What is a Singularity in Geometrized Newtonian Gravitation?

James Owen Weatherall

Department of Logic and Philosophy of Science
University of California, Irvine, CA 92697

Abstract

I discuss singular spacetimes in the context of the geometrized formulation of Newtonian
gravitation. I argue first that geodesic incompleteness is a natural criterion for when a model
of geometrized Newtonian gravitation is singular, and then I show that singularities in this
sense arise naturally in classical physics by stating and proving a classical version of the
Raychaudhuri-Komar singularity theorem.

1. Introduction

Spacetime singularities are often described as a puzzling and strikingly non-classical feature

of general relativity. This is in part because, as John Earman (1995) has emphasized,

singularities are closely connected to indeterminism and acausality in relativity theory. But

in a sense the puzzles arise before one gets to such issues. Even identifying when a spacetime

is singular has proved to be a major foundational problem. And, perhaps most remarkably of

all given these difficulties, one apparently cannot avoid singularities in the general relativity:

the celebrated Penrose-Hawking-Geroch singularity theorems show that singular spacetimes

are generic among the physically reasonable models of the theory.1

The problem is not that singularities do not arise in other theories. They do. For

instance, there are solutions to Maxwell’s equations on which the electromagnetic field di-

Email address: weatherj@uci.edu (James Owen Weatherall)
1For more on the conceptual issues of characterizing and classifying singularities, see Geroch (1968), Ear-

man (1995), and Curiel (1999). For a history of singularity theorems leading to the Penrose-Hawking-Geroch
theorems, see Earman (1999). The classic resource on spacetime singularities in the physics literature is
Hawking and Ellis (1973), but see also Geroch (1970), Ellis and Schmidt (1977), Tipler et al. (1980), and
Clarke (1993). Wald (1984) and Choquet-Bruhat (2009) provide more systematically pedagogical presenta-
tions.

Draft of August 7, 2013 PLEASE DO NOT QUOTE OR CITE!



verges at some point(s) of spacetime.2 Such solutions are naturally characterized as singular,

with singularities occurring at the points of divergence. In general relativity, however, one

cannot reason in this way, since the dynamical fields themselves determine the geometry

of spacetime. Thus if one of these fields fails to be defined at a point, that point must be

“removed” from the spacetime. For this reason, there is now broad consensus that the best

way to think of singularities in general relativity is in terms of “incomplete” spacetimes—

specifically, spacetimes that are (timelike or null) geodesically incomplete (I will make this

precise below). But the situation remains murky, and questions remain concerning both (1)

whether geodesic incompleteness is a necessary (or even sufficient) condition for a spacetime

to be singular and (2) how to understand the relationship between geodesic incompleteness

and the divergence of some physical quantity.

In its standard formulation, Newtonian gravitation (conceived as a field theory) is es-

sentially similar to electrodynamics. In this sense, general relativity is starkly different

from classical gravitational theory with regard to singular structure, which justifies the view

that spacetime singularities of the sort that lead to foundational problems are a distinctly

non-classical feature of relativity theory. However, there is a reformulation of Newtonian

gravitation due to Élie Cartan (1923, 1924) and Kurt Friedrichs (1927) known as geometrized

Newtonian gravitation (or often, Newton-Cartan theory). Though in various senses equiva-

lent to Newtonian gravitation,3 geometrized Newtonian gravitation shares many important

qualitative features with general relativity. Most importantly, in geometrized Newtonian

gravitation, as in general relativity, the dynamical fields determine the geometry of space-

2One needs to be careful here. The electromagnetic field is most naturally understood as a tensor
field, Fab. Thus it is ambiguous to say the field “diverges,” since the components of a tensor field might
diverge in some coordinate systems but not others. To be precise, then, one should say the electromagnetic
field is singular just in case an appropriate scalar field diverges. Two natural candidates are FabFab and
�abcdF

abFcd—or, in perhaps more familiar notation, the invariant quantities 2(||E||2 − ||B||2) and EaBa,
where Ea and Ba are the electric and magnetic fields relative to some observer. Both of these capture, in
different ways, the “magnitude” of the electromagnetic field at a point.

3For more on the equivalence of Newtonian gravitation and geometrized Newtonian gravitation, see
Glymour (1970), Weatherall (2013), and Knox (2013).

2



time.

