Two Views on Time Reversal∗

(Philosophy of Science 75 (2008): 201-223)

Jill North†

Abstract

In a recent paper, Malament (2004) employs a time reversal
transformation that differs from the standard one, without explicitly
arguing for it. This is a new and important understanding of time
reversal that deserves arguing for in its own right. I argue that it
improves upon the standard one. Recent discussion has focused
on whether velocities should undergo a time reversal operation. I
address a prior question: What is the proper notion of time reversal?
This is important, for it will affect our conclusion as to whether
our best theories are time-reversal symmetric, and hence whether
our spacetime is temporally oriented.

1. Introduction

In a recent paper, David Malament (2004) employs a time reversal trans-
formation that differs from the standard one. Aside from noting the
naturalness and general applicability of this transformation, Malament
does not offer explicit argument for rejecting the standard account. This is
because his purpose is to argue against David Albert’s view (2000, Chapter
1) view that classical electromagnetism is not time reversal invariant.
∗For comments and discussion, I am indebted to Frank Arntzenius, Hartry Field,

Stephen Leeds, David Malament, Ted Sider, students in my seminars at NYU and
Yale, audience members at the Paci�c APA in 2006, and an anonymous referee for this
journal.

†Department of Philosophy, Yale University: jill.north@yale.edu.

1



I think Malament has been too modest. He has proposed a new
and important understanding of time reversal that deserves arguing for
in its own right. In this paper, I argue that Malament’s time reversal
transformation improves upon the standard one.

Recent discussion of time reversal has focused on whether velocities
should undergo a time reversal operation. I address a prior question:
What is the proper notion of time reversal? The answer to this question
is important, for it will affect our conclusion as to whether our best
theories are time reversal symmetric, and hence whether the spacetime
of our world is temporally oriented.1

2. Why does it matter?

What I call the standard transformation is the one that is generally as-
sumed by physicists and philosophers. I will need to motivate my lumping
together under the heading ‘the standard view’ such disparate positions as
those of the physics books and of philosophers like Albert (2000, Chapter
1) and Paul Horwich (1987, Chapter 3), who argue against much of what
the books say. I will do so in a moment.

First let us ask the obvious question: why should we care what the
proper time reversal transformation is? Why not assume the one the
books do, and leave it at that? Forget the details of this or that particular
operation. De�ne our time reversal transformation, and a corresponding
sense of time reversal symmetry, any way we like.

The reason we cannot be so nonchalant is that there is something
important we want the transformation to do for us. We want it to tell us
about the spacetime structure of our world.

In applying any transformation to a theory, we hope to learn about
the symmetry of the theory, and of the world that theory describes. We
do this by comparing the theory with what happens to it after the trans-
formation. If the theory remains the same after the transformation—if it
is invariant under the transformation—then it is symmetric under that
operation. This indicates that the theory “says the same thing” regardless

1I limit discussion to classical theories. Though the general conclusions remain, the
proper time reversal transformation could change when we take into account quantum
�eld theory (Arntzenius, 2004, 2005; Arntzenius and Greaves, 2007).

2



of how processes are oriented with respect to the structure underlying the
transformation. We conclude that a world described by the theory lacks
the structure that would be needed to support an asymmetry under the
operation. For example, from the space-translation invariance of the laws,
we infer that space is homogeneous, that there is no preferred location in
space.

As for transformations and their related symmetries generally, so
too for the time reversal transformation and time reversal symmetry
speci�cally. We are not interested in the time reversal transformation per
se. What we are ultimately after is the nature of time itself. We want to
know whether there is an objective distinction between past and future: a
temporal orientation on the spacetime manifold.2

We cannot directly observe whether spacetime has this structure. Nor
do the phenomena suf�ce to tell us this. Things are asymmetrically
distributed in space, but we do not conclude from this alone that space
itself is asymmetric, possessing structure picking out a preferred spatial
direction. Similarly, there may be asymmetries in the way things are laid
out in time, but this alone does not indicate that time itself is asymmetric.3

We can learn about the structure of time a different way: by means
of the fundamental dynamical laws. For the following inference seems
reasonable: Non-time-reversal-invariant laws would give us reason to
believe that spacetime is temporally oriented.

