on the explanation of inertia

ARTICLE

On the Explanation of Inertia

Adan Sus

� Springer Science+Business Media Dordrecht 2014

Abstract In General Relativity (GR), it has been claimed that inertia receives a

dynamical explanation. This is in contrast to the situation in other theories, such as Special

Relativity, because the geodesic principle of GR can be derived from Einstein’s field

equations. The claim can be challenged in different ways, all of which question whether the

status of inertia in GR is physically different from its status in previous spacetime theories.

In this paper I state the original argument for the claim precisely, discuss the different

objections to it and then propose a formulation that avoids the problems the original claim

encounters. My conclusion is that one can say meaningfully that inertia is dynamically

explained in GR. There are two senses in which the derivation of geodetic motion can be

said to provide a (more) dynamical explanation of inertia in GR: it holds for any material

test body that is a source of the gravitational field; and it is derivable without assuming

inertial structures that are fixed independently of matter.

Keywords General Relativity � Inertia � Geodesic principle

1 Introduction

The law of inertia is one of the foundational pillars of modern physics. It states that a body

will remain in its state of motion unless an external force is applied to it. In the system of

Newtonian physics this law occupies a predominant position as the first of the laws of

motion and implies the existence of a class of coordinate systems in relation to which all

free bodies move in a coordinated manner: with constant velocity and in straight lines.

From our modern geometrical perspective, one can take the law of inertia to be the claim

that free bodies move along the geodesics of the affine connection that four-dimensional

A. Sus (&)Departamento de Filosofıa, Facultad de Filosofıa y Letras, Universidad de Valladolid, Plaza delCampus s/n, 47011 Valladolid, Spaine-mail: [email protected]

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J Gen Philos SciDOI 10.1007/s10838-014-9246-8

Galilean spacetime is endowed with. This latter perspective allows us to provide an

equivalent to the law of inertia in gravitational physics; in particular, in General Relativity

(GR). This is what has been called the geodesic principle and states that free-falling bodies

move along timelike geodesics of the spacetime metric.

In the standard presentation of Newtonian mechanics, the law of inertia is taken as a

postulate of the theory: it is a brute fact that free bodies move the way they do. None-

theless, one may consider that the phenomenon of inertial motion is still in need of

explanation, even though this formulation of the theory does not provide one.

Things seem rather different in GR. Right from the very first discussions after the

publication of Einstein’s theory, several authors (including Einstein himself) argued that

inertial motion is dynamically explained in GR. This claim is based on the fact that in GR

the geodesic principle can be derived from Einstein’s field equations (EFE) and it is thus a

theorem instead of a postulate. The most recent expression of this can be found in Harvey

Brown’s Physical Relativity:

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, 163).

Since the publication of Brown’s book, several objections have been raised against the

idea that GR contains a dynamical explanation of inertia. In my opinion, however, those

objections may leave one with the wrong impression. In one way or another, the criticism

is aimed at diminishing the uniqueness of the explanation of inertia in GR by indicating an

alleged derivation of inertial motion in other theories that would have similar features. So

one might think that the right moral to extract from this debate is that the status of inertia in

GR is in no way substantially different from its status in previous theories [i.e., Newtonian

mechanics and Special Relativity (SR)]. In this paper I argue quite the opposite: there are

singular features of the derivation of inertial motion in GR that makes it worthy of being

described as providing a dynamical explanation (different from those found in other the-

ories); although those unique features are not captured by the claim as it was originally

formulated. My argument will show: first, to what extent the objections to the original

formulation of the claim are fair; and, second, in what sense it is still possible to say that

the explanation of inertia in GR is different from the alleged explanations in previous

theories. As a result, I believe we can arrive at a new understanding of the differential

status of inertia in GR.

The original proposal as expressed by Brown is simply that inertia can be derived as a

theorem in GR. This implicitly assumes that such a derivation provides a dynamical

explanation of inertia and that this is a distinctive feature of GR that allows us to distin-

guish the theory from others. Nonetheless, Brown’s proposal does not specify in what

sense such a derivation is more explanatory than others that can be devised for the same

purpose in different theories. A discussion of the objections to Brown’s original proposal

shows that one needs to work harder to explicate the sense in which inertia is explained in

GR; but also that it is indeed possible to give such an account.

The objections that I will present to the original claim are found in recent work by

Malament (2012), Pooley (2013) and Weatherall (2011b). What these objections bring to

the fore is that if one wants to defend the claim that GR contains a dynamical explanation

of inertia, it is not enough to say that it does so because the geodesic principle can be

derived from EFE. In Sect. 3 below, I will discuss the objections and indicate in what sense

they pose problems for the original idea. Before that (in Sect. 2) I will try to formulate that

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idea as clearly as possible. My discussion of the objections leads to what I believe to be the

proper way of analysing the alleged explanation of inertia (Sect. 4) This will be supported

by comparing, in Sect. 5, the explanations available in GR and geometrised Newtonian

gravity (Newton–Cartan theory). Finally (Sect. 6), I will introduce a more general dis-

cussion of the explanation of ‘‘inertial structure’’ that is crucial in marking out the dif-

ference between the possible explanations available in GR and, say, in special relativistic

theories. That GR satisfies a strong version of Einstein’s Equivalence Principle (EEP), and

to some degree at least what is often called ‘‘Mach’s Principle’’, will be seen to be aspects

of the stronger sense in which inertia is explained in Einstein’s theory.

Before entering into details, let me advance the main line of my argument. My proposed

elucidation will move along two fundamental axes.

The first is the idea that GR provides a more satisfactory explanation of inertia than

other theories because it contains a general justification of the response equations (the

vanishing of the covariant divergence of the energy-momentum tensor) for any sources of

the gravitational field. Inertial motion can then be derived for test bodies from this result,

under certain conditions. This contrasts with the status of the response equations in other

theories, where they can be seen, in principle, as a consequence of a particular set of

equations (the field equations of a particular matter field). We will see that the sense in

which GR contains an explanation has the following structure: given GR, every free

material body (any body that can be considered a source of the gravitational field) must

travel (approximately) along timelike geodesics (if superluminal energy propagation is

forbidden). In contrast, special relativistic theories can be said at most to contain an

explanation in the following sense: given the field equations of the theory, bodies made of

such-and-such fields travel (approximately) along timelike geodesics. We must note the

different scope of the two kinds of explanation, which is evident from the need to make

reference to the composition of bodies in the second type. Furthermore, there are a couple

of subtleties that oblige me to be initially cautious and write ‘‘at most’’ when referring to

the explanation in special relativistic theories. The first of these concerns whether, from the

canonical form of the field equations in those theories, one can derive the response

equations and if so, what presuppositions come into play when doing so. The second has to

do with whether, in reality, when one talks about material bodies one should be allowed to

consider them as blobs of a particular field; a supposition that seems to be needed if one

wants the derivation to go through in these cases. In the course of the discussion I will

consider under what conditions special relativistic theories can be said to contain an

explanation of inertia that is as general as that claimed to be contained in GR and I will

expand on the above mentioned subtleties.

The second conceptual axis of my explication of the status of inertia in GR is based on

there being a sense in which the explanation in GR is more dynamical than that allegedly

found in other theories, due to the fact that the spacetime structures that are interpreted as

inertial in GR are not fixed a priori, while they are (partially or completely) in other

theories.

So, in the account that I propose, the explanation found in GR is both dynamical and

distinctive. However, the contrast between it and the explanations provided by other

theories can be envisaged using two different criteria: the status and scope of the derivation

of inertial motion (whether it is proved in its full generality or only for certain kinds of

matter and as resulting from a nomological fact or from a brute fact); and the dynamical

character of the inertial structures. These criteria will allow us to conceptualise the dif-

ferences between GR and special relativistic theories (or theories that are silent about why

the inertial structures are the way they are), on the one hand; and also to distinguish GR


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from other gravitational theories that have a less dynamical spacetime structure, on the

other. We will then see that, with respect to the explanation of inertia, GR occupies a

prominent place both in relation to matter theories that do not provide a dynamics for

spacetime and to other possible geometrisations of the gravitational field (Newton–Cartan

theory, for instance).

