On Gödel and the Ideality of Time


On Gödel and the Ideality of Time

John Byron Manchak*y

Gödel’s remarks concerning the ideality of time are examined. In the literature, some of
these remarks have been somewhat neglected while others have been heavily criticized.
In this article, I propose a clear and defensible sense in which Gödel’s work bears on the
question whether there is an objective lapse of time in our world.

1. Introduction. The cosmological model given by Gödel (1949a) is an
exact solution of Einstein’s equation in which matter takes the form of a
pressure-free perfect fluid. Its peculiar causal properties (e.g., a global time
function fails to exist) have been of considerable interest to philosophers of
time since the properties seem to imply the nonexistence of an objective time
lapse. But it is not clear how the peculiar features of the Gödel model bear
on the nature of time in our own universe. This thought Gödel explicitly con-
sidered. He writes (1949b, 561–62): “It might, however, be asked: Of what
use is it if such conditions prevail in certain possible worlds? Does that
mean anything for the question interesting us whether in our world there
exists an objective lapse of time?” Gödel offers two remarks in response
to the questions (562):

I think it does. For: (1) Our world, it is true, can hardly be represented by
the particular kind of rotating solutions referred to above (because the so-
lutions are static and, therefore, yield no red-shift for distant objects); there
exist however also expanding rotating solutions. In such universes an ab-
solute time might fail to exist, and it is not impossible that our world is a
universe of this kind. (2) The mere compatibility with the laws of nature
of worlds in which there is no distinguished absolute time . . . throws some

*To contact the author, please write to: 3151 Social Science Plaza A, University of Cal-
ifornia, Irvine, Irvine, CA 92697; e-mail: jmanchak@uci.edu.

yI wish to thank Thomas Barrett, Gordon Belot, David Malament, Steve Savitt, and Jim
Weatherall for helpful discussions. Special thanks to Thomas Barrett for presenting this
article a few days before the birth of baby June. The article is dedicated to baby June.

Philosophy of Science, 83 (December 2016) pp. 1050–1058. 0031-8248/2016/8305-0034$10.00
Copyright 2016 by the Philosophy of Science Association. All rights reserved.

1050

This content downloaded from 128.200.138.056 on November 17, 2016 10:51:03 AM
All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).



light on the meaning of time in those worlds in which an absolute time can
be defined. For, if someone asserts that this absolute time is lapsing, he ac-
cepts as a consequence that whether or not an objective lapse of time ex-
ists . . . depends on the particular way in which matter and its motion are
arranged in the world. This is not a straightforward contradiction; neverthe-
less a philosophical view leading to such consequences can hardly be con-
sidered as satisfactory.

Concerning remark 1, Gödel has been somewhat neglected (Yourgrau 1991;
Savitt 1994; Earman 1995; Dorato 2002; Belot 2005; Smeenk and Wüthrich
2011). Concerning remark 2, Gödel has been heavily criticized (Savitt 1994;
Earman 1995; Dorato 2002; Belot 2005; Smeenk and Wüthrich 2011). What
follows is intended to be a straightforward defense of remark 1 and charita-
ble reconstruction of remark 2 that, together, serve to clarify the significance
of Gödel’s work for the nature of time in our world.

2. Preliminaries. We begin with a few preliminaries concerning the rel-
evant background formalism of general relativity.1 An n-dimensional, rel-
ativistic space-time (for n ≥ 2) is a pair of mathematical objects (M, gab),
where M is a connected n-dimensional manifold (without boundary) that
is smooth (infinitely differentiable). Here, gab is a smooth, nondegenerate,
pseudo-Riemannian metric of Lorentz signature (2, 1,:::, 1) defined on
M. Two space-times (M, gab) and (M

0, g0ab ) are isometric if there is a dif-
feomorphism f : M → M 0 such that f

*
(gab) 5 g

0
ab.

