What Is a Physically Reasonable Space-Time?


Philosophy of Science, 78 (July 2011) pp. 410–420. 0031-8248/2011/7803-0005$10.00
Copyright 2011 by the Philosophy of Science Association. All rights reserved.

410

What Is a Physically Reasonable
Space-Time?*

John Byron Manchak†‡

Cosmologists often use certain global properties to exclude “physically unreasonable”
cosmological models from serious consideration. But, on what grounds should these
properties be regarded as physically unreasonable if we cannot rule out, even with a
robust type of inductive reasoning, the possibility of the properties obtaining in our
own universe?

1. Introduction. Recent results show one sense in which the global struc-
ture of space-time cannot be fully established (Manchak 2009a). It seems
that, excluding certain pathological examples, every cosmological model
is empirically underdetermined; no amount of observational data we could
ever (even in principle) accumulate can force one and only one cosmo-
logical model on us. Additionally, one can show that even under the
assumption of an inductive principle—that the physical laws we determine
locally are applicable throughout the universe—these general epistemo-
logical difficulties remain.

However, it may be that we are able to make partial determinations
concerning important global properties. For example, it might be possible
to discover whether our universe is causally well behaved in some respect.
The first task of this article will be to demonstrate that for many global
properties of interest, even this partial determination is impossible. This
certainly exacerbates the epistemic plight of the cosmologist. But, in ad-

*Received September 2010; revised November 2010.

†To contact the author, please write to: Department of Philosophy, 361 Savery Hall,
Seattle, WA 98195; e-mail: manchak@uw.edu.

‡I am grateful to Jeff Barrett, Bob Geroch, David Malament, John Norton, and Kyle
Stanford for comments on an earlier draft. I also wish to thank audiences at Pittsburgh
and Oxford for valuable discussions.

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WHAT IS A PHYSICALLY REASONABLE SPACE-TIME? 411

dition, these underdetermination results seem to have implications for
what counts as a “physically reasonable” cosmological model.

Cosmologists often use certain global properties (e.g., those relating to
causal misbehavior) to exclude some models from serious consideration.
But, on what grounds should these properties be regarded as physically
unreasonable if we cannot rule out, even with a robust type of inductive
reasoning, the possibility of the properties obtaining in our own universe?
The second task of this article is to carefully investigate this question.

2. Background Structure. We start by reviewing the relevant background
formalism of general relativity.1 A relativistic space-time is a pair of math-
ematical objects , where is a connected four-dimensional man-(M, g ) Mab
ifold (without boundary) that is smooth (infinitely differentiable). Here,

is a smooth, nondegenerate, pseudo-Riemannian metric of Lorentzgab
signature defined on . Each point in the manifold rep-(�, � , � , �) M
resents an “event” in space-time. For each point in the manifold, thep
metric assigns a light cone structure in the tangent space . Any tangentMp
vector in will be timelike (if ), null (if ), ora a b a by M g y y 1 0 g y y p 0p ab ab
spacelike (if ). Null vectors create the “cone” structure. Timelikea bg y y ! 0ab
vectors are inside the cone, while spacelike vectors are outside. A time-
orientable space-time is one that has a continuous timelike vector field on

. A time-orientable space-time allows us to distinguish between theM
future and the past lobes of the light cone. In what follows, we assume
that space-times are time orientable.

For some interval , a smooth curve is timelike if theI P R g:I r M
tangent vector at each point in is timelike. Similarly, a curve isay g[I ]
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 , is to the timelike future of (written ) if therep, q � M q p p !! q
exists a timelike, future-directed curve from to . A future-directedg p q
curve from to that is either timelike or null indicates that is to thep q q
causal future of (written ). These relations allow us to define thep p ! q
following sets of points in : , ,� �M I ( p) p {q:q !! p} I ( p) p {q:p !! q}

, and . The set will be used ex-� � �J ( p) p {q:q ! p} J ( p) p {q:p ! q} I ( p)
tensively throughout the article and is called the observational past of .p
This set represents those points that can possibly be observed from .2p

A point is a future endpoint of a future-directed causal curvep � M
if, for every neighborhood of , there exists a pointg:I r M O p t � I0

1. Details can be found in Hawking and Ellis (1973) and Wald (1984).

2. One uses the set instead of to represent the observational past of for� �I ( p) J ( p) p
reasons of mathematical convenience.

