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Forthcoming in the British Journal for the Philosophy of Science 

 

There is No Conspiracy of Inertia 

Ryan Samaroo 

Department of Philosophy, University of Bristol, Bristol BS6 6JL, UK 

ryan.samaroo@bristol.ac.uk 

 

Abstract 

I examine two claims that arise in Brown’s account of inertial motion in Physical Relativity 

(2005). Brown claims there is something objectionable about the way in which the motions of 

free particles in Newtonian theory and special relativity are coordinated. Brown also claims that 

since a geodesic principle can be derived in Einsteinian gravitation the objectionable feature is 

explained away. I argue that there is nothing objectionable about inertia and that, while the 

theorems that motivate Brown’s second claim can be said to figure in a deductive-nomological 

explanation, their main contribution lies in their explication rather than their explanation of 

inertial motion. 

 

1 Introduction 

2 The alleged conspiracy 

3 The conspirators unmasked 

3.1 The action-reaction principle 

3.2 Global coordinate systems as an artifice of thought 

3.3 Inertia as a concept in need of explanation 

4 If inertia is conspiratorial, should a broad class of theories be so characterised? 

5 The alleged explanation of inertia by Einsteinian gravitation 

6 Conclusion 

 

1 Introduction 

Conceptual analysis, at least in the analytic tradition since Frege, is the practice of identifying 

central features of a concept by revealing the assumptions on which use of the concept depends.1 

                                                
1This way of expressing the basic idea of conceptual analysis is due to Demopoulos (2000, p. 220). 



 2 

This approach to conceptual analysis has also been a part of the foundations of physics, at least 

since Newton. Conceptual analysis in physics is responsible to the body of theory and practice in 

which the concept is situated and in which it is interconnected with other concepts both physical 

and mathematical. The identification and explication of these connections, therefore, is a main 

objective of an analysis. 

 

There is, however, an older tradition in which conceptual analysis does not proceed in this 

way; concepts are analysed with reference to systems of metaphysical and methodological 

enquiry. The intuitions that drive this kind of enquiry are held to bear decisively on the analysis 

of physical theories and the concepts they comprise. This tradition, at least so far as the theory of 

space and time is concerned, is exemplified in certain arguments offered by Leibniz, Huyghens, 

Berkeley, Mach, and Einstein.2 This is not to say that these thinkers did not also pursue 

conceptual analysis in the sense characteristic of the analytic tradition, but their most famous and 

influential criticisms of Newton’s theory reflect underlying conceptions of substance, action, and 

causality that are taken at least as seriously as their views about physics. These conceptions, 

furthermore, are bound up with views about how knowledge of the structure of the world is 

gained, about the nature of scientific explanation, and about what an empirical theory may 

legitimately postulate. Harvey Brown’s Physical Relativity (2005) belongs to this tradition, and 

those claims that I will examine are motivated by a number of intuitions about inertial motion 

that have their source in Einstein. Like Einstein, Brown holds that there is something 

objectionable about inertial motion in Newtonian theory and special relativity. He calls the 

objectionable feature ‘the conspiracy of inertia’, and he claims that the conspiracy is explained 

away by Einstein’s theory of gravitation.3 

 

In this essay I will examine Brown’s account of inertial motion in Newtonian theory and 

special relativity. I will argue that there is nothing objectionable about inertia in these theories, 

                                                
2This is not to say that Newton’s views on space, time, and motion are free of philosophical intuitions. But Newton 
stands out among these thinkers for offering principles that constitute these concepts independently of any 
philosophical intuitions one may have about them. In this way, he ensures that the concepts are insulated from such 
intuitions, even his own. 
3These claims are separable from the principal claim of Physical Relativity, namely, that length contraction and clock 
retardation are in need of a dynamical explanation, and that such an explanation must come from a ‘constructive 
theory’, that is, a theory of the forces of cohesion that maintain a body’s configuration. This view has been criticised 
by Norton (2008), Hagar (2008), Janssen (2009), DiSalle (2012), and Nerlich (2013, Chapter 5). 



 3 

and I will argue that, while there is a sense in which Einsteinian gravitation explains inertia, the 

main contribution of the theorems that motivate Brown’s second claim lies in their explication 

rather than their explanation of inertia. 

 

The structure of the essay is as follows. In Section 2 I will begin by presenting Brown’s 

account of space-time structure and the metaphor of a conspiracy. In Section 3 I will explore a 

number of metaphysical principles or intuitions that seem to underlie the conspiracy, and I will 

argue that they are inimical to our understanding of inertia in Newtonian theory and special 

relativity. In Section 4 I will consider an implication of accepting Brown’s claim that there is 

something conspiratorial about inertia: I will consider the suggestion that, for a conspiracy 

theorist, all physical theories are conspiratorial. And I will argue that, even if such a view is 

defensible, there is little to gain by explaining away the conspiracy of inertia by appealing to 

Einsteinian gravitation, for one can point to conspiratorial features even in that framework. In 

Section 5 I will turn to Brown’s claim that inertia is explained by Einsteinian gravitation because 

a geodesic principle can be derived from the field equations. I will review Weatherall’s (2011b) 

challenge to Brown’s claim. Weatherall argues that, if there is any sense in which Einsteinian 

gravitation can be said to explain inertia, then geometrised Newtonian gravitation explains it at 

least as well. While I agree with Weatherall, I will argue that there is a better way of thinking 

about the geodesic theorems. Their main contribution lies not in their explanation of inertial 

motion but in their explication of it. 

 

2 The alleged conspiracy 

There is a view that can be found in Einstein (e.g., 1922 [1950]; 1924 [1991]) according to which 

space-time structure explains the motion of free particles. Free particles and light rays run along 

the ‘ruts’ and ‘grooves’ of the affine geodesics of space-time, much as trains run along tracks. On 

this view, there is a causal inference from the phenomena of motion to space-time structure. This 

view is sometimes called ‘the space-time explanation’ or ‘the causal-explanatory view’.4 

 

                                                
4For an entirely different account of space-time explanation, one that is explicitly non-causal, see Nerlich’s The 
Shape of Space (1994b), What Spacetime Explains (1994a), and Einstein’s Genie (2013, Chapter 8). 



 4 

Brown’s account of space-time structure is set against this view. The structure of space-

time is determined by the equivalence-class structure of some particular dynamical theory. For 

example, the structure of the space-time of Newtonian mechanics is the structure determined by 

the invariance of the laws of motion under the Galilean group; the structure of the space-time of 

special relativity is the structure determined by the invariance of Maxwell’s electrodynamics 

(and, as would later be discovered, the laws governing all non-gravitational interactions) under 

the Poincaré group. In both the Newtonian and special-relativistic frameworks, ‘geometric 

objects’ or ‘space-time structures’, as they are sometimes called, encode the equivalence-class 

structures determined by the dynamics. For example, an affine geodesic codifies the trajectory of 

a free particle. 