This feature of geometrized Newtonian gravitation raises a cluster of interesting questions

in connection with singularities. There is a sense in which the difficulties associated with

singularities in relativity theory arise from the geometrical character of the theory, and

geometrized Newtonian gravitation is geometrical in a very similar sense. But there is also an

important difference between the theories: in a wide variety of cases, one can “translate” back

and forth between models of geometrized Newtonian gravitation and of standard Newtonian

gravitation, where singularities are well understood. This suggests that one might gain

some insight into singular structure in general relativity by investigating singularities in

geometrized Newtonian gravitation.

This paper is a first step in this investigation. I hope to make two modest points. The first

is that geodesic incompleteness is a natural way of thinking about singularities is geometrized

Newtonian gravitation, and indeed, the route to geodesic incompleteness is more direct than

in relativity theory. The second point is that singularities arise naturally in Newtonian

gravitation. To support this claim, I will state and prove a Newtonian singularity theorem

analogous to a historically important singularity theorem due to Raychaudhuri (1955) and

Komar (1956).4 The upshot of this theorem is that all homogenous and isotropic Newtonian

cosmological models with non-vanishing matter content exhibit either an initial (big bang)

or final (big crunch) singularity, or both.

4For more on the history of this theorem, see Earman (1999). The theorem and its significance is also
discussed in Sachs and Wu (1977) and Senovilla (1997).

3



2. Classical spacetime structure and geometrized Newtonian gravitation

I will begin by briefly reviewing the geometrical structures I will work with.5 A model of

general relativity is a relativistic spacetime, which is an ordered pair (M,gab), where M is a

(smooth)6 four-dimensional paracompact Hausdorff manifold and gab is a Lorentzian metric,

with signature (+1,−1,−1,−1). Associated with any such metric is a unique torsion-free

covariant derivative operator, ∇, satisfying ∇agbc = 0. We call a tangent vector ξa at a

point p timelike if gabξ
aξb > 0, spacelike if gabξ

aξb < 0, and null if gabξ
aξb = 0. A vector

field is timelike (resp. spacelike or null) if it is timelike at every point; likewise, a curve

is timelike if its tangent field is at every point. Matter is represented by its (symmetric)

energy-momentum tensor Tab, which is related to the Ricci curvature of ∇ by Einstein’s

equation, Rab − 12Rgab = 8πTab. Timelike curves are the possible trajectories of massive

point particles; in the absence of an external force, massive point particles traverse timelike

geodesics.

Models of both standard and geometrized Newtonian gravitation, meanwhile, are clas-

sical spacetimes. These are ordered quadruples, (M,tab,h
ab,∇), where M is as above, but

tab and h
ab are degenerate “metrics” with signatures (1, 0, 0, 0) and (0, 1, 1, 1) respectively

and ∇ is a torsion-free derivative operator. The metrics are required to satisfy habtbc = 0,

and the derivative operator is required to be compatible with the metrics in the sense that

∇atbc = 0 and ∇ahbc = 0. (We have to stipulate a derivative operator since there are in

general many that are compatible with the degenerate metrics.) The field tab can be thought

of as a temporal metric: given a vector ξa at a point p, the temporal length of ξa is given

by ξaξbtab. If ξ
aξbtab > 0, we say the vector is timelike; otherwise it is spacelike. Similar

5For reasons of space, I take for granted that the reader is familiar with the theories in question. This
section is only a short overview, for completeness and to fix notation. For a more systematic discussion of
general relativity, see Hawking and Ellis (1973) and Wald (1984); for (geometrized) Newtonian gravitation,
see Trautman (1965). Malament (2012) discusses both theories in a unified language. Here and throughout,
I use the “abstract index” notation for tensorial objects, which is explained in Malament (2012).

6In what follows, I will assume that any object that is a candidate to be smooth is so.

4



considerations apply to vector fields and curves. The field hab, meanwhile, can be under-

stood as a spatial metric in the following extended sense: given a spacelike vector ξa at a

point p, the spatial length of ξa is given by (habσaσb)
1/2, where σa is a (sufficiently unique)

covector at p such that habσb = ξ
a. We say a classical spacetime is temporally orientable

if there exists a smooth (globally defined) field ta, called a temporal orientation, such that

tab = tatb. In what follows, it will be convenient to restrict attention to temporally orientable

classical spacetimes and work directly with a choice of temporal orientation instead of the

full temporal metric.