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412 JOHN BYRON MANCHAK

such that for all . A past endpoint is defined similarly. Forg(t) � O t 1 t0
any set , we define the past domain of dependence of (writtenS P M S

) to be the set of points such that every causal curve with�D (S ) p � M
past endpoint and no future endpoint intersects . The future domainp S
of dependence of (written ) is defined analogously. The entire�S D (S )
domain of dependence of (written ) is just the set .� �S D(S ) D (S ) ∪ D (S )

A set is achronal if no two points in can be connected by aS O M S
timelike curve. The edge of a closed, achronal set is the collectionS O M
of points such that every open neighborhood of contains ap � S O p
point , a point , and a timelike curve from to that� �q � I ( p) r � I ( p) r q
does not intersect . A set is a slice if it is closed, achronal, andS S O M
without edge. A set is a spacelike hypersurface if is an three-S O M S
dimensional submanifold such that every curve in is spacelike.S

Two space-times and are isometric if there is a diffeo-′ ′(M, g ) (M , g )ab ab
morphism such that . For ease of presentation,′ ′f:M r M f (g ) p g∗ ab ab
we will sometimes say that two manifolds and are isometric when′M M
it is clear which metrics are associated with and . Two space-times′M M

and are locally isometric if, for each point , there′ ′(M, g ) (M , g ) p � Mab ab
is an open neighborhood of and an open subset of such that′ ′O p O M

and are isometric and, correspondingly, with the roles of′O O (M, g )ab
and interchanged.′ ′(M , g )ab

3. Observational Indistinguishability. We are now prepared to consider the
notion of observationally indistinguishable space-times. Intuitively, one
space-time is observationally indistinguishable from another if no observer
in the first space-time has grounds for deciding which of the two she
inhabits.3 Formally, we have the what follows.

Definition 1. Let and be space-times. We say′ ′(M, g ) (M , g ) (M, g )ab ab ab
is observationally indistinguishable from if, for every point′ ′(M , g )ab

, there is a point such that and are iso-′ ′ ′� �p � M p � M I ( p) I ( p )
metric.

It should be clear from the definition that, given that a space-time (M, g )ab
is observationally indistinguishable from another space-time , no′ ′(M , g )ab
empirical data could (even in principle) allow an observer in to(M, g )ab
distinguish between the two models.

It has been shown that, except for certain pathological models, not only
is every space-time observationally indistinguishable from some other, but
the result holds even if we require the two space-times to have identical

3. There are various definitions of observational indistinguishability found in the lit-
erature. Here, we restrict our attention to one given by Malament (1977). For others,
see Glymour (1972, 1977).

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WHAT IS A PHYSICALLY REASONABLE SPACE-TIME? 413

local properties. In other words, cosmologists inherit a serious epistemic
predicament even under the inductive assumption that “the normal phys-
ical laws we determine in our space-time vicinity are applicable at all other
spacetime points” (Ellis 1975, 246). In order to present the result, consider
the following definition.

Definition 2. A property on a space-time is local if, given any twoP
locally isometric space-times and , has if′ ′(M, g ) (M , g ) (M, g ) Pab ab ab
and only if has . A property is global if it is not local.′ ′(M , g ) Pab

Under this definition, it should be clear that, for example, the property
of satisfying the standard energy conditions counts as local, while the
property of being stably causal is classified as global.4

Now, consider a special class of space-times with the property(M, g )ab
that for some point in , . Such space-times necessarily�p M I ( p) p M
contain closed timelike curves and, in addition, have the property that
all of space-time may be observed from one point. We will call these
space-times causally bizarre.

Now we are ready to introduce the underdetermination result discussed
above.5

Proposition 1. Let be any space-time having any set of local(M, g )ab
properties . If is not causally bizarre, then there exists a� (M, g )ab
space-time such that (i) has the set of local prop-′ ′ ′ ′(M , g ) (M , g ) �ab ab
erties, (ii) is observationally indistinguishable from ,′ ′(M, g ) (M , g )ab ab
and (iii) and are not isometric.′ ′(M, g ) (M , g )ab ab

So, even if one fixes the local structure of space-time completely, there is a
sense in which the global structure of every model is underdetermined. This
poses quite a problem for the cosmologist. But, as we are about to see, the
epistemic situation is even worse than the theorem above seems to suggest.

