
one's basic spatiotemporal and material commitments (see article
 David Lewis
 for further discussion). Now, smooth coordinate changes applied to
such a description will generate distinct mathematical representations
of that Humean mosaic, given using distinct coordinatizations of the
field-stuff. It might happen that, among all such representations,
there is a subclass of coordinate systems which are such that (i) when
the scalar field is described using a member of the class, it turns
out that its values at spacetime points satisfy some simple/elegant
mathematical equation; and moreover, (ii) the members of the class are
related by a nicely-specifiable symmetry group. If this is so, then
the simple/elegant equation can be taken as expressing a dynamical law
for the world of this mosaic (understood as a statement: "There
are frames in which ..."), and the symmetry group of the
law can be seen as capturing the *derivative*, not intrinsic,
spacetime structure of the world. If the symmetry group is the
Poincare group, for example, then the field behaves 'as
if' it were embedded in a spacetime with Minkowski geometry. But
all this means is that the dynamics is representationally equivalent
to a theory with an autonomous Minkowski geometry. From the point of
view of the dynamical approach, such a theory is merely an
interesting, and perhaps useful, representation of the real facts: and
it's a mistake to take every feature of a representation to
correspond to something in reality (Brown & Read 2020,
SS5).



Even granting that this regularity relationist understanding of the
dynamical approach goes through, three outstanding issues for the
dynamical approach deserve to be mentioned. First: given that the
proponent of the view seeks to excise metrical (more generally:
geometrical) spacetime structure, one might ask: why stop there? Is
there not something unnatural about excising fixed metric structure,
while taking topological structure to be primitive? Such a concern was
raised by Norton (2008), to which two responses have been offered in
the literature: (I) In direct response to Norton, Pooley points out
that "the project was to reduce chronogeometric facts to
symmetries, not to recover the entire spatiotemporal nature of the
world from no spatiotemporal assumptions whatsoever" (2013, p.
57). (II) Menon (2019) has argued that the machinery of
'algebraic fields' can be deployed in order to reduce
topological structure to facts about matter, thereby, if successful,
meeting Norton's challenge head-on. Second: how is one to extend
the dynamical approach, understood as a form of Huggett's
regularity relationism, to theories of dynamical space-time such as
GTR? Here, the lack of spacetime symmetries in the theory has posed
problems for the successful implementation of any such account
(Stevens 2014), although arguably initial progress in this regard has
been made by Vassallo and Esfeld (2016). Third: to which symmetries of
the laws is the dynamical approach supposed to be sensitive? In the
philosophy of physics, it is common to draw a distinction between
'internal' and 'external' symmetries: examples
of the former include U(1) gauge transformations in electromagnetism;
examples of the latter are coordinate transformations, such as
Galilean boosts in Newtonian mechanics. But there are many questions
here, such as: (i) how, precisely, is the distinction between internal
and external symmetries to be drawn? (ii) why should the proponent of
the dynamical approach stop at external symmetries? For discussion of
these questions, see (Dewar 2020).


### 4.2 Space-time and Explanation on the Dynamical Approach



We have already seen how the dynamical approach, *qua*
programme of ontological reduction, is supposed to play out in the
context of theories with fixed spacetime structure, including both
Newtonian theories and STR. We have also witnessed Brown and
Pooley's concerns about the ability of a substantival spacetime
to explain facts about the behavior of matter. These concerns are
motivated by apparent problem cases, in which the symmetries of a
substantival spacetime seem to come apart from those of the dynamical
laws governing matter. Such cases include: (i) Newtonian mechanics set
in Newtonian spacetime (Read 2020a); (ii) the Jacobson-Mattingly
theory (Jacobson & Mattingly 2001), in which dynamical symmetries
are a subset of spacetime symmetries, as a result of the presence of
an additional (dynamical) symmetry-breaking vector field (Read, Brown
& Lehmkuhl 2018).