In standard Newtonian gravitation, one considers a flat (i.e., Rabcd = 0) classical space-

time along with a scalar field ϕ representing the gravitational potential and a symmetric

tensor field Tab called the mass-momentum tensor, which represents the matter content of

spacetime. These together satisfy Poisson’s equation, ∇a∇aϕ = 4πρ, where ρ = Tabtatb is

the mass density. The (spatial) gradient of the gravitational potential, ∇aϕ = hab∇bϕ, is

the gravitational field. Timelike curves are again the possible trajectories of massive point

particles. In the absence of any external force, including gravitation, massive point particles

traverse timelike geodesics; in the presence of a gravitational field (and no other forces), a

massive point particle will follow a trajectory with acceleration ξn∇nξa = −∇aϕ, where ξa

is the tangent field to the trajectory.

In geometrized Newtonian gravitation, meanwhile, there is no gravitational potential.

Instead one considers classical spacetimes in which the derivative operator is curved in

the presence of matter, with Ricci curvature given by the geometrized Poisson equation,

Rab = 4πρtatb. (As in standard Newtonian gravitation, T
ab and ρ are the mass-momentum

tensor and mass density fields, respectively.) And in the absence of external forces (where

gravitational influences are no longer conceived as external forces), massive point particles

traverse timelike geodesics. Thus in geometrized Newtonian gravitation, gravitation is a

manifestation of spacetime curvature, in much the same way as in relativity theory; similarly,

5



the geometrical structure of spacetime—specifically, the derivative operator—depends on the

distribution of matter. It is in this setting that I will presently explore singularities.

3. Singular spacetimes in geometrized Newtonian gravitation

As I noted above, the standard rough-and-ready characterization of singularities in general

relativity is as follows: one says that a spacetime is singular if it is geodesically incomplete.

To make this precise, let M be a manifold endowed with a covariant derivative operator

∇. Given any point p ∈ M and any tangent vector ξa at p, there always exists a unique

maximal geodesic γ : I → M satisfying γ(0) = p and whose tangent vector at p is equal to

ξa, where by maximal I mean that given any other geodesic γ′ : I′ → M with γ′(0) = p and

whose tangent vector at p is ξa, I′ ⊆ I and γ′(s) = γ(s) for all s ∈ I′. A geodesic is said

to be incomplete if it is maximal but its domain is not all of R. A relativistic spacetime

(M,gab) is (timelike, null, or spacelike) geodesically incomplete if it admits an incomplete

(timelike, null, or spacelike) geodesic. Otherwise, it is geodesically complete.

The definition of a geodesic puts a strong constraint on the parametrization of the

curve, which permits the following interpretation of geodesic incompleteness: a spacetime

is geodesically incomplete if there exists a maximal geodesic that “ends” after only finite

distance (as measured by the affine parameter). In particular, for timelike geodesically

incomplete spacetimes, there is a trajectory that in principle could be followed by a non-

accelerating observer that extends into the future or the past for only finite time. Any

such spacetime clearly exhibits pathological behavior, which is what justifies the idea that

geodesic incompleteness is at least sufficient for a relativistic spacetime to be singular.7

What is the situation in geometrized Newtonian gravitation? If one begins by thinking

7Some authors take a slightly weaker condition—known as b-incompleteness—to (also) be sufficient for a
relativistic spacetime to be singular. Roughly, a spacetime is b-incomplete if there is a timelike or null curve,
not necessarily a geodesic, that is incomplete in the sense that it is maximal but ends after finite parameter
distance, where one requires the curve to be parametrized by a “generalized affine parameter”. For more on
this condition, see Ellis and Schmidt (1977), Earman (1995), and Curiel (1999).

6



exclusively about the geometrized theory, one confronts many of the same problems that

arise when one tries to characterize singularities in general relativity. These considerations

might well push one to geodesic incompleteness as a rough-and-ready definition of singular

spacetimes in this context as well. But here we have an additional resource: models of

geometrized Newtonian gravitation are systematically related to models of the standard

theory, where singularities are comparatively well understood. In particular, given any

model of Newtonian gravitation, there exists a unique corresponding model of geometrized

Newtonian gravitation. This claim is made precise by the following theorem, due to Andrzej

Trautman.