4. Underdetermination of Global Properties. In this section, we restrict
our attention to four global properties of interest: inextendibility, isotropy,
global hyperbolicity, and hole-freeness. Let us define each of them in turn.

We say a space-time is inextendible if it is not possible to(M, g )ab
properly embed it isometrically into another space-time. So, the property
of inextendibility ensures that space-time is “as large as it can be.”

A space-time is (spatially) isotropic if there is a timelike vector(M, g )ab
field on such that, for all and any two unit vectors anda ay M p � M j1

at that are orthogonal to , there is an isometry thata aj p y J:M r M2
leaves and the field fixed but rotates into . Intuitively, isotropica a ap y j j1 2

4. See Wald (1984) for details concerning these properties.

5. The proof for this claim is given in Manchak (2009a).

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414 JOHN BYRON MANCHAK

space-times, sometimes called the “Friedmann models,” have no preferred
spatial directions.6 A space-time is globally hyperbolic if there is(M, g )ab
a slice in such that . Roughly, a globally hyperbolic space-S M D(S ) p M
time allows one to determine, from initial conditions on , the physicalS
situation on all of . For this reason, global hyperbolicity is usuallyM
taken to be a necessary condition of Laplacian determinism.

A space-time is hole-free if, for any spacelike hypersurface(M, g ) Sab
in there is no isometric embedding into another space-time′M v:D(S) r M

such that . So, in a hole-free space-time, the′ ′(M , g ) v(D(S)) ( D(v(S))ab
Cauchy development of every spacelike surface is “as large as it can be.”

With these definitions in place, we are ready to strengthen the under-
determination result from above. Let be the set of global properties�
consisting of inextendibility, isotropy, global hyperbolicity, and hole-free-
ness. We have what follows.

Proposition 2. Let be any space-time having any set of local(M, g )ab
properties . If is not causally bizarre, then there exists a� (M, g )ab
space-time such that (i) has all of the properties in′ ′ ′ ′(M , g ) (M , g )ab ab

, (ii) has none of the properties in , (iii) is ob-′ ′� (M , g ) � (M, g )ab ab
servationally indistinguishable from , and (iv) and′ ′(M , g ) (M, g )ab ab

are not isometric.′ ′(M , g )ab

Proof. Let be any space-time that is not causally bizarre(M, g )ab
having the set of local properties . Now construct ac-′ ′� (M , g )ab
cording to the method outlined in Manchak (2009a). Next, remove
any point in the portion of the manifold . It is easily′u M(1, b) M
verified that the resulting space-time, call it , is such that′ ′ ′ ′(M , g )ab
(i) has all of the properties in , (ii) has none of′ ′ ′ ′ ′ ′ ′ ′(M , g ) � (M , g )ab ab
the properties in , and (iii) is observationally indistinguish-� (M, g )ab
able from , and (iv) and are not isometric.′ ′ ′ ′ ′ ′ ′ ′(M , g ) (M, g ) (M , g )ab ab ab
QED

We can understand the theorem to be saying that, not only is it im-
possible to fully establish the global structure of space-time, but we cannot
even make partial determinations concerning a handful of space-time
properties of interest. It seems that, although our universe may be inex-
tendible, isotropic, globally hyperbolic, and hole-free, we can never know
that it is. We mention in passing, however, that if it turns out that our
universe fails to have the properties in , it may be possible to determine�
that this is the case (see Malament 1977).

6. Note that, under this definition, spatial isotropy implies spatial homogeneity. See
Ellis (2007, 1225).

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WHAT IS A PHYSICALLY REASONABLE SPACE-TIME? 415

5. Physical Reasonableness. Some may insist that the result just presented
has little or no physical significance.7 One view is to hold that the space-
time constructed in the proof cannot be physically reasonable because it
was assembled via a seemingly artificial “cut-and-paste” process. But, such
an objection must be formulated carefully because any space-time can be
fabricated by such a construction (see Geroch 1971b, 78).

Some conditions for ruling out manufactured examples have been pro-
posed. But, none seem to be entirely satisfactory. Inextendibility is not
strong enough to forbid many cut-and-paste examples (see Earman 1995,
98). However, the property of local inextendibility, introduced by Hawking
and Ellis (1973, 59), was later shown to be much too strong (see Beem
1980). And although the condition of hole-freeness seems to be adequate
for many purposes, as we will see below, it carries with it some significant
problems as well.

