## 1. Modern Spacetime Theories: A Beginner's Guide



Virtually all modern spacetime theories are now built in the same way.
The theory posits a manifold of events and then assigns further
structures to those events to represent the content of spacetime. A
standard example is Einstein's general theory of relativity. As
a host for the hole argument, we will pursue one of its best known
applications, the expanding universes of modern relativistic
cosmology.



This one example illustrates the core content of the hole argument.
With only a little further effort, the argument can be made more
precise and general. This will be done concurrently in these
 notes,[1]
 intended for readers with some background in differential geometry
and general relativity.



Here are the two basic building blocks of modern, relativistic
cosmology: a manifold of events and the fields defined on it.



**Manifold of Events**. Consider our universe, which
relativistic cosmologies attempt to model. Its spacetime is the
entirety of all space through all time. The events of this spacetime
are generalizations of the dimensionless points of ordinary spatial
geometry. A geometric point in ordinary spatial geometry is just a
particular spot in the space and has no extension. Correspondingly, an
event in spacetime is a particular point in a cosmological space at a
particular moment of time.



So far, all we have defined is a set of events. To be a
four-dimensional manifold, the set of events must be a little bit more
organized. In a real spacetime, we have the idea that each event sits
in some local neighborhood of events; and this neighborhood sits
inside a larger neighborhood of events; and so on. That extra
organization comes from the requirement that we can label the events
with four numbers--or at least we can do this for any
sufficiently small chunk of the manifold. These labels form coordinate
systems. The fact that four numbers are just sufficient to label the
events makes the manifold four-dimensional. We can now pick out
neighborhoods of some event as the set of all points whose spacetime
coordinates differ from our starting event by at most one unit; or two
units; or three units; etc.. That gives us the nested neighborhoods of
events. Figure 1 illustrates how a set of events may be made into a
two dimensional manifold by assigning "\(x\)" and
"\(y\)" coordinates to the events.



![Figure shows a small, upended bag containing events, represented as dots, being emptied into a pile, labeled 'a set of events'. An arrow labeled 'Label smoothly with numbers' points to the site of the dots redistributed over a two-dimensional grid. It has horizontal and vertical straight lines, representing lines of constant Cartesian coordinates. The lines are labeled 'x=0', 'x=1', 'x=2', 'x=3' and 'x=4'; and 'y=0', 'y=1', 'y=2', 'y=3' and 'y=4'. The grid is labeled 'A manifold of events'.](1.svg)

Figure 1. Forming a manifold of
events




**Metrical Structure and Matter Fields**. In specifying
that events form a four dimensional manifold, there is still a lot we
have *not* said about the events. We have not specified which
events lie in the future and past of which other events, how much time
elapses between these events, which events are simultaneous with
others so that they can form three dimensional spaces, what spatial
distances separates these events, and many more related
properties.



These additional properties are introduced by specifying the metric
field. To see how this field provides that information, imagine a
curve connecting a given pairs of events in spacetime. The information
about times elapsed and spatial distances is given by the times
elapsed and distances along such curves. See Figure 2:



![A rectangle is labeled 'a manifold of events'. The events are shown as dots. An up arrow labeled 'time' indicate that time is represented as advancing in the up-vertical direction and a two-headed horizontal arrow labeled 'space' indicates that space is spread across horizontally. A constant time surface, seen side on as a parallelogram, has two points representing events connected by a line. The surface is labeled 'Events at the same time form a surface in the manifold that is a snapshot of space at one instant'. and 'Metric measures distance along curves in space that connect events at the same time'. A vertical line connects two events, represented as dots, labeled 'earlier' and 'later'. The line is labeled 'Metric measures time elapsed along worldliness between earlier and later events'.](2.svg)

Figure 2. The function of the metric
field.




That information could be supplied by a huge catalog that specifies
the spatial or temporal distance between pairs of events along curves
connecting them. Such a huge catalog would be massively redundant,
however. If we know the distance from \(A\) to \(B\) and
from \(B\) to \(C\) along some curve, then we know the
distance from \(A\) to \(C\) along that curve as well. The
minimum information we need is the temporal and spatial distance
between each event and all those (loosely speaking) infinitesimally
close to it. That information is what the metric field provides. It is
a "field" since that information belongs just to one
event. We can then piece together temporal and spatial distance along
any curve just by summing all the distances between successive
infinitesimally close points along the curve. 



