Presentism and black holes


Euro Jnl Phil Sci (2017) 7:1–15
DOI 10.1007/s13194-015-0135-8

ORIGINAL PAPER IN PHILOSOPHY OF PHYSICS

Presentism and black holes

Geurt Sengers1

Received: 24 April 2015 / Accepted: 18 December 2015 / Published online: 23 April 2016
© The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract In a recent publication in the European Journal for Philosophy of Sci-
ence (Romero and Pérez, European Journal for Philosophy of Science, 4, 293–308,
2014), Romero and Pérez claim to reveal new trouble for the already difficult life
of presentism in relativistic spacetimes. Their argument purports to demonstrate the
impossibility of postulating a viable present in the presence of black holes, in par-
ticular the Schwarzschild geometries. I argue that their argument is flawed, and that
the Schwarzschild geometries they consider offer no novel threats to presentism.
However, if we consider more general black holes, different and more potent threats
may be lurking in the dark. I discuss these threats and sketch how a presentist may
respond, thus painting a more balanced picture of the way the existence of black
holes might impact presentist metaphysics. At present there is no decisive verdict
from black holes, but it turns out a close scrutiny of the developments in the relevant
physics will be needed to see whether their import will remain so inconclusive.

Keywords Presentism · General relativity · Black holes · Philosophy of time

1 Introduction

Presentism may still be the metaphysics presumed by common-sense. In the philo-
sophical literature however, it leads a troubled life. The thesis can be understood as
the claim that only events in the present exist, while those in future and past do not.

� Geurt Sengers
geurtsengers@gmail.com

1 Faculty of Philosophy, Erasmus University Rotterdam, Burg. Oudlaan 50, Room H5-09, 3062
PA Rotterdam, The Netherlands

http://crossmark.crossref.org/dialog/?doi=10.1007/s13194-015-0135-8&domain=pdf
mailto:geurtsengers@gmail.com


2 Euro Jnl Phil Sci (2017) 7:1–15

The formulation is subject to many subtleties that I will not go into here (see Crisp
2003 or Bourne 2006 for proposals). I will focus here only on a requirement that the
more robust and traditional of such formulations share: that of a global, space-like
foliation of our spacetime, with each slice representing the world at a moment in time.

To many minds, special relativity alone decisively refutes presentism already.
Unattractive as it may be, however, it remains possible to consistently add a present
to special relativity (see for instance Rakic 1997). So for the purposes of this paper,
I will ask to suspend disbelief and focus only on some further problems that general
relativity brings.

That not all spacetimes of general relativity can harbour such a present has been
known at least since Gödel (1949) offered a solution of Einstein’s field equations
that is not everywhere so foliable. But, since it was clear that these solutions do
not describe the actual universe, the troubles were at least a modal step removed
(Earman 1995; Dorato 2002), and the presentist might still find a sense of safety in
that distance.

However, there are other problematic solutions in general relativity that promise
to bring similar trouble closer to actuality. Candidate troublemakers are the solutions
thought to describe black holes—by now uncontroversially part of the viable models
of our actual universe. A recent paper by Romero and Pérez, ’Presentism meets Black
Holes’, argues that there can be no viable presentism in such geometries. Romero
and Pérez should be praised for putting black holes on the agenda in the presentism
debate, but I will argue that their argument is flawed.

In brief, Romero and Pérez argue that at the horizon of a black hole, the only viable
choice for a present is its (light-like) horizon itself. Since different events on this hori-
zon are co-present with temporally separated distant events, we run into a problem:
we must now say that temporally separated events are each co-present with events on
the horizon which are mutually co-present. Then, by transitivity of co-presentness,
the temporally separated events must also be co-present. This is incompatible with
presentism.

First I will show how their argument fails. However, this does not mean presentism
is safe. I will dive deeper into black holes, encounter further trouble, and discuss
how a presentist might respond. In the exposition that follows, I will limit my focus
to the novel problems presentism encounters in black hole geometries. This means
that, even if I do sometimes touch upon them, I will not pursue in any depth many
problematic issues for presentism in these geometries in so far as they are already
present in relativistic context more generally, since then they are not specific to the
black hole geometries considered here.