 

Einsteinian gravitation differs from these theories because the space-times corresponding 

to solutions of Einstein’s equations are not given a priori, but depend on the distribution of mass 

and energy in the universe. For this reason, the elements of space-time structure that figure in 

these accounts and the symmetries they admit are not given a priori. In much the same way as in 

Newtonian mechanics and special relativity, however, elements of space-time structure codify 

certain important features of physical processes. For example, time-like and null geodesics with 

respect to the Lorentzian metric codify the trajectories of free massive test particles and light 

rays.5 In all of these frameworks, space-time structures do not explain but codify important 

features of physical processes.6 

 

Brown’s view is significant for its rejection of the space-time explanation account.7 But it 

has another aspect that is just as problematic as the latter view: Newtonian theory and special 

relativity commit us to accepting something questionable. This is illustrated with a number of 

                                                
5Brown (2005, pp. 161-168) appeals to the geodesic theorem to establish that time-like geodesics encode the 
trajectories of free massive test particles. He notes (2005, pp. 169-177) that the strong equivalence principle must be 
assumed for null geodesics to encode the trajectories of light rays. The strong equivalence principle is not merely 
Einstein’s equivalence principle, according to which it is impossible to distinguish locally between a homogeneous 
gravitational field and uniform acceleration. Nor is it that principle together with the further claim that all non-
gravitational fields couple to the gravitational field. It is the conjunction of these principles and the principle of 
minimal coupling, according to which no terms of the special-relativistic equations of motion contain the Riemann 
curvature tensor. The strong principle ensures that in the neighbourhood of any event the structure of space-time is 
approximately locally Lorentzian so long as tidal gravitational effects can be ignored. 
6For a detailed account of how space-time structures encode certain important features of physical processes in 
Einsteinian gravitation, see Malament (2012, §2.1-2.3). 
7See also DiSalle (1995; 2006a) for a critique of the space-time explanation account. 



 5 

metaphors, among them, that of a conspiracy among the free particles of the universe. The 

metaphors can be found in a number of passages, of which the following are representative: 

Inertia, before Einstein’s general theory of relativity, was a miracle. I ... mean the ... 
postulate that force-free (henceforth free) bodies conspire to move in straight lines at 
uniform speeds while being unable, by fiat, to communicate with each other. (Brown, 
2005, pp. 14-15) 

 
A kind of highly non-trivial pre-established harmony is being postulated, and it takes the 
form of the claim that there exists a coordinate system xµ and parameters τ such that 
[d2xµ/dτ2 = 0] holds for each and every free particle in the universe. (Brown, 2005, p. 17) 

 
... there is a prima facie mystery as to why objects with no antennae should move in an 
orchestrated fashion. That is precisely the pre-established harmony, or miracle, that was 
highlighted above. (Brown, 2005, p. 24) 
 
... force-free particles have no antennae ... they are unaware of the existence of other 
particles. That is the prima facie mystery of inertia in pre-GR theories: how do all the free 
particles of the world know how to behave in a mutually coordinated way such that their 
motion appears extremely simple from the point of view of a family of privileged frames? 
(Brown, 2005, p. 142) 
 

I propose the following as a synthesis of these and other passages that exemplify what I call 

Brown’s alleged conspiracy: As a matter of definition, the free particles of the universe 
are non-interacting, and thus cannot detect other objects or even determine whether there 
are any.8 Yet, they seem to move in a mutually coordinated way. How do they know to 
move in the way that they do? Newtonian theory and special relativity commit us to 
thinking that there is a conspiracy among them. These theories assert that there exists a 
coordinate system xµ and parameters τ associated with each particle such that the equation 
d2xµ/dτ2 = 0 holds. 

 
To put the idea another way, one could say that the free particles of the universe agree not to 

accelerate and to follow geodesics of the space-time. Furthermore, particles that are themselves 

composites must satisfy the conservation of momentum. That is, the forces among the constituent 

particles must be equal and opposite, failing which the composite particles, by their internal 

forces, will accelerate of their own accord. Therefore, one could say that free composite particles 

must also conspire to maintain a state of equilibrium.9 (I will address this in detail in Section 5.) 

                                                
8Note that we are considering here only the framework of the laws of motion. In Newtonian gravitation, there is an 
interaction among all of the particles of the universe—every particle attracts every other with a force that is 
proportional to the product of their masses and inversely proportional to the square of the distance between them. 
9Although the alleged conspiracy may be understood completely in these terms, it is worth noting that, for Brown, 
there must be more than four particles for there to be a conspiracy. This idea is motivated by a Lange-style path-
construction proposed by Pfister (2004). While Brown’s discussion of Pfister’s construction goes part of the way 



 6 

 

It is worth noting that the conspiracy metaphor is awkward. Free particles seem to 

conspire in spite of the fact that they cannot communicate. This, presumably, is what is 

‘miraculous’. But, pressing on with the conspiracy metaphor, one could still ask why the particles 

are prohibited from conspiring to move according to some law—for example, a law relating their 

motion to the distribution of mass-energy.10 Perhaps the metaphor of a pre-established harmony 

is more apt. 

 

There may be better metaphors or better ways of fleshing out the existing ones with the 

relevant physics. But I am not at all concerned with the metaphors themselves—I am concerned 

with the view that underlies them. I will argue that the view in question is driven by a number of 

metaphysical intuitions that are inimical to our understanding of inertia. 

 

3 The conspirators unmasked 

I will examine three metaphysical principles or intuitions that seem to underlie the allegation of a 

conspiracy. I will show that Brown’s view, though set against the causal-explanatory account of 

space-time, is reminiscent of Einstein’s view of inertia in Newtonian theory and special relativity. 

I wish to emphasize, however, that these are plausibility arguments—all of which have textual 

support, but none of which may account for the alleged conspiracy. 

 

3.1 The action-reaction principle 

The most notable of these principles can be extracted from Einstein’s account of inertial frames 

in Newtonian theory and special relativity as ‘factitious causes’ of inertial effects. This is 

exemplified in his (1916 [1952]) illustration involving two fluid bodies S1 and S2 that are alike in 

size and constitution. The bodies are sufficiently far apart from each other and from other masses 

that they are subject only to those gravitational forces that arise among their constituent parts. S2 

rotates with constant velocity about the line joining them. S1 is perfectly spherical while S2 

                                                                                                                                                        
towards explicating the mathematical requirements for inertia, I will argue that it distracts from what really underlies 
the alleged conspiracy, namely a set of metaphysical principles or intuitions. 
10Or, pressing the conspiracy metaphor further, do free particles conspire or is it the ‘parts of space’ that conspire to 
have certain symmetries? 



 7 

bulges at the equator in the manner of a body subject to a centrifugal force. Einstein asks for the 

explanation of the difference between these bodies and he replies: 

Newtonian mechanics does not give a satisfactory answer to this question. It pronounces 
as follows: The laws of mechanics apply to the space R1, in respect to which the body S1 
is at rest, but not to the space R2, in respect to which body S2 is at rest. But the privileged 
space R1 of Galileo, thus introduced, is a merely factitious cause, and not a thing that can 
be observed. (Einstein, 1916 [1952], pp. 112-115) 
 

Newtonian theory fails to give a satisfactory answer because the space R1—namely, the inertial 

frame—is invoked as the cause of the difference between the two bodies. This is philosophically 

objectionable because something unobservable is being granted a causal role and because this 

cause acts without being acted upon. Because of these features, Einstein holds the inertial frame 

to be a factitious cause, one that must be replaced by a genuine cause like the fixed stars. 

Einstein’s illustration bears out what some have called ‘the action-reaction principle’: For 

something to be a physical entity it cannot act without being acted upon.11 Passages supportive of 

this idea can also be found in Relativity: The Special and the General Theory (1916 [1939], pp. 

171-173), The Meaning of Relativity (1922 [1950], pp. 54-55), ‘On the Ether’ (1924 [1991], pp. 