The principle behind the inference is this. If the fundamental laws
cannot be formulated without reference to a particular kind of structure,
then this structure must exist in order to support the laws—“support” in
the sense that the laws could not be formulated without making reference

2A temporal orientation τa is an everywhere continuous timelike vector �eld on the
(temporally orientable) spacetime manifold that contains only vectors which lie in one
lobe of the light cones. τ determines a temporal orientation on the manifold: it picks
out, at each spacetime point, which direction is to the future of that point, and which is
to the past.

This is not to say the question is whether there is an objective fact as to which direction
is past and which is future. The question is whether there is a structural difference
between the two directions, regardless of which we call or experience as ‘past’ and which
‘future’.

3As Huw Price puts it (1996, Chapter 1), we are interested in the asymmetry of time,
above the asymmetry of things in time.

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to that structure. In general, we infer that spacetime has a certain structure
on the basis of corresponding (non-)invariances in the fundamental laws
of physics. Consider the inference to a lack of preferred spatial location. If
the laws were not invariant under spatial translations, then we would infer
that space is inhomogeneous. For we would need to include a preferred-
point spatial structure in giving our theory of the world. The laws could
not be formulated without it.

Similarly here. If the laws are not invariant under time reversal, then
we could not state them without presupposing a temporal orientation
on the spacetime manifold—an objective distinction between the two
temporal directions, indicating in which one things are allowed to evolve
and in which they are not. The laws themselves make reference to the
distinction. If these are the fundamental laws, then we have reason to
infer that the world has the structure needed to support the distinction. If
the laws are symmetric under time reversal, then they do not presuppose a
temporal orientation. They say the same thing regardless of the direction
things evolve in. If these are the fundamental laws, then we would not
infer a temporal orientation. In giving our theory of the world, we would
not need to include this structure in its spacetime. This inference is
not conclusive; but it is reasonable. (And it is generally endorsed: see,
e.g., Savitt (1996); Arntzenius (1995, 1997). Still, there could be highly
nonlocal laws or asymmetric boundary conditions rather than asymmetric
spacetime structure. The inference must ultimately consider our best
fundamental theory; see note 4.)

We want to know whether there is a distinction between past and
future that is independent of everything other than the structure of time
itself (independent of our experience in time, of an arbitrarily chosen
reference frame, and so on). The best way to learn about this is to look
for relevant invariances in the fundamental laws. And the time reversal
transformation is the mathematical gadget that we use to see whether the
laws have these invariances.

Thus, our time reversal transformation should be able to tell us
whether the fundamental laws are noninvariant in a way that suggests a
direction of time. This, in turn, means that the inference to a direction
of time is what should guide us in locating the proper transformation. So
this is our test: the proper time reversal transformation should be such

4



that, when we compare a theory with its time reverse, we can reasonably
infer whether spacetime has a temporal orientation.

Note that we should have in hand our best, most fundamental theory
when making this inference. There are temporally asymmetric, nonfunda-
mental theories that do not suggest an underlying structural asymmetry;
thermodynamics is an example. The idea is that, once we have in hand
what we take to be our best fundamental theory, operating on it with the
proper transformation should reveal what temporal structure is needed
to support the theory, and therefore what structure we should infer the
spacetime to have.4

4Even then, the inference will not be conclusive. An anonymous referee offered this
example. Consider two worlds, each containing a single one-dimensional object. In
one world, the fundamental law says the object doubles its length per unit time. In the
other, the fundamental law says the object halves its length per unit time. The law for
each world is time asymmetric, so that according to our guiding inference, there is a
temporal orientation in each world. But consider a different law for each of these worlds
(or rather, if this is the law, then there really is no genuine distinction between the two
worlds): the single object changes its length at a uniform rate per unit time. This law
is time symmetric, so that according to our guiding inference, there is no temporal
orientation in either world. The symmetric law seems just as much a candidate best
theory for each of these worlds; both theories save the phenomena. But then how could
the phenomena ever indicate whether the symmetric or asymmetric theory is correct,
and therefore whether time is temporally oriented—not only in simple worlds like these,
but more generally as well?