For each point p ∈ M, the metric assigns a cone structure to the tangent
space Mp. Any tangent vector y

a in Mp will be time-like (if gaby
ayb < 0), null

(if gaby
ayb 5 0), or space-like (if gaby

ayb > 0). Null vectors create the cone
structure; time-like vectors are inside the cone, while space-like vectors are
outside. A time orientable space-time is one that has a continuous time-like
vector field on M. A time orientable space-time allows us to distinguish be-
tween the future and past lobes of the light cone. In what follows, it is as-
sumed that space-times are time orientable.

For some interval I ⊆ R, a smooth curve g : I → M is time-like if the tan-
gent vector ya at each point in g½I" is time-like. Similarly, a curve is null
(respectively, space-like) if its tangent vector at each point is null (respec-
tively, space-like). A curve is causal if its tangent vector at each point is ei-
ther null or time-like. A causal curve is future directed if its tangent vector
at each point falls in or on the future lobe of the light cone. A time-like
curve g : I → M is closed if there are two distinct points s1, s2 ∈ I such that
g(s1) 5 g(s2).

1. The reader is encouraged to consult Wald (1984) and Malament (2012) for details.

ON GÖDEL AND THE IDEALITY OF TIME 1051

This content downloaded from 128.200.138.056 on November 17, 2016 10:51:03 AM
All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).



For any two points p, q ∈ M, we write p ≪ q if there exists a future-
directed time-like curve from p to q. We write p < q if there exists a future-
directed causal curve from p to q. These relations allow us to define the
time-like and causal pasts and futures of a point p: I2( p) 5 fq : q ≪ pg,
I1( p) 5 fq : p ≪ qg, J2( p) 5 fq : q < pg, and J1( p) 5 fq : p < qg. We
say a space-time (M, gab) admits a global time function if there is a smooth
function t : M → R such that, for any distinct points p, q ∈ M, if p ∈ J 1(q),
then t( p) > t(q). The function assigns a “time” to every point in M such
that it increases along every (nontrivial) future-directed causal curve. Fol-
lowing the literature (e.g., Earman 1995; Dorato 2002), we take the exis-
tence of a global time function to be a necessary (but not sufficient) condition
for the objective lapse of time.

3. Concerning Remark 1. Gödel’s first consideration relating to the ideal-
ity of time and our universe is clear: despite the empirical data collected by
cosmologists (e.g., data suggesting an expanding universe), there remains
the epistemic possibility that our universe is one in which absolute time can-
not be defined. Concerning 1, Yourgrau (1991), Savitt (1994), and Dorato
(2002) are silent. Earman (1995) and Belot (2005) do consider Gödel’s claim
but swiftly find it unconvincing.

Belot states that Gödel himself grants that the more adequate models of
our cosmos support an absolute time (2005, 270). But the text certainly does
not lead us to this conclusion. And we know that, even late in his life, Gödel
had still not given up on the possibility that we inhabit a Gödel-type model.
Indeed, he would remain intensely interested in the collection of all astro-
nomical data relevant to this possibility (Bernstein 1991). Earman (1995,
199) claims that “we have all sorts of . . . experiences which lend strong
support to the inference that we do not inhabit a Gödel type universe but
rather a universe that fulfills all of the necessary conditions for an objec-
tive lapse of time.” However, Earman does leave open the possibility that
Gödel’s remark 1 can be defended. To do so, it is sufficient to show that
“there are cosmological models that (i) lack the features necessary for an
objective time lapse, but (ii) reproduce the redshift, etc., so that they are ef-
fectively observationally indistinguishable from models that fit current as-
tronomical data and have the spatiotemporal structure needed to ground an
objective lapse of time” (199).

Earman strongly doubts that one can find “models which allow for time
travel and which are observationally indistinguishable from non-time travel
models” (1995, 200). But this remark is puzzling since one need not find
models that are so causally misbehaved as to allow for time travel—it
would suffice to find models that lack some feature or other necessary for an
objective time lapse. And it has already been shown by Malament (1977,
79–80) that the relation of observational indistinguishability as introduced

1052 JOHN BYRON MANCHAK

This content downloaded from 128.200.138.056 on November 17, 2016 10:51:03 AM
All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).



by Glymour (1977) and used by Earman (1995) does not always preserve
some properties necessary for an objective time lapse (in particular, the ex-
istence of a global time function). Also, the relation of observational indistin-
guishability as introduced by Glymour is symmetric and allows observers to
live eternally. But, these conditions can be justifiably softened. Indeed, Ma-
lament (1977) has introduced the weaker relation: the space-time (M, gab) is
weakly observationally indistinguishable from the space-time (M 0, g0ab) if
for every point p ∈ M there is a point p0 ∈ M 0 such that I2( p) and I2( p0)
are isometric.