15-18), and ‘The Mechanics of Newton’ (1926 [1954], p. 260).12 

 

It is significant that Einstein’s account of inertial frames in Newtonian theory and special 

relativity is based on a serious distortion of Mach’s and especially Newton’s views on rotation.13 

                                                
11Brown (2005, pp. 140-142) points out that a similar principle can be distilled from Leibniz’s philosophy: For 
something to be a substance it cannot act without being acted upon. See, for example, the Discourse on Metaphysics 
(section 14) and Monadology (proposition 61). But Brown and Lehmkuhl (forthcoming, p. 3) question whether the 
action-reaction principle can be accurately attributed to Leibniz. So, in the end, it is left undecided whether or not the 
idea has anything to do with Leibniz. 
12Further mention of the idea can be found in Einstein’s correspondence. See Brown and Lehmkuhl (forthcoming) 
for references. 
13The distortion to which I am referring can be found in Einstein (1916) and elsewhere (e.g., Rindler, 2006, p. 8). On 
the distorted reading of Newton, it is the inertial frames (sometimes, the absolute space) of Newtonian mechanics 
and special relativity that cause the rotating fluid body to bulge at the centre. On the distorted reading of Mach, it is 
the fixed stars that cause the bulge. But Newton did not regard inertial frames or absolute space as the cause of the 
bulging—he took this ‘endeavour to recede’ from the axis of motion as a criterion of rotation. In Mach’s work we 
certainly do find the suggestion that some theory other than Newton’s might provide a more fundamental or more 
encompassing account of inertia. This is his suggestion that the fixed stars be treated as causally relevant to inertial 
motion (Mach, 1901 [1902], p. 234). But his criticism of Newton’s concepts of absolute space and time is distinct 
from this suggestion. He argued that claims about absolute motion tacitly refer to the fixed stars and not to absolute 
space; claims about the uniform flow of absolute time tacitly refer to the motion of some body that travels equal 
distances in equal times, and Mach proposed that we take as equal time intervals those in which the Earth turns 
through equal angles. For a careful analysis of Newton’s and Mach’s views on rotation, see Stein (1967) and DiSalle 
(2002; 2006a). 



 8 

However, the principle that is motivated by this account may be considered in its own right. I will 

consider three objections one might raise against it—independently of anything Brown has 

written—before considering its bearing on Brown’s view. 

 

To begin with, one might argue that the action-reaction principle is neither a metaphysical 

criterion of physicality nor an epistemological criterion of legitimate postulation but an arbitrary 

invention or ‘mere hypothesis’. One might object, as Norton (1993, pp. 848-849) and Pitts (2006, 

p. 349) have, that a spurious necessity has been attributed to a principle that derives from a 

vaguely Leibnizian or Aristotelian metaphysics, and thus is not empirically constrained. Second, 

one might argue that to think of an inertial frame in Newtonian theory or special relativity as 

something that acts without being acted upon is to misunderstand its role in these theories. 

Neither Newtonian space-time nor the inertial frame is being postulated as a theoretical entity 

that is the cause of inertial effects, for such a theoretical entity would go against Newton’s theory 

in which one is always dealing with interactions in which the participants enter reciprocally. 

Rather, Newtonian space-time is the structure that is implicit in Newton’s account of causal 

influence; Minkowski space-time is the structure that is implicit in special relativity. To be sure, 

Newtonian space-time and Minkowski space-time express constraints on the possible evolution 

of fields—in just the same way that the Hilbert space structure of quantum mechanics and the 

configuration space of classical mechanics express constraints on possible states of systems. But 

such constraints do not represent the action of these structures on the fields in Einstein’s sense.14 

Third, one might point out that the action-reaction principle amounts to a principle that excludes 

a priori the possibility that space-time is flat. But whether or not it makes sense to think of space-

time as flat ought to be an empirical question. For example, the equivalence principle—it is 

impossible to distinguish locally between a homogeneous gravitational field and uniform 

acceleration—provides a basis for arguing that space-time is not flat. Therefore, there is certainly 

a basis for arguing that it does not make sense to think of space-time as unaffected by matter, but 

that argument is founded on an empirical hypothesis and not an a priori demand. Without the 

equivalence principle, one would have in Newtonian theory and special relativity space-time 

theories that are empirically unexceptionable. In such a case, the action-reaction principle would, 

                                                
14The recognition of this important point can be found in DiSalle (2002, p. 182), Brown (2005, p. 139), Pitts (2006, 
p. 349), Nerlich (2013, Chapter 8), and Brown and Lehmkuhl (forthcoming, pp. 2-3). 



 9 

strictly speaking, express nothing but a metatheoretical or metaphysical preference for a different 

sort of theory—a sort of theory that, in the absence of the equivalence principle or something like 

it, would be difficult to motivate empirically. 

 

Brown’s view of inertial motion is bound up with the action-reaction principle, though the 

precise bearing of the principle needs to be carefully identified. A few positions can be found: 

endorsement (Anandan and Brown, 1995), but without any suggestion that the principle is 

violated by Newtonian mechanics and special relativity; a more measured stance towards the 

principle (Brown, 1996; Brown, 2005); most recently, a neutral historical study of the role played 

by the principle in Einstein’s thought (Brown and Lehmkuhl, forthcoming). What is common to 

these works is the view that the global affine and conformal structures that are presupposed by 

inertial frames in Newtonian theory and special relativity do not play a causal role as Einstein 

claimed, and so do not violate the action-reaction principle. These structures are ‘a codification of 

certain key aspects of the behaviour of particles and fields’ (Brown, 2005, p. 142). 

 

It is interesting to contrast Brown’s view of space-time structures as a codification with 

Weyl’s view. While Einstein held that inertial structure in Newtonian theory and special relativity 

is a factitious cause, Weyl held that once inertial structure is understood to be inseparable from 

gravitation it must be recognised as something that ‘not only exerts effects upon matter but in 

turn suffers such effects’ (Weyl, 1927 [1949], p. 105). He referred to the inertial structure of 

space-time as ‘the guiding field’ in analogy with other physical fields, notably fluids. Brown 

remarks that ‘To appeal ... to the action of a background space-time connection in which the 

particles are immersed—to what Weyl called the “guiding field”—is arguably to enhance the 

mystery, not to remove it.’ (Brown, 2005, p. 142) Weyl’s account of the guiding field, with 

Brown’s emphasis on its fluid aspect, enhances the mystery because free particles do not know 

what kind of space-time they are immersed in—as Brown has put it, ‘they just do what they do’ 

(personal communication).15 Though Brown does not acknowledge Weyl’s view except in these 

few remarks, it is safe to say that he dismisses the guiding field because it has a measure of 

explanatory power that is reminiscent of Einstein’s causal-explanatory account and that goes 

                                                
15But there is another reading of Weyl that focuses on his account of the ‘world structure’ that is exhibited in inertial 
motion rather than on his account of the guiding field. See, e.g., DiSalle (2006a, pp. 137-149; 2006b). 



 10 

against his view that space-time structure should be regarded as a codification or representational 

framework. His use of ‘codification’ in fact represents a deliberate deflation of inertial structure 

as something with explanatory power. 

 

On Brown’s account, the global affine and conformal structures of Newtonian theory and 

special relativity do not figure in a causal explanation of inertial motion. But his account has 

another aspect: Brown emphasises that these structures are absolute or global—they do not 

couple to matter. It is this lack of coupling that makes the world-lines of free particles ‘wholly 

unexplained’ (Brown, 1996, p. 186) and that ‘enhance[s] the mystery’ (Brown, 2006, p. 142). So, 

though Brown claims that the action-reaction principle is satisfied by Newtonian theory and 

special relativity, we may ask whether there is nonetheless a sense in which the principle is 

violated: Absolute or global background-structures constrain the evolution of fields without being 

themselves influenced by them. The proper account of this constraint, if not in terms of ‘acting’ 

or ‘causing’ as usually understood, is an open problem and one that has been examined at length, 

notably by Nerlich (e.g., 1994a, Chapter 7; 1994b, Chapter 2; 2013, Chapters 8 and 9). But, 

setting aside the question of how we should understand the relation of space-time structures to 

physical processes, there is at least a defensible interpretation of Brown’s view on which the 

action-reaction principle is violated by Newtonian theory and special relativity and satisfied by 

Einsteinian gravitation. What is questionable, if the above arguments are on the mark, is whether 

the principle, in any sense, should be defended at all. 