The fact that in GR these two different senses of explanation are present (while being

absent in most other theories) is remarkable and suggests that they might be related. The

discussion in Sect. 6 will help to show why this is the case. It can be argued that GR

implements what I call an active version of the Equivalence Principle, expressed in capsule

form by the dictum ‘inertia is gravity’. Other theories, in order to have at least some

minimum possibility of containing an explanation of inertia similar to that found in GR,

must implement some kind of EP expressed by the rule: ‘‘comma changes to semicolon‘‘. I

will argue that those passive forms of EP are not by themselves explanatory, but are instead

condensed expressions of the limited sense in which inertia is explained in such theories.

Before we look at the initial claim that the geodesic law receives its first real expla-

nation in GR, a few words about explanation, and what makes one explanation stronger

than another, are perhaps in order. Without advocating a particular theory of scientific

explanation in general, certain points about comparative explanatory power can be made

which most philosophers (and physicists) would probably find intuitively correct.

In mathematical physics any non-trivial deduction involving physical theories and

entities may be considered an explanation, or putative explanation. However, at least three

things can be said about features of an explanation that physicists usually take to be virtues;

things that make an explanation stronger (ceteris paribus) than a comparable explanation

that lacks them:

1. The presence of a causal story positing interactions that have at least some range of

invariance [as in the account of explanation defended by Woodward (2003)]; ceteris

paribus, the wider the range of invariance, the better.

2. The absence of ad hoc elements; i.e., elements that are introduced to perform a certain

explanatory role without having further consequences (observational or interactional).

3. Extendability of the explanatory story to a range of other phenomena or situations;

ceteris paribus, the greater the range of other situations, the more powerful the

explanation.

These three criteria all speak in favour of explanations in physics that are based on

genuinely dynamical natural laws: laws that show how one field or variable quantity

mathematically relates to another (or several others), over a range of different possible

cases, with none being ‘‘absolute’’ in the ‘‘acts-on-but-is-not-acted-upon-in-turn’’ sense

that Einstein objected to, which is characteristic of both Newtonian and special relativistic

physics.

2 Is Inertial Motion Explained in GR?

The claim that inertial motion is explained in GR has been equated with the statement that

the geodesic principle in GR is a theorem instead of a postulate; or, in a less condensed

way, that such a principle is derivable from EFE. In order to evaluate the plausibility of this

idea, we first need to spell it out carefully. We will see that once we start to look into the

details, the formulation of this idea is not completely straightforward.

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In more detail, the explanatory story is as follows. In GR, EFE imply the response

equations (which express the vanishing of the covariant divergence of the energy-

momentum tensor) without the need to add any additional conditions. From the response

equations, by applying different theorems available in the literature, one can derive the

geodesic principle; which states that (non-rotating, spinless) test bodies move along geo-

desics of the spacetime metric, and that massive bodies do so if we stay within certain

limits of size. This can be taken to be an explanation of inertia in the following sense: EFE

encode the dynamics of the metric (the gravitational field) and its dynamics by itself

constrains force-free material bodies (irrespective of their composition) to move approx-

imately along geodesics. This is supposed to explain inertial motion in a sense not found in

other theories; but how exactly does it do so? The answer, one might initially say, is that

while in GR only EFE are needed in order to arrive at the geodesic principle, in other

theories one has to introduce some presuppositions about the matter fields. So the differ-

ence, and the explanatory power in GR, seems to be rooted in the idea that somehow EFE

are more explanatory than whatever conditions on matter fields are needed in the other

theories. And this story, of course, presupposes that EFE are sufficient to arrive at the

geodesic principle.

As we will see, there are at least two sources of problems for this story. First, one has to

justify the idea that the process by which the response equations are derived in GR is

essentially different from an arguably equivalent process in theories such as Newtonian

mechanics or SR. Second, one needs to make clear the limits and presuppositions of the

theorems used to deduce geodetic motion from the response equations and their conse-

quences for the project just described. These two problems will be dealt with in turn in the

next section, but let us first break down the argument.

2.1 Derivation of the Response Equations

The matter response equations are the closest one can get to a conservation law in GR.

They are a direct consequence of EFE; starting with EFE, they can be derived as a

consequence of the contracted Bianchi identities, mathematical identities expressing the

vanishing of the covariant divergence of the left-hand side of the equations:

Rlm � 1

2glmR ¼ jTlm ð1Þ

Glm;m ¼ Rlm � 1

2glmR

� �;m

¼ 0 Contracted Bianchi identities ð2Þ

Tlm;m ¼ 0 ð3Þ

It is well known that when this theory is written out using the Lagrangian formalism, the

response equations can be deduced as a consequence of the general covariance of the

action by means of Noether’s second theorem (NST). It is worth recalling that in

Lagrangian GR, one can follow two different routes to arrive at the same result1: the first

takes into account the variational properties of the matter action and extracts the response

equations by assuming that the matter field equations hold; the second deals with the

gravitational action and assumes EFE to obtain the same result.

Let us take Hilbert’s action plus an action term for the matter fields, /i, as the action of

GR:

1 See, for instance: Carroll (2004), Brown (2005), Brading and Brown (2003).


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S ¼ Sg þ Sm ð4Þ

Sg ¼Z

d4xffiffiffiffiffiffiffi�gp

R ð5Þ

Sm ¼Z


Lmðglm;/i; . . .Þ ð6Þ

Route 1 starts from the invariance of Sm under general coordinate transformations of the

form:

xl0 ¼ xl þ �l ð7Þ

d0/i ¼ ail�l þ bia

mom�a ð8Þ

d0glm ¼ �glaom�a � gmaol�

a � glm;a�a; ð9Þ

where a and b are coefficients for the Lie drags of the fields involved. According to NST,

we can derive the following identities:

1

2

ffiffiffiffiffiffiffi�gp

oagqmTqm þ aia

dLm

d/i

¼ orffiffiffiffiffiffiffi�gp

Tra

� �þ om bm

iadLm

d/i

� �; ð10Þ

where we have introduced the stress-energy tensor, Tlm, via the definition:

Tlm ¼ � 2ffiffiffiffiffiffiffi�gp

dffiffiffiffiffiffiffi�gp

Lm

dglm: ð11Þ

Assuming that the field equations for the matter fields hold ðdLm

d/i¼ 0Þ and using the

identity:

orffiffiffiffiffiffiffi�gp

Tra

� �� 1

2

ffiffiffiffiffiffiffi�gp

Tlmoaglm ¼ ffiffiffiffiffiffiffi�gp rrTr

a ; ð12Þ

we obtain the response equations, rm Tlm = 0.

Route 2 starts with the fact that Sg is invariant under general coordinate transforma-

tions. Now the Noether identities take the following form:

oaglmdLg

dglm¼ 2or grl dLg

dgla

� �: ð13Þ

Using the gravitational field equations,dLg

dglm¼ � ffiffiffiffiffiffiffi�gp

Tlm: the definition of the energy-

momentum tensor (11); and identity (12), we obtain the matter response equations again.

That is the situation in GR. To be able to compare it with what happens in other

theories, however, the vital question is: under what conditions in a given spacetime theory

can the response equations be derived from the theory’s gravitational field equations? This

question arises in the context of the Lagrangian formulation of modified theories of gravity,

which allow other fields, apart from the metric, to enter the gravitational part of the action.

We must bear in mind that in such a context we assume the existence of a matter action

(from which the matter field equations can be derived, and which provides the definition of

the matter stress-energy tensor) and a separate gravitational action. Examples of these

theories abound in the literature and include scalar-tensor theories (Brans et al. 1961, for

instance), tensor-vector theories (Zlosnik et al. 2006), tensor–vector–scalar theories

(Bekenstein 2004) and bimetric theories (Rosen 1980), although other examples could be

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devised. We divide the Lagrangian into its gravitational and matter parts as before,

assuming that the former depends on the metric and other gravitational fields while the

latter takes the same form as above:

S ¼ Sg þ Sm

Sg ¼Z


Lgðglm;Ai; . . .Þð14Þ

It is clear now that we can apply NST using the two routes discussed above. When using

the first, nothing changes and we obtain the response equations straightaway. The second

route produces the following noether identity:

�oaglmdLg

dglmþ aia

dLg

dAi

¼ �2or grl dLg

dgla

� �þ om bm

iadLg

dAi

� �ð15Þ

Assuming that the gravitational field equations hold, we obtain an expression for the

covariant divergence of the energy-momentum tensor:

rmTma ¼ �aia

dLg

dAi

þ om bmia

dLg

dAi

� �ð16Þ

Now, if we impose the condition that all the gravitational fields are dynamical (in the sense

of being subject to Hamilton’s principle) we again recover the matter response equations.