So, one might now wonder whether there are cosmological models that
fit current astronomical data that are weakly observationally indistinguish-
able from models that lack features necessary for an objective time lapse (in
particular, the existence of a global time function). And it turns out that there
are. In fact, one can show that every cosmological model is weakly observa-
tionally indistinguishable from some model that lacks features necessary for
an objective time lapse.

Proposition 1. Every space-time (M, gab) is weakly observationally indis-
tinguishable from a space-time (M 0, g0ab) that fails to have a global time
function.

It should be clear that the proposition (a proof is given in the appendix) pro-
vides support for Gödel’s remark 1. Indeed, it remains an epistemic possibil-
ity, just as Gödel claimed it was, that we inhabit a world that has no objective
time lapse. Further, even in the face of any (as yet uncollected) astronomical
data, this epistemic possibility remains. One final comment concerning the
proposition: it makes precise the heavily criticized statement made else-
where by Gödel that “the experience of the lapse of time can exist without
an objective lapse of time” (1949b, 561).

4. Concerning Remark 2. Gödel’s second remark is sometimes interpreted
to be an argument that time in our universe is ideal (Savitt 1994; Earman 1995;
Dorato 2002). But this reading seems to be a bit strong. Gödel only states that
“the mere compatibility with the laws of nature of worlds in which there is no
distinguished absolute time . . . throws some light on the meaning of time in
those worlds in which an absolute time can be defined.” In other words, the
existence of Gödel-type solutions simply have implications regarding the na-
ture of time for all cosmological models. However, it seems “there is a con-
sensus that even this modest conclusion is not warranted” (Smeenk and Wüth-
rich 2011, 597).

Now we have already shown that the nonexistence of a global time func-
tion in certain models does have epistemic implications for all models. But
Gödel seems to have something more in mind concerning remark 2. Indeed,

ON GÖDEL AND THE IDEALITY OF TIME 1053

This content downloaded from 128.200.138.056 on November 17, 2016 10:51:03 AM
All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).



of great importance seems to be that fact that “whether or not an objective
lapse of time exists . . . depends on the particular way in which matter and
its motion are arranged in the world.” And it is unclear how this statement
leads to any general implications concerning all models. Here, we provide
one way to spell out some of the details.

Consider an arbitrary model (M, gab) that has all the geometric proper-
ties necessary for absolute time. In particular, assume it admits a global time
function. Following Gödel, consider an observer at some point p ∈ M who
“asserts that this absolute time is lapsing.” Since time is lapsing objectively,
this means that at p—the event of the assertion—all events in the set I1( p)
have yet to “come into existence” (Gödel 1949b, 558). In other words, at p,
if one takes the idea of an objective time lapse seriously, one is led to con-
sider as fixed not the space-time (M, gab) but rather merely a portion of it.
It is then natural to wonder whether there is a sense in which this leaves
open from the perspective of the observer making the assertion the nomo-
logical possibility that, after the time of the assertion, matter and its motions
might be (re)arranged in such a way that a global time function can no lon-
ger be defined. This question whether a cosmological model can “start out”
with well-behaved causal structure but not “end up” that way was, in a
sense, asked some time ago by Stein (1970, 594). “Consider either an arbi-
trary given cosmological model, or a model having the structure of one of
the sorts assumed to hold in the real world. Then: is it ((a) ever, (b) always)
possible to introduce into such a model a continuous deformation of the
structure, leading through intermediate states, all compatible with Einstein’s
theory, to a state in which Gödel-type relationships occur?”