It is not obvious that these critiques are fair to proponents of a
geometrical view. One might take their position not to be that a
certain piece of geometrical structure (e.g., the Minkowski metric of
STR) invariably constrains matter, whenever it is present in a theory,
to manifest its symmetries (a claim which seems to be false, in light
of the above cases). Instead, one might take their claim to be
conditional: *if* one has matter which couples to this piece of
geometrical structure in such-and-such a way, *then* that
geometrical structure can explain why the laws have the such-and-such
symmetries. In (Read, 2020a), the (arguably) straw man version of a
geometrical view critiqued by Brown and Pooley is dubbed the
'unqualified geometrical approach', in contrast with this
more nuanced and defensible version of the view, which is dubbed the
'qualified geometrical approach'. (Brown might still
reject the qualified geometrical approach on the grounds that it makes
explanatory appeal to objects which violate the 'action-reaction
principle', which states that every entity physical should both
act on, and react to, other physical entities (Brown 2005, p. 140). If
so, that this is the real reason for the rejection deserves to be
flagged; moreover, it remains open whether the objection succeeds
against the non-substantivalist versions of the geometrical view which
are discussed below.)



Focussing on the qualified geometrical approach, there are also
questions regarding the particular sense in which spacetime structure
can be said to be explanatory of dynamical symmetries. One notion of
explanation discussed in this literature is that of a
'constructive explanation'.This is derivative on
Einstein's distinction between 'principle theories'
and 'constructive theories' (Einstein 1919): for detailed
discussion, see (Brown 2005, SS5.2). In brief, a constructive
explanation is one in which phenomenological effects are explained by
reference to real (but possibly unobservable) physical bodies. (For
further discussion of how to understand constructive theories and
explanations, see (Frisch 2011).) With the idea of a constructive
explanation in mind, one can say this: if a proponent of a geometrical
view hypostatizes spacetime, then they can give constructive
explanations of certain physical effects by appeal to that spacetime
structure; otherwise, they cannot. That said, even if one does not
hypostatise spacetime, and so concedes that spacetime cannot offer
constructive explanations of the behaviour of matter, it is not
obvious that spacetime cannot still facilitate *other* kinds of
explanation. For discussions of these issues, see (Acuna 2016;
Dorato & Felline 2010; Frisch 2011; Read 2020b).


### 4.3 The Dynamical Approach and General Relativity



As we have already seen in section 2, spacetime in GTR is dynamical.
This leads Brown to maintain that there is no substantial conceptual
distinction between the metric field of GTR and matter fields:
"Gravity is different from the other interactions, but this
doesn't mean that it is *categorically* distinct from,
say, the electromagnetic field" (Brown 2005, p. 159). In this
sense, Brown is a relationist about GTR, and counts authors such as
(Rovelli, 1997) as allies. However, much caution is needed concerning
this use of the term 'relationism'. In particular, in the
context of GTR - and in significant contrast with his approach
to theories such as STR - Brown makes no claim that the metric
field should be ontologically reduced to properties of (the laws
governing) matter fields; rather, in light of its dynamical status,
the metric field of GTR "cries out for reification"
(Brown, personal communication). Indeed, even if Brown did not
maintain this, we have already registered above that there are
technical problems with attempting to apply the dynamical approach,
understood as a version of regularity relationism, to theories such as
GTR.



In light of these issues, when considering GTR, Brown (2005, ch. 9)
focuses entirely on thesis (2), presented in the introduction to this
section: no piece of geometrical structure has its
'chronogeometric significance' of necessity - that
is, no piece of geometrical structure is necessarily surveyed by
physical bodies; rather, in order to ascertain whether such is the
case, one must pay detailed attention to the dynamics of the matter
fields constituting those physical bodies. This, indeed, should
already be evident in light of the examples discussed in the previous
subsection, such as the Jacobson-Mattingly theory, in which matter
does not 'advert' to the designated piece of spacetime
structure.



This thesis (2) should be uncontroversial. There are, however,
concerns that the thesis is *so* uncontroversial that any
distinction between the dynamical approach and its opponents in the
context of theories such as GTR (and, in particular, without the
regularity relationist approach to ontological reduction applied in
the case of theories with fixed spacetime structure) has been effaced
(Pooley 2013; Read 2020a). Even setting this aside, there are also
disagreements regarding how exactly a piece of structure in a given
theory *is* to acquire its 'chronogeometric
significance' - that is, for the intervals which it
determines to be accessible operationally to physical bodies and
measuring devices. Brown's preferred answer to this question
(Brown 2005, ch. 9) makes appeal to the 'strong equivalence
principle'. There are a great many subtleties and technical
difficulties which need to be overcome in order to attain a clear
understanding of this principle (Read, Brown & Lehmkuhl 2018;
Weatherall 2020), but, roughly speaking, it states that, in local
regions in GTR, matter fields can be understood to obey Lorentz
covariant dynamical equations, just as in STR (we have already seen
something of this in section 2 above). Absent further details,
*pace* Brown, it is not clear why this is sufficient to secure
the 'chronogeometric significance' of the metric field in
GTR. Even setting this aside, there are questions regarding whether
the strong equivalence principle is *necessary* for
chronogeometric significance. For example, an alternative approach
might make appeal to the results of (Ehlers, Pirani & Schild,
1972), in which the authors demonstrate that the trajectories of
massive and massless bodies are sufficient to reconstruct the metric
field in GTR (cf. (Malament 2012, SS2.1)). These issues are raised
in (Read 2020a), but much work remains to be done in uncovering the
full range of ways in which a given piece of structure might come to
have chronogeometric significance.