Theorem 3.1 (Trautman Geometrization Theorem). Let (M,ta,h
ab,

f

∇) be a flat clas-
sical spacetime. Let ϕ and ρ be smooth scalar fields on M satisfying Poisson’s equation,
f

∇a
f

∇aϕ = 4πρ. Finally, let
g

∇ = (
f

∇,Cabc), with C
a
bc = −tbtc

f

∇aϕ.8 Then (M,ta,hab,
g

∇) is
a classical spacetime;

g

∇ is the unique derivative operator on M such that given any timelike
curve with tangent vector field ξa,

ξn
g

∇nξa = 0 ⇔ ξn
f

∇nξa = −
f

∇aϕ; (G)

and the Riemann curvature tensor relative to
g

∇,
g

R abcd, satisfies
g

Rab = 4πρtatb.

This theorem allows us to ask what characterizes the geometrized equivalent of a singular

model of standard Newtonian gravitation.

In what follows, suppose that (M,ta,h
ab,∇) is a flat classical spacetime, and let ρ and ϕ

be scalar fields representing the mass density and gravitational potential on M, respectively.

Suppose, too, that ρ and ϕ together constitute a solution to Poisson’s equation. Given

such a model, there is a natural candidate for the field whose divergence would signal a

“gravitational singularity”: namely, the gravitational field ϕa = ∇aϕ—or more precisely,

the energy density of the field, which is proportional to the scalar (∇aϕ)(∇aϕ).9 Were

8The notation ∇′ = (∇,Cabc) is explained in Malament (2012, Prop. 1.7.3).
9One might think a divergent gravitational potential would also produce a singularity, but since the

7



this quantity to become infinite, it would indicate that a point particle passing through the

region where the field diverged would experience unbounded acceleration.

Of course, neither the gravitational potential nor the gravitational field appears in a

model of geometrized Newtonian gravitation. But the Trautman geometrization theorem

makes clear what role the gravitational field plays in determining a model of the geometrized

theory: it enters directly in the definition of the curved derivative operator,
g

∇. More

specifically, the gravitational field is used to determine the Cabc field that specifies the action

of
g

∇ in terms of the action of
f

∇ on arbitrary smooth tensor fields. Thus, given a point (or

region) on which the gravitational field becomes infinite (or otherwise undefined), one has

a failure of geometrization, insofar as it proves impossible to define a curved derivative

operator satisfying the conditions of the Trautman geometrization theorem at that point.

This indicates that these points would need to be excised from a model of the geometrized

theory, since (by definition) a classical spacetime consists of a manifold along with the

specification of the two degenerate metrics and a compatible covariant derivative operator

satisfying certain conditions, and no such derivative operator exists at the points in question.

Hence singularities in this context can be thought of as points that have been removed from

a non-geometrized model precisely because a certain part of the geometrical structure could

not be defined at those points.

Once the points are removed, one encounters a number of problems that are familiar

from the relativistic case. But, given the nature of the failure of geometrization, there

is nonetheless a direct way of thinking about what pathological features the geometrized

model should be expected to exhibit: the excised points correspond to regions where the

derivative operator could not be defined. In general, though, a covariant derivative operator

gravitational potential is only defined up to a function that is constant on each spacelike surface—i.e., a
function ψ : M → R such that ∇aψ = 0—one can choose the gravitational potential in such a way as to
diverge wherever one likes. For instance, pick a coordinate system and let ϕ = ln t, where t is the time
coordinate in that coordinate system. Then ϕ will diverge as t → 0, even though the gravitational field (and
the gravitational force experienced by any point particle) vanishes everywhere.

8



both fully determines and is fully determined by a collection of geodesics. Indeed, one

can think of the covariant derivative operator in Newtonian gravitation (or even relativity

theory) as equivalent to a standard of (non-)acceleration. This means that a failure of

the derivative operator to be defined can be understood as a breakdown in the standard

of geodesy associated with the spacetime. This suggests that the natural way of thinking

about the divergence of the gravitational field in geometrized Newtonion gravitation is in

terms of incomplete geodesics.