Before adding some comments, I must stress that this result is not new, although it

seems to have been largely ignored by the recent literature. It is included in a theorem

contained in a 1974 paper by Lee et al. (1974) (the LLN theorem, henceforth) which states

the following:

The matter response equations of a Lagrangian-based, generally covariant theory of

gravity follow from the gravitational field equations if and only if there exist no

absolute variables in the theory.

My derivation above of the response equations for modified theories of gravity fol-

lowing Route 2 is a simplified proof of the ‘‘if’’ part. (Note that the LLN paper does not

provide an actual proof of the ‘‘only if’’ part, although the rest of the discussion does not

presuppose this part.)

While in GR both these two routes are available, it is clear that this is not going to be the

case in general for other theories. In conclusion, while Route 1 is always going to be

available; for Lagrangian theories, the availability of Route 2 will depend on whether any

non-matter fields are dynamical. The real significance of this difference for our discussion

can be summed up as follows: the availability of Route 2 in GR can be seen as being at the

root of the special status of inertia in that theory. Remember that the original argument is

based on the idea that the derivation of inertial motion from the gravitational field equa-

tions is explanatory and dynamical. In the next section, we will see, however, the limi-

tations of such a characterisation.

2.2 Response Equations and Geodetic Motion

The argument that GR provides a dynamical explanation of inertia depends on the extent to

which geodetic motion can be derived from the response equations. If it is the case that the

response equations imply geodetic motion for test particles—and this can be generalised


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for sources of the gravitational field—then, given that in GR the response equations can be

obtained completely from the gravitational field equations, we have a very powerful result.

It was originally assumed that in GR, as in other field theories, the motion of the sources

was independent of the field equations; it was therefore thought to be necessary to intro-

duce a separate postulate specifying that test bodies moved on geodesics of the gravita-

tional field.2 However, having the motion of sources restricted to being geodetic by the

field equations seems to make such a geodesic postulate unnecessary, and can arguably be

considered as providing a dynamical explanation of inertia in the sense that it makes the

property a corollary of the dynamical laws of the gravitational field. This is exactly what

Brown (2005) argues.

So, we need to look briefly at the strength of the results that take us from the response

equations to the geodesic principle. Attempts to prove such a connection date back to Weyl

(1952) and follow different strategies3 but all of them seem to face the same initial

difficulty: how are we to model a body associated with the energy-momentum tensor of the

theory? One of the most promising attempts to prove the result was proposed by Geroch

and Jang in 1975. Their formulation is based on the idea of taking the limit of a family of

actual material bodies that satisfy the response equations and whose size and mass

diminish; and then proving that in the limit, the resulting material points move on a

timelike geodesic in spacetime. It must be noted that the limit of such series of material

bodies is not itself a material body and, in this sense, this method only proves that the

motion of material test bodies is approximately geodetic.

Malament (2012) notes that these theorems use further assumptions that go beyond the

mere use of the response equations. Specifically, the theorem proved by Geroch and Jang

needs the condition that every body in the limiting series satisfies a certain energy con-

dition; the strengthened dominant energy condition (SDEC): given any future-directed

timelike covector na at any point on the spacetime manifold, either Tab = 0 or Tabna is

timelike. As Malament puts it: It asserts, in effect, that, whatever else is the case, energy

propagates within the body at velocities that are timelike.

The importance of this fact might be fatal for the attempt to find an explanation of

inertia in GR that is different from that available in rival theories. If the only thing that is

different about the status of inertia in GR is that geodetic motion is supposed to be a direct

consequence of EFE, and then we find out that actually, in order to derive geodetic motion

we need to impose certain conditions that depend, so to speak, directly on the matter field

equations as well, then we seem to have lost sight of the original idea.

I now turn to this objection, and two others that can be raised against Brown’s point of

view.

3 Three Objections

My next task, taking into account the contents of the previous section, will be to question

the claim that the derivation of geodetic motion in GR is more explanatory than derivations

available in other theories. To do so, I will consider three different, but related, (types of)

objections that one can raise against the original idea. It will be useful to keep in mind the

2 Soon after the formulation of GR, it was noted that it is a consequence of the theory that the gravitationalfield equations constrain the dynamics of the matter fields. See, for instance, Weyl (1952).3 For a brief review of the different approaches and difficulties associated with them see Geroch and Jang(1975).

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argument supporting the explanation story, together with its presuppositions, as presented

in Sect. 2.

The first objection is an attempt to undermine the physical significance of the derivation

of the response equations in GR from EFE. The second, as already mentioned above, is

against the implicit claim that EFE are sufficient to derive geodetic motion; it functions by

noting that further assumptions are needed to take us from the response equations to

geodetic motion. Finally the third, suggested by Pooley (2013), questions a putative

implicit assumption made by defenders of the existence of a better explanation in GR: that

EFE are more explanatory than, say, matter field equations. I think that all three objections

highlight important issues; they show that the original idea is either too vague, or wrong (if

it is taken as the claim that in GR inertia is explained dynamically because only EFE are

needed to derive geodetic motion). At the same time, however, the objections do not

actually undermine the essence of Brown’s claim. Instead, the objections can show us the

way to formulate the claim correctly.

3.1 Objection 1

Part of the initial appeal of the derivation of geodetic motion in GR is that it proceeds from

the EFE (field equations for dynamical fields, i.e. g) and apparently it uses only such

dynamical conditions. In a nomological account of explanation, any derivation of the

phenomenon starting from a scientific law would be explanatory; but, as mentioned above,

that the derivation follows putative causal paths and that it does not introduce ad hoc

conditions to permit the derivation is usually taken as a sign of a real substantive expla-

nation. This is precisely what the qualification of dynamical applied to explanation tries to

capture. Questioning, then, the substantive character of the derivation undermines the idea

of the superiority of the explanation in GR; and this can easily be achieved in the following

way. The LLN theorem mentioned above seems to capture the reason why the GR deri-

vation is more explanatory than others: it is given via the gravitational field equations and

this, according to the theorem, is only possible if no non-dynamical fields are present in the

theory. Therefore, such a derivation can be taken as providing a dynamical explanation,

and the theorem as providing a substantive criterion to distinguish when an explanation is

of a strongly dynamical type: when no absolute variables exist in the theory. Things are

not, however, that simple: for other theories, it is easy to devise a formal trick to transform

an absolute variable into a dynamical one (in the variational sense). This implies that

thinking that the derivation available in GR provides a distinctive dynamical explanation

because it proceeds from gravitational field equations in which no absolute variables

appear, simpliciter, will not work; we need a thicker notion of being dynamical.

3.2 Objection 2

The second criticism focuses on a fact that is left out of the original idea. The derivation of

geodetic motion from the response equations needs an extra condition: as we saw, Ma-

lament shows that SDEC is a necessary premise for the derivation. Now, from this fact one

can easily criticise the original claim as it is sometimes expressed by Harvey Brown: that

what means that inertial motion is properly explained in GR is the fact that the geodesic

principle is a theorem in the theory. As Malament’s note shows, it is not correct to call it a

theorem, if by that one means that it is derivable from the gravitational field equations

alone. And if one is ready to be more permissive and allow that something like SDEC can

be part of the explanatory derivation, then the defender of the superiority of the GR


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explanation seems to be in trouble. On the one hand, one might seem to be betraying the

original project. Part of the idea was that GR was peculiar in that it allowed us to derive

geodetic motion without imposing any conditions on the matter fields; and Malament’s

point shows that this is not the case. On the other hand, this also seems to question the

possibility of drawing a clear-cut contrast between GR and other theories. If one allows

conditions on matter such as SDEC as part of the derivation, why declare an explanation

that starts from EFE better than one that begins with the response equations (taken as an

independent postulate for the matter fields of a given theory)? So, unless one can show that

SDEC is more natural than the response equations themselves, the superiority of the GR

explanation cannot be defended.