2 Trouble on the horizon?

2.1 A simple black hole

In general relativity, motion under gravity alone is described as geodesic trajecto-
ries in spacetime geometries that are dynamically determined by their matter and
energy content. These geometries are described by metrics that are solutions of the



Euro Jnl Phil Sci (2017) 7:1–15 3

Einstein field equations. Since light traces null geodesics, the metric determines the
behaviour of the lightcones familiar from special relativity: at each event there is a
future cone that encloses all events that could be influenced from that event, respect-
ing the speed of light as an upper bound on the propagation of these influences;
similarly, there is a past lightcone comprising all events that could have influenced
the event. It is between these two cones that, locally, a three-dimensional space-
like slice might be sought that could represent a present. The metric determines
how these cones fit together in a larger spacetime. As such their behaviour is deter-
mined by matter content—giving rise to gravity as we know it. In the presence of a
black hole, the lightcones tilt toward its centre. At the black hole’s horizon, the tilt
becomes such that any outgoing light-ray aligns with this horizon. Since the trajec-
tories of both light and massive objects are confined to this cone, neither light nor
massive objects can move fast enough to escape this horizon, as this would require
a speed greater than that of light. Such is it that a black hole begets its colour, or
lack thereof.1

2.2 Romero and Pérez’s arguments

Romero and Pérez offer arguments that in the presence of a Schwarzschild black
hole, the only viable option for an observer crossing the horizon is to choose
the present on the Schwarzschild horizon itself. One can see how this would
lead to trouble for presentism. According to the in-falling presentist, the whole
of the horizon should exist. Distant presentists will not agree. They will postu-
late a present that slices the horizon in a particular way. They might, for instance,
consider themselves at some time co-present with the event of the first observer
passing the horizon. But then, they will maintain part of the horizon still lies in
the nonexistent future. So we’ve run into a contradiction, and presentism takes
the fall.

But why do Romero and Pérez claim that the present is to be chosen on the hori-
zon? Their claim is initially suggested by inspecting the behaviour of the lightcones
at the Schwarzschild horizon in Schwarzschild coordinates. In these coordinates
(setting c = G = 1), the metric reads

ds
2 = −

(
1 − 2M

r

)
dt

2 +
(

1 − 2M
r

)−1
dr

2 + r 2d�2. (2.1)

We can calculate the behaviour of the lightcones relatively easily by imposing the
null condition, (ds = 0), obeyed by radial, (d� = 0), light-rays. We obtain the
following condition.

0 = ds2 = −
(

1 − 2M
r

)
dt

2 +
(

1 − 2M
r

)−1
dr

2 (2.2)

1A somewhat less simplistic sketch would attribute this to the redshift effect.



4 Euro Jnl Phil Sci (2017) 7:1–15

Fig. 1 Behaviour of the lightcones approaching the horizon in Schwarzschild coordinates

From Eq. 2.2 we can easily derive an expression for the behaviour of the slopes
dt/dr, in these coordinates, of the lightcones as we approach, and go beyond, the
horizon:

dt

dr
= ±

(
1 − 2M

r

)
(2.3)

This leads to the familiar diagram in Fig. 1.
In this diagram, we see that the lightcones from the outside in thin out (right), and

from the inside out flatten out (left), both leading to a seeming alignment of both the
in and outgoing light-rays with the horizon. Now Romero and Pérez take this to imply
that any present in between these lightcones must be squashed onto the horizon, as
indicated in their Fig. 2.

The figure is suggestive, but it is not all Romero and Pérez offer in support of their
claim. Let me quote their arguments:

There is not much to do with [the claim that all events on the Schwarzschild
horizon must be simultaneous] since it follows from the condition that defines
the null surface: ds = 0;2 [...]. A presentist might refuse to identify ’the present’
with a null surface. After all, in Minkowskian space-time or even in a globally
time-orientable pseudo-Riemannian space-time the present is usually taken as
the hyperplane perpendicular to the local time [...]. But in space-times with
black holes, the horizon is not only a null surface, it is also a surface locally
normal to the time direction. This can be appreciated in Fig. 2, where the angle
θ is the angle between the null surface and the hyperplane of the present. In
Minkowskian space-time such an angle is 45 deg when the speed of light is
measured in natural units (c = 1). In such a space-time, certainly the plane of
the present is not coincident with a null surface. However, close to the event
horizon of a black hole, things change, as indicated by Eq. 2.3. As we approach
the horizon, θ goes to zero, i.e. the null surface matches the plane of the present.
On the horizon, both surfaces are exactly coincident: θ → 0 when r → rSchw.
A presentist rejecting the identification of the present with a closed null surface
on an event horizon should abandon what is perhaps her most cherished belief:
the identification of ’the present’ with hypersurfaces that are normal to a local
time direction.