The matter of the universe is represented by matter fields. The
simplest form of matter--the big lumps that make
galaxies--can be represented by worldlines that trace out the
history of each galaxy through time. In standard models, the galaxies
recede from one another and this is represented by a spreading apart
of the galactic worldlines as we proceed to later times. See Figure
3:



![In a rectangle, an up arrow labeled 'time' indicate that time is represented as advancing in the up-vertical direction and a two-headed horizontal arrow labeled 'space' indicates that space is spread across horizontally. An instant of space is seen side on as a parallelogram. Tiny sketches of five galaxies are laid out across the parallelogram. Five nearly vertical lines are drawn through the galaxies such that they spread apart as they proceed up the figure. They are labeled 'Worldlines of galaxies'.](3.svg)

Figure 3. Galaxies in an expanding
universe.



## 2. The Freedom of General Covariance



Newton's first law states that force-free bodies move on
straight-line trajectories through spacetime. Clearly, this law as
stated cannot be true in all coordinate systems, for imagine such a
system accelerating with respect to one in which the law holds true:
in the accelerating coordinate system, the force-free bodies will also
appear to accelerate! So Newton's first law is not--to use
a piece of terminology introduced by Einstein in his quest for the
general theory of relativity in the 1910s--'generally
covariant'. When Einstein finally arrived at his completed
theory of general relativity in 1915, one of its novel features was
its general covariance: unlike Newton's first law in its
formulation above, if the Einstein field equations of general
relativity hold in one coordinate system, then they hold true in all
coordinate systems related to that original system by smooth but
otherwise arbitrary transformations. General relativity was, indeed,
the first spacetime theory in which one was free to use arbitrary
spacetime coordinate systems in this way. This feature is now shared
by virtually all modern formulations of spacetime theories, including
modern versions of special relativity and (perhaps surprisingly!)
Newtonian spacetime theory (on the latter, see e.g. Friedman
(1983)).



In its original form, general covariance was understood
"passively"; that is, as a freedom to describe structures
in spacetime by means of arbitrarily chosen coordinate systems. That
freedom is closely related to another freedom, known as
"active" general covariance. According to active general
covariance, we are licensed to spread structures like metric fields
over the manifold in as many different ways as there are coordinate
transformations, and the resulting redistributions of said fields on
the manifold will preserve solutionhood of the theory under
consideration. Passive general covariance is not equivalent to active
general covariance defined in this way, for a theory can be passively
generally covariant and yet arbitrary smooth redistributions of the
fields on the manifold will in general fail to preserve solutionhood:
the details are not important here, but see Pooley (2017) for further
discussion. (For a more extensive account of the relationship between
active and passive covariance, see also the supplementary document:
 Active and Passive Covariance.)[2]
 What is important is that every 'local' spacetime theory
(to use terminology of Earman & Norton (1987)) is actively
generally covariant, and so will be subject to a version of the hole
argument--this includes both special relativity and certain
formulations of Newtonian mechanics. (One view is that this goes too
far, and that general relativity is distinct from many other spacetime
theories in that its spacetime geometry has become dynamical and it is
only in such theories that the hole argument should be mounted: for
further discussion, see Earman (1989, Ch.9, Section 5), Stachel
(1993), and Iftime and Stachel (2006).)



In what follows, we'll focus on active general covariance, which
is indeed a property possessed by general relativity (as well as other
local spacetime theories). The essential manipulation of the hole
argument involves exercising this freedom. Figure 4 illustrates one
way that we might spread the metrical structure and matter fields over
the manifold of spacetime events:



![In a rectangular sheet with overturned top edge, the expanding system of galaxies of Figure 3 is shown with two spatial surfaces connected by vertical lines. It is labeled 'Metric and matter fields'. Arrows indicate that the sheet is to be superimposed on a second rectangle, labeled 'manifold', in which some events are represented as dots. A duplicate of the rectangular sheet is shown superimposed on the manifold.](4.svg)

Figure 4. One way to spread metric and
matter over the manifold.




Figure 5 illustrates a second way:



![The figure is a duplication of the previous Figure 4 with one change. In the center of the sheet superimposed on the manifold is a dotted ellipse, labeled 'THE HOLE'. Inside the ellipse the spatial surface and vertical lines are displaced slightly up and to the right, while still connecting smoothly to the other parts of the unaltered figure at the edge of the ellipse.](5.svg)

Figure 5. Another way to spread metric
and matter over the manifold.