I will argue against this type of objection using two arguments: first, that SDEC is as

necessary for the derivation in GR as for the derivation in any relativistic theory of matter;

second, that SDEC is clearly much more general than any particular set of matter field

equations and, I will argue, more natural than the condition that ensures derivation of

inertial motion for any relativistic material test body without assuming EFE. The challenge

posed by Malament’s result is clear: the contrast in relation to inertial motion between GR

and other theories cannot simply be that in GR it can be derived without imposing further

conditions on material fields. One needs to show to what extent, even with the use of

SDEC, the derivation in GR is more explanatory.

3.3 Objection 3

The third objection questions the idea that, if it were true that in GR a derivation of inertial

motion from EFE alone was available, then this would make an alleged explanation based

on such a derivation better in any sense (more explanatory or more dynamical). As it is

suggested in Pooley (2013), in a Lagrangian theory, a derivation following Route 1 is

always available. This means that if we are ready to regard the derivation in GR as

dynamically explanatory because it starts from EFE, then by the same token we should

consider the derivation in special relativistic theories of matter as explaining dynamically

because they start from matter field equations. On the other hand, if one thinks that inertial

motion has the status of a postulate in SR, then that would also apply to the situation in GR.

This shows that saying that inertia is explained in GR because it can be derived from EFE

cannot be the whole story. A problematic feature of the original proposal that Pooley’s

criticism brings to the fore is that, if it is taken as a complete account of the difference

between the explanations of inertia in GR and in other relativistic theories, then it would

imply some kind of dynamical superiority of EFE over matter field equations. If that is

what Brown’s proposal amounts to, I think that the criticism would be justified and would

show that it is not plausible to say that inertia is explained in a theory just because it is

derivable from some equations/laws rather than others.

I am going to take the three objections as a demand for further clarification of the

original project. There are two especially pressing tasks at hand. The first is to show that

the necessity of SDEC in the GR derivation does not undermine the idea of inertial motion

having a especial status in GR (thus responding to objection 2). The second consists of

dissolving the apparent symmetry that exists between the derivations in GR and special

relativistic theories (thus responding to objections 1 and 3). I hope that success in these two

tasks will also show in what sense the derivation of inertial motion in GR can be an

especially dynamical explanation of inertia. As a warm up exercise, before facing the

objections, let us return to the intuitions that support the original idea that there is a certain

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asymmetry between the status of inertial motion in GR and other theories, and the effect of

the three objections on such intuitions.

In GR there is a derivation of inertial motion from the gravitational field equations; this

provides a dynamical explanation of inertia. Objection 1 questions that this is indeed a

dynamical explanation. Objection 2 points out that it is only true if one adds certain

restrictions to the matter fields. Objection 3 reminds us that in other theories there are

similar derivations starting from matter field equations. Why did we think that the deri-

vation in GR was especially explanatory? Probably we, and the physicists who discovered

the GR derivation, thought that the fact that we can derive inertial motion from gravita-

tional field equations is meaningful because:

• It means that EFE are field equations that constrain the motion of their sources.4 And

this is a rather peculiar situation.

• As a consequence of what is stated above, a derivation from EFE is going to be more

general than one involving particular matter field equations: it will apply to any source

of the gravitational field.

• A derivation that starts from EFE does not assume any fixed spacetime structure while

a (Route 1) derivation starting from matter field equations takes such structures as fixed

a priori; and such fixed structures arguably play some or all of the role that Newton’s

absolute space played with regard to inertial motion—a mysterious role, to some at

least.

Naturally these are only intuitions and in the rest of the paper we will need to see whether

they can survive the previous objections.

4 The Sense(s) in Which GR Explains More

We have seen so far that an attempt to base the claim that inertia is explained in GR on the

fact that one can derive geodetic motion from EFE alone fails in three different aspects: it

is not strictly true in GR, it does not demarcate GR from other theories (insofar as this

constitutes an explanation in GR it also does so in special relativistic theories) and it does

not seem to capture the feature that makes EFE the basis of a dynamical explanation (at

least, it does not constitute a generalisable substantive criterion). At the same time, the

discussion of the problems with the original proposal strengthens the intuition that there is

a peculiar and more satisfactory sense in which inertia is explained in GR than in other

theories. In this section I intend to spell out the different characteristics that make the

explanation of inertia in GR special.

A putative explanation of inertia (in the sense of being able to specify inertial motion)

has two distinct components: (a) it gives an account of the existence of a preferred set of

spacetime paths; and (b) it derives the motion of force-free bodies in relation to this path

structure. We can compare different theories according to how they fare in relation to these

two aspects of the explanation.

4 I am aware that the use of the term source applied to the material part of EFE is very problematic. Strictlyspeaking, neither matter fields nor the stress-energy-momentum tensor are sources of the gravitational fieldin the same sense that charges can be said to be sources of the electromagnetic field. The dissimilarity istraceable to the non-linearity of EFE, which makes the same gravitational field also a source of itself. I usethe term source here, for historical reasons, to mean, in general, non-gravitational fields (and in particular,material particles) that are partially responsible for the shape of the gravitational field.


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In relation to the first one, GR seems to be in a prominent position. Einstein’s theory

revolutionised physics by converting the fixed canvas that space and time were in previous

theories (including their combination into flat spacetime in Einstein’s SR) into a non-

absolute dynamical player. It became a physical field governed by differential equations

that is not the same in all the models of the theory, and that is affected by the matter

content. In this sense, there does not seem to be much doubt concerning the superiority of

GR in providing a dynamical account of inertial structures when compared to theories with

fixed space and time. Nonetheless, even here one must be careful with two potential

caveats. The first, related to the first previous objection, is that being subjected to field

equations is an ambiguous account of being dynamical; the condition of flat spacetime can

be expressed through a differential condition that is fulfilled by the spacetime structures. A

better strategy to identify the non-absolute character of spacetime in GR is given by the

idea that absolute objects do not change from one model to another of the theory.5 Leaving

aside some subtleties, one can say that GR is distinguished from previous spacetime

theories by the geometrical objects that represent spacetime not being absolute in that

sense. The second caveat has to do with the contrast between GR and other theories of

gravitation. Even in Newtonian gravitation, not to mention modified theories of gravity,

one can argue that space and time can be seen as non-fixed in a similar way to how they are

in GR. Hence, GR is not alone in providing a dynamical explanation of inertial structures.

An account of the peculiarity of the GR explanation will have to deal with this fact.

As a consequence of this, it is possible to explicate part of the superiority of the

explanation of inertia in GR by making reference the non-absolute character of inertial

structures: the derivation of inertial motion in some theories is silent on the origin of

inertial structures, while in others there is some account of their fundamental properties.

This, at least, seems to establish a clear-cut distinction between GR and non-gravitational

theories (special relativistic theories in particular). But it is not enough to separate GR,

without further work, from Newtonian gravity and modified relativistic theories of gravity.

And it is not fully satisfactory because it leaves out an essential property of spacetime in

GR (and other theories of gravity): that, on top of not being absolute, it is somehow and to

some extent affected by matter. In order to capture these features and see how they affect

the explanatory import of the derivations of inertial motion, we need to say something else

about how the spacetime structures are determined in the different theories. I will do that in

Sect. 6.

Let us focus now on the second aspect of the explanation: deriving the motion of free

bodies.6 As we have seen, we can view the derivations as divided into two parts. The

second one, concerned with deriving geodetic motion for material free bodies from the

response equations, is arguably the same for every relativistic field theory. Some of the

geodesic theorems (the version of the Geroch and Jang theorem used by Malament, for

instance) only require us to start from a relativistic spacetime and would require the same

energy conditions for all of them. Therefore, no difference can be established in this part of

the derivation between GR and other relativistic theories. The different explanatory import

of the derivation, at this point, will therefore have to proceed from how the response

equations are established.