2The authors add a footnote: “ds = cdτ = 0 → dτ = 0, where dτ is the proper temporal separation.”
The authors do not specify the proper time separation between what events; but the separation is only zero
for events both on the lightcone.



Euro Jnl Phil Sci (2017) 7:1–15 5

Fig. 2 Figure provided in Romero and Pérez (2014) p.304. Original caption: “Light cones aperture angles
at different distances from the horizon of a Schwarzschild black hole. On the horizon the null surface is
coincident with the hyperplane of present.”

So we can discern two distinct arguments. First, the claim that the lightcones de
facto open up onto the horizon upon approach, taking the present with them. Second,
the claim that the choice of the present on the horizon is enforced by the requirement
of the present’s orthogonality to the time direction.

Consider the first argument. First observe that much is made of the behaviour
of lightcones in, specifically, Schwarzschild coordinates and their behaviour at the
horizon. Let us remember that expression (2.1) for the Schwarzschild metric looks
different in different coordinates. This version of the metric becomes degenerate at
the horizon r = 2M, as limr→2M grr is infinite, where grr = (1 − 2M/r)−1 in
Eq. 2.1. The degeneracy, however, is long known to be a mere coordinate artefact.
Sometimes it is referred to as a coordinate-singularity, distinguished from real singu-
larities by the fact that they are removable by transforming to different coordinates.
The nature of real singularities in general relativity is still under much discussion,
but the nature of the Schwarzschild singularity is not. An in-falling observer would
locally note nothing strange as he passes the horizon—except ever increasing tidal
forces. When using Schwarzschild coordinates, care must be taken not to draw onto-
logical conclusions about space-time that derive from the singular behaviour of
coordinates at the horizon.

Now, what is immediately suspicious about the argument is the basic fact of
general relativity that we can always move to local inertial frames in which the
metric takes the standard Minkowskian form familiar from special relativity. But
in these coordinates there remains, of course, plenty of wriggling room in between
the lightcones. It is true that for an infalling presentist, the outgoing lightray would
lie along the horizon (this is what makes it a horizon; nothing may move faster
than this ray). But there is also an infalling lightray that does not, and there is
simply a region of spacetime in between them. An in-falling presentist would
then have no trouble finding a local spatial present at the horizon. In fact, this
is exhibited by any coordinate system that remains regular at the horizon; in the
oft-used Eddington-Finkelstein coordinates for instance, the behaviour would look
like Fig. 3.

We must take good care to avoid drawing conclusions that depend upon
Schwarzschild coordinates, and extrapolation of their use into a region where they
are known to become degenerate, namely the horizon. For instance, while it is true to
say that dt/dr approaches ±∞ as r approaches the horizon, it does not makes sense



6 Euro Jnl Phil Sci (2017) 7:1–15

Fig. 3 Behaviour of the lightcones approaching at the horizon in Eddington-Finkelstein coordinates

to use expression (2.2) at the horizon itself, and claim that both ingoing and outgoing
light-rays, and hence the lightcones, align with it. The fact that the Schwarzschild-t
coordinate tends to infinity for any object passing the horizon, including that of ingo-
ing and outgoing light-rays, makes it hard to see that in fact ingoing and outgoing
rays remain distinct there.

The above considerations show that the angle between the lightcones simply is
not a coordinate invariant—the expression for the slope is manifestly coordinate
dependent. Romero and Peréz do argue correctly that the spatiotemporal character
(a distance’s or surface’s being time-, space-, or light-like) is a coordinate invariant
feature. But the invariant light-likeness of the horizon does not suffice for the invari-
ance of the angle. So nor does it suffice for the alignment of both (past and future)
lightcones with the horizon. In Minkowski spacetime the cones are also null, but they
certainly do not coincide.3