It is worth emphasizing the difference between this situation and the one in general rel-

ativity, where the option of direct translation between a “geometrized” theory and some

alternative theory on which singularities seem less troublesome is not available. There, one

gets to geodesic incompleteness via a circuitous route of trying other, (historically speaking)

more “obvious” alternatives. And at least on some accounts (for instance Geroch (1968)),

the ultimate motivation for using geodesic incompleteness as the characteristic feature of

singular spacetimes is that it is an indication that something has gone wrong with a space-

time “at finite distance”, and not that it is a direct way of capturing the kind of pathology

we were originally interested in. In geometrized Newtonian gravitation, meanwhile, geodesic

incompleteness captures exactly the kind of pathology one was originally interested in, in-

sofar as geodesic incompleteness is a natural way of characterizing the failure of a derivative

operator—and hence, a pathology in the gravitational field.

Indeed, one might take the naturalness of geodesic incompleteness as a way of char-

acterizing singular models of geometrized Newtonian gravitation as a new motivation for

adopting geodesic incompleteness10 as the standard condition in general relativity. After all,

the analogies between general relativity and geometrized Newtonian gravitation are strong,

and in the latter case it is clear that it is the failure of the derivative operator to be well-

defined that corresponds most closely with the divergence of the gravitational field. And so,

10Or perhaps a related condition, such as b-incompleteness.

9



there is a sense in which even in general relativity geodesic incompleteness is the condition

that best captures the intuition about what constitutes a singular solution inherited from

electromagnetism.

As a final remark, in the context of relativity theory, one sometimes considers cases

where in addition to admitting an incomplete geodesic, a spacetime also has some scalar

measure of curvature, or even the physical components of the curvature tensor expressed in

a parallel propagated orthonormal frame, diverge along an incomplete curve (or geodesic).

When this occurs, one speaks of the spacetime as admitting a curvature singularity ; some

physicists have urged that the existence of a curvature singularity somehow emphasizes, or

makes more physical, the singular nature of a geodesically incomplete spacetime.

I do not want to comment on what, if anything, is added once one observes that a

geodesically incomplete spacetime also exhibits a curvature singularity. But I do want to

point out that one can also consider curvature singularities in geometrized Newtonian grav-

itation. In that context, the so-called Ricci curvature scalar, R = Raa, always vanishes, so

that is not a good candidate for a potentially-divergent scalar curvature. But there is an-

other natural candidate: namely, if the geometrized Possion equation holds, then one gets a

scalar quantity by contracting the Ricci curvature tensor Rab with an arbitrary unit timelike

vector, ξa. This quantity is independent of the choice of timelike vector—indeed, it is just

Rabξ
aξb = 4πρ. In other words, a model of geometrized Newtonian gravitation exhibits a

curvature singularity of this kind just in case the mass density diverges along an incomplete

geodesic. So curvature singularities have a clear interpretation in this context: they corre-

spond to singularities in the source matter of a spacetime, whereas geodesic incompleteness

simpliciter corresponds to singularities in the gravitational field. Of course, in many cases

of physical interest, both conditions will be met, though they are in principal conceptually

distinct.

10



4. A Newtonian singularity theorem

As we have seen, there is a sense in which the Trautman geometrization theorem provides

a more direct route to geodesic incompleteness as a characterization of singular spacetimes

than is available in general relativity. But one might worry that there remains an important

difference between the two theories. Perhaps one can introduce singularities in geometrized

Newtonian gravitation if one wants. But the Penrose-Hawking-Geroch theorems show that

one is forced to reckon with singular spacetimes in general relativity, and the same cannot

be said about geometrized Newtonian gravitation.

I want to resist this line, however. There are certainly senses in which singularities are

endemic to general relativity, but not to Newtonian gravitation. Nonetheless, singularities

do appear, naturally and universally, in a broad class of realistic models of Newtonian grav-

itation. In particular, one can prove a singularity theorem directly analogous to a theorem

of relativity due to Raychaudhuri (1955) and Komar (1956) that shows that essentially all

homogeneous and isotropic cosmological models of Newtonian gravitation are singular, in

the sense that they are geodesically incomplete.11

To state and prove the theorem, I first need to introduce some background material

concerning cosmological models of geometrized Newtonian gravitation.12 The models I have

in mind are the Newtonian analogue to Friedman-Lemâıtre-Robertson-Walker spacetimes,

which are the standard cosmological models of relativity theory (setting aside inflation).