5 For the Anderson–Friedman program to characterise the peculiarity of GR, see, for instance, Piits (2006).6 For now, I am going to concentrate on relativistic theories, for which our previous Route 1 is available inthe Lagrangian formulation (or something analogous if a Lagrangian is not available). The comparisonbetween the GR and Newtonian gravity derivations is left for Sect. 5.

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Let us assume, to make the contrast with GR easier, that we have a Lagrangian for-

mulation of a relativistic field theory. In GR we can use either of the two routes mentioned

earlier to derive the response equations. The use of Route 2 is independent of which matter

fields appear in the matter part of the Lagrangian. This means that the validity of the

response equations is highly general: any form of matter, with its associated energy-

momentum tensor, will meet the response equations. The only requirement that enters into

the derivation is the validity of EFE. From this it follows that, for any form of matter

subjected to SDEC, the geodesic principle will hold. In contrast, in a theory with a fixed

metric, where only a matter Lagrangian governs the dynamics and, therefore, only Route 1

is available, the derivation of the response equations only holds for particular types of

matter fields (those that enter into the given Lagrangian). An explanation based on such a

derivation is thus less general: given such-and-such field equations/equations of motion (or

Lagrangian), the geodesic principle is valid for the type of matter that is made of such-and-

such fields (and meets SDEC). And it should be clear by now that such an explanation is

weaker than the one available in GR. First, the scope of the explanation is restricted by the

type of matter theory under consideration: while in GR one can say that force-free bodies

move on timelike geodesics (with the proviso of SDEC), in the matter field theory one

would say that a certain kind of mater moves in such-and-such a way. Nonetheless, this

might seem fully satisfactory as the latter theory provides an explanation for the type of

matter that it is concerned with. So the explanation is narrower, but equally powerful. But

even this claim is not fully defensible. If GR provides an explanation of inertial motion, it

does so without assuming anything (apart from what might be involved in SDEC) about the

nature of matter. A matter field theory needs to assume that the material bodies for which

the geodesic principle holds are composed of the fields for which the theory provides a

dynamical description. And this seems quite an unnatural assumption for a matter field

theory to make; field theories provide field equations for the fields but usually do not

contain assumptions about the nature of the sources of the fields, so their inertial motion

would fall outside their scope.7 This would be the core of the reply to objection 3: theories

for which only Route 1 is available need to make much stronger assumptions about the

composition of bodies in order to be able to state anything like the geodesic principle.

One way to escape the previous conclusion is suggested by the structure of the deri-

vation of the response equations in the Lagrangian context. Route 1 provides a recipe to

derive the response equations given a general covariant Lagrangian for a matter field. So

perhaps we should consider the formal conditions required for the derivation to be those

that support the weight of the explanation and delimit its scope. Apart from the covariance

of the Lagrangian, one needs to assume something about how the matter fields couple to

the metric. A minimal assumption is that every matter field couples to the same metric (and

no other non-material fields appear in the Lagrangian, apart from the metric).8 So, in order

to derive the response equations from a general Lagrangian theory of matter, these are the

two conditions that must be imposed: general covariance and universal coupling.

This might seem no big deal, but we must check whether there are good physical

motivations for such conditions to hold. Recall that we are no longer in the context of a

given physical theory, in which such requirements can be seen as properties of the

dynamical equations; if we were in GR, for instance, general covariance of the Lagrangian

7 It might be useful to think of electromagnetic theory in this context. Maxwell’s equations are fieldequations for the electromagnetic field but do not constrain the motion of the sources (electrons, forinstance) of the fields.8 This will be further discussed in Sect. 6 under the label ‘‘‘universal coupling’’’.


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could just be seen as a property of the Lagrangian that renders the right field equations. In

the present situation, these conditions play the role of meta-principles that any matter

theory needs to meet, if we want the response equations to have the same status as they

have in GR. And no merely formal condition seems suitable to do the job.9 If we think of

an arbitrary theory of matter, I cannot see any general argument that would defend either a

particular symmetry requirement for the Lagrangian or a unique spacetime field to which

matter fields should couple. But this might be seen as a very artificial starting point. We

should be starting from a theory of matter that is locally special relativistic: in the end, the

reason for carrying out a contrast between GR and theories with a given fixed metric is that

SR belongs to the latter class of theories. So we need to see whether, from the point of view

of SR, such assumptions are justified. It is difficult to see how the requirement of writing a

generally covariant Lagrangian might be justified in the context of SR. Nonetheless, the

other condition is where the physically meaningful requirements enter. The condition being

considered now is that the Lagrangian renders locally special relativistic laws, i.e., that

they are locally Lorentz covariant. And even if there is no unique way of writing a

generally covariant Lagrangian that meets such a condition (different Lagrangians with

non-minimal coupling can be written), the introduction of other non-material fields is

excluded and, therefore, this is sufficient to produce a Lagrangian from which the response

equations are derivable through Route 1.10

So, the situation is now the following. We have found a way of deriving the response

equations for special relativistic theories that is as general as the way in GR and that

furthermore does not mention the composition of material bodies. We have achieved this

through the imposition of some general conditions on the Lagrangian from which the

matter field equations are derived: those conditions boil down to general covariance and

local Lorentz covariance. So, if we single out SR as the theory providing such criteria, we

can say that SR provides an explanation of inertial motion, in relation to its second

component (the derivation of motion for force-free bodies), equivalent to the one present in

GR. But this is so only if one takes local Lorentz covariance as a brute unexplainable fact

of every physical law, which again raises the asymmetry between the explanations

available in the two theories. In both theories some dynamical equations are necessary to

carry out the derivations (EFE or matter field equations). Nevertheless, while in GR the

field equations are sufficient to derive the response equations, in SR this is only the case if

the condition of Lorentz covariance is part of the definition of the theory, which is a

reasonable and fairly standard way of individuating SR. But even so, we must admit that

the condition of Lorentz covariance, equivalent to the conservation of the energy-

momentum tensor, plays an essential role in the special relativistic explanation, making the

general relativistic one more parsimonious (it uses fewer conceptual resources).

9 We must be careful here. Even if we thought that formal conditions (such as general covariance) couldsomehow be justifiably introduced, they would prove to be insufficient to play a substantive role in anyattempted dynamical explanation.10 It is interesting to introduce a consideration here regarding the possibility of non-minimal coupling. Evenif non-minimal coupling does not, by itself, preclude the derivation of the response equations, the energy-momentum tensor for which they would hold would be a different one. This prompts a question about thephysical interpretation of that tensor. In principle, in a special relativistic theory, the Hilbert energy-momentum tensor obtains its physical meaning from its connection to the symmetrical version of thecanonical energy-momentum tensor. But such a connection is only available for Lagrangians with minimalcoupling (see Leclerc 2006). So, allowing for Lagrangians with non-minimal coupling makes the responseequations as general as in GR but it leaves the energy-momentum tensor without a natural interpretation.

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4.1 The Challenge of SDEC

Behind the previous discussion there lurks an implicit assumption; that an explanation

involving fewer assumptions regarding matter fields is better. This seems a natural thing to

presume if we think that the less that is presupposed about the matter fields, the wider the

scope of the explanation will be. At the same time, however, Malament’s point regarding

the necessity of an energy condition seems to undermine the idea that the explanation in

GR is a better one. Once we allow conditions on matter fields in the derivation, the defense

of the superiority of the GR explanation is not as straightforward as it might originally have

seemed. After having revealed in what precise sense we can claim some superiority for the

GR explanation, I must now face the full implication of objection 2.11

The objection can be stated in the following way. We will focus on the derivation of the

motion of force-free bodies: the second aspect of the explanation of inertia. I have argued

that the sense in which GR does better than SR is a consequence of the nature of the

suppositions that enter into the derivation of the response equations: in GR only EFE are

necessary, in SR one needs to assume local Lorentz covariance (or the response equations

themselves). So, the superiority of GR can be regarded as being a consequence of the fact

that the conditions entering the derivation are fully dynamical (field equations) and they

affect any matter fields; while some conditions participating in the SR derivation are non-

dynamical and physically unmotivated as a meta-principle.12 One way of seeing this is that

the GR derivation is more general because it does not say anything about matter fields

(beyond their having energy-momentum) while the SR derivation needs to assume

something like local Lorentz covariance for matter fields; so either this is taken as a

selection rule (restrictive) or as a general meta-principle (arguably unmotivated). And here

is where objection 2 comes in and reminds us of the necessity of SDEC to derive geodetic

motion from the response equations. SDEC is an energy condition on matter fields, with a

status similar to that imposed in SR, which needs to be imposed in any case to derive

geodetic motion; therefore, any superiority that GR might have in the first part of the

derivation is washed away by SDEC in the second part.