We shall call the transformation between the two spreadings a
"hole transformation." The dotted region is The Hole. The
first distribution of metric and matter fields is transformed into the
second in a way that


* leaves the fields unchanged outside the hole;
* within the hole they are spread differently;
* and the spreadings inside and outside the hole join
 smoothly.[3]



Importantly, both the pre- and post-transformed states in Figure 5 are
solutions of Einstein's field equations of general
relativity--meaning that they both represent possible ways for
the world to be, according to the theory.


## 3. The Preservation of Invariants


### 3.1 Invariants



The two different spreadings share one vital characteristic upon which
the hole argument depends: the two spreadings agree completely on all
invariant properties.



These invariant properties are, loosely speaking, the ones that are
intrinsic to the geometry and dynamics, such as distance along spatial
curves and time along worldlines of galaxies, the rest mass of the
galaxy, the number of particles in it, as well as a host of other
properties, such as whether the spacetimes are metrically flat or
curved.



The invariant properties are distinguished from non-invariant
properties. Best known of the non-invariant properties are those
dependent on a particular choice of coordinate system. For example,
just one event in a two-dimensional Euclidean space lies at the origin
of a coordinate system, that is, at \(x=0,\) \(y=0\). But
which event that is changes when we change our coordinate system. So
"being at the origin" is not an invariant. The spatial
distance between two points, however, is that same no matter which
coordinate system is used to describe the space. It is an
invariant.



While the fields are spread differently in the two cases, they agree
in all invariant properties; so, in invariant terms, they are the
same.


### 3.2 Invariants and Observables



There is a special relationship between the invariants of a spacetime
theory and its observables, i.e. those quantities which are accessible
to observational verification:



> 
> All observables can be reduced to invariants.
> 



For example, if one makes a journey from one galaxy to another, all
observables pertinent to the trip will be invariants. These include
the time elapsed along the journey, whether the spaceship is
accelerating or not at any time in its journey, the age of the galaxy
one leaves at the start of the trip and the age of the destination
galaxy at the end and all operations that may involve signaling with
particles or light pulses.



Therefore, since the two spreadings or distributions of metric and
matter fields of a hole transformation agree on invariants, they also
agree on all observables. They are observationally
indistinguishable.


## 4. What Represents Spacetime?



Recall our original concern: we want to know whether we can conceive
of spacetime as a substance--that is, as something that exists
independently of the material events which unfold within it. To do
this, we need to know what in the above structures represents
spacetime. One popular answer to that question is that the manifold of
events represents spacetime. This choice is natural since modern
spacetime theories are built up by first positing a manifold of events
and then defining further structures on them. So, on a realist view of
physical theories on which they are to be understood
'literally', it is very natural to regard the manifold as
being an independently existing structure that bears properties, and
which thereby plays the role of a container just as we expect that
spacetime
 does.[4]



One might wonder whether some of the further structures defined on the
manifold represent further properties of spacetime rather than what is
contained within spacetime. In particular, the metric field contains
important information on spatial distances and times elapsed. Ought
that not also to be considered a part of the containing spacetime as
opposed to what is contained within spacetime?



Against this line of thought, some would argue that general relativity
makes it hard to view the metric field simply as being part of the
containing spacetime. For, in addition to spatial and temporal
information, the metric field also represents the gravitational field.
Therefore it also carries energy and momentum--the energy and
momentum of the gravitational field (although a notorious technical
problem in general relativity precludes identifying the energy and
momentum *density* of the gravitational field at any particular
event in spacetime). This energy and momentum is freely interchanged
with other matter fields in spacetimes. To carry energy and momentum
is (the thought goes) a natural distinguishing characteristic of
matter contained within spacetime.



So, the metric field of general relativity seems to defy easy
characterization. We would like it to be exclusively part of spacetime
the container, or exclusively part of matter the contained. Yet it
seems to be part of both. But in any case, the crucial point to note
here (contrary to some historical writing on the hole argument) is
that one need not settle this issue in order to get the hole argument
off the ground! As long as one is dealing with a theory which is
actively generally covariant in the sense articulated above, the hole
argument will rear its head, as we will now see.