Now consider the second argument, that the choice of the present on the horizon
is the only one satisfying a requirement of orthogonality to a local time direction. It
is hard to make out what can be meant by “the” time direction since this of course
differs per frame. One might suggest the temporal axis to be that of Schwarzschild
coordinate time; after all this is one direction orthogonal to the horizon (null surfaces
have the curious property that their normals are also tangent to them). But, being
a light-like direction, no (massive) observer can identify the temporal direction of
his restframe with this direction. A more plausible reading would be that, since the
plane orthogonal to any local time direction lies in between the cones, it seems to be
squashed onto the horizon (as suggested in Fig. 2) no matter what temporal direction
we choose. But there is an easy way to see that this second reading cannot work. The
light-like character of a surface is defined by its normal being everywhere light-like.
The space-like character of a surface, in turn, is defined by its normal being every-
where time-like. Hence, there cannot be a time-like direction normal to a light-like

3If one needs conviction, simply consider that if the rays were to coincide, being light-like their inner
product would be zero. But this inner product is a coordinate invariant, and it is easily seen not to be zero
in a local inertial frame.



Euro Jnl Phil Sci (2017) 7:1–15 7

horizon—the surface would have to be both light-like and space-like. And indeed,
if we take proper heed of the non-degeneracy of the lightcones at the horizon as
emphasised above, we can see that there is ample room for a spatial plane locally
orthogonal to any temporal direction there—the situation is locally no different from
the Minkowskian one.

Finally, let me digress briefly on this orthogonality requirement. I argued here only
that local orthogonality to any single temporal direction can be had at the horizon.
Of course, if we allow for crossing observers, no one hypersurface will be orthogo-
nal to both of their worldlines at the event of crossing. And even if we do not allow
for crossing observers, it is generally impossible to foliate a congruence of world-
lines in an everywhere orthogonal way if rotation is present. (This is a special case
of Froebenius’ theorem; Malament (1995) explains this intuitively as the impossi-
bility of slicing a rope in a way orthogonal to every fiber.) So even independent of
considerations from black holes, the presentist might want to rethink the desideratum
of worldline-orthogonality. Gödel found his infamous universe searching for solu-
tions that exhibit this failure, where the rotation is that of the heavy bodies. Usually
when the orthogonality condition is used to define a cosmic time, it is the heavy bod-
ies the slicing is sought orthogonal to, not just any congruence of possible observer
trajectories.

2.3 How to slice a black hole

If the above shows that there one can unproblematically find a present locally at
the horizon, we may still wonder whether the same holds globally. Recall that
we are investigating a traditional presentism here that demands a global spatial
present. In fact, the Eddington-Finkelstein system mentioned above is everywhere
regular and allows for a global spatial foliation. But the possibility for such a
foliation is easier to see in yet another well-known coordinate system adopted to
the Schwarzschild geometry: Kruskal-Szekeres coordinates. Let us show by con-
struction that a global spatial slicing of a Schwarzschild black hole is indeed
possible.

When we inspect the Schwarzschild metric in Kruskal-Szekeres coordinates we
obtain the following:

ds
2 = 32M

3

r
e−r/2M

(
−dv2 + du2

)
+ r 2d�. (2.4)

The Schwarzschild geometry is depicted in terms of these coordinates as in Fig. 4.
From Eq. 2.4 we immediately see that for radial directions (d� = 0), setting dv =

±du yields ds = 0. So a great benefit of these coordinates is that radial, light-like
geodesics are always 45 degree lines in two-dimensional Kruskal-Szekeres diagrams
(see Fig. 4). It can also be seen that we can foliate the whole of this geometry in a
way such that every hyper-surface lies everywhere (including at the horizon) between
the local future and past lightcones.

For instance, we may choose a v = constant slicing. Inspecting the metric, we
see that the induced metric on these hyper-surfaces (dv = 0) is everywhere positive.
The slicing is therefore space-like. A presentist postulating this slicing will avoid



8 Euro Jnl Phil Sci (2017) 7:1–15

Fig. 4 The Schwarzschild geometry in Kruskal-Szekeres coordinates

the troubles that Romero and Peréz imagine. The choice is by no means unique:
any slicing that is everywhere less than 45 degrees with respect to the u-axis in the
diagram will also be everywhere space-like.