The first thing to do is to remind ourselves that SDEC is needed equally in GR and SR.

This means that in the final count of the conditions needed to derive geodetic motions, SR

still wins: GR is still more parsimonious than SR. Nonetheless, this might seem small

comfort; using fewer conditions does not by itself establish a substantive difference. To see

the differences we need to go into the interpretation of the energy conditions. SDEC

implies two things: the energy density is positive; and any observer will see that the

energy-momentum associated with the body in question propagates along timelike curves.

As Malament notes, this excludes energy-momentum tensors associated with either light or

tachyonic matter. Under this interpretation, it is easy to take SDEC as a natural energy

condition on relativistic matter; in fact, when we talk about material test bodies we refer to

forms of energy-momentum that propagate in a timelike way. So, in GR the theorem

involves material test bodies (not light or tachyons) moving along timelike geodesics. The

idea is then to take SDEC as part of the definition of what relativistic matter is. Can we do

something similar with the extra condition present in SR? Does the requirement of Lorentz

11 In this section I will explore the implication of such an objection for the contrast between GR and SR; itseffects on the comparison with Newtonian theory are considered in the next section.12 Again, one could say that Lorentz covariance needs no further motivation than being taken as a propertythat matter fields happen to have. But in that case, the explanation loses generality by not being applicable toall matter fields.


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covariance have a natural physical interpretation? Such a condition is equivalent to the

conservation of energy-momentum (the response equations for flat spacetime). Hence the

interpretation is transparent: what one is imposing with local Lorentz covariance is the

conservation of energy-momentum; a stronger energy condition but an energy condition

nonetheless. And this seems to dissolve the substance of the difference between GR and

SR.

Our original claim must now be tempered. GR contains a dynamical explanation of

force-free motion for non-tachyonic matter because a geodesic principle can be derived for

it from EFE alone. In contrast, SR contains an explanation of force-free motion for non-

tachyonic matter because a geodesic principle can be derived from matter field equations,

insofar as one takes conservation of the energy-momentum tensor for material fields as a

defining feature of the theory. The explanation is less general (if one contemplates the

possibility of non-conservative matter fields) and less dynamical (as one of the essential

conditions needed for the derivation that supports the explanation has to be assumed). If

one wants to regard such a condition as dynamical because it is a property of matter field

equations, one needs to give up the aim of maximal generality (see note 15). And this

without mentioning the first aspect of the explanation: that in GR the spacetime structures

are dynamically determined.

5 Explanation of Inertia and Newtonian Theory

Up to now, I have contrasted the explanation of inertia available in GR with its counterpart

in other relativistic theories. Nonetheless, the evaluation of the strength and singularity of

the explanation in GR would not be complete without comparing it with the situation in its

gravitational precursor (Newton’s theory of gravity).13

Initially one might be tempted to declare inertial motion a postulate of Newtonian

theory in agreement with its axiomatic structure; Newton’s first law states that force-free

bodies move in straight lines. Instead of going into the subtle job of interpreting Newton’s

first law in order to compare the status of inertia in Newtonian gravitation and GR, it will

be useful to refer to the geometrised version of Newtonian gravitation, aka Newton–Cartan

theory. So, although it might be the case that the two-step derivation available in rela-

tivistic theories does not have a direct translation into Newtonian theories, under this

perspective the strategy is to find an expression in Newtonian theory that is equivalent to

the response equations in the following sense: from it, under certain conditions that will

have to be determined, a result to the effect that force-free test bodies move along geo-

desics can be derived. With this to hand, one can compare the status of inertia in the two

theories by: first, contrasting the status of the response equations and their equivalent in

Newtonian theory; and, second, evaluating the conditions that allow us to derive the

geodesic principle in both cases. Fortunately, the technical part of this work has already

been carried out in Weatherall (2011a).

The setting for geometrised Newtonian gravitation is a classical spacetime

(M, ta, hab, r), where M is a smooth, connected, four-dimensional manifold, ta defines a

temporal metric tab = ta tb, hab is a symmetric field that can be taken as the spatial metric

13 Note that for other relativistic theories of gravity the previous scheme of analysis, namely comparing thestatus of the response equations, can be applied. If we do so, we will find that different justification ofinertial structures plus a different scope of applicability of the response equations results from it; as aconsequence of which inertia does not receive the same level of explanation in all of them.

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(and is orthogonal to the temporal one) and r is a derivative operator on M compatible

with tab and hab. Given the derivative operator, one can define a Riemann tensor (Rbcda ) and

a Ricci tensor (Rab). Flatness is given by Rbcda = 0 and spatial flatness by

Rabcd = Rarsth

brhcshdt ¼ 0 or equivalently by Rab = hashbtRst = 0. Given a point particle

(represented by its worldline) we can define its four velocity (a smooth unit vector field na

tangent to the worldline) and a four-momentum field pa = mna. We can associate, with any

matter field, a symmetric field Tab encoding the mass and momentum density.

Let us express Newtonian dynamics in this language. The force on a massive test

particle of mass m and with four-velocity na is defined by Fa = mnbrbna. From this it is

clear that if the external force on a particle is zero, it is going to move along a geodesic.

Now we must distinguish two different cases. In standard Newtonian mechanics, r is

taken to be flat: in this case, then, geodetic motion gives us Newton’s first law. In geo-

metrised Newtonian theory, r can be curved and it can be proved that gravitation places a

constraint on r. In particular, it can be proved that, starting with a flat derivative operator

and the Poisson equation, ra ra / = 4pq (an equation that relates the mass density and

the scalar gravitational potential), there is a unique curved r(g) whose geodesics corre-

spond to the trajectories of test particles under the gravitational force ðnnrðgÞnna ¼ 0$nnrnn

a ¼ �ra/Þ: The Riemann curvature tensor associated with this derivative operator

satisfies the geometrised Poisson equation, Rab = 4pq ta tb and Rcdab = 0.

In this context, Weatherall proves a theorem (his Theorem 4.4) that states the conditions

under which geodetic motion can be derived in geometrised Newtonian gravitation. They

are:

• two curvature conditions: spatial flatness (Rabcd = 0) and Rabcd ¼ 0

• the mass condition (positive mass): whenever Tab= 0, Tabta tb [ 0

• the conservation condition: raTab = 0

As Weatherall notes, spatial flatness is a consequence of the geometrised Poisson

equation and the other curvature condition is what would allow us to recover standard

Newtonian gravitation from the geometrised version. He also seems to think that it would

be possible to relax this last condition and still obtain a geodesic theorem, in which case it

would be proved for a more general class of geometrised Newtonian theories.

We now have all we need to compare the status of inertia in Newtonian theory and GR.

Starting from the response equations (or the conservation condition) we can derive geo-

detic motion in both theories assuming either the dominant energy condition, in GR, or the

mass condition, in Newtonian geometrised theory. I argued above that SDEC can be seen

as part of a natural definition of relativistic matter; in a similar way, the mass condition is

also part of the definition of matter. So far things seem on a par but geometrised Newtonian

theory requires, on top of this, the curvature conditions; so if we concentrate on this step of

the derivation it seems to be in a worse position.14 Now we must remember that our interest

is in the extent to which inertia receives a dynamical explanation in both cases. In GR, not

needing more conditions on top of EFE to derive the response equations was essential in

arguing for the explanation story. In geometrised Newtonian gravitation the situation

differs in two senses. The first one is that even having the conservation condition one needs

the curvature conditions to get the theorem running. While the spatial flatness can be

accommodated into a dynamical explanation as it is a consequence of the Newtonian

14 Since the first version of the present paper, Weatherall (2011b) has published his own view with respectto the comparative status of inertial motion in geometrised Newtonian gravity and GR. There he argues thatthe weak energy condition needed in the Newtonian context is less demanding than SDEC for GR.