The short answer is that our guiding inference is not conclusive; but this does not
mean it is unreasonable. Our conclusion about time’s arrow will depend on what we take
to be the best overall fundamental theory. And which theory we take to be “best” will
involve the usual cost-beneÞt analyses, which may themselves be inconclusive. In the
example here, if we accept the symmetric theory, then we do away with some temporal
structure, but at the cost of a highly nonlocal theory (the theory says something like:
the object uniformly changes its length in the direction in which it had been changing
all along). Or we could chalk up the observed asymmetry to asymmetric boundary
conditions, which may or may not, depending on one’s view, be sufÞcient grounds to
infer a temporal orientation. (Many think not; Maudlin (2007) is a dissenting view.) If
we accept the asymmetric law, on the other hand, then we get rid of the nonlocality and
the positing of special boundary conditions (assuming they are not needed for other
reasons) at the expense of additional spacetime structure. Which is the best theory, all
things considered? It is not clear in this case. Still, there is reason to be optimistic that
worlds like ours will provide enough evidential considerations to pick out one of the
candidate best theories, which we can then plug into our guiding inference.

5



Summing up: What makes for a good time reversal transformation?
The proper transformation should link up in the right way with the
inference to a direction of time. We can indeed de�ne other kinds of
“time reversal” transformations. But only this test will locate the one to
use in �guring out whether our spacetime is temporally oriented. A good
transformation is hard to �nd.

3. Active and passive transformations

There are two ways of understanding any transformation—actively or pas-
sively. It will be worthwhile for what comes later to review the difference
between the two.

Geometric objects (points, vectors, tensors, and the like) exist inde-
pendently of any particular coordinate system. They take on different
coordinate-dependent representations, in terms of different components,
depending on the coordinate system being used. But the objects them-
selves are invariant under coordinate changes. Think of a vector in a
two-dimensional Euclidean plane: rotate coordinate axes, and the com-
ponents of the vector will change; the vector itself (think of its length
given by the Pythagorean theorem) will not.

There are two kinds of transformation that we can apply to theories
about these objects. An active transformation leaves the coordinate system
alone and transforms the coordinate-independent objects to alter their
coordinate-dependent descriptions. The transformation is “active” in
that it acts directly on the objects to yield their different coordinate
descriptions. A passive transformation changes the coordinates, altering
objects’ coordinate-dependent descriptions. It is “passive” in that it
leaves the objects alone, transforming the coordinate system around
them. Actively, a solution is mapped to another solution; passively, the
coordinatization is changed. Thus, actively, a vector is rotated, changing
its components. Passively, the coordinate axes are rotated (in the opposite
direction), changing the components in the same way.

For any such transformation, there is typically both a passive and
an active version: the passive transformation yields the same changes
to objects’ coordinate-dependent representations as the corresponding
active transformation does. It seems we would like this to be true of our

6



time reversal transformation as well. If a time reversal transformation,
understood actively, changes solutions to a theory into nonsolutions, then
the corresponding passive transformation should do the same. After all,
if the transformations did not agree in this way, that could spell trouble
for our guiding inference. Why should we infer a temporal orientation
structure if the passive transformation allows us to make this inference,
but the active version of the transformation does not?

Our inference can be put either way. Passively: non-time-reversal-
invariant laws do not remain the same under a change in time coordinate
from t to −t (equivalently, under an inversion of the time axis about the
temporal coordinate origin). Actively: take a solution to the theory—a
sequence or history of states that is possible according to the laws of the
theory—and apply the time reversal transformation directly to it. The
theory is not time reversal invariant if doing so transforms the solution
into a nonsolution (a process that is no longer possible). Either way, if
this is the proper time reversal transformation, then we can reasonably
infer whether spacetime is temporally oriented, according to the theory.

4. The standard transformation

The standard time reversal transformation is captured by the image of a
movie playing backward: any �lm of an allowable process on a symmetric
theory also depicts an allowable process when we run that �lm in reverse.
In other words, for any sequence of states allowed by the theory, the
temporally inverted sequence of states is also allowed. The running of a
�lm backward is the playing out of the time-reversed sequence of states.
Watch a movie of an allowable process, and if the theory is temporally
symmetric, we would not be able to tell whether the movie is playing
forward or backward.

This is an intuitive understanding of time reversal, what we initially
have in mind when we think of what it means for the direction of time to
matter or not to a given theory. If the time-reversed sequence of states
were not allowed, then the processes governed by the theory are sensitive
to the direction of time in which they occur. We would be able to tell
which way the movie is playing.

Physics texts also allow a time reversal operator to act on the instan-

7



taneous states comprising the time-reversed process. This alters the
understanding of time reversal a bit. Now we say that a theory is time
reversal invariant just in case, for any sequence of instantaneous states
. . . , S(t0), S(t1), S(t2), ... allowed by the theory, the reverse sequence
of time-reversed states, . . . , ST(t2), ST(t1), ST(t0), . . . , is also allowed,
where T is the appropriate time reversal operator (time runs left to right).5
(T inverts velocities in Newtonian mechanics, for example.6)

Not everyone agrees with this. Albert (2000, Chapter 1) and Horwich
(1987, Chapter 3) disagree with the books on what happens to objects
under time reversal. They argue that the proper time reversal transfor-
mation should convert a given sequence of states into the temporally
inverted sequence of the very same states. No time reversal operations on
the instantaneous states themselves are allowed.7

Yet all parties agree on the basic action of the time reversal transfor-
mation as a mirroring of states across a time. To get the time reverse of
a process, �ip the states that comprise the process across the temporal

5More precisely, for any history t ↦ S(t) allowed by the theory, the time-reversed
history t ↦ TS(t) is also allowed.

6T will vary from theory to theory. In Newtonian mechanics, T inverts velocities.
Think of a given Newtonian process run backward. If velocities are included in the
instantaneous states (as they standardly are, though see below), they must �ip direction
in order for the backward process to be possible (indeed, for it to be coherent). In
classical electromagnetism, T inverts B �elds. One reason to question the standard view
is its lack of a principle on the basis of which to determine the appropriate time reversal
operator, for any given theory. The worry is that the resultant notion of time reversal
will be trivial. For an argument that completely free rein in de�ning our time reversal
operators results in a trivial notion of time reversal, see Arntzenius (2004, pp. 32-33).

7Hence Albert’s unorthodox view that classical electromagnetism is not time reversal
invariant. The magnetic �eld would have to invert in the time-reversed process for the
theory to be time reversal invariant. Yet Albert only allows time reversal operations on
non-fundamental quantities, such as velocities, that are time derivatives of fundamental
ones, such as particle positions in Newtonian mechanics. Horwich treats the magnetic
�eld similarly, though he concludes that electromagnetism is time reversal invariant. His
own view suggests he should agree with Albert on this point, however: see Arntzenius
(2004, note 4). Albert and the books both say that particles and scalar �eld values get
shifted to their temporal mirror image locations, and that the directions of the velocities
get �ipped. But their justi�cation for this differs. For Albert, since velocities are de�ned
as time derivatives of positions, they will invert by virtue of taking the time-reversed
sequence of positions, which for him are fundamental quantities in classical theories.

8



origin. A theory is time reversal symmetric when this mirroring operation
always transforms a possible process into another possible process.8

In agreement with Albert and Horwich, the textbook view consid-
ers the time-reverse of a process to be the original sequence of states
(albeit acted on by T) shifted to their temporal mirror-image locations.
The difference between Albert and Horwich, on the one hand, and the
textbooks, on the other, lies in what kinds of objects get transformed
under this action—not in the basic mirroring action of the transformation
itself. Henceforth, I take the ‘standard view’ to be the transformation
that mirrors objects across a time, and consider this to encompass the
position of the textbooks and of Albert and Horwich. This will allow us
to focus on the primary innovation of Malament’s proposal.

Note that the standard transformation, as just stated, is an active
one: it shifts objects an equal temporal distance across the time origin.
Though the books focus on this understanding of the transformation,
they note a corresponding passive version. This is simply a change in time
coordinate from t to−t, or an inversion of the time axis about the temporal
origin. This will yield the same changes to objects’ coordinate-dependent
descriptions that the standard active transformation does. (At least for
objects in classical space; in Section 7 we will see a problem for this idea
applied to objects in spacetime.) On both versions of the transformation,
particles and �eld values get shifted an equal temporal distance across the
time origin, for example: actively, the objects themselves get shifted that
way; passively, they get the correspondingly shifted time coordinate.

5. The standard transformation, revised and updated

Malament approaches the issue of time reversal by means of the invariant
formulation of a given theory. This is an illuminating way of going about
things. I wish to follow suit.

One might wonder why we should take this approach; discussions in
physics books generally do not. The reason is simple. Our theories of
physics are about coordinate-independent objects, things like particles

8This criterion must be modi�ed a bit for indeterministic theories. See Arntzenius
(1997) for discussion and a clari�cation of the standardly cited criterion.

9



and �elds. These objects can take on different descriptions depending on
the coordinate system we use, but they are physical entities that exist in the
world independently of how we choose to describe them. What is more,
our best physical theories tell us that different choices of coordinates can
yield equally legitimate ways of describing these objects. This