Here is one way to formulate this question precisely. Let (M, gab) be any
space-time that admits a global time function. For any point p ∈ M, is
there is a space-time (M, g0ab) that (i) fails to admit a global time function
and (ii) is such that g0ab 5 gab on the region M 2 I

1( p)? When (M, gab) is
at least three-dimensional, yes.

Proposition 2. Let (M, gab) be any space-time of dimension n ≥ 3 that ad-
mits a global time function. For any point p ∈ M, there is a space-time
(M, g0ab) that (i) fails to admit a global time function and (ii) is such that
g0ab 5 gab on the region M 2 I

1( p).

It should be clear how the proposition (proof given in the appendix) can be
used to understand Gödel’s remark 2. It is philosophically unsatisfying (al-
though not a contradiction) for one to assert that time is objectively lapsing
in one’s universe when from the perspective of the observer making the
assertion there remains the nomological possibility that, after the time of
the assertion, matter and its motions might be smoothly (re)arranged in such
a way so as to prohibit an objective time lapse.

1054 JOHN BYRON MANCHAK

This content downloaded from 128.200.138.056 on November 17, 2016 10:51:03 AM
All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).



5. Conclusion. Despite the above propositions, one might insist that “we
do not inhabit a Gödel type universe but rather a universe that fulfills all
of the necessary conditions for an objective lapse of time” (Earman 1995,
199). We close with one final word of caution. It seems reasonable that “in
order to be physically significant, a property of space-time ought to have
some form of stability, that is to say, it should also be a property of ‘nearby’
space-times” (Hawking and Ellis 1973, 197). There are a number of differ-
ent ways to understand the notion of nearby space-times—none entirely sat-
isfactory (Geroch 1971; Fletcher 2015). Here we simply note one sense in
which causally well-behaved space-times can be “close” to space-times that
are not.

Consider the one-parameter family of space-times (M, gab(l)) where l ∈
½0, 1", M 5 R4, and gab(l) 5 2 ∇a t ∇b t 1 ∇a x ∇b x 2 (1=2)exp(2lx) ∇a y
∇b y 2 2exp(lx) ∇(a t ∇b) y 1 ∇az ∇b z. One can easily verify that (M, gab(l))
is Gödelian for all l ∈ (0, 1". But what about (M, gab(0))? Surprisingly,
one finds it is a Minkowski space-time.2 Thus, there is sense in which a
model satisfying all of the necessary conditions for an objective lapse of
time is “close” to a set of models that do not satisfy these conditions. Should
this fact not give us pause?

Appendix
Lemma 1. Let (M, gab) be any space-time and let O be any open set in
M. There is an open set Ô in O and a space-time (M, g0ab) such that g

0
ab is

flat on Ô and g0ab 5 gab on M 2 O.
Proof. Let (M, gab) be any two-dimensional space-time (one can gen-

eralize to higher dimensions), and let O be any open set in M. Consider
a chart (O 0, J) such that (i) O0 ⊂ O; (ii) for some e > 0, J½O0" is the open
ball Be(0, 0) centered at the origin in R

2 with radius e; and (iii) the coordi-
nate maps t : O0 → R and x : O0 → R associated with (O 0, J) are such that gab
at the point J21(0, 0) is 2 ∇at ∇bt 1 ∇ax ∇b x. We can now express gabjO0
as f ∇a t ∇b t 1 g ∇a x ∇b x 1 2h ∇(a t ∇b) x for some smooth scalar fields
f : O0 → R, g : O0 → R, and h : O0 → R.

Let hab 5 f
0 ∇a t ∇b t 1 g

0 ∇a x ∇b x 1 2h
0 ∇(a t ∇b) x be a flat (Lorentzian)

metric on O 0 for some smooth scalar fields f 0 : O0 → R, g0 : O0 → R, and
h0 : O0 → R such that hab at the point J

21(0, 0) is 2 ∇at ∇b t 1 ∇a x ∇b x.
Since f 5 f 0 < 0 < g 5 g0 at the point J21(0, 0), we can find a d ∈ (0, e)

such that f < 0 < g and f 0 < 0 < g0 on all of J21½Bd(0, 0)". Let O00 ⊂ O0 be
this set J21½Bd(0, 0)". Now we divide O00 into three disjoint regions: U, V, W.
For convenience, let r be the scalar function on O00 defined by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2 1 x2

p
. Let

U be the region where r < d=3; V the region where d=3 ≤ r < 2d=3; W the
region where 2d=3 ≤ r < d.