## 5. The Price of Spacetime Substantivalism



So far we have characterized the substantivalist doctrine as the view
that spacetime has an existence independent of its contents. This
formulation conjures up powerful if vague intuitive pictures, but it
is not clear enough to be deployed in the context of the
interpretation of physical theories. If we represent spacetime by a
manifold of events, how do we characterize the independence of its
existence? Is it the counterfactual claim that were there no metric or
matter fields, there would still be a manifold of events? That
counterfactual is automatically denied by the standard formulation
which posits that all spacetimes have at least metrical structure.
That seems too cheap a refutation of manifold substantivalism. Surely,
there must be an improved formulation. Fortunately, we do not need to
wrestle with finding it. For present purposes we need only consider a
consequence of the substantivalist view and can set aside the task of
giving a precise formulation of that view.



In their celebrated debate over space and time, Leibniz taunted the
substantivalist Newton's representative, Clarke, by asking how
the world would change if East and West were switched. For Leibniz
there would be no change since all spatial relations between bodies
would be preserved by such a switch. But the Newtonian substantivalist
had to concede that the bodies of the world were now located in
different spatial positions, so the two systems were physically
distinct.



Correspondingly, when we spread the metric and matter fields
differently over a manifold of events, we are now assigning metrical
and material properties in different ways to the events of the
manifold. For example, imagine that a galaxy passes through some event
E in the hole. After the hole transformation, this galaxy might not
pass through that event. For the manifold substantivalist, this must
be a matter of objective physical fact: either the galaxy passes
through E or not. The two distributions represent two physically
distinct possibilities.



![The figure shows two copies of the rectangular sheet of Figure 4 with the ellipse labeled 'THE HOLE' shown on both. In the first, a dot on the central vertical line within the ellipse is labeled 'E'. In the second copy, the distortions of Figure 5 within the ellipse are shown. The dot labeled E is in its original position, so that the vertical line no longer passes through it.](6.svg)

Figure 6. Does the galaxy pass through
event \(E\)?




That is, manifold substantivalists must (it seems) deny an equivalence
inspired by Leibniz' taunt which is thus named after
 him:[5]



> 
> **Leibniz Equivalence**. If two distributions of fields
> are related by a smooth transformation, then they represent the same
> physical systems.
> 



The supplementary document
 Visualizing Leibniz Equivalence Through Map Projections
 illustrates the essential idea of Leibniz Equivalence through an
analogy with different map projections of the Earth's
surface.


## 6. Unhappy Consequences



We can now assemble the pieces above to generate unhappy consequences
for the manifold substantivalist. Consider the two distributions of
metric and material fields related by a hole transformation. Since the
manifold substantivalist denies Leibniz Equivalence, the
substantivalist must hold that the two systems represent distinct
physical systems. But the properties that distinguish the two are very
elusive. They escape both (a) observational verification and (b) the
determining power of cosmological theory.



(a) *Observational verification*. The substantivalist must
insist that it makes a physical difference whether the galaxy passes
through event \(E\) or not. But we have already noticed that the
two distributions are observationally equivalent: no observation can
tell us if we are in a world in which the galaxy passes through event
\(E\) or misses event \(E\).



It might be a little hard to see from Figure 6 that the two
distributions are observationally equivalent. In the first
distribution on the left, the middle galaxy moves in what looks like a
straight line and stays exactly at the spatial midpoint between the
galaxies on either side. In the second distribution on the right, all
that seems to be undone. The galaxy looks like it accelerates in
taking a swerve to the right, so that it moves closer to the galaxy on
its right.



These differences that show up in the portrayal of Figure 6 are all
non-invariant differences. For the right hand distribution, the galaxy
does veer to the right in the figure, but at the same time, distances
between events get stretched as well, just as they get stretched in
the various map projections shown in the supplement,
 Visualizing Leibniz Equivalence Through Map Projections.
 So the galaxy always remains at the spatial midpoint of the galaxies
on either side; it just doesn't look like it is at the spatial
midpoint from the way the figure is drawn.



Similarly, an acceleration vector along the galaxy's worldline
determines whether the galaxy is accelerating. That acceleration
vector is an invariant. So, if the galaxy in the left hand
distribution has a zero acceleration vector, then the corresponding
galaxy in the right hand distribution will also have a zero
acceleration vector. Remember, a hole transformation preserves
invariants. So if a galaxy is unaccelerated in the left hand
distribution, it is also unaccelerated in the right hand
distribution.