It must be remembered, too, that the theorem can have physical rele-
vance, even if the particular model constructed in the proof does not.
Geroch (1971b, 78) explains: “The space-times obtained by cutting and
patching are not normally considered as serious models of our universe.
However, the mere existence of a space-time having certain global features
suggests that there are many models—some perhaps quite reasonable
physically—with similar properties.” Some have held that any space-time
that fails to have one or more of the properties in is ipso facto physically�
unreasonable. In other words, the conditions of inextendibility, isotropy,
global hyperbolicity, and hole-freeness have all been taken, at one time
or another, to be satisfied by all reasonable models of our universe. But
how does one justify such a position? We know that, given the theorem
above, this justification cannot be due to any observational data we have
collected or likewise any considerations of the local structure of space-
time. In the remainder of this section, we will briefly examine the rationale
in support of each of the properties under investigation. We hope to show
that, in each case, the justification is dubious in certain respects.

Inextendibilty. A space-time that fails to be inextendible is often thought
to be physically unreasonable for metaphysical reasons (Earman 1995,
32). In particular, Leibniz’s principles of plenitude and sufficient reason
seem to be at work. Geroch (1970, 262) asks, “Why, after all, would Nature
stop building our universe . . . when She could just as well have carried
on?” Others use similar reasoning (see Penrose 1969, 253; Clarke 1976,
17).

But, however compelling the metaphysics, it is sometimes problematic

7. See Norton (2008) for a related discussion of various ways a system can be regarded
as “unphysical.”

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416 JOHN BYRON MANCHAK

to insist on inextendibility. For example, Clarke (1976, 20) has shown
that not every well-behaved space-time admits a well-behaved inextendible
extension. Should we cling to inextendibility at the expense of other de-
sirable space-time properties? The answer is far from clear. Additionally,
a space-time does not always have just one inextendible extension (Clarke
1993, 9). Thus, the principle of sufficient reason can actually be used to
argue against the property of inextendibility. After all, why should one
extension be preferred over another?

Isotropy. The Copernican principle is often used to support the claim
that realistic models of the universe are (on sufficiently large scales) iso-
tropic. Although precise formulations vary, this principle is generally
taken to be the statement that we do not occupy a privileged position in
space-time. Since observational data seem to indicate no preferred spatial
direction from our vantage point, the Copernican principle implies an
isotropic universe (see Wald 1984, 94). However, some have questioned
the Friedmann models. Wainwright and Ellis (1996) have shown that
“intermediate isotropisation” can occur in which space-time exhibits
(highly approximate) isotropic behavior for arbitrarily long cosmic times,
despite being extremely anisotropic at very early and very late epochs. In
addition, we know that under the assumption of some formulations of
cosmic inflation, we would not even expect our universe to be isotropic
(Ellis 2007, 1227).

Elsewhere, it has been argued that although the Copernican principle
seems to modestly deny us a special status, this “seeming modesty is belied
by the immodest use to which the principle is put in justifying an inductive
extrapolation” (Earman 1995, 129). Indeed, induction on such large scales
would seem to be suspect, given that we are able to observe only a “neg-
ligibly small region” of the universe (see Wald 1984, 91).

Global Hyperbolicity. Motivated largely by considerations of causal
determinism, some insist on the (strong) cosmic censorship hypothesis—
the statement that all physically reasonable space-times are globally hy-
perbolic (see Joshi 1993; Earman 1995). To support such a position, Pen-
rose (1979, 626) maintains that cosmological models that fail to be globally
hyperbolic are unstable under certain types of perturbations. However,
such a claim is difficult to express precisely (see Geroch 1971a). And,
although some evidence does seem to indicate that instabilities are present
in nonglobally hyperbolic space-times, still other evidence suggests oth-
erwise (see Chandrasekhar and Hartle [1982] and Morris, Thorne, and
Yurtsever [1988] for opposing perspectives).

A precise formulation of the cosmic censorship hypothesis due to Wald
(1984, 304) eschews any reference to stability and, instead, simply forbids

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WHAT IS A PHYSICALLY REASONABLE SPACE-TIME? 417

known counterexamples. But, one must be careful with such an approach.
As Earman (1995, 80) has emphasized, the term “physically unreasonable”
should not be used as an “elastic label that can be stretched to include
any ad hoc way of discrediting putative counterexamples.” Finally, even
if a particular formulation can be agreed on, it seems that a proof of
cosmic censorship is still a “long way” off (Penrose 1999, 245).

