Can we know the global structure of

spacetime? ⋆

John Byron Manchak

Abstract

Here, we briefly review the notion of observational indistinguishability within the

context of classical general relativity. We settle a conjecture given by Malament
(1977) concerning the subject and then strengthen the result considerably. The
upshot is this: There seems to be a robust sense in which the global structure of
every cosmological model is underdetermined.

Key words: Epistemology, Observational Indistinguishability,

Underdetermination, General Relativity, Cosmology

1 Introduction

Here, we consider some epistemological issues arising within the context of the
general theory of relativity. The principal thesis to emerge is this: It appears
to follow from relativity theory itself that we can know very little about the
global structure of spacetime.

More specifically, our goal is to show that (1) excluding certain pathological
examples, every cosmological model of our universe is empirically underde-
termined; no amount of observational data we could ever (even in principle)
accumulate, can force one and only one cosmological model upon us. Addi-
tionally, we also claim that (2) even if one assumes a principle of uniformity
– that the physical laws we determine locally are applicable throughout the
universe – these general epistemological difficulties remain.

We begin with a pair of general observations to frame the technical results

below. First, in discussions of underdetermination within the philosophy of
⋆ I wish to thank Jeffrey Barrett, John Earman, Robert Geroch, Clark Glymour,
John Norton, and Kyle Stanford for valuable discussions on this topic. I am espe-
cially grateful to David Malament for his help and encouragement.

Preprint submitted to Elsevier 14 August 2008

science, one sometimes encounters assertions of the following kind: If we are
willing to sufficiently revise our background physical theories, then no amount
of evidence will ever compel us to embrace a particular scientific claim. How-
ever, it is absolutely imperative to note that there is a different sort of assertion
altogether which states that similar epistemological predicaments may be in-
herited without renouncing or revising our best physical theories. Of course, it
is this latter type of assertion that concerns us in this paper.

Second, a discussion of observationally indistinguishable cosmological models

is sometimes used to buttress some broad philosophical stance. 1 We are cer-
tainly interested in the implications of our work for these extensive projects.
But we want to make it very clear, however, that we have no special stake
in any one of these more general positions. The motivations for this paper
have primarily stemmed from simply considering the observer’s epistemologi-
cal situation within our best large-scale physical theory. We feel it is of some
interest that general relativity is the sort of physical theory that allows for
a wide variety of cosmological models but that, due to structure internal to
the theory itself, does not allow us to determine which of these models best
represents our universe.

2 Background structure

We begin with a few preliminaries concerning the relevant background for-

malism of general relativity. 2 A relativistic spacetime is a pair of mathemat-
ical objects (M, gab ). M is a connected four-dimensional manifold (without
boundary) that is smooth (infinitely differentiable). Here, gab is a smooth,
non-degenerate, pseudo-Riemannian metric of Lorentz signature (+, −, −, −)
defined on M. Each point in the manifold represents an “event” in spacetime.
For each point p in the manifold, the metric assigns a light cone structure
in the tangent space Mp . Any tangent vector ξ a in Mp will be timelike (if
gab ξ a ξ b > 0), null (if gab ξ a ξ b = 0), or spacelike (if gab ξ a ξ b < 0). Null vec-
tors create the “cone” structure. Timelike vectors are inside the cone while
spacelike vectors are outside. A time orientable spacetime is one that has a
continuous timelike vector field on M. A time orientable spacetime allows us
to distinguish between the future and past lobes of the light cone. In what
follows, it is assumed that spacetimes are time orientable.

1 For example, see Glymour (1980), van Fraassen (1980), Sklar (1985), Earman
(1993), Leplin (1997), Stanford (2006), and Norton (2008).
2 The reader is encouraged to consult Hawking & Ellis (1973) and Wald (1984)

for details. An outstanding (and less technical) survey of the global structure of
spacetime is given by Geroch & Horowitz (1979).