(b) *Determinism*. The physical theory of relativistic
cosmology is unable to pick between the two cases. This is manifested
as an indeterminism of the theory. We can specify the distribution of
metric and material fields throughout the manifold of events,
excepting within the region designated as The Hole. Then the theory is
unable to tell us how the fields will develop into The Hole. Both the
original and the transformed distribution are legitimate extensions of
the metric and matter fields outside The Hole into The Hole, since
each satisfies all the laws of the theory of relativistic cosmology.
The theory has no resources which allow us to insist that one only is
admissible.



It is important to see that the unhappy consequence does not consist
merely of a failure of determinism. We are all too familiar with such
failures and it is certainly not automatic grounds for dismissal of a
physical theory. The best known instance of a widely celebrated,
indeterministic theory is quantum theory, where, in the standard
interpretation, the measurement of a system can lead to an
indeterministic collapse onto one of many possible outcomes. Less well
known is that it is possible to devise indeterministic systems in
classical physics as well. Most examples involves oddities such as
bodies materializing at unbounded speed from spatial infinity, so
called "space invaders." (Earman, 1986a, Ch. III; see also
 determinism: causal)
 Or they may arise through the interaction of infinitely many bodies
in a supertask
 (Supertasks).
 More recently, an extremely simple example has emerged in which a
single mass sits atop a dome and spontaneously sets itself into motion
after an arbitrary time delay and in an arbitrary direction (Norton,
2003, Section 3).



The problem with the failure of determinism in the hole argument is
not the fact of failure but the way that it fails. If we deny manifold
substantivalism and accept Leibniz Equivalence, then the indeterminism
induced by a hole transformation is eradicated. While there are
uncountably many mathematically distinct developments of the fields
into the hole, under Leibniz Equivalence, they are all physically the
same. That is, there is a unique development of the physical fields
into the hole after all. Thus the indeterminism is a direct
consequence of the substantivalist's metaphysical commitments.
Similarly, if we accept Leibniz Equivalence, then we are no longer
troubled that the two distributions cannot be distinguished by any
possible observation. They are merely different mathematical
descriptions of the same physical reality and so should agree on all
observables.



So, the anti-substantivalist conclusion invited by the hole argument
is this. We can load up any physical theory with objects or properties
(here: spacetime events) that cannot be fixed by observation. If their
invisibility to observation is not already sufficient warning that
these properties are illegitimate, then finding that they visit
indeterminism onto a theory that is otherwise deterministic ought to
be warning enough. Therefore, such objects or properties (again, here
spacetime events) should be discarded along with any doctrine that
requires their retention.


## 7. The Hole Argument in Brief



In sum, the hole argument amounts to
 this:[6]


1. If one has two distributions of metric and material fields related
by a hole transformation, then manifold substantivalists must maintain
that the two systems represent two distinct physical systems (i.e.,
they must deny Leibniz Equivalence).
2. This physical distinctness transcends both observation and the
determining power of the theory since:
	* The two distributions are observationally identical.
	* The laws of the theory cannot pick between the two developments of
	the fields into the hole.
3. Therefore, the manifold substantivalist advocates a problematic
metaphysics which should be discarded.


## 8. The History of the Hole Argument


### 8.1 Einstein Falls into the Hole...



The hole argument was created by Albert Einstein late in 1913 as an
act of desperation when his quest for his general theory of relativity
had encountered what appeared to be insuperable obstacles. Over the
previous year, he had been determined to find a gravitation theory
that was generally covariant in the sense introduced above. He had
even considered essentially the celebrated generally covariant
equations he would settle upon in November 1915 and which now appear
in all the textbooks.



Unfortunately, Einstein had at least initially been unable to see that
these equations were admissible. Newton's theory of gravitation
worked virtually perfectly for weak gravitational fields. So it was
essential that Einstein's theory revert to Newton's in
that case. But try as he might, Einstein could not see that his
equations and many variants of them could properly mesh with
Newton's theory. In mid 1913 he published a compromise: a sketch
of a relativistic theory of gravitation that was not generally
covariant. (For further details of these struggles, see Norton
(1984).)



His failure to find an admissible generally covariant theory troubled
Einstein greatly. Later in 1913 he sought to transform his failure
into a victory of sorts: he thought he could show that no generally
covariant theory at all is admissible. Any such theory would violate
what he called the Law of Causality--we would now call it
determinism. He sought to demonstrate this remarkable claim with the
hole argument.