The conclusion for the Schwarzschild case is that a presentist here is indeed free
to postulate a foliation in spacelike hypersurfaces such as his theory requires. In
fact, there are many distinct foliations, equally suitable. This does lead to a certain
embarrassment of riches: on what basis does the presentist choose one? Indeed there
remains an air of arbitrariness, even ad hocness, attached to the postulation of a foli-
ation where the physical theory requires none, more so because it requires breaking
local relativistic symmetries by introducing a preferred spatial slicing. It could be
argued then, that the presentist has work to do in the justification of the slicing of
choice.4

3 Trouble beyond the horizon?

If the Schwarzschild metric is indeed, as argued above, compatible with a space-like
foliation, does that mean that black holes pose no novel threat to presentism? That
conclusion would be too quick. The Schwarzschild case is only the simplest of a

4Sometimes surfaces of constant mean curvature are considered (CMC) in the context of presentism, see
Monton (2006) and Wüthrich (2010). I will not discuss these proposals because they are not specific to
black holes, but note that CMC-foliations of Schwarzschild geometries seem available, see Brill et al.
(1980). I thank an anonymous referee for pointing this out to me.



Euro Jnl Phil Sci (2017) 7:1–15 9

range of geometries that can harbour some uncomfortable surprises. In the present
section, I briefly sketch some of these surprises, bring out the issues they cause for
presentism, and, leaning heavily on the authority of others, outline the way these
issues are evaluated in current physics. Doing so will paint a more balanced view of
the way the existence of black holes might impact presentist metaphysics.

3.1 Rotating black holes: the Kerr geometry

The Schwarzschild metric (2.1) can be seen as a limiting case of a more general
geometry describing rotating black holes. The vacuum solution, known as the Kerr
metric, was the result of a search for metrics that are not orthogonally foliable (Carter
1968), so we might expect a conflict with a desideratum of a present orthogonal to
time. But the geometry turned out to contain a few more surprising features, which
indeed pose a more direct threat to presentism. These features are relatively easily
exhibited.

The Kerr metric in Boyer-Lindquist coordinates reads:

ds
2 =

(
−1 + 2Mr

ρ2

)
dt

2 + ρ
2

�
dr

2 + ρ2dθ 2

+
(

r
2 + a2 + 2Mra

2 sin2 θ

ρ2

)
sin2 θ dϕ2 − 4Mra sin

2 θ

ρ2
dtdϕ. (3.1)

with
ρ

2 = r 2 + a2 cos2 θ (3.2)
and

� = r 2 − 2Mr + a2. (3.3)
Here a can be interpreted as the angular momentum of the black hole and M

as its mass. For a = 0, this expression reduces to the Schwarzschild metric. The
Kerr geometry is a strange beast. The exterior is used successfully in the modelling
of astrophysical black holes, and is thought to describe their exterior final state
uniquely.5 But the interior can become quite extravagant, as we will see below.

At first glance, the Boyer-Lindquist coordinates look somewhat like standard
spherical coordinates plus a time direction on R4. But looks can be deceiving. First
note that the metric expression diverges in two cases: when � = 0, and when ρ = 0.
If first we look at expression (3.3) for �, we see that it equals zero at two values for
r.6 This turns out to give us not one, but two horizons. Like in the Schwarzschild

5This expectation derives from the famous “no-hair theorems”. Note that the theorems are specific to the
matter content considered, and there is no proof of a completely general uniqueness conjecture (Chrusciel
et al. 2012). Note also that avoiding these theorems—say by modifications of general relativity, or by
relaxing the stationarity assumption—may not solve the problems for the presentist discussed below, as
closed time-like curves may still occur (Johannsen 2013).
6At least when a < M, or “slow Kerr”, arguably the case of most physical interest (Carroll 2004). Fast
Kerr, with a > M, is often considered unphysical because it contains naked singularities—famously
conjectured away by Penrose. The problems for presentism would be similar to the ones discussed below
for slow Kerr, only worse: there are no horizons and all of spacetime would be causally vicious.



10 Euro Jnl Phil Sci (2017) 7:1–15

case, these degeneracies can be removed by transforming to other coordinates, and
the metric can be extended to cover these horizons.

By contrast, the case ρ = 0 gives us a true singularity. If we inspect expression
(3.2) for ρ, we see that r = 0 does not suffice to have ρ = 0: the metric only fails
at r = 0 when also θ = π/2. Suppose we were to stick to the interpretation of the
Boyer-Lindquist coordinates as spherical coordinates on R4. Then, if geodesics were
to approach r = 0 from an angle other than θ = π/2, the curvature on them would not
blow up; they would in fact be extendible. This shows the untenability of interpreting
metric (3.1) as describing a geometry on R4 in standard spherical coordinates. In that
interpretation, we would not have covered the whole manifold; indeed the manifold
would be extendible. In fact r = 0 does not describe a point but a disk, as r = 0 does
in ellipsoidal coordinates (one may write Minkowski spacetime in such coordinates;
see Wald 1984). The metric fails only at the edge of the disk, for r = 0 and θ = π/2.
Since the metric is not singular at the inside of the disk, we may extend geodesics
through it.