 

3.2 Global coordinate systems as an artifice of thought 

Another philosophical intuition about inertial structure is found in Einstein’s view that global 

coordinate systems are an artifice of our thought. Nature is indifferent to our choice of coordinate 

systems and does not single out certain kinds.16 Einstein writes: 

                                                
16This intuition is bound up with the principle of general covariance, according to which the possible laws of physics 
should be restricted to those that admit a coordinate-independent formulation. The satisfaction of the principle of 
general covariance was supposed to be a philosophical advantage of Einsteinian gravitation, one that eliminated the 
‘epistemological defect’ peculiar to Newtonian theory and special relativity with their global inertial frames. When 
Kretschmann showed in 1917 that Einsteinian gravitation is not unique in this respect, Einstein (1918 [2002], p. 242; 
1951, p. 69) proposed an alternative to the principle that he took to capture the theory’s characteristic feature and to 
surmount Kretschmann’s objection: The possible laws of physics should not only admit coordinate-independent 
formulations but these formulations should also be the simplest and most transparent ones available to them. Einstein 
claimed that this methodological principle has ‘significant heuristic force’. The notion of ‘theories that are not the 
simplest and most transparent in generally-covariant form’ means ‘theories that, in addition to being generally 



 11 

What makes this situation appear particularly unpleasant is the fact that there should be 
infinitely many inertial systems, moving uniformly and without rotation with respect to 
one another, that are distinguished from all other rigid systems. (Einstein, 1951, pp. 27-
29) 

 
There are a number of objections to this intuition. First, as DiSalle (2002, pp. 178-180) 

has argued, saying that it is inherently mysterious that nature should single out certain kinds of 

inertial frames and their associated coordinate systems amounts to saying that it is inherently 

mysterious that space-time should have non-trivial symmetries. It may be that the existence of 

such symmetries and the dynamical laws that exhibit them are themselves mysterious, but the 

sense of mystery derives from a philosophical view, whatever that might be. Even if one were 

committed to such a view—one more akin to a form of apriorism than empiricism—it would be 

equally problematic that nature distinguishes conservative systems from all other physical 

systems. Second, one might point out that this intuition is bound up with a mischaracterisation of 

the relation between dynamical laws and coordinate systems, namely the idea that Newton’s laws 

only ‘hold’ in special coordinate systems. This idea can be found in the work of various authors 

(e.g., Einstein, 1951, p. 27; van Fraassen, 1985, p. 116; Cushing, 1998, p. 98), and there are 

passages in which Brown appears to be making such a claim: ‘Inertial coordinate systems are 

those special coordinate systems relative to which the above conspiracy, involving rectilinear 

uniform motions, unfolds.’ (Brown, 2005, p. 15) To put this another way, a class of special 

coordinate systems is being postulated in which the laws of motion—the laws that determine the 

alleged conspiracy—hold. But this is to put the cart before the horse. Newton’s laws do not hold 

in special coordinate systems—they assert the possibility of coordinate systems in which all 

accelerations depend on impressed forces. The possibility of such systems is asserted by 

Newton’s laws; those systems are not prerequisites for the laws of motion.17 

 

There are passages of Physical Relativity that are reminiscent of the Einsteinian intuition 

that nature does not single out certain kinds of coordinate systems. There are also passages in 

which the existence of global coordinate systems seems to be conceptually antecedent to, or 

virtually the same as, the laws of motion and the states of motion they determine: ‘… we have 

                                                                                                                                                        
covariant, have other non-trivial symmetries’. In this way, we are returned again to the a priori demand to eliminate 
theories that assert the possibility of global structure, theories whose status one would prefer to think of as an 
empirical question. 
17See also DiSalle (2006a; 2002) in this regard. 



 12 

been looking at the nature of the inertial coordinate system xµ, the existence of which was claimed 

to be tantamount to Newton’s first law of motion.’ (Brown, 2005, p. 26) At the very least, these 

passages raise the question whether global coordinate systems are part of what motivates the 

conspiracy allegation. 

 

To this, Brown might reply that there could hardly be anything objectionable about global 

coordinate systems since, even in Einsteinian gravitation, space-time is approximately locally 

Lorentzian, and so a rigid coordinate system can be associated with it. If anything is 

objectionable, therefore, it is not these coordinate systems but the privileged state of inertial 

motion in Newtonian theory and special relativity that permits their use—a state of motion that, 

before Einsteinian gravitation, had to be postulated; it could not be derived from some deeper 

theory. The claim that this state of motion is unexplained and in need of an explanation will be 

examined in the next section. 

 

3.3 Inertia as a concept in need of explanation 

There is another assumption underlying the allegation of a conspiracy that should be singled out, 

namely that inertia in Newtonian theory and special relativity is in need of an explanation. This 

assumption is expressed in many ways in Physical Relativity (2005). See, for example, pp. 14-15 

and p. 141. 

 

The idea that a proper theory of inertia should say something about the ‘dynamical origins 

of inertia’, and in this way explain the concept, is a persistent one. Such a theory would give an 

account of how inertial motion arises as the effect of some underlying interaction—for example, 

it might tie inertia in some way to the stability of atoms. But, if this is what is needed, it is 

significant that inertia is no more explained by Einsteinian gravitation than by Newtonian theory 

and special relativity. 

 

Leaving aside the question of a dynamical origin in this particular sense, what seems to 

underlie the allegation of a conspiracy is the idea that inertia is unexplained because there is no 

more fundamental theory from which it can be derived. But this idea is at best unmotivated. It is 

useful to recall how the concept of inertia arises in Newton’s account of causal interaction. The 



 13 

first law defines an ideal force-free trajectory, one from which a particle can be deflected by the 

action of some force. In this way, the law provides the basis for an account of force. The second 

law then defines and interprets the concepts of force and mass—the acceleration of mass is the 

measure of the action of some force. Because the second law expresses a criterion for 

distinguishing free particles from particles acted upon by a force, it might be argued that it alone 

is sufficient for articulating the concept of inertia: The first law is a limiting case of the second 

(where F = 0). The first law, however, serves to coordinate or associate the state of motion of a 

particle unacted upon by forces with a geometric notion, namely a straight line. The first and 

second laws, therefore, are interdependent. But the third law—and so the conservation of 

momentum—is also necessary to distinguish free particles from those acted upon by a force. That 

is, it is necessary provided that we are interested not only in ideal point-particles but actual 

composite bodies.18 What this should establish, if it is not already evident, is that the concept of 

inertia relies on all three laws for its articulation.19 Furthermore, it should be evident that the laws 

of motion are mutually complementary. Only taken together do they constitute the Newtonian 

dynamics.20 

 

The idea that there is something objectionable about inertia because there is no more 

fundamental theory from which it can be derived rests on a demand for an explanation where 

none is needed. With the laws of motion Newton is offering mathematically formulated criteria 

for applying the concepts of force, mass, and inertia.21 It is enough that the concept of inertia 

behaves according to these criteria and that the theory founded on that concept succeeds in 

                                                
18In Section 5 I address directly the use of the notion of point-particle in interpreting Newton’s account of inertia. 
19For this reason, it does not make sense to ask: How is it that Newton can assert the first law of motion—before he 
has introduced a law that defines and interprets the concepts of inertial mass and force and the relation between 
them? This question can be found in analyses of Newton’s account of inertia by such authors as Rigden (1987), 
according to whom that account is a ‘logician’s nightmare’, and Pfister (2004), who calls it ‘logically fallacious’. 
There are passages of Physical Relativity (2005), for example, pp. 14-16, in which the question seems just below the 
surface. 
20This important point has been made by many authors. See, e.g., Torretti (1990, Chapter 3). 
21What is more, the laws of motion express criteria for explicating and applying concepts that were already in use. 
For example, Huyghens saw clearly that determining places and velocities, accelerations and rotations implicitly 
presupposes a privileged state of uniform rectilinear motion relative to which they can be referred. But it was 
Newton who first offered systematic criteria for the application of the concept, as part of an empirically adequate 
dynamical theory. Or take, for example, the pre-theoretical concept of force as something determined by pushing, 
pulling or pounding some mass. The third law, too, is implicitly presupposed in the work of Huyghens, Wallis, and 
Wren on collisions. 