In its original incarnation, Einstein considered a spacetime filled
with matter excepting one region, the hole, which was matter-free. (So
in this original form, the term "hole" makes more sense
than in the modern version.) He then asked if a full specification of
both metric and material fields outside the hole would fix the metric
field within. Since he had tacitly eschewed Leibniz Equivalence,
Einstein thought that the resulting negative answer sufficient to damn
all generally covariant theories.


### 8.2 ...and Climbs out Again



Einstein struggled on for two years with his misshapen theory of
limited covariance. Late in 1915, as evidence of his errors mounted
inexorably, Einstein was driven to near despair and ultimately
capitulation. He returned to the search for generally covariant
equations with a new urgency, fueled in part by the knowledge that
none other than David Hilbert had thrown himself into analysis of his
theory. Einstein's quest came to a happy close in late November
1915 with the completion of his theory in generally covariant
form.



For a long time it was thought that Hilbert had beaten Einstein by
five days to the final theory. New evidence in the form of the proof
pages of Hilbert's paper now suggests he may not have. More
importantly, it shows clearly that Hilbert, like Einstein, at least
temporarily believed that the hole argument precluded all generally
covariant theories and that the belief survived at least as far as the
proof pages of his paper. (See Corry, Renn and Stachel 1997.)



While Einstein had tacitly withdrawn his objections to generally
covariant theories, he had not made public where he thought the hole
argument failed. This he finally did when he published what John
Stachel calls the "point-coincidence argument." This
argument, well known from Einstein's (1916, p.117) review of his
general theory of relativity, amounts to a defence of Leibniz
Equivalence. He urges that the physical content of a theory is
exhausted by the catalog of the spacetime coincidences it licenses.
For example, in a theory that treats particles only, the coincidences
are the points of intersection of the particle worldlines. These
coincidences are preserved by transformations of the fields. Therefore
two systems of fields that can be intertransformed have the same
physical content; they represent the same physical system.



Over the years, the hole argument was deemed to be a trivial error by
an otherwise insightful Einstein. It was John Stachel (1980) who
recognized its highly non-trivial character and brought this
realization to the modern community of historians and philosophers of
physics. (See also Stachel, 1986.) In Earman and Norton (1987), the
argument was recast as one that explicitly targets spacetime
substantivalism. For further historical discussion, see Howard and
Norton (1993), Janssen (1999), Klein (1995) and Norton (1987). A
thorough, synoptic treatment in four volumes is Renn (2007); for a
history of the philosophical revival of the hole argument, see
Weatherall (2020).



For an account of the appropriation and misappropriation of
Einstein's point-coincidence argument by the logical
empiricists, see Giovanelli (2013).


## 9. Responses to the Hole Argument



There are at least as many responses to the hole argument as authors
who have written on it. In this section, we regiment the literature by
considering five broad classes of response to the argument since it
was revitalised in the philosophical literature of the 1980s. In the
course of scrutinizing the argument, by now virtually all its aspects
have been weighed and tested.


### 9.1 Reject substantivalism



One line of thought simply agrees that the hole argument makes
acceptance of Leibniz Equivalence compelling. It seeks to make more
transparent what that acceptance involves by trying to find a single
mathematical structure that represents a physical spacetime system
rather than the equivalence class of intertransformable structures
licensed by Leibniz Equivalence. One such attempt involves the notion
of a "Leibniz algebra" (see Earman, 1989, Ch. 9, Sect. 9).
It is unclear that such attempts can succeed. Just as
intertransformable fields represent the same physical system, there
are distinct but intertransformable Leibniz algebras with the same
physical import. If the formalisms of manifolds and of Leibniz
algebras are intertranslatable, one would expect the hole argument to
reappear in the latter formalism as well under this translation. (See
Rynasiewicz, 1992.)



Another approach seeks to explain Leibniz Equivalence and demonstrate
the compatibility of general relativity with the hole argument through
the individuation of spacetime points by means of "Dirac
observables" and an associated gauge fixing stipulation (Lusanna
and Pauri, 2006). More generally, we may well wonder whether the
problems faced by spacetime substantivalism are artifacts of the
particular formalism described above. Bain (1998, 2003) has explored
the effect of a transition to other formalisms (including but not
limited to Leibniz algebras).


### 9.2 Accept the indeterminacy



An alternative response to the hole argument is to accept that
generally covariant theories of space and time such as general
relativity are indeterministic. Perhaps (the thought goes) this
indeterminism is not troubling, because it is an indeterminism only
about which objects instantiate which properties and not about which
patterns of properties are instantiated. It is not obvious, however,
that this is sufficient to defuse worries about indeterminism: at the
very least, if another response to the argument were available, they
would seem to be preferable.