A further complication arises because the metric fails to be smooth across the
coordinate disk enclosed by this ring. A smooth extension across this disk can be
achieved by glueing two copies of R4 with the disk removed together, identifying the
top of the removed disk of the one with the bottom of the removed disk of the other.
Details of this extension can be found in Hawking and Ellis (1973). This way we can
allow r to take negative values, indeed range from −∞ to +∞. Whereas this would
not make sense in the Minkowskian case, here we have a different manifold that
can be meaningfully charted with a negative radius. We have found, then, a solution
that lives on a manifold topologically quite different from, say, the Schwarzschild or
Minkowskian case. One way to visualise this manifold is as in Fig. 5.

3.2 Closed timelike curves

But in these extended regions, an immediate problem for presentism is found. If
we inspect the metric, we find that the longitudal direction ∂ϕ can become timelike
in regions with small negative r. This can be seen relatively easily. The char-
acter of ∂ϕ is determined by gϕϕ : it is timelike iff gϕϕ < 0. The expression
for gϕϕ ,

gϕϕ =
(

r
2 + a2 + 2Mra

2 sin2 θ

ρ2

)
sin2 θ, (3.4)

reduces on the equatorial plane
(
θ = 12 π

)
to

gϕϕ = r 2 + a2 +
2Ma2

r
. (3.5)

This is negative for small negative r. The direction, then, is time-like there. But
integral curves for the direction ∂ϕ are closed curves circling the axis of rotation. For
small negative r, then, these are closed time-like curves. The situation is often visu-
alised as the lightcones tilting over in direction of rotation, eventually sufficiently so



Euro Jnl Phil Sci (2017) 7:1–15 11

Fig. 5 Visualisation of the Kerr interior; adapted from O’Neill (2014). The r = 0 disk here is the surface
of a sphere, with each side of the disk a hemisphere. The extension through the disk is included as a
negative-r sphere, with r drawn exponentially. So, r = −∞ lies at the centre, and the region is as large as
the positive r region

to include this direction in their futures (but see Andréka et al. (2008) for qualifica-
tion of this picture). Following the direction ∂ϕ around, a curve can enter into its past
once again.

Note that these curves are not “unrollable”, that is, they cannot be removed by
going to a larger covering space. By contrast, we could obtain closed time-like curves
by “rolling up” Minkowski spacetime along a space-like direction. It may be argued
that presentism could live with such closed time-like curves—we would obtain a
cyclic time that could still parametrise a space-like foliation hospitable to presentism.
Any such curve intersects the space-like hypersurfaces only once, and, as Monton
(2003) argues, this could be interpreted as an eternal-recurrence scenario not neces-
sarily in conflict with presentism. Technically, the saving grace is that these more
benign curves are not contractible to a point (not homotopic to zero). But those in
the Kerr case are. Carter (1968) considers the options one would have for prevent-
ing this contractibility. One might want to remove the ring-singularity from the space
to obtain a space that is not simply connected, and allows curves around the ring-
singularity that are not contractible. But the closed time-like curves in our geometry
do not wind around the ring-singularity, so this would be of no help. To prevent the



12 Euro Jnl Phil Sci (2017) 7:1–15

example curve from being contractible, one would have to remove (for instance) the
axis—but there is no physical reason to do so.7

Perhaps it is intuitively obvious that such closed time-like curves are hard to swal-
low for the presentist, but one can make the trouble explicit in the following way.
Start from the observation (Carter 1968) that every contractible closed time-like curve
must cross a global space-like surface an even number of times. We can parametrise
the presentist’s foliation by such surfaces with a global time coordinate T . We can
also parametrise the closed time-like curve with a parameter λ. Now if at one cross-
ing point, the curve will cross in positive time direction (dT /dλ > 0), there will be
another where it will cross in negative global time direction (dT /dλ < 0).

If this—processes along the curve will evolve in reverse global time—does not
spell enough trouble, consider this. If the curve is sufficiently smooth (as our ear-
lier circular example curve is), then somewhere between the crossing points, dT /dλ
will have to go through zero to change sign. That means that at some point, global
time T will not increase along the curve. But that means that its tangent there is
space-like: it means that the tangent there lies in a surface assumed to be space-like.
But by assumption the tangent was time-like (it is a time-like curve). So we have a
contradiction.