## 5. Conclusion



This entry, and its companion on
 classical theories,
 have been concerned with tracing the history and philosophy of
'absolute' and 'relative' theories of space
and motion. Along the way we have been at pains to introduce some
clear terminology for various different concepts (e.g.,
'true' motion, 'substantivalism',
'absolute space'), but what we have not really done is say
what *the* difference between absolute and relative space and
motion is: just what is at stake? Rynasiewicz (2000) argued that there
simply are no constant issues running through the history from
antiquity through general relativity theory; that there is no stable
meaning for either 'absolute motion' or 'relative
motion' (or 'substantival space' vs
'relational space'). While we agree to a certain extent,
we think that nevertheless there are a series of issues that have
motivated thinkers again and again. Rynasiewicz is probably right that
the issues cannot be expressed in formally precise terms, but that
does not mean that there are no looser philosophical affinities that
shed useful light on the history and on current theorizing.



Our discussion has revealed several different issues, of which we will
highlight three as components of the 'absolute-relative
debate'. (i) There is the question of whether all motions and
all possible descriptions of motions are equal, or whether some are
'real' - what we have called, in Seventeenth Century
parlance, 'true'. There is a natural temptation for those
who hold that there is 'nothing but the relative positions and
motions between bodies' to add 'and all such motions are
equal', thus denying the existence of true motion. However,
arguably - perhaps surprisingly - no one we have discussed
has unreservedly held this view (at least not consistently): Descartes
considered motion 'properly speaking' to be privileged,
Leibniz introduced 'active force' to ground motion
(arguably in his mechanics as well as metaphysically), and
Mach's view seems to be that the distribution of matter in the
universe determines a preferred standard of inertial motion. In
general relativity there is a well-defined distinction between
inertial and accelerated motion, given by the spacetime metric, but
Einstein initially hoped that the metric itself would be determined in
turn by relative locations and motions of the matter distribution in
spacetime.



That is, relationists can allow 'true' motions if they
offer an analysis of them in terms of the relations between bodies.
Given this logical point, we are led to the second question: (ii) is
true motion definable in terms of relations or not? (And if one hopes
to give an affirmative answer, what kinds of relations are acceptable
to use in the reductive definition?) It seems reasonable to call this
the issue of whether *motion* is absolute or relative.
Descartes and Mach were relationists about motion in this sense, while
Newton was an absolutist. In the case of Einstein and GTR we linked
relational motion to the satisfaction of Mach's Principle, just
as Einstein did in the early years of the theory. Despite some
promising features displayed by GTR, and certain of its models, we saw
that Mach's Principle is certainly not fully satisfied in GTR as
a whole. We also noted that in the absence of absolute simultaneity,
it becomes an open question what relations are to be permitted in the
definition (or supervience base) - spacetime interval relations?
Instantaneous spatial distances and velocities on a 3-d hypersurface?
The shape dynamics program comes at this question from a new
perspective, starting with momentary slices of space (with or without
matter contents) which are given a strongly relational - as
opposed to absolute - interpretation. However, we argued that it
ultimately remains unclear whether this approach vindicates
Mach's Principle.



The final issue we have discussed in this article is that of (iii)
whether spacetime structures are substantial entities in their own
right, metaphysically speaking not grounded on facts about dynamical
laws, or whether instead it is best to think of the reality of
spacetime structures as dependent upon, and explained by, facts about
the world's dynamical laws, as advocates of the dynamical
approach maintain. The debate here is not the same as that between
classical relationism and substantivalism, although there are clear
affinities between the dynamical approach and classical relationism.
We explored how this issue takes quite different forms in the context
of special relativistic (Lorentz covariant) physical theories and in
the context of general relativistic theories.