A related response here is to *redefine* determinism, and to
argue that, in the relevant sense, theories such as general relativity
are deterministic after all, in spite of the hole argument.
Modifications to the definition of determinism in light of the hole
argument have been explored by Belot (1995b), Brighouse (1994, 2020),
Butterfield (1989), Melia (1999), and Pooley (2021).


### 9.3 Metric essentialism



In his own highly original response to the hole argument, Maudlin
(1990) urges that each spacetime event carries its metrical properties
essentially; that is, it would not be that very event if (after
redistribution of the fields) we tried to assign different metrical
properties to it. As a result, although there appears to be a class of
distinct possible worlds associated with each class of
intertransformable solutions of general relativity (or whatever other
actively generally covariant theory in which one is interested), in
fact only one such world is metaphysically possible, and thereby the
hole argument is vitiated. Teitel (2019) has explored a refined
version of this essentialist response but concludes that it fails to
improve on standard modal responses to the hole argument. Butterfield
(1989) portrays intertransformable systems as different possible
worlds and uses counterpart theory to argue that at most one can
represent an actual spacetime. (For an updated version of
Butterfield's appeal to counterpart theory, see Butterfield
& Gomes (2023a, 2023b).)


### 9.4 Sophisticated substantivalism



Maudlin's metrical essentialism is an example of
'sophisticated substantivalism'. This term was
introduced--in a somewhat pejorative sense--by Belot &
Earman (2001) to refer to a class of views according to which it is
legitimate after all for a substantivalist to deny that systems
related by hole transformation represent distinct possibilities,
thereby side-stepping the hole argument. Another version of
sophisticated substantivalism is anti-haecceitist substantivalism,
according to which physical spacetime points do not possess
trans-world identities. The distinctness of worlds in which the
material content of the universe is shifted from its distribution in
the actual world presupposes such identities. Thus the apparent
indeterminism of general relativity and other actively generally
covariant theories is avoided by denying such identities. This
position is currently a popular response to the hole argument: for
discussion e.g. Hoefer (1996) and Pooley (2006b); for some concerns
that the position is metaphysically obscure, see Dasgupta (2011).



Another related version of sophisticated substantivalism has it that
spacetime is better represented not by the manifold of events alone
but by some richer structure, such as the manifold of events in
conjunction with metrical properties. (See, for example, Hoefer,
1996.) What motivates this escape is the idea that the manifold of
events lacks properties essential to spacetime. For example, there is
no notion of past and future, of time elapsed or of spatial distance
in the manifold of events. Thus one might be tempted to identify
spacetime with the manifold of events *plus* some further
structure that supplies these spatiotemporal notions. Thereby, the
thought might go, it is the metrical structure which individuates
spacetime points non-rigidly. This escape from the hole argument
sometimes succeeds and sometimes fails. In certain important special
cases, alternative versions of the hole argument can be mounted
against the view: see Norton (1988).


### 9.5 Mathematical/formalist responses



Perhaps the simplest possible challenge to the hole argument maintains
that Leibniz Equivalence is a standard presumption in the modern
mathematical physics literature and suggests that even entertaining
its denial is a mathematical blunder unworthy of serious attention.
But in response: while acceptance of Leibniz Equivalence is widespread
in the physics literature, it is not a *logical truth* that can
only be denied on pain of contradiction. That it embodies non-trivial
assumptions whose import must be accepted with sober reflection is
indicated by the early acceptance of the hole argument by none other
than David Hilbert himself. (See
 Section 8.2
 above.) If denial of Leibniz Equivalence is a blunder so egregious
that no competent mathematician would make it, then our standards for
competence have become unattainably high, for they must exclude David
Hilbert in 1915 at the height of his powers.



The question has nevertheless been reopened recently by Weatherall
(2018), who aargues that intertransformable mathematical structures
are taken in standard mathematical practice to be the same structure.
Thus they should represent the same physical system, precluding the
denial of Leibniz Equivalence. Curiel (2018) argues for a similar
triviality conclusion as Weatherall but on a different basis: there is
no *physical* correlate to the hole transformation in standard
physical practice. For a response to arguments of this kind, defending
the view that this kind of 'mathematical structuralism' is
insufficient to block the hole argument, see Pooley & Read (2021);
likewise, Roberts (2020) has responded that Nature--not
mathematical practice--should decide whether two mathematical
structures represent the same physical system.