When we look back at the visualisation in Fig. 5, we now see that it cannot be
understood as representing a spatial slice at a moment in time: some of the distances
in this figure are time-like. And since there will be such time-like distances in at least
some slices of any foliation of the Kerr geometry, this geometry cannot accommodate
a presentist metaphysics that postulates an everywhere space-like foliation.

3.3 Threat evaluation for presentists

How problematic for presentism the presence of closed time-like curves in a solution
is, depends of course on how seriously we should take that solution. The causality
violating region in Kerr spacetime lies in an extension through the singularity to neg-
ative r values.8 This is a region as large as the positive r region, indeed a parallel
universe, with peculiar features such as a reversal of the metric’s gravitational field
(the region inside the inner horizon is sometimes referred to as a “white hole” region).
And indeed it is hard to imagine moving from a manifold with Minkowski-like J −
(distant past) in which a star begins its collapse, to a topologically different one—one
might have to postulate a pre-established harmony of coalescing parallel universes9

(for speculation on a different, wormhole-like scenario, see Charlton and Clarke
1990). The maximal extension of Kerr spacetime becomes stranger still. The horizons
divide the spacetime in so-called Boyer-Lindquist blocks, and by a copy-and-paste
process we can assemble these into an endless patchwork of—traversable—parallel

7Furthermore, Carter argues that removing the axis would not even cure the spacetime of all closed time-
like curves: we could still obtain small such curves that briefly dip in and out of the time-machine region–
“thus making up for lost time”–without circling around the axis.
8However, for charged black holes, described by the family of Kerr-Newman metrics, Carter (1968)
mentions that the problematic region may spill over into positive r.
9See Misner et al. (1973) for visualisations of the analogous case in the maximal Schwarzschild extension.



Euro Jnl Phil Sci (2017) 7:1–15 13

universes that together form the maximally extended Kerr spacetime (see O’Neill
2014 for exposition).

It is not surprising then, that many authoritative voices echo the sentiment that
somewhere in these regions, the Kerr metric must depart from actuality (see Misner
et al. (1973) p883, Carroll (2004) p266, O’Neill (2014) p103, Wald (1984) p317).
Perhaps the presentist may find strength in such allies, operating on a methodological
principle that metaphysics need only conform to actual physics. Discussion of such
principles are familiar from the context of Gödelian universes (Dorato 2002), so I
will here focus on the physical arguments against actuality in this case. Some of
the reasons for not taking the problematic region as a realistic description of the
actual universe are the following. For starters, wherever matter fields are present,
the solution will depart from the given vacuum solution. The Schwarzschild case
is also maximally extendible into a spacetime containing a white hole and a non-
traversable wormhole connecting to a parallel universe. These regions, however, are
usually not considered realistic because they will be replaced by the matter of the
collapsing star as it turns into a black hole. The Kerr case is more difficult. It is not
spherically symmetrical, and only in the asymptotic future will the geometry settle
down to a stationary state. Whereas there are arguments to the effect that the exterior
solution will eventually resemble Kerr,10 these do not hold for the interior. Wald
compares the case tentatively to the spherically symmetric case of a charged non-
rotating black hole (Reissner-Nordström geometry). In that case, there remains a non-
negligible possibility that for certain choices of parameters, the collapsing matter will
leave some of the problematic regions uncovered. But there are reasons to suspect
that physical fields will diverge at the inner horizon which will produce singular
behaviour there preventing such further evolution. Similar phenomena are thought to
occur in a realistic Kerr-like collapse.11

There are more general reasons to object to the development of closed time-like
curves than merely a taste for presentism. Because such acausalities bring with them
a host of problems not just for the traditional metaphysician, but also for the work-
ing physicist, their elimination has also long been on that physicist’s wish-list. These
issues range from the existence and nature of consistency constraints on the fields
living on such curves (see Earman 1995) to energy conditions thought to be violated
in their development (see Smeenk and Wüthrich 2011). There is an important con-
nection to ongoing research on singularities in general relativity (and beyond, see
Earman and Wüthrich 2013). If closed time-like curves do develop in a region of

10See footnote 5.
11Sometimes the existence of a Cauchy horizon, necessarily associated with the development of acausal-
ities, is cited as a ground for not taking the causality violating region seriously within general relativity
(Visser 2003). While it is off course a nontrivial issue how to even formulate the dependence of the acausal
region beyond a Cauchy horizon on the causal region before it, the problem does not seem to be insur-
mountable, and it is too quick to write off the existence of the extensions on these grounds (See Smeenk
and Wüthrich 2011 for discussion). Penrose, however, in 1968 went further and first argued for an infi-
nite blueshift at the inner horizon preventing the development of the acausal region. See Israel (2003) for
an informal account; the exact nature of the behaviour at the Cauchy horizon is still under debate (see for
example Dafermos 2008).