In both general relativity and geometrized Newtonian gravitation, these are models char-

acterized by assumptions of spatial homogeneity and isotropy relative to some choice of

11The work of Raychaudhuri and Komar is discussed from a historical perspective in Earman (1999).
Note, though, that the present characterization of the theorem is anachronistic: Raychaudhuri and Komar
(who worked independently) did not characterize singularities in terms of geodesic incompleteness. The
present version of the theorem is based on more modern reconstructions, as presented by Sachs and Wu
(1977) and Senovilla (1997).

12The treatment here follows Malament (2012) closely. For additional sources on Newtonian cosmology,
and particularly FLRW-like spacetimes, see the classic Heckmann and Schücking (1955), as well as Ruede
and Straumann (1997) and Senovilla et al. (1998).

11



timelike vector field. This assumption can be made precise as follows. A classical spacetime

(M,ta,h
ab,∇) is homogeneous and spatially isotropic relative to a timelike vector field ξa if,

given any point p ∈ M and any two unit spacelike vectors µa and νa at p, there exists an

open set O containing p and a (derivative operator preserving) isometry ϕ : O → O that

preserves p and ξa and that maps µa to νa.

One can show that if a classical spacetime is spatially homogenous and istropic relative

to a vector field ξa, then it must be a perfect fluid solution with velocity vector field ξa—that

is, it must be a spacetime whose matter field has an associated mass-momentum field of the

form Tab = ρξaξb + phab, where ρ is the mass density of the fluid and p is an (isotropic)

pressure term. Moreover, ξa must be geodesic (i.e., ξn∇nξa = 0 everywhere), irrotational

(i.e., ∇[aξb] = 0), and shear-free (i.e., ĥm(aĥb)n∇mξn = 13θĥab, where θ = ∇aξ
a and ĥab is the

unique smooth symmetric field such that (a) ĥabξ
a = 0 and (b) ĥanh

nb = δa
b−taξb). Finally,

ρ must be constant on spacelike surfaces (i.e., ∇aρ = 0). Note that the requirements that

ξa is geodesic, irrotational, and shear-free jointly imply that ∇aξb = 13 (δa
b − taξb)θ.

Definition 4.1. An FLRW-like classical spacetime is a sextuple (M,ta,h
ab,∇,ξa,Tab), where

ξa is a timelike vector field on M and (M,ta,h
ab,∇) is a classical spacetime that is spatially

homogeneous and isotropic relative to ξa and which, together with the (perfect fluid) mass-
momentum field Tab, satisfies the geometrized Poisson equation.

There are two additional standard assumptions about the matter fields associated with

FLRW-like classical spacetimes. First, we assume that the conservation condition is satisfied,

i.e., that ∇aTab = 0. For a perfect fluid, the conservation condition implies the continuity

equation, i.e., ξa∇aρ = −ρ(∇aξa), which is a statement of the conservation of mass. Second,

we assume the mass condition is satisfied, i.e., Tabtatb = ρ ≥ 0. The mass condition plays

the role of an energy condition in geometrized Newtonian gravitation.

The classical analogue to the Raychaudhuri-Komar theorem concerns FLRW-like classical

spacetimes.

12



Theorem 4.2. Suppose that (M,ta,h
ab,∇,ξa,Tab) is a FLRW-like classical spacetime sat-

isfying the conservation and mass conditions. Then either ρ = 0 or the spacetime is timelike
geodesically incomplete.

Proof.13 In what follows, suppose that ρ > 0 (the theorem trivially holds if ρ = 0, and the

mass condition prevents ρ < 0). We will begin by deriving a special case of Raychaudhuri’s

equation for the perfect fluid Tab.14 Consider that ξa∇aθ = ξa∇a∇bξb = −ξaRbcabξc +

ξa∇b∇aξb = −Rabξaξb + ∇b(ξa∇aξb) − (∇bξa)(∇aξb), where the penultimate equality is

a consequence of the definition of the Riemann tensor and the last equality follows from

Leibniz’s rule. Now, since ξa is geodesic, irrotational, and shear-free, we know that ξn∇nξa =

0 and that ∇aξb = 13 (δa
b − taξb)θ. Moreover, since an FLRW-like spacetime is required to

satisfy the geometrized Poisson equation, we know that Rabξ
aξb = 4πρ. Together, these

imply that ξa∇aθ = −4πρ− 13θ
2. Since ρ > 0, it follows that ξa∇aθ < 0 everywhere.

Now pick a point p ∈ M and consider the integral curve γ : I → M of ξa passing through

p. Since ξa∇aθ < 0 everywhere, there can be at most one point x ∈ I such that θ = 0 at

γ(x). This means that it is always possible to reparameterize (and possibly truncate) γ

so that either γ : [0,a) → M and θ0 = θ|γ(0) < 0 at γ(0), or else γ : (a, 0] → M and

−θ0 = −θ|γ(0) < 0 at γ(0). Without loss of generality, we can assume the first case holds.

We can now define a new function f = θ◦γ : [0,a) → R. A quick calculation shows that
df
dt

= (ξa∇aθ) ◦γ; moreover, by the derivation above, we know that dfdt < −
1
3
f2 everywhere.

Since f(0) = θ0 < 0, and since
df
dt
< 0, it follows that f < 0 everywhere on [0,a). Since

f 6= 0, it follows that 1/f is well defined, and more, that d
dt

(1/f) = −df
dt
/f2 > 1

3
. If we

integrate both sides of this inequality on [0,a) from 0 to T < a, we find,

∫ T
0

(
d

dt
(1/f)

)
dt >

∫ T
0

1

3
dt ⇒ 1/f(T) −θ0 >

1

3
T.

13This proof essentially follows that of Sachs and Wu (1977, Prop. 4.3.4).
14In fact, the full version of Raychaudhuri’s equation holds for perfect fluid models of geometrized New-

tonian gravitation. See Ruede and Straumann (1997).

13



This, in turn, implies that f(T) < 3θ0/(Tθ0 + 3). But now note that, since θ0 < 0 by

construction,

lim
T→3/|θ0|

3θ0/(Tθ0 + 3) = −∞⇒ lim
T→3/|θ0|

f(T) = −∞.

It follows that a ≤ 3/|θ0|, since by construction f was well-defined everywhere on [0,a).

Thus there is an upper bound on the parameter value for the geodesic γ, and so γ must be

incomplete. It follows that the spacetime is geodesically incomplete. �

Several remarks are in order. The first is that, although the present version of the theorem

is effective at making my point, it can be strengthened in several ways. For instance,

the theorem states that there is an incomplete geodesic. It is clear, though, from the

present argument that in fact every maximal integral curve of ξa is an incomplete geodesic.

Moreover, one can show that if there exists a geodesic γ : [0,a) → M for which a takes the

maximal value of 3/|θ0|, then there is a curvature singularity in the sense given above: if ξa

is the tangent field to such a curve, then limx→a(Rabξ
aξb)◦γ = ∞. Finally, the assumptions

of spatial homogeneity and isotropy are of special interest because of their connection to

standard cosmological models in general relativity. But in fact, the argument would go

through for any perfect fluid solution whose flow vector is geodesic and irrotational.15

Regarding the interpretation of the theorem: note that in the proof, I identified two cases.

In one, the scalar θ is always negative moving forward along a (half) geodesic; in the other,

the scalar is negative moving backward along a (half) geodesic. Although mathematically

these cases are equivalent, they have rather different interpretations. In the first case, one

has a situation in which the cosmic fluid is everywhere collapsing into itself. This will

ultimately (i.e., in finite proper time as measured by an observer co-moving with the fluid)

lead to a big crunch singularity. In the other case, meanwhile, is the time-reverse of this

15Note, however, that it seems one cannot drop the requirement that the fluid be irrotational. This fact
is sometimes identified—see, for instance, Hawking and Ellis (1973)—as a significant difference between
relativity and Newtonian gravitation. However, there is an important caveat to such claims that space
constraints prevent me from developing here, and so I will defer it to future work.

14



situation—i.e., a spacetime corresponding to a big bang singularity. In some cases, one will

have both initial and final singularities (those are the cases where θ passes through 0); in

all cases where ρ > 0, however, one will get at least one of these.

5. Conclusion

As I noted in the introduction, this paper had two (modest) goals. One was to argue that an

analysis of singularities in geometrized Newtonian gravitation provides a more direct route

to geodesic incompleteness as a natural characterization of singularities. Though hardly

dispositive with regard to whether geodesic incompleteness is necessary or sufficient for a

spacetime to be singular in relativity theory, these considerations provide a new perspective

on that question and on the physical interpretation of singular behavior more generally.

The second goal was to show that singular spacetimes do arise naturally in geometrized

Newtonian gravitation. These considerations suggest that further study of singular classical

spacetimes may be a fruitful way to explore singularities in geometric theories of gravitation.

Acknowledgements

Thank you to my fellow symposiasts, Erik Curiel and John Manchak, to Jeremy Butterfield,

who chaired the session, and for the audience at the PSA 2012 biennial meeting for their

helpful comments and questions. I am especially indebted to David Malament for helpful

discussions and for detailed comments on a previous draft.

References

Cartan, E., 1923. Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première
partie). Annales scientifiques de l’École Normale Supérieure 40, 325–412.

Cartan, E., 1924. Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie)
(suite). Annales scientifiques de l’École Normale Supérieure 41, 1–25.

Choquet-Bruhat, Y., 2009. General Relativity and the Einstein Equations. Oxford University Press, New
York.

Clarke, C. J. S., 1993. The Analysis of Space-Time Singularities. Cambridge University Press, New York.
Curiel, E., 1999. The analysis of singular spacetimes. Philosophy of Science 66, S119–S145.
Earman, J., 1995. Bangs, Crunches, Whimpers, and Shrieks. Oxford University Press, New York.

15



Earman, J., 1999. The penrose-hawking singularity theorems: History and implications. In: Goenner, H.,
Renn, J., Ritter, J., Sauer, T. (Eds.), The Expanding Worlds of General Relativity. Birkhäuser, Boston,
pp. 235–267.

Ellis, G. F. R., Schmidt, B. G., 1977. Singular space-times. General Relativity and Gravitation 8 (11),
915–953.

Friedrichs, K. O., 1927. Eine invariante formulierun des newtonschen gravitationsgesetzes und der
grenzüberganges vom einsteinschen zum newtonschen gesetz. Mathematische Annalen 98, 566–575.

Geroch, R., 1968. What is a singularity in general relativity? Annals of Physics 48 (3), 526–540.
Geroch, R., 1970. Singularities. In: Relativity. Plenum Press, New York, pp. 259–291.
Glymour, C., 1970. Theoretical equivalence and theoretical realism. PSA: Proceedings of the Biennial Meet-

ing of the Philosophy of Science Association 1970, 275–288.
Hawking, S. W., Ellis, G. F. R., 1973. The Large Scale Structure of Space-Time. Cambridge University

Press, New York.
Heckmann, O., Schücking, E., 1955. Bemerkungen zur Newtonschen Kosmologie. Zeitschrift für Astrophysik

38, 95–109.
Knox, E., 2013. Newtonian spacetime structure in light of the equivalence principle, forthcoming in The

British Journal for Philosophy of Science.
Komar, A., 1956. Necessity of singularities in the solution of the field equations of general relativity. Physical

Review 104 (2), 544–546.
Malament, D. B., 2012. Topics in the Foundations of General Relativity and Newtonian Gravitation Theory.

University of Chicago Press, Chicago.
Raychaudhuri, A., 1955. Relativistic cosmology. i. Physical Review 98 (4), 1123–1126.
Ruede, C., Straumann, N., 1997. On Newton-Cartan cosmology. Helvetica Physica Acta 70, 318–335.
Sachs, R. K., Wu, H., 1977. General Relativity for Mathematicians. Springer-Verlag, New York.
Senovilla, J. M. M., 1997. Singularity theorems and their consequences. General Relativity and Gravitation

29 (5), 701–848.
Senovilla, J. M. M., Sopuerta, C. F., Szekeres, P., 1998. Theorems on shear-free perfect fluids with their

Newtonian analogues. General Relativity and Gravitation 30 (3), 389–411.
Tipler, F., Clarke, C. J. S., Ellis, G. F. R., 1980. Singularities and horizons—a review article. In: Held, A.

(Ed.), General Relativity and Gravitation. Vol. 2. Plenum Press, New York, pp. 97–206.
Trautman, A., 1965. Foundations and current problem of general relativity. In: Deser, S., Ford, K. W.

(Eds.), Lectures on General Relativity. Prentice-Hall, Englewood Cliffs, NJ, pp. 1–248.
Wald, R. M., 1984. General Relativity. University of Chicago Press, Chicago.
Weatherall, J. O., 2013. Gauge transformations, theoretical equivalence, and the interpretation of physical

theories, unpublished manuscript.

16