Recently, some authors have argued that modifying the foundations of
mathematics from the set-theoretic orthodoxy would be sufficient to
block the hole argument: see Ladyman & Presnell (2020) and
Dougherty (2020) for discussions of such arguments in the particular
context of homotopy type theory. Separately, Halvorson and Manchak
(2021) have argued that because there is a unique metric-preserving
map ('isometry') relating two solutions of general
relativity related by the hole transformation, the hole argument is
thereby blocked; for sceptical responses to this argument, see Menon
& Read (2023).



Belot (2018) argues against a single decision univocally in favor or
contrary to Leibniz Equivalence. While allowing that hole
transformations relate systems that are physically the same, he argues
that in some sectors of general relativity, some transformations that
preserve the metric may relate physically distinct systems.


## 10. Broader Significance of the Hole Argument



The hole argument has had a broader significance in the philosophy of
science, pertaining *inter alia* to realism about theoretical
entities, to theories of quantum gravity, and to the issue of gauge
freedoms in our physical theories. We discuss all three of these
relations in this section; there is also a supplementary document
which expands upon the third. 


### 10.1 A Limit to Scientific Realism



The hole argument has presented a new sort of obstacle to the rise of
scientific realism. According to that view, we should read the
assertions of our mature theories literally. So, if general relativity
describes a manifold of events and a metrical structure, then that is
literally what is there in the view of the strict scientific realist.
To think otherwise, it is asserted, would be to leave the success of
these theories an unexplained miracle. If spacetime does not really
have the geometrical structure attributed to it by general relativity,
then how can we explain the theory's success?



Appealing as this view is, the hole argument shows that some limits
must be placed on our literal reading of a successful theory. Or at
least that persistence in such literal readings comes with a high
price. The hole argument shows us that we might want to admit that
there is something a little less really there than the literal reading
says, lest we be forced to posit physically real properties that
transcend both observation and the determining power of our
theory.


### 10.2 The Hole Argument and the Quantization of Gravity



One of the most tenacious problems in modern theoretical physics is
the quantization of gravity. While Einstein's 1915 general
theory of relativity produced a revolutionary new way of thinking of
gravitation in terms of the curvature of spacetime, it is generally
agreed now that it cannot be the final account of gravity. The reason
is that it is still a classical theory. It does not treat matter in
accord with the quantum theory.



The problem of bringing quantum theory and general relativity together
in a single theory remains unsolved. (See
 Quantum Gravity.)
 There are many contenders, notably string theory and loop quantum
gravity. One of the issues that has been raised is that the hole
argument has shown us (so goes the claim) that no successful theory of
quantum gravity can be set against an independent, container
spacetime. John Stachel was an early proponent of this outcome of the
hole argument. See Stachel (2006). This issue has often been raised by
loop quantum gravity theorists specifically as a criticism of string
theoretic approaches, for string theoretic approaches have such a
background spacetime. See Rovelli and Gaul (2000) and Smolin (2006);
for philosophical engagement with these issues of "background
independence", see Pooley (2017) and Read (2023).



In a related development, Gryb and Thebault (2016) have argued
that the problem of the hole argument and the "problem of
time" of quantum gravity are essentially the same, given
suitable assumptions. For more, see
 Problem of Time
 in the article on quantum gravity.


### 10.3 The Hole Argument and Gauge Freedoms



The hole argument has played a role in the growing recognition in
philosophy of physics of the importance of gauge transformations:
transformations acting on certain degrees of freedom in our physical
theories which are supposed to have no correlates in physical reality.
The analysis of the hole argument provides philosophers of physics
with a convenient template when they are trying to decide whether or
not something is a gauge freedom. As mentioned in the introduction, we
learn from the hole argument that the identification of surplus
mathematical structure cannot be achieved by any a priori or purely
mathematical rule (at least on the assumption that the
mathematical/formalist responses to the hole argument discussed above
do not succeed). Rather, some physical grounds are needed. The hole
argument provides two grounds that can be used: (i)
verifiability--changes in the candidate surplus structure make no
difference to what can be verified in observation; and (ii)
determinism--the laws of the theory are unable to fix the
candidate surplus structure. For more detailed discussion on the hole
argument and gauge freedoms, see the supplement
 The Hole Argument as a Template for Analyzing Gauge Freedoms.