14 Euro Jnl Phil Sci (2017) 7:1–15

spacetime that is otherwise causal, the spacetime will have a Cauchy horizon: a hori-
zon beyond which prediction from initial data as specified on an earlier hyper-surface
fails. Since such failure is also a feature of spacetimes with singularities, certain argu-
ments against singularities (strong cosmic censorship, as opposed to weak censorship
of naked singularities, which would only hide the pathologies behind a horizon)
would also work against closed time-like curves (see Smeenk and Wüthrich 2011).
Other theorems purport to show that any spacetime has a maximal extension without
closed time-like curves in the future, thus blocking the necessity of their development
(the theorem is Krasnikov’s, and is contested by Manchak, as discussed in Smeenk
and Wüthrich 2011). Most directly, Hawking’s 1992 chronology protection conjec-
ture argues precisely against the existence of Cauchy horizons, and hence closed
timelike curves.12 If successfully developed into a theorem, this conjecture would
make the universe a safer place not only for historians, but also for presentists. Thus
far, none of these attempts are thought to be entirely successful (see Earman et al.
2009). But the presentist might draw courage from the fact that his fight is part of a
larger battle where he may find forces to join against a common enemy.

4 Conclusion

Reconciliations between presentism and modern spacetime physics, even if possible
in the letter, often smack contrary to the relativistic spirit. It seems unlikely that rel-
ativity and presentism will ever enter into a happy marriage.13 However, we have
seen above that the case for a definitive split on the basis of the encounter with black
holes is inconclusive. First we have seen that, contrary to arguments by Romero and
Pérez, Schwarzschild black holes pose no new independent threat to presentism. The
case of rotating black holes proved more intricate. There may be insurmountable
trouble lurking in their interior, but we have seen that it is as yet hard to evaluate its
eventual metaphysical impact, precisely because the physicist’s verdict is still out.
Strange things happen in black holes, and much of it is inextricably connected to the
future developments of physics.14 We have seen that the fate of presentism is likewise
intimately connected to these developments. It is, then, too soon to write off presen-
tism as a viable metaphysical candidate, but a close scrutiny of developments on the
frontiers of physics will be needed to see whether it will remain a live possibility.

Acknowledgments Let me thank all in the project ‘The Structure of Reality and the Reality of Struc-
ture’, Gijs Leegwater, Stefan Wintein and F.A. Muller, for helpful discussion, and the Netherlands
Organisation for Scientific Research (NWO) for funding that project. I also thank two anonymous referees
for useful comments.

12See Visser (2003) for an informal review of arguments leading to the conviction that a full understanding
of chronology horizons will require a quantum theory of gravity; see also Earman and Wüthrich (2013).
13There have been arguments for Monton (2006) and against Wüthrich (2010) the hope of finding a more
hospitable environment for presentism in theories of quantum gravity. While it is true that there are pro-
grams employing a foliation (see for instance Hořava 2009), it is hard to place any reliable bet in this
field.
14Of course the black hole physics I have discussed here is relatively old-hat: speculation of firewalls and
fuzzballs closer to the frontiers may, when more firmly established, require us to rethink my evaluation in
unforeseeable ways.



Euro Jnl Phil Sci (2017) 7:1–15 15

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http://creativecommons.org/licenses/by/4.0/
http://arxiv.org/abs/gr-qc/0401121v1
http://arxiv.org/abs/gr-qc/1304.7786v3

	Presentism and black holes
	Abstract
	Introduction
	Trouble on the horizon?
	A simple black hole
	Romero and Pérez's arguments
	How to slice a black hole

	Trouble beyond the horizon?
	Rotating black holes: the Kerr geometry
	Closed timelike curves
	Threat evaluation for presentists

	Conclusion
	Acknowledgments
	Open Access
	References