2. Thanks to David Malament for this example.

ON GÖDEL AND THE IDEALITY OF TIME 1055

This content downloaded from 128.200.138.056 on November 17, 2016 10:51:03 AM
All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).



Next, we define a field gab on each of the three regions. On region W, let
gab 5 gab. On region U, let gab 5 hab. On region V, let gab be as follows:
(vf 1 (1 2 v) f 0 ) ∇a t ∇b t 1 (vg 1 (1 2 v)g

0 ) ∇a x ∇b x 1 2(vh 1 (12v)h
0 )

∇(at ∇b) x, where v : V → R is given by

v(r) 5

ð3(r2d=3)=d

0
exp½2(z22 1 (z 2 1)22)"dz

ð1

0
exp½2(z22 1 (z 2 1)22)"dz

:

By inspection, one can see that gab is smooth field on O
00 (cf. Geroch

1968, 536). We will work to show that it is a metric. Clearly, it is every-
where symmetric and is nondegenerate on U and W. We claim it is non-
degenerate on V as well. For convenience, let f 00 5 vf 1 (1 2 v)f 0, g00 5
vg 1 (1 2 v)g0, h00 5 vh 1 (1 2 v)h0. Let p be any point in V, and let ya

be any vector at p. We can express ya as a(∂=∂t)a 1 b(∂=∂x)a for some
a, b ∈ R. Consider gaby

a. It must come out as ( f 00( p)a 1 h00( p)b) ∇b t 1
(h00( p)a 1 g00( p)b) ∇b x. Now suppose that gaby

a 5 0. This implies that
f 00( p)a 1 h00( p)b 5 0 and h00( p)a 1 g00( p)b 5 0. It follows that a( f 00

( p)g00( p) 2 h00( p)2) 5 0 and b( f 00( p)g00( p) 2 h00( p)2) 5 0. So either a 5
b 5 0 or f 00( p)g00( p) 5 h00( p)2. But the latter case cannot obtain: because
f ( p) < 0 < g( p), f 0( p) < 0 < g0( p), and v( p) ∈ ½0, 1", we know f 00( p) <
0 < g00( p). So a 5 b 5 0, and thus ya 5 0. So gab is nondegenerate on
V. So, gab is a smooth metric on O

00. Since gab is Lorentzian at J
21(0, 0)

and O00 is connected, gab is Lorentzian on all of O
00.

Now, consider the space-time (M, g0ab), where g
0
ab 5 gab on M 2 O

00 and
g 0ab 5 gab on O

00. By construction, g0ab is smooth. Also by construction, there
is an open set Ô in O such that g0ab is flat on Ô. Just take Ô 5 U. QED

Lemma 2. Let (M, gab) be any space-time of dimension n ≥ 3 and let O
be any open set in M. There is a space-time (M, g0ab) such that there are
closed time-like curves contained in O and g0ab 5 gab on M 2 O.

Proof. Let (M, gab) be any three-dimensional space-time (one can gen-
eralize to higher dimensions) and let O be any open set in M. By the lemma
above, there is an open set Ô in O and a space-time (M, g00ab) such that g

00
ab is

flat on Ô and g00ab 5 gab on M 2 O.
Next, consider a chart (U, J) such that (i) U ⊂ Ô for some d > 0,

(ii) J½U" is the open ball Bd(0, 0, 0) centered at the origin in R
3 with radius

d, and (iii) the coordinate maps t : U → R, x : U → R, and y : U → R asso-
ciated with (U, J) are such that the (flat) metric g00ab on U can be expressed as
the (flat) metric: 2 ∇at ∇b t 1 ∇a x ∇b x 2 (1=2) ∇a y ∇b y 2 2 ∇(at ∇b) y.