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counterpart of EFE, the other curvature condition would not be dynamically justified (it is

clearly a condition that fixes part of the spacetime structure that is both non-dynamical and

independent from matter but that would be necessary to get geodetic motion).15 So, in this

respect, only if it is shown that the last curvature condition is in the end unnecessary, the

situation would be equivalent to GR.

With respect to the first part of the derivation, the analogy between the two theories

breaks down: while in GR the response equations are a consequence of EFE, in Newtonian

theory the conservation condition must be assumed independently. There is obviously a

way to justify this assumption by showing that it is a consequence of Newtonian dynamics,

the force law, but then it is clear that this constitutes an independent dynamical assump-

tion. We can say then that in geometrised Newtonian gravitation, there is an explanation of

inertia that comes from two independent dynamical principles: an equation for the grav-

itational potential plus a force law for material bodies; this thereby concedes that the

second curvature condition is not needed after all. But this differs from GR in an essential

point: in GR the field equations for the gravitational field make the force law (an inde-

pendent law for the motion of force-free material bodies) unnecessary.16 Note that it is

precisely this feature that the physicist who first became aware of this property found

remarkable: that the field equations constrained the motion of the sources.17

Furthermore, standard Newtonian dynamics does not provide any account of the

existing inertial structure, so it presupposes the existence of a privileged set of paths. Given

this, one can say that Newton’s first law is derivable from his second law. In this theory,

then, we would have a dynamical explanation of inertia in this limited sense: in a way

similar to what happened in the context of SR, but with Newton’s second law playing the

role that Lorentz covariance played in the special relativistic context.

To sum up, the contrast between GR and geometrised Newtonian gravity with respect to

the two aspects of an explanation of inertial motion comes down to the following:

• In both theories there is a dynamical account of the putative inertial structures.

Although it is an open question whether Newtonian gravity introduces a larger amount

of absolute structure.

• The derivation of geodetic motion needs an energy condition in both cases (arguably

interpretable as part of the definition of ponderable matter).

• While in GR the response equations are a consequence of EFE, in Newtonian

geometrised gravity their counterpart must be assumed.

This means, as I understand it, that GR is in a better position to provide a dynamical

explanation of inertial motion.

15 I stress here that Weatherall believes that this condition can be relaxed, in which case this is not the placewhere to look for a difference between GR and geometrised Newtonian theory. The main difference comesfrom what I say below.16 Weatherall regards this difference between GR and the Newtonian case as irrelevant. His first reason forthis view is that in GR, even assuming that the response equations hold, one needs an extra assumptionregarding the nature of test bodies. I cannot see the strength of this concern if one understands test matter asa limiting case of ordinary matter. His second reason is that the conservation condition is better seen as ameta-principle both in Newtonian theory and relativistic theories of matter. I agree with him with respect toit playing such a role historically; but this by itself does not undermine my main argument: even if Einsteintook energy conservation as a requisite for his sought-after theory, it is physically meaningful that uniquelyin GR such a condition is a consequence of the gravitational field equations.17 The situation in geometrised Newtonian gravitation can be thought of as analogous to that in electro-magnetism where the field equations (Maxwell’s equations) do not restrict the motion of charged bodies andone must add the Lorentz force law.

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6 The View from the Equivalence Principle: Inertia and Gravity

There are two senses in which the explanation of inertia present in GR is more satisfactory

than the one allegedly found in other theories.

• It is maximally general. The response equations, from which geodetic motion for force-

free bodies can be derived, hold for any kind of matter whatsoever and the geodesic

theorem resulting from them can be interpreted as constraining the motion of free

bodies to be timelike geodetic without any concern about the constitution of the bodies.

The only restriction on the generality of the result comes from the SDEC.

• The theory contains a dynamical account of the spacetime structures that determine the

paths followed by bodies in inertial motion (what we may call the inertial structure).

Even better, it contains an account that relates this dynamics to the material content

present in the models of the theory.18

These two senses, although different, are not independent in GR; the fact that the

response equations have this general character in the theory is a consequence of them

holding as a consequence of EFE, without requiring matter field equations, and EFE are

precisely what contain a dynamical account of inertial structure. In an abbreviated way,

one can say that in GR inertia is dynamically explained (better than in other theories)

because inertial motion can be derived without assuming anything about the material

composition of bodies or any inertial structure that is fixed independently from matter. In

this section, I will argue that the convergence of these two senses is a meaningful feature of

the general relativistic explanation.

In order to do so, we need to return to the contrast between the alleged explanations of

inertia found in GR and in special relativistic theories. In the latter case, we distinguished

two different situations: special relativistic material theories and SR itself, understood as a

set of principles that any theory of matter must meet. Let me sketch, in turn, how an

explanation of inertia can be provided in these two contexts.

In the case of a special relativistic theory of matter (let us assume, for the sake of the

argument, that we have a Lagrangian field theory), as I discussed above, geodetic motion

can be derived as a direct consequence of the field equations; the validity of this derivation

is then restricted to matter that obeys a particular set of field equations. Nonetheless, we

can view this derivation as emanating from a more general feature of the theory: that its

dynamics is derivable from a generally covariant Lagrangian. This indicates that, in such

theories, the weight of a general explanation must lie elsewhere. This then takes us to the

second theoretical context to consider: SR understood as a framework within which the-

ories of matter can be formulated.

The usual way of individuating SR is through viewing it as imposing a local symmetry

condition: Lorentz covariance implies that the energy-momentum tensor is conserved in the

ordinary sense, and geodetic motion can be derived from it for any spacial relativistic theory.

Based on this, we have an explanation that is as general as the one in GR, but we must ask:

what is actually doing the explanatory work here? Put this way, it might seem that it is the

symmetry principle that is doing the substantive explanatory work but, at the same time, that it

needs to be assumed as a brute fact. So the degree of generality is achieved by taking Lorentz

18 In principle it is arguable that seeing this as a positive feature of the explanation is somehow arbitrary.Nonetheless, it seems difficult to imagine how a condition that fixes spacetime structure in a way that isindependent of matter can have an impact on the derivation of the response equations without diminishingits generality.


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covariance (and therefore the conservation condition) as a given. For someone seeking a

dynamical explanation, this situation does not seem desirable. Alternately, the defender of the

special relativistic explanation may think that the derivation of the response equations in SR

proceeds in a similar way to how it does in GR (through our previous Route 1). If this is the

case, then local Lorentz covariance can be seen as part of a more general principle: the

dynamics of the theory must be derivable from a generally covariant Lagrangian that yields

the special relativistic laws in local freely-falling frames. Then the explanation has the same

generality as in GR and it needs the explicit specification of the matter field equations; it is

therefore a good candidate dynamical explanation. In what follows, I offer an alternative take

on why still to think that GR is in a better explanatory position.

The general principle to which I referred above, in the overall context of metric theories

of gravitation, is one of the forms that an expression of the Equivalence Principle can take.

Will (1981) argues that a theory that implements EEP must be a metric theory of gravity,

which thereby satisfies the following postulates (Will 1981, 22):

1. the spacetime of the theory is endowed with a metric g;

2. the worldlines of test bodies are geodesics of that metric;

3. in local freely-falling frames, called local Lorentz frames, the non-gravitational laws

of physics are those of SR.

This is also called universal coupling and describes the geometrisation of gravity. In the

context of our discussion of the derivation of inertial motion for Lagrangian theories of

matter, we could substitute the second condition by SDEC; and now, from these conditions

we would be able to derive inertial motion. So a theory that meets these three conditions

would be a theory that contains a derivation of inertial motion that may be regarded as

explanatory. In this sense, the Equivalence Principle can be thought of as being behind the

explanation of inertial motion.