 14 

explaining the phenomena that it sets out to explain. Inertia no more than force or mass is in need 

of further explanation. 

 

In each of the foregoing lines of argument I have tried to expose and refute an assumption 

underlying the alleged conspiracy. But even if Brown were to reject some or all of these 

assumptions or to question the relative support that they lend to his total view, the burden is still 

on him to explain the philosophical basis that supports the allegation of a conspiracy—in other 

words, to explain why a paragon of an empirically-successful theoretical concept is in any way 

lacking in its explanatory credentials. 

 

4 If inertia is conspiratorial, should a broad class of theories be so characterised? 

Leaving aside the philosophical basis that seems to motivate the allegation of a conspiracy, one 

might ask: If in fact there is something conspiratorial about inertial motion in Newtonian theory 

and special relativity, should a broad class of theories be so characterised? As a way of making 

sense of why one might read a conspiracy into the motion of free particles, one might suggest 

that, for a conspiracy theorist, all physical theories are conspiratorial to varying degrees. But I 

will argue that this suggestion, in a sweeping sense at least, deflects attention from the ideas that 

truly motivate the conspiracy allegation. 

 

It is helpful to consider a few examples that do not bear out the correct sense of 

‘conspiracy’. For example, one might say: If there is something conspiratorial about inertia, it is 

remarkable that there is nothing conspiratorial about the conservation of linear momentum, 

according to which the total momentum in an isolated system is conserved.22 Take, for example, 

a system of billiard balls in free space. The total momentum of the balls, before and after a 

collision, is conserved. One could impute to a conspiracy theorist the view that the balls conspire 

to interact only with each other and not with their environments, and to transfer momentum 

among themselves such that their total momentum is conserved. 

 

                                                
22One might think that the conservation of linear momentum is in fact an excellent example since, in the case of an 
isolated system, the principle of conservation of linear momentum simply is the first law. 



 15 

Or, if inertia is a conspiracy, why is there nothing conspiratorial about the motion of non-

interacting charged particles in electromagnetic fields? Every electron interacts with a given 

electromagnetic field in exactly the same way. One could suggest that a conspiracy theorist might 

say that the electrons conspire to act in this way. But the conspiracy theorist might reply that the 

field is a common cause to which their motion can be attributed, and so the alleged conspiracy is 

explained away.23 On this line of reasoning, it is the presence of a field that charged particles can 

‘feel’ that distinguishes their movement from the conspiratorial behaviour of free uncharged 

particles. 

 

To take another example, if inertia is a conspiracy, there is equal reason to think of 

equilibrium as arising from a conspiracy. There are many ways in which one might formulate 

such a conspiracy. To take a simple example, consider a rod in uniform translatory motion whose 

particles are in a stable equilibrium configuration. We might then Lorentz-boost the rod so it 

travels faster. The rod undergoes an acceleration before settling into a new stable equilibrium 

configuration. One could imagine that a conspiracy theorist might say that the particles conspire 

to reassemble themselves into the Lorentz-contracted rod. But no doubt a conspiracy theorist 

thinks of equilibrium as explicable by locally-acting forces and would therefore reject the 

comparison. 

 

Though one might attempt to make sense of the conspiracy allegation in this way, this 

suggestion trivialises Brown’s view. Not one of these examples captures the sense of 

‘conspiracy’ or ‘pre-established harmony’ at issue for him. The principles and intuitions we have 

considered above reveal a view about absolute or global background-structures, structures that 

constrain the possible states of a system without being themselves influenced by the system’s 

evolution. 

 

To take an example that does seem to capture the correct sense of ‘conspiracy’, we might 

look to the theory of weak interactions. Consider chiral or ‘handed’ processes, that is, processes 

                                                
23For the same reason, there is nothing conspiratorial about the motion of a pair of harmonic oscillators—of similar 
constitution that are isolated from, and thus unable to communicate with, each other—oscillating at the same 
frequency. Presumably, there is a common cause in their past for the synchrony of the forces that produce the 
oscillations. 



 16 

whose theoretical account displays a left-right asymmetry. If inertia is a conspiracy, there is 

equally good reason for seeing something conspiratorial in the handedness exhibited by parity 

violation in the theory of weak interactions. One could ask, why isn’t there anything 

conspiratorial about the beta-decay of, e.g., cobalt 60 atoms? How do all of the cobalt atoms in 

the universe know to exhibit handedness in the same sense when they are oblivious to one 

another? One could say that they conspire to do so.24 This example seems to have the salient 

feature: The relevant phenomenon, handedness, is tied to a global space-time structure, namely 

orientation. The natural reply to the conspiracy theorist is, of course, that cobalt 60 atoms display 

handedness not because they conspire but because parity-violating experiments are part of the 

evidentiary basis for orientation, one that we have specified in Minkowski space-time.25 But, for 

a conspiracy theorist, the orientation of Minkowski space-time ought to be just as problematic as 

the global affine structure. 

 

What seems to underlie Brown’s view is the idea that absolute or global background-

structures are what might be called ‘unexplained foundations’. They are unexplained in the sense 

that they cannot be derived from more general assumptions. But the idea that there is a problem 

with unexplained foundations is itself problematic; it seems to reflect a foundationalism that is 

difficult to motivate empirically. Even if Brown acknowledges that ‘all explanation must stop 

somewhere’, and so his view is not susceptible to any sort of regress, he still has to establish that, 

e.g., the global affine structures of Newtonian theory and special relativity are in need of an 

explanation—a view I have argued against in Section 3.3. Furthermore, if the notion of an 

unexplained foundation is indeed at the root of the conspiracy allegation, then there can hardly be 

much gain in explaining away the conspiracy of inertia by appealing to Einsteinian gravitation, 

for one can point to conspiratorial features even in that framework. 

 

One could regard the global topology, metric signature, orientability, and temporal 

orientation, among other features of Einsteinian gravitation, as having the marks of a conspiracy, 

                                                
24This example is suggested by Brown (2005, p. 142; personal communication). A detailed discussion of parity 
violation in the beta-decay of a cobalt 60 isotope, in the philosophical literature, can be found in Huggett (2000) and 
Pooley (2003). 
25For details on specifying the orientability and orientation of a manifold, see Malament (2012, §2.1-2.2). 



 17 

in this more specific sense.26 Consider the Lorentzian signature of the pseudo-Riemannian metric. 

The Lorentzian signature of the metric does not come from the field equations—it is a separate 

assumption, one motivated by the Lorentz covariance of the non-gravitational interactions.27 

Temporal orientation must also be specified. By examining these and other features of 

Einsteinian gravitation, we find that, while the affine and conformal structures are determined by 

the distribution of mass-energy, the theory requires the postulation of a number of quantities that 

do not come from the field equations alone. If there is any sense in which Newtonian theory and 

special relativity are conspiratorial, then certain features of Einsteinian gravitation can be said to 

be no less conspiratorial. 