Now we divide U into three disjoint regions: U1, U2, U3. For conve-
nience, let r be the scalar function on U defined by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2 1 x2 1 y2

p
. Let

1056 JOHN BYRON MANCHAK

This content downloaded from 128.200.138.056 on November 17, 2016 10:51:03 AM
All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).



U1 be the region where r < d=3; U2 the region where d=3 ≤ r < 2d=3; U3
the region where 2d=3 ≤ r < d.

Next, we define a metric g0ab on M with the desired properties. On regions
U3 and M 2 U, let g0ab 5 g

00
ab. On region U1, let g

0
ab be Gödelian:2∇at ∇b t 1

∇a x ∇b x 2 (1=2)exp(2ax) ∇a y ∇b y 2 2exp(ax) ∇(a t ∇b) y, where a > 0 is
large enough that closed time-like curves exist in U1. On region U2, let g0ab
be as follows: 2 ∇a t ∇b t 1 ∇a x ∇b x 2 (1=2)exp(2ax(1 2 v)) ∇a y ∇b y 2
2exp(ax(1 2 v)) ∇(a t ∇b) y, where v : U2 → R is just the function v(r) given
in the proof of the above lemma. By inspection, one can see that g0ab is a
smooth metric on M (cf. Geroch 1968, 536). By construction, the space-
time M 2 U is such that there are closed time-like curves contained in O
and g 0ab 5 gab on M 2 O. QED

Proposition 1. Every space-time (M, gab ) is weakly observationally in-
distinguishable from a space-time (M 0, g 0ab ) that fails to have a global time
function.

Proof. Let (M, gab ) be a two-dimensional space-time (one can general-
ize to higher dimensions). If there is a point p ∈ M such that I2( p) 5 M,
then (M, gab) fails to have a global time function. Suppose there does not
exist a p ∈ M for which I2( p) 5 M. Construct the space-time (M 0, g0ab )
according to the method outlined in Manchak (2009). Next, consider any
open set O in the M(1, b) portion of the manifold M 0 that is disjoint from
the set O1 [ O2. From the first lemma, there is an open set Ô in O and a
space-time (M 0, g00ab ) such that g

0
ab is flat on Ô and g

0
ab 5 gab on M 2 O.

Consider a chart (O0, J) such that (i) O0 ⊂ Ô; (ii) for some e > 0, J½O0"
is the open ball Be(0, 0) centered at the origin in R

2 with radius e; and
(iii) the coordinate maps t : O0 → R and x : O0 → R associated with (O0, J)
are such that g00ab is 2 ∇at ∇b t 1 ∇a x ∇b x. Now, excise two sets of points
from O: S1 5 f(t, x) : t 5 e=2, 2e=2 ≤ x ≤ e=2g and S2f(t, x) : t 5 2e=2,
2e=2 ≤ x ≤ e=2g. Identify the bottom edge of S2 with the top edge of S1
and the top edge of S2 with the bottom edge of S1 (cf. Hawking and Ellis
1973, 58–59). The resulting space-time, call it (M 00, g00ab ), contains closed
time-like curves. By construction, (M, gab ) is weakly observationally in-
distinguishable from (M 00, g00ab). Of course, the nonexistence of a global time
function follows from the existence of closed time-like curves. QED

Proposition 2. Let (M, gab ) be any space-time of dimension n ≥ 3 that
admits a global time function. For any point p ∈ M, there is a space-time
(M, g0ab) that (i) fails to admit a global time function and (ii) is such that
g0ab 5 gab on the region M 2 I

1( p).
Proof. Let (M, gab) be any space-time of dimension n ≥ 3 that admits

a global time function. Let p be any point in M. By the second lemma, we
know (since I1( p) is an open set) there exists a space-time (M, g0ab ) such that
there are closed time-like curves in I1( p) and gab 5 g

0
ab on M 2 I

1( p). Of

ON GÖDEL AND THE IDEALITY OF TIME 1057

This content downloaded from 128.200.138.056 on November 17, 2016 10:51:03 AM
All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).



course, the nonexistence of a global time function follows from the exis-
tence of closed time-like curves. QED

REFERENCES

Belot, G. 2005. “Dust, Time, and Symmetry.” British Journal for the Philosophy of Science 56:
255–91.