2
For some interval I ⊆ R, a smooth curve γ : I → M is timelike if the tangent
vector ξ a at each point in γ[I] is timelike. Similarly, a curve is null (respectively,
spacelike) if its tangent vector at each point is null (respectively, spacelike). A
timelike curve is future-directed if its tangent vector at each point lies in the
future lobe of the light cone. For any two points p, q ∈ M, q is to the timelike
future of p (written p << q) if there exists a timelike, future-directed curve γ
from p to q. A future-directed curve from p to q that is either timelike or null
indicates that q is to the causal future of p (written p < q). These relations
allow us to define the following sets of points in M: I − (p) = {q : q << p},
I + (p) = {q : p << q}, J − (p) = {q : q < p}, and J + (p) = {q : p < q}.
The set I − (p) will be used extensively throughout the paper and is called the
observational past of p. This set represents those points which can possibly be
observed from p. 3

Two spacetimes (M, gab ) and (M ′ , gab ′

) are isometric if there is a diffeomor-
phism φ : M → M such that φ∗ (gab ) = gab
′ ′
. For ease of presentation, we will
sometimes say that two manifolds M and M ′ are isometric when it is clear
which metrics are associated with M and M ′ . Two spacetimes (M, gab ) and
(M ′ , gab
′
) are locally isometric if, for each point p ∈ M, there is an open neigh-
borhood O of p and an open subset O ′ of M ′ such that O and O ′ are isometric,
and, correspondingly, with the roles of (M, gab ) and (M ′ , gab ′
) interchanged.

Finally, to facilitate the proof of the theorem below, we note the following
lemma.

Lemma. Let (M, gab ) be any spacetime and let p be any point in M. If
I − (p) ⊂ M then I − (p) ⊂ M. 4

Proof. Assume I − (p) ⊂ M. So there exists an s ∈ M such that s ∈ / I − (p).

Let r be any point in I + (s). So s ∈ I − (r). But, we know that for any points
p, r, s ∈ M, if r ∈ I − (p) and s ∈ I − (r), then s ∈ I − (p). Hence, r ∈
/ I − (p). So
I (p) ⊂ M. 
−

3 Observational indistinguishability

Now we consider what it would mean to say that two spacetimes are observa-
tional indistinguishable. The development of this subject can be traced back
to Clark Glymour (1972, 1977) and David Malament (1977) and the reader
is invited to study their papers. In what follows, we will restrict our atten-

3 One uses the set I − (p) instead of J − (p) to represent the “observational past” of
p for reasons of mathematical convenience.
4 Here, ⊂ should be interpreted as “proper subset”.

3
tion to Malament’s somewhat weaker (asymmetric) definition. 5 We have the
following.

Definition. Let (M, gab ) and (M ′ , gab

′
) be spacetimes. We say (M, gab ) is ob-
servationally indistinguishable from (M ′ , gab
′
) if, for every point p ∈ M, there
is a point p ∈ M such that I (p) and I (p ) are isometric.
′ ′ − − ′

So, given that a spacetime (M, gab ) is observationally indistinguishable from

another spacetime (M ′ , gab ′
), an observer at any point in M cannot know which
of the two spacetimes she inhabits. No amount of observational data she could
ever, even in principle, accumulate, allow her to distinguish between the mod-
els (M, gab ) and (M ′ , gab
′
). We are now prepared to consider an open question
concerning observationally indistinguishable spacetimes.

Consider spacetimes (M, gab ) with the property that for some point p in M,
I − (p) = M. Such spacetimes necessarily contain closed timelike curves and,
in addition, have the property that all of spacetime may be observed from
one point. We will call these spacetimes causally bizarre. Malament (1977)
has conjectured that only causally bizarre spacetimes do not have a distinct
(non-isometric) observationally indistinguishable counterpart.

Conjecture. (Malament) Let (M, gab ) be any spacetime which is not causally
bizarre. There exists another spacetime (M ′ , gab ′
) (one that is not isometric to
(M, gab )) such that (M, gab ) is observationally indistinguishable from (M ′ , gab
′
).

Malament’s argument sketch in support of such a claim has been dubbed

the “clothesline construction” and may be summarized as follows. 6 Let {pi }
S
be a countable sequence of points in M such that {I − (pi )} = M. 7 Like
a “clothesline”, it would seem that these I − (pi ) can be “strung out” with
an appropriately chosen “spacetime filler” such that they are isometrically
embedded (disjointly) in some other spacetime (M ′ , gab ′
). The spacetime filler
in between the embedded I (pi ) is subject only to the constraint that the
−

metric gab ′
be smooth. With great flexibility, then, it is possible to choose
the filler so that (M, gab ) is not isometric to (M ′ , gab
′
). But, because of the
way (M , gab ) was constructed, every point p ∈ M will be in some I − (pi )
′ ′

5 What we simply call “observational indistinguishability” Malament calls “weak

observational indistinguishability” to avoid confusion with Glymour’s stronger
(symmetric) formulation. For a discussion of the relationship between the defi-
nitions of Glymour and Malament and for numerous examples of observationally
indistinguishable spacetimes, see Malament (1977).
6 As an argument sketch, we have intentionally left some of the steps somewhat

imprecise. The reader is encouraged to consult the theorem below for a detailed
proof of the claim.
7 Such a countable set of points exists in any spacetime. See Malament (1977, p.

80).

4
that is isometrically embedded in (M ′ , gab
′
). So, (M, gab ) is observationally
indistinguishable from (M , gab ).
′ ′

Although it seemed likely, it was not certain that Malament’s intuitive argu-
ment sketch would eventually materialize into an honest proof. But, as we will
shortly see, the conjecture is indeed true. So, there is a sense in which, exclud-
ing causally bizarre spacetimes, every observer in every spacetime inherits a
serious epistemological predicament.

4 An underdetermination theorem

Of course, Malament’s claim considers arbitrary spacetimes – no attempt is

made to differentiate between cosmological models which are, in some sense,
“physically reasonable” and those which are not. Thus, there is room to argue
that there may not be any significant underdetermination at work after all;
if the cosmologist takes seriously only “physically reasonable” spacetimes, it
may be that her epistemological plight vanishes entirely. Here we provide a
response to this line of reasoning.

If one is interested in realistic cosmological models, one may restrict one’s

attention to spacetimes satisfying any number of constraints. For example,
Einstein’s equation along with some reasonable matter source is often imposed
to rule out seemingly artificial models. 8 The precise formulation of Einstein’s
equation need not concern us here. What is important to note, however, is that
it, along with many of the other conditions usually considered, are “local” in
the sense that they are conserved under local isometries.

Formally, we say that a condition C on a spacetime is local if, given any two
locally isometric spacetimes (M, gab ) and (M ′ , gab
′
), (M, gab ) satisfies C if and
only if (M , gab ) satisfies C. We now strengthen Malament’s claim considerably
′ ′

by showing that it holds even under the imposition of any local constraints on
spacetime. We have the following.

Theorem. Let (M, gab ) be any spacetime which is not causally bizarre satis-
fying any set C of local conditions. There exists another spacetime (M ′ , gab ′
)
(one that is not isometric to (M, gab )) such that (i) (M , gab ) satisfies the set C
′ ′

of local conditions and (ii) (M, gab ) is observationally indistinguishable from

(M ′ , gab
′
).

Proof. Let (M, gab ) be any spacetime which is not causally bizarre satisfying
any set C of local conditions. Let {pi } be a countable sequence of points in M
8 The interested reader can consult Hawking & Ellis (1973) or Wald (1984) for a
formal discussion of Einstein’s equation.

5
 S
such that {I − (pi )} = M. By assumption and the lemma above, we know that
for all i, I − (pi ) 6= M. So, associate with each point pi a point qi ∈
/ I − (pi ) and
neighborhood Oi of qi such that Oi ∩ I (pi ) = ∅. By the Hausdorff condition,
−

for all i the points qi and qi+1 are such that we can make their associated
neighborhoods Oi and Oi+1 disjoint.

Now, let Ki+ be a closed three-dimensional spacelike surface such that Ki+ ⊂
Oi . Similarly, let Ki− be a closed three-dimensional spacelike surface such that
Ki− ⊂ Oi and Ki+ ∩ Ki− = ∅. It is important to note that, by construction,
Ki+ ∩ Ki+1
−
= ∅ for all i.

Oi Oi
Ki+ Ki+
pi Ki−

Ki+1
−
Oi+1
I (pi )
−

(M(i, α), gab ) (M(i, β), gab )

Fig. 1. The spacetimes (M (i, α), gab ) (for all i 6= 1) and (M (i, β), gab ) (for all i).
Note that (M (1, α), gab ) (not shown) is constructed without removing the set K1− .

Now, for each i 6= 1, let M(i, α) be the manifold M − (Ki+ ∪ Ki− ). Let
M(1, α) = M −K1+ . Next, for all i let M(i, β) be the manifold M −(Ki+ ∪Ki+1 −
).
Consider the spacetimes (M(i, α), gab ) and (M(i, β), gab ) (see figure 1). We
make the following identifications: For all i, we identify the upper edge of
Ki+ in (M(i, α), gab ) with the lower edge of Ki+ in (M(i, β), gab ) (excluding
the boundary points 9 ) and the lower edge of Ki+ in (M(i, α), gab ) with the
upper edge of Ki+ in (M(i, β), gab ). Similarly, we identify the upper edge of
− −
Ki+1 in (M(i, β), gab ) with the lower edge of Ki+1 in (M(i + 1, α), gab) and
10
vice versa. The identifications produce a “chain” in which (M(1, α), gab ) is
attached to (M(1, β), gab ) which is attached to (M(2, α), gab ) which is attached
to (M(2, β), gab ) and so on (see figure 2). By definition, this resulting “chain”
is a spacetime – call it (M ′ , gab
′
). We now show that (M, gab ) and (M ′ , gab′
)
satisfy properties (i) and (ii).
9 A 2-sphere makes up the boundary of the excised 3-disk. After the identifications,
this 2-sphere is necessarily “left out” of the resulting spacetime. See Hawking & Ellis
(1973, p. 58-59) for further discussion.
10 For other examples of this type of construction (by which one produces a space-

time by “cutting” and “pasting” together regions of other spacetimes), see Hawking
& Ellis (1973, p. 58-59) and Geroch (1977, p. 89-90). A short discussion of the
procedure can be found in Geroch & Horowitz (1979, p. 220-221).

6
 (M(1, α), gab ) (M(2, α), gab )

p1 K1+
p2
. . .

K2+
I (p1 )
−
K2− I − (p2 )

K1+

. . .
K3−
K2+
K2−

(M(1, β), gab ) (M(2, β), gab )

Fig. 2. The spacetime (M ′ , gab′ ). Here, for all i, we identify the upper edge of K +
i
in (M (i, α), gab ) with the lower edge of Ki+ in (M (i, β), gab ) (excluding boundary
points) and the lower edge of Ki+ in (M (i, α), gab ) with the upper edge of Ki+ in
−
(M (i, β), gab ). Similarly, we identify the upper edge of Ki+1 in (M (i, β), gab ) with
−
the lower edge of Ki+1 in (M (i + 1, α), gab ) and vice versa.

To see that condition (i) holds, consider any point q ∈ M. Clearly, q ∈ I − (pi )
for some i. Because I − (pi ) ⊂ M(i, α) one can always find a neighborhood O of
q and an open set O ′ ⊂ M(i, α) ⊂ M ′ such that O and O ′ are isometric. Next
consider any point q ′ ∈ M ′ . By construction, q ′ ∈ M(i, α) or q ′ ∈ M(i, β) for
some i. But either way q ′ ∈ M (because M(i, α), M(i, β) ⊂ M). So clearly,
one can always find a neighborhood O ′ of q ′ and an open set O on M such
that O and O ′ are isometric. So the two spacetimes are locally isometric and
therefore (M ′ , gab
′
) satisfies the set C of local conditions. So, condition (i) is
satisfied.

To see that condition (ii) holds, consider any point q ∈ M. Clearly I − (q) ⊂
I − (pi ) for some i. But consider any I − (pi ). By construction, I − (pi ) ⊂ M(i, α) ⊂
M ′ . So there will always be an isometric counterpart to I − (pi ) in M ′ . Thus,
condition (ii) is satisfied and we are done. 

7
5 Conclusion

The theorem above seems to indicate that we cannot know the global structure
of spacetime even if we make the unverifiable assumption, as do the cosmolo-
gists, that “the normal physical laws we determine in our spacetime vicinity
are applicable at all other spacetime points.” 11 In other words, there is a
sense in which, even if we restrict our attention to “physically reasonable”
spacetimes, we face a substantial epistemological predicament.

One may wonder about the role of matter fields in our discussion of observa-
tional indistinguishability. It could be argued, for example, that when these
fields are taken into consideration, one may be able, from the observations of
such fields, to distinguish between “observationally indistinguishable” space-
times. But we suggest here that the “clothesline construction” utilized in our
proof can rule out any objections of this kind. 12 So, it seems that there is a
robust sense in which the global structure of every spacetime is underdeter-
mined.

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