This offers a new perspective on the explanation available in SR; it can be justified by

viewing a special relativistic matter theory as a local particularisation of a matter field theory

with a general metric for which the EEP holds. In such an explanation, the EEP would be an

essential ingredient, but the justification of the EEP itself belongs to the context of GR.

The Equivalence Principle played a very important, and convoluted, heuristic role in the

conception of GR. Starting from a particular situation in which a uniform gravitational field

can be seen as an inertial effect, Einstein sought to generalise this original intuition. Although

there is no unique way of extending the Equivalence Principle to include arbitrary gravita-

tional fields,19 there is no doubt that GR is in part the result of Einstein’s efforts in that

direction. Apart from its heuristic role, the principle concerns the interrelation between

matter and spacetime that can be embodied to different degrees in spacetime theories. GR

and other metric theories implement the version of the Equivalence Principle sketched

above, and this can be used as a unitary account of the explanation of inertia for a certain class

of relativistic theories; the explanatory story in this situation would run from the matter

Lagrangians through Route 1. Nonetheless, as we have seen, such a story is redundant in GR;

EFE (plus SDEC) are enough to derive geodetic motion. This can be taken as an indication

that GR implements a stronger version of the Equivalence Principle.20

19 See Norton (1985).20 The idea is that while GR and other theories meet Will’s notion of EEP, GR also contains a kind ofexplanation of it. This might be thought of as being a consequence of EFE and seen as a fuller imple-mentation of the equivalence between inertia and gravity. I leave the precise version of such a principle to bederived in future work elsewhere.

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In this setting, we can see the singularity of the explanation available in GR; EFE suffice

to derive the response equations. They restrict the way in which the matter fields interact

with the spacetime structures; it must be in such a way that the response equations hold.

And EFE are, at the same time, the set of equations that dictates the dynamics of the

gravitational field. In GR, then, we find the following singular situation: the same equations

that are responsible for the dynamics of the gravitational field suffice to derive the response

equations (from which geodetic motion can be derived). In contrast, in a special relativistic

theory of matter, one can allude to the EP as part of the explanation of inertial motion; as a

way of justifying the condition of local Lorentz covariance. Nonetheless, by doing so one

is invoking a principle that has no justification in such a context, or whose justification can

be seen as a consequence of a deeper principle. GR contains a candidate for being such a

principle: that the spacetime structures that we call inertial are, at the same time, part of the

gravitational field and, as such, affected by matter. The significance of this can be

expressed as follows:

• In GR, compared to other relativistic spacetime theories, we reduce to the minimum the

number of principles necessary to explain inertial motion. One set of equations

accomplishes the two tasks: it fixes the set of geodesics and constrains matter to move

on them.

• In GR this can be thought of as providing a physical justification for the Equivalence

Principle; why material bodies in free fall follow geodesics of a field that can be

interpreted as the gravitational field. In other theories, such as SR, the justification of

the Equivalence Principle can be seen at most as being subsidiary to that in GR.

Reference to the Equivalence Principle can also illuminate the differences, seen in the

previous section, between the explanations available in GR and in geometrised Newtonian

gravity. The similarities are stressed by Weatherall: Newtonian gravity contains a geodesic

principle, if one assumes an energy condition, and the inertial structures are partly

determined by matter content. At the same time, the difference is striking: in Newtonian

gravity these two things come from independent principles (the Poisson equation and the

conservation condition) while in GR they emanate from the same condition: EFE. From the

perspective of the equivalence principle, we can say that in the Newtonian context the

principle from which inertial motion follows needs to be assumed independently of the

field equations: that inertia is gravity is an independent condition instead of a consequence

of the field equations. In this sense, geometrised Newtonian gravity is closer to special

relativistic theories than to GR.

7 Conclusion

We started with a claim to the effect that inertia, for the first time, stops being a mystery

and receives a dynamical explanation in GR. This claim was based on the fact that in GR

inertial motion can be derived from EFE and this was supposed to be qualitatively different

from what happens in other theories. A claim such as this is twofold: it involves asserting

that the way in which inertial motion is derived in GR constitutes a dynamical explanation

of inertia and, also, that this derivation is substantially different from what is available in

contrasting theories. As we have seen, in its original formulation the claim is ambiguous

enough to fall prey to three different types of criticism. The first questions the dynamical

status of a derivation of the type allegedly available in GR because it can easily be arrived

at through a formal trick. The second points to the fact that in reality, in GR we need


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something more than EFE to derive inertial motion, and this undermines the idea that the

possible differential status of inertia in GR is based on the fact that only non-matter

conditions are used to derive inertial motion. The third attacks both the dynamical char-

acter of the derivation in GR and its uniqueness; it points out that while in GR inertia can

be derived from EFE, in other theories it can be claimed to be derived from matter field

equations. Moreover, if this latter fact is seen as a non-dynamical feature, one must

remember that it is also true for GR.

I think that the three types of criticism are fair and show the initial proposal to be

insufficient; but they do not show the original idea to be invalid. I believe that there is a

genuine sense in which it can be claimed that inertia is dynamically better explained in GR

than in other theories such as SR. I defend this by proposing an elucidation of the initial

claim that meets the three criticisms, and does well in both justifying the dynamical status

of the explanation and showing the uniqueness of GR. It can be expressed in the following

twofold way:

• In GR, inertial motion is derived by using field equations that do not assume that

inertial structures are fixed a priori.

• In GR, inertial motion is derived for any kind of matter (source of the gravitational

field) irrespective of its composition.

This is different from the way in which inertial motion is justified in some other theories

(where one needs to assume a conservation condition to achieve the same level of gen-

erality) and it is more explanatory because of its generality together with the dynamical

justification of the structures involved in the derivation. To be more specific in the contrast,

one can say that GR combines generality with a lack of assumptions beyond the field

equations (leaving aside the dominant energy condition), while other theories strike a

worse compromise:

• Special relativistic matter theories provide a dynamical explanation of inertia (based

only on field equations) only for particular types of matter and are silent about the

origin of inertial structures.

• SR only explains inertia either by incorporating an energy conservation condition (or a

Newtonian-type force law) or by assuming a symmetry principle (such as local Lorentz

covariance) and is again silent regarding inertial structures.

• Geometrised Newtonian theory (Newton–Cartan theory) provides an explanation by

using two independent principles: the gravitational equation (Poisson equation) plus

energy conservation. Alternatively, one can say that it explains through two

independent dynamical principles: the geometrised gravitational equation plus

Newton’s second law. Moreover, inertial structures are partly justified by the equation

for the gravitational field. In a sense this situation is similar to that in GR because part

of the conditions needed to derive geodetic motion are a consequence of the

gravitational field equation, but it differs in that the main condition (conservation)

needs to be assumed independently.

I argue that this way of formulating the claim meets all the criticisms and allows us to

give a general recipe for deciding when inertia is dynamically explained. The main idea of

the argument is that the derivation of inertial motion based on the gravitational field

equations is special because it does not need to use any condition that fixes spacetime in a

way that is independent of matter and this is connected to the range of applicability of the

geodesic postulate. This provides unambiguous meaning for the expression ‘‘‘gravitational

field equation’’’ (in answer to criticism 1), makes it irrelevant whether some energy

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conditions are necessary to derive the geodesic principle (in answer to criticism 2) and

shows a sense in which having this route to the geodesic principle available is differential

and more explanatory than having only the derivation through the matter field equations (in

answer to criticism 3). Furthermore, this way of understanding the claim connects with a

key principle in the formulation of GR: the Equivalence Principle. The discussion of this

connection shows that if one understands the Equivalence Principle properly, in a stronger

sense than usual (where it is usually reduced to universal coupling) then the provision of a

dynamical explanation of inertia is at the very core of the theory.

Acknowledgments My special thanks go to Carl Hoefer for his help, and patience, which led to majorimprovements in the substance and form of this paper. I am also grateful to Toffa Evans and Laura Fellinewho read, commented and suggested useful changes on different drafts of the paper. Needless to say, theremaining errors are only mine.

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