Bernstein, J. 1991. Quantum Profiles. Princeton, NJ: Princeton University Press.
Dorato, M. 2002. “On Becoming, Cosmic Time, and Rotating Universes.” In Time, Reality, and

Existence, ed. C. Callender, 253–76. Cambridge: Cambridge University Press.
Earman, J. 1995. Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Rel-

ativistic Spacetimes. Oxford: Oxford University Press.
Fletcher, S. 2015. “Similarity, Topology, and Physical Significance in Relativity Theory.” British

Journal for the Philosophy of Science, forthcoming.
Geroch, R. 1968. “What Is a Singularity in General Relativity?” Annals of Physics 48:526–40.
———. 1971. “General Relativity in the Large.” General Relativity and Gravitation 2:61–74.
Glymour, C. 1977. “Indistinguishable Space-Times and the Fundamental Group.” In Foundations

of Space-Time Theories, Minnesota Studies in the Philosophy of Science 8, ed. J. Earman,
C. Glymour, and J. Stachel, 50–60. Minneapolis: University of Minnesota Press.

Gödel, K. 1949a. “An Example of a New Type of Cosmological Solutions of Einstein’s Field Equa-
tions of Gravitation.” Review of Modern Physics 21:447–50.

———. 1949b. “A Remark about the Relationship between Relativity Theory and Idealistic Phi-
losophy.” In Albert Einstein: Philosopher-Scientist, ed. P. A. Schilpp, 557–62. La Salle, IL:
Open Court.

Hawking, S., and G. Ellis. 1973. The Large Scale Structure of Space-Time. Cambridge: Cambridge
University Press.

Malament, D. 1977. “Observationally Indistinguishable Space-Times.” In Foundations of Space-
Time Theories, Minnesota Studies in the Philosophy of Science 8, ed. J. Earman, C. Glymour,
and J. Stachel, 61–80. Minneapolis: University of Minnesota Press.

———. 2012. Topics in the Foundations of General Relativity and Newtonian Gravitation Theory.
Chicago: University of Chicago Press.

Manchak, J. 2009. “Can We Know the Global Structure of Spacetime?” Studies in History and Phi-
losophy of Modern Physics 40:53–56.

Savitt, S. 1994. “The Replacement of Time.” Australasian Journal of Philosophy 72:463–74.
Smeenk, C., and C. Wüthrich. 2011. “Time Travel and Time Machines.” In The Oxford Handbook

of Philosophy of Time, ed. C. Callender, 577–632. Oxford: Oxford University Press.
Stein, H. 1970. “On the Paradoxical Time-Structures of Gödel.” Philosophy of Science 37:589–

601.
Wald, R. 1984. General Relativity. Chicago: University of Chicago Press.
Yourgrau, P. 1991. The Disappearance of Time: Kurt Gödel and the Idealistic Tradition in Philos-

ophy. Cambridge: Cambridge University Press.

1058 JOHN BYRON MANCHAK

This content downloaded from 128.200.138.056 on November 17, 2016 10:51:03 AM
All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).

http://www.journals.uchicago.edu/action/showLinks?crossref=10.1007%2FBF02450519
http://www.journals.uchicago.edu/action/showLinks?crossref=10.1093%2Fbjps%2Faxi116
http://www.journals.uchicago.edu/action/showLinks?crossref=10.1080%2F00048409412346261
http://www.journals.uchicago.edu/action/showLinks?crossref=10.1103%2FRevModPhys.21.447
http://www.journals.uchicago.edu/action/showLinks?crossref=10.1016%2F0003-4916%2868%2990144-9
http://www.journals.uchicago.edu/action/showLinks?system=10.1086%2F288328
http://www.journals.uchicago.edu/action/showLinks?crossref=10.1016%2Fj.shpsb.2008.07.004
http://www.journals.uchicago.edu/action/showLinks?crossref=10.1016%2Fj.shpsb.2008.07.004