 

To such a challenge, a conspiracy theorist might reply that, whenever we can explain the 

conspiratorial features of a theory by showing how they emerge from dynamics at a lower level, 

we have improved our understanding to a certain degree, we have shown that something 

miraculous at one level has a deeper reason. For example, Einsteinian gravitation explains 

remarkable structural correspondences that were previously taken for granted. If indeed all 

physical theories are conspiratorial—in the specific sense of having unexplained foundations—

such a strategy may be available to Brown. But this still fails to address the more important 

question of why we should regard the features in question as problematic. For example, though 

the Lorentzian metric signature, orientability, and temporal orientation in Einsteinian gravitation 

do not derive from the field equations, they are not brute posits—their application is controlled 

by empirical criteria. If the notion of an unexplained foundation is what is driving the conspiracy 

allegation, it is better by far to argue that there are no conspiracies at all. 

 

5 The alleged explanation of inertia by Einsteinian gravitation 

In this final section I will address Brown’s claim that inertial motion is explained by Einsteinian 

gravitation. I will begin by presenting that claim as well as Weatherall’s challenge to it. I will 

then propose another way of thinking about the theorems that drive the claim. 

                                                
26A discussion of the metric type (pseudo-Riemannian) and signature can be found in Brown (1997). See also Brown 
(2009, p. 9), who acknowledges that we might regard the universality of local Lorentz covariance as ‘mysterious’. 
27The strong equivalence principle, which incorporates the principle of minimal coupling, is also one of the 
assumptions that ensures that the structure of space-time is approximately locally Lorentzian—at least to the extent 
that we disregard matter-free vacuum solutions, where the principle is inapplicable. See Ehlers (1973) for a careful 
discussion. 



 18 

 

Brown claims: ‘GR is the first in a long line of dynamical theories ... that explains inertial 

motion.’ (Brown, 2005, p. 141) Further on, he writes: 

Inertia, in GR, is just as much a consequence of the field equations as gravitational waves. 
For the first time since Aristotle introduced the fundamental distinction between natural 
and forced motions, inertial motion is part of the dynamics. It is no longer a miracle. 
(Brown, 2005, p. 163) 

 
Brown’s claim rests on the fact that a geodesic principle—free massive test particles traverse 

timelike geodesics—can be derived from Einstein’s field equations together with other 

assumptions.28 The claim seems to presuppose a deductive-nomological scheme: One can take 

the field equations, energy and conservation conditions, and the resulting geodesic principle as 

explanans, then derive the motion of a free particle as explanandum. 

 

There are various geodesic theorems, but Geroch and Jang’s (1975) has a claim to being 

the most perspicuous, and I will limit my attention to it. Their theorem has the advantage of 

avoiding specific assumptions about the nature of the free massive test particle; it also has the 

advantage of showing that the particle traverses a curve in space-time rather than a line 

singularity. In any case, if any geodesic theorem can be said to figure in a deductive-nomological 

explanation of inertial motion, the Geroch-Jang theorem can be said to do so. 

 

 Brown’s claim that inertial motion is explained by Einsteinian gravitation in a distinctive 

way was challenged by Weatherall (2011a), who showed that a geodesic principle can be derived 

in geometrised Newtonian gravitation. With this theorem in hand, Weatherall observes of inertial 

motion in geometrised Newtonian and Einsteinian gravitation: ‘if either theory can be thought to 

explain inertial motion, then both do, in much the same way’ (Weatherall, 2011b, p. 280). 

 

A line of objection is available to Brown: Both Geroch and Jang’s and Weatherall’s 

theorems proceed from the conservation condition ∇aTab = 0 which is assumed to hold at a 

                                                
28This is not quite Brown’s claim. Brown claims that the geodesic principle follows directly from the field equations, 
a claim that Malament (2012), in light of the result of Geroch and Jang (1975), has shown to be not so 
straightforward. Brown grants this (Brown and Lehmkuhl, forthcoming, pp. 19-20) and maintains nonetheless that 
inertial motion is explained by Einsteinian gravitation. 



 19 

point.29 But, in Einsteinian gravitation, the conservation condition follows from Einstein’s field 

equations; in geometrised Newtonian gravitation, it is an independent assumption. This line of 

objection is undermined by Weatherall, who argues that the conservation condition is a 

background assumption in both theories. It is an assumption that is more general than Newtonian 

and Einsteinian gravitation, an assumption about a general feature of the world that these theories 

and others respect. 

 

Weatherall (2011b) offers several arguments for regarding the conservation principle as a 

background assumption. But, looking at his analysis more generally, it is important to understand 

this characterisation in his total view. Weatherall defends the idea that space-time theories ought 

to be seen as arising from a set of assumptions or constraints—as Weatherall (2011b, p. 280) has 

put it, from an ‘interconnected network of mutually dependent principles’. On this view, 

Einstein’s field equations are not attributed any fundamentality or priority as compared with the 

conservation principle, nor is their novelty—their non-linear character with its important 

implications—diminished.30 These equations of mutual constraint are not taken as the 

‘fundamental dynamical equations’, but as an element of a set of constraints, none of which, 

taken alone, is sufficient to give an account of the physical systems and processes of interest in 

gravitational physics.31 For a sustained argument for this view and a careful study of the relation 

of energy conditions to the field equations, see Curiel (forthcoming). 

 

                                                
29There is disagreement about whether ∇aT

ab = 0 has a legitimate claim to being a conservation principle. Some (e.g., 
Weatherall, 2011a, 2011b; Malament, 2012) refer to it as a conservation principle or condition without qualification, 
which is not to suggest that they are insensitive to the difficulties surrounding energy-momentum conservation in 
Einsteinian gravitation. Others (e.g., Torretti, 1983; Brown, 2005) have stressed that it is not a conservation principle 
in the usual sense, namely an integral conservation principle. Brown (2005, p. 141) notes that the equation is ‘as 
close as anything is’ to a conservation principle in Einsteinian gravitation; further on (2005, p. 161) he observes that 
‘it was appreciated from the very beginning that the presence of the covariant derivative and not the simple partial 
derivative in the equation makes this reading strictly untenable in curved space-time’. For this reason, ∇aT

ab = 0 is 
sometimes (e.g., Stephani, 2004; Pitts, 2010) referred to as a ‘balance equation’. See Byers (1999) for a particularly 
clear survey of the historical background to the problem of energy-momentum conservation in Einsteinian 
gravitation and the relation to Noether’s theorems. See Weiss and Baez (2012) for a clear survey of the technical 
issues. 
30The nature of the constraint on matter fields that is effected by Einstein’s field equations is discussed by Brown 
(2005, p. 163), who includes references for further reading. 
31It is noteworthy that, at least in the case of the Geroch-Jang theorem, the matter field is not required to satisfy 
Einstein’s field equations. As Weatherall (2011b, pp. 277-278) notes, ‘The absence of such a condition indicates that 
the matter described in the theorem is test matter, i.e., it is not a source term in Einstein’s equation.’ So, while the 
field equations do in general constrain the account of matter fields in Einsteinian gravitation, the theorem’s 
restriction to test matter removes the difficulties that this particular constraint would otherwise raise. 



 20 

I agree with Weatherall’s observation that, if there is any sense in which Einsteinian 

gravitation can be said to explain inertial motion, then geometrised Newtonian gravitation can be 

said to explain it at least as well. But I will argue that the main contribution of these theorems lies 

not in their explanation but in their explication of inertial motion. By ‘explication’, I mean the 

clarification afforded by these theorems of the conceptual structure of Einstein’s theory—of a 

certain account of matter, of the assumptions required for describing the evolution of that matter, 

and of the interdependence of these conditions—rather than any importance they might have in 

some or another philosophical account of explanation. This analysis does not diminish the fact 

that inertia can be brought under a deductive-nomological scheme, but it offers a deeper 

understanding of the concept than that scheme can provide. 

 

The geodesic theorems make explicit an assumption that Newton makes in his own 

account of inertial motion. Current discussions of inertial motion in old-fashioned Newtonian 

theory focus on the laws of motion and the corollaries to the laws. But there is an 

underappreciated discussion in the Scholium to the Laws in which Newton shows that the third 

law—and thus the conservation of momentum—is necessary for the first law to apply to systems 

that are subject to attractive forces. The passage of interest to us is Newton’s demonstration of 

the third law of motion for attractions. The proof is straightforward. Take any two bodies A and B 

that attract each other. Place between them an obstacle that impedes their coming together. 

Suppose, for reductio, that A is more attracted to B than B is to A. That is, suppose that FB on A ≠ 

FA on B. Bodies A and B will move towards each other, both eventually reaching the obstacle. The 

obstacle will be pressed more strongly by body A than by body B, and so will not remain in 

equilibrium between them. The stronger pressure of A against the system comprising the obstacle 

and B will make the entire system of the three touching bodies move straight forward in the 

direction from A to B. In empty space, the system will go on indefinitely with a motion that is 

always accelerated. But this contradicts Law 1. Hence, our supposition that FB on A ≠ FA on B must 

be false. Hence, FB on A = FA on B.32 

 

                                                
32Newton is anticipating the application of the third law to the Solar System. He envisages the steps that he will take 
to show that the Solar System is effectively isolated. 



 21 

Though this demonstration of the third law focuses on a system of bodies, it is significant 

that the law applies also to a single body that is itself a composite system. The first law, together 

with the second law which expresses a criterion for determining which particles are force-free, is 

satisfied only in the case of point-particles. For bodies that are themselves composed of particles, 

the third law is a necessary condition for inertial motion. That is, the system of particles making 

up a single body must interact in such a way that every force is balanced, failing which the body 

will accelerate by its own internal forces and violate the first law. Therefore, the notion of 

equilibrium—as it pertains to a single, composite body as well as to a system of bodies—enters 

Newtonian theory only via the third law. 

 

Now one might object that Newton writes ‘body’ in the first law and not point-particle.33 

Further on, for example, in the proof of Proposition 1 of Book 1 and elsewhere in Book 1, 

Newton also uses ‘body’ and ‘law 1’ in the same breath. But even though he writes ‘body’, 

Newton is tacitly treating bodies as point-masses, at least in the early sections of Principia. 34 

Furthermore, Newton’s discussion in the Scholium to the Laws makes it clear that only the third 

law guarantees that a composite body—a composite of two or more simple particles held together 

by some force of cohesion—will obey the first law. So, though Newton does not always clearly 

identify where he is treating bodies as point-particles, the point-particle simplification is integral 

to his thinking and introducing it into the analysis of Principia illuminates the account of inertia. 

 

Though it is often overlooked that the third law is a necessary condition for the first law 

to apply to systems held together by attractive forces, it was well understood by Newtonians in 

the eighteenth and nineteenth centuries with whom it was further elaborated and clarified. There 

are too many to consider individually, but it is important to mention d’Alembert, who, in his 

Traité de Dynamique (1743 [1967]), proposed a rational mechanics founded on laws of impact 

between perfectly hard bodies. Though d’Alembert was manifestly a Newtonian, the influence of 

Descartes on d’Alembert’s thought can be clearly seen. D’Alembert sought to deduce the laws of 

mechanics from ‘certain dispositions of size, figure and motion’, in other words, from a purely 

                                                
33I thank a referee for raising this line of objection. 
34This is a feature of what I B Cohen calls ‘stage one’, before Newton gradually introduces, in ‘stage two’, further 
properties of physical systems. See the discussion of this in A Guide to Newton’s Principia (Cohen, 1999, pp. 159-
160). 



 22 

geometrical account. From such a clearly and distinctly known geometrical basis, his laws of 

motion would be necessary truths and his mechanics would be a genuine metaphysical discovery. 

This view led d’Alembert to propose laws of motion that are close to Newton’s laws.35 In spite of 

its Cartesian aspect, however, d’Alembert’s mechanics is a restriction of Newtonian mechanics to 

the mechanics of rigid bodies.36 

 

D’Alembert understood clearly that Newton’s third law, and therefore the conservation of 

momentum, must be assumed to give an account of the transfer of motion from one body to 

another in a collision. Newton’s third law enters d’Alembert’s mechanics as ‘the principle of 

equilibrium’, and the concept of equilibrium is the core of his ‘general principle’, which we now 

know as ‘d’Alembert’s principle’.37 The principle asserts that ‘all the forces acting on points of 

the system form, with the reactions against acceleration, an equilibrating set of forces on the 

whole system’ (Thomson and Tait, 1867 [1879], p. 248). This is the culmination of the Traité de 

Dynamique; it represents d’Alembert’s attempt to reduce the laws of mechanics to a single 

principle. 

 

With the general principle in hand, d’Alembert deduced three theorems. The first, which 

is of greatest interest to us, asserts that ‘[t]he state of motion or rest of the centre of gravity of 

several bodies does not change by the mutual action of these bodies among themselves, provided 

that the system is completely free’ (d’Alembert, 1743 [1967], Part II, Ch. 2, Theorem I). In this 

way, the principle of the conservation of the centre of gravity is recovered from his general 

principle. He deduced a second theorem, according to which ‘if weight or an accelerative force—

constant for each body and different, if one wants, for each of them—acts on these bodies 

following parallel lines, the centre of gravity or rather the common centre of mass will describe 

the same curve that it would have described if these bodies had been free’ (d’Alembert, 1743 

                                                
35D’Alembert was reluctant to write of forces. He eschewed the lingering notion of inherent cause and the vis viva 
controversy. He insisted that ‘force’ is only that quantity with which we are acquainted through its effects. 
36D’Alembert proposes to focus on bodies that act on one another by ‘immediate impulse, as in the case of an 
ordinary impact’ or by ‘the interposition between them of some body to which they are attached’ (d’Alembert, 1743 
[1967], p. 49). He considers attractions to have been sufficiently well examined by Newton, and so sets these actions 
aside. 
37D’Alembert’s own statement of the principle (1743 [1967], p. 51) is not straightforward, but clear statements of its 
essential content can be found in the work of his successors. Thomson and Tait’s statement is one such; other 
statements are found in Mach (1901 [1902], pp. 335-337). For a good, recent discussion of the principle, see Lanczos 
(1970). 



 23 

[1967], Part II, Ch. 2, Theorem II). This theorem generalises the first to encompass those 

situations in which an isolated system is acted upon by a force that is sufficiently distant for the 

system to be treated like an isolated or ‘near enough’ isolated system.38 A third theorem 

generalises the first still further to encompass systems subject to a constraint. In the Scholia to the 

Theorems d’Alembert notes that these theorems are equally true for attractions; so, though he 

deliberately restricts his attention to rigid-body mechanics, he acknowledges that his principle has 

wider applicability. D’Alembert’s laws of motion and his general principle establish clearly that 

the total ‘quantity of motion’ or ‘momentum’ in isolated systems of interacting bodies is 

conserved. 

 

The centrality of the conservation principle to Newtonian theory was equally well 

understood by Thomson and Tait in their Treatise on Natural Philosophy (1867 [1879]). In their 

discussion of Newton’s laws, they observe that 

Of late there has been a tendency to split the second law into two, called respectively the 
second and third, and to ignore the third entirely, though using it directly in every 
dynamical problem; but all who have done so have been forced indirectly to acknowledge 
the completeness of Newton’s system, by introducing as an axiom what is called 
D’Alembert’s principle, which is really Newton’s rejected third law in another form. 
Newton’s own interpretation of his third law directly points out not only D’Alembert’s 
principle, but also the modern principles of Work and Energy. (Thomson and Tait, 1867 
[1879], p. 240) 
 

That the conservation of momentum is a necessary condition for the inertial motion of composite 

systems was noted in the same year by Maxwell in Matter and Motion (1867 [1888]): 

... Newton goes on to point out the consequence of denying the truth of [the third law of 
motion]. For instance, if the attraction of any part of the earth, say a mountain, upon the 
remainder of the earth were greater or less than that of the remainer of the earth upon the 
mountain, there would be a residual force, acting upon the system of the earth and the 
mountain as a whole, which would cause it to move off, with an ever-increasing velocity, 
through infinite space. (Maxwell, 1876 [1888], p. 48) 

 

                                                
38Theorem II reveals d’Alembert’s understanding of Newton’s Corollary VI to the laws of motion. What is puzzling, 
however, is his suggestion that the forces may be different for each body. It may be that d’Alembert states Theorem 
II in the way that he does because he wants to acknowledge that Corollary VI contains an explicit (restrictive) 
hypothesis ‘If bodies are ... urged by equal accelerative forces along parallel lines...’ that is never strictly satisfied, 
except in the trivial case of zero accelerative forces. This reading seems to be supported by Sklar’s (2013, pp. 120-
121) interpretation of Theorem II as a generalisation of the principle of the conservation of the centre of gravity ‘to 
include systems of particles all subject to the same external accelerating force, either constant and acting along 
parallel lines or directed to a point and distance-dependent’. 



 24 

This vivid illustration of the application of the third law to a body that is itself a composite 

system establishes in still another way the fundamental role of the conservation of momentum. 

 

What we find in the work of D’Alembert, Thomson and Tait, Maxwell, and others is a 

deliberate attempt to give a perspicuous account of the necessity of the conservation of 

momentum for the inertial motion of composite systems, a relation that is manifest in Newton’s 

work but not sufficiently appreciated. As the Geroch-Jang and Weatherall theorems bear out, this 

relation is equally essential to Einsteinian gravitation and geometrised Newtonian gravitation. In 

old-fashioned Newtonian theory no less than in geometrised theories the account of inertial 

motion is not determined by any single principle susceptible of separate explanation but by an 

interdependence of physical principles that must be assumed together. Old-fashioned Newtonian 

theory and the geometrised theories are in fact analogous in their accounts of inertial motion: The 

third law of motion is to old-fashioned Newtonian theory as the conservation principles are to the 

geometrised theories. This analogy is clearly exhibited by the geodesic theorems, and it 

highlights the sense in which their contribution to our understanding does not lie in their 

explanation of inertial motion but in their explication of it. 

 

One might object that the analogy is strained.39 One might argue that, in the Geroch-Jang 

theorem, it is assumed that ∇aTab = 0 represents the total energy-momentum only at a point. But 

what this assumption reflects is not a straining of the analogy, but the recognition of the inherent 

limitations of the theorem and an appreciation of the idealisations that it requires. What we want 

to ensure is that the free massive test particle that figures in the theorem is indeed free, that is, 

that its internal energy-momentum is conserved, that it is not exchanging energy-momentum with 

external fields, and that the background space-time metric can be kept fixed. To be sure, this 

represents a severe restriction on allowable models for the theory. But noting this restriction does 

not undermine the dependency of inertia on the conservation of energy-momentum. 

 

The sense of ‘explication’ at issue has nothing to do with our ability to derive a previously 

unprovable proposition from a new theory, though, in the cases that concern us, the proofs 

contribute to that explication. Nor does this sense of explication have anything to do with any 

                                                
39I thank a referee for raising this line of objection. 



 25 

particular philosophical account of scientific explanation, and so it is independent of the success 

or failure that attaches to such an account. Rather, the explication is the outcome of an analysis 

that began with the question, on what assumptions does our use of the concept of inertia depend? 

The analysis reveals that, in both old-fashioned Newtonian theory and in geometrised theories, 

inertia depends fundamentally on the conservation of momentum. Far from a concern with 

explaining the causal or dynamical origin of inertia, the geodesic theorems explicate the concept 

of inertia by explaining the connections between it and other concepts. 

 

6 Conclusion 

I set out to evaluate Brown’s account of inertial motion in Newtonian theory and special 

relativity, in particular his claim that there is something objectionable—something 

conspiratorial—about inertia in these theories. I presented and clarified the conspiracy allegation, 

and I argued that it is motivated by a commitment to a number of philosophical principles or 

intuitions that are reminiscent of Einstein’s view, namely the action-reaction principle, the idea 

that global coordinate systems are an artifice of thought, and the idea that inertia in Newton’s 

theory is in need of an explanation. These principles reveal that the conspiracy allegation is 

bound up with a view according to which there is something problematic about absolute or global 

space-time structures. I argued that, even if Brown were to reject some or all of these principles, 

the onus would still be on him to explain why there is anything problematic about inertial motion 

in Newtonian theory and special relativity. 

 

I proceeded to ask, if there is something conspiratorial about inertia, should a broad class 

of theories be so characterised? I considered the seemingly natural suggestion that, for a 

conspiracy theorist, all physical theories are conspiratorial. I examined and rejected a sweeping 

sense of ‘conspiracy’ that trivialises Brown’s view. I then examined a narrower sense that is 

bound up with the notion that absolute or global background-structures are an unexplained 

foundation, and I pointed out that Einsteinian gravitation also has such features. I argued that, if 

indeed the conspiracy allegation is driven by this idea, then it is better to argue that there are no 

conspiracies at all. 

 



 26 

Last, I addressed Brown’s claim that inertia is explained by Einsteinian gravitation 

because a geodesic principle can be derived from the field equations. I reviewed Weatherall’s 

(2011b) challenge to Brown’s claim. Weatherall argues that, if there is any sense in which 

Einsteinian gravitation can be said to explain inertia, then geometrised Newtonian gravitation 

explains it at least as well. While I agree with Weatherall, I argued that there is a better way of 

thinking about the geodesic theorems. That is, their main contribution lies not in their explanation 

of inertial motion but in their explication of it. This explication is independent of any 

philosophical account of explanation under which inertia can be subsumed; it is concerned with 

clearly exhibiting the assumptions on which our use of the concept depends. 

 

I argued that the geodesic theorems of Geroch and Jang (1975) and Weatherall (2011a) 

explicate inertial motion by making perspicuous the dependency of inertial motion on the 

conservation of momentum. This is manifest, though under-appreciated, in Newton’s own 

account of inertia, and I argued that the work of his successors—notably d’Alembert, Thompson 

and Tait, and Maxwell—represents a deliberate attempt to establish the fundamental importance 

of the conservation principle. In spite of their important differences, old-fashioned Newtonian 

theory, geometrised Newtonian gravitation, and Einsteinian gravitation are analogous in their 

accounts of inertial motion. 

 

Acknowledgements 

I wish to thank Harvey Brown, Jeremy Butterfield, Erik Curiel, Michael Friedman, James 

Ladyman, Stuart Presnell, Chris Smeenk, David Wallace, Jim Weatherall, and especially Robert 

DiSalle, Bill Demopoulos, and Wayne Myrvold. I also thank two anonymous referees for the 

British Journal for the Philosophy of Science. This work was supported by the Social Sciences 

and Humanities Research Council of Canada. 

 

 

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