Hole-Freeness. The preservation of causal determinism also seems to
be the motivating force behind the assumption that all physically rea-
sonable space-times are hole-free. According to Clarke (1976, 17), hole-
freeness is needed to ensure that predictions are “not falsified by the
spontaneous appearance of uncaused singularities.” Additionally, Geroch
(1977, 87) has suggested that, in order to uphold certain theorems con-
cerning causal determinism, we may want to modify general relativity
such that only hole-free space-times are permitted (see also Ellis and
Schmidt 1977, 927). But, Earman (1995, 98) has argued that this impo-
sition of hole-freeness amounts to little more than question begging. In-
deed, it seems impermissible to justify the modification of one’s physical
theories (so as to maintain determinism) merely because not doing so
would put determinism in jeopardy.

But there are other, more serious, problems with hole-freeness. We
know that hole-freeness can fail to mesh well with other properties of
interest: not every hole-free space-time admits an inextendible hole-free
extension (Clarke 1976, 20). And it has recently been shown that some
inextendible, globally hyperbolic space-times are not hole-free (Manchak
2009b). In other words, hole-freeness is not necessarily a property of some
maximal Cauchy developments. In fact, Krasinkov (2009) was able to
show that even Minkowski space-time is not free of holes.

Under a more complicated alternative formulation—hole-freeness*—
it does follow that every inextendible, globally hyperbolic space-time,
including Minkowski space-time, is hole-free* (Manchak 2009b). But,
given the concerns regarding inextendibility and global hyperbolicity men-
tioned above, it is still not clear whether this property of hole-freeness*
must satisfied by all physically reasonable space-times. Indeed, there exist
some (presumably physically reasonable) models of spherically symmetric,
radiating stars, which are neither hole-free nor hole-free* (Steinmüller,
King, and Lasota 1975).

6. Conclusion. It seems that one can certainly find principled reasons for
believing that all physically reasonable cosmological models are inexten-
dible, isotropic, globally hyperbolic, and hole-free. Indeed, metaphysical
considerations, the Copernican principle, and a desire to preserve causal

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418 JOHN BYRON MANCHAK

determinism all lead one to certain conclusions concerning the global
structure of space-time.

But as we have seen, these guiding principles, as well as the conclusions
drawn from them, are far from uncontroversial. It is our position that
the existence of this controversy, coupled with the underdetermination
result above, requires a certain modesty with regard to the situation. One
should simply be open to the possibility that our own universe is not best
represented by an inextendible, isotropic, globally hyperbolic, hole-free
model. Of course, this possibility demands that we apply great care when
labeling as “physically unreasonable” space-times that fail to have some
or all of these properties.

We close with some final thoughts concerning physical reasonableness
and underdetermination. We have proceeded under the rather basic as-
sumptions that space-time is temporally orientable, the manifold is Haus-
dorff, and so on. But, do we know that all physically reasonable space-
times possess these properties? It seems that we do not—a construction
similar to the one used above shows this.

So, it may be that our universe is not time orientable, or Hausdorff, and
so on (see Earman 2002, 2008). But, the point is that we need not relax
these standard assumptions of physicality in order to arrive at our findings.
Indeed, we have attempted to strengthen our underdetermination results
as much as possible by greatly limiting the class of admissible cosmological
models and then showing that the epistemological predicament persists.

A few matters are still unresolved. Which global properties can be
assumed while maintaining the underdetermination of space-time struc-
ture? In other words, for which sets of global properties will the fol-�
lowing statement be true?

Proposition 3. Let be any space-time having any set of local(M, g )ab
properties . If is not causally bizarre and has all of the� (M, g )ab
properties in , then there exists a space-time such that′ ′� (M , g )ab
(i) has all of the properties in and , (ii) is′ ′(M , g ) � � (M, g )ab ab
observationally indistinguishable from , and (iii) and′ ′(M , g ) (M, g )ab ab

are not isometric.′ ′(M , g )ab

One can straightforwardly show that if p {inextendibility, stable cau-�
sality}, the proposition holds. What about other properties thought to be
characteristic of physically reasonable space-time? In particular, if con-�
tains hole-freeness* or global hyperbolicity, does the underdetermination
result fall apart? These questions remain open.

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WHAT IS A PHYSICALLY REASONABLE SPACE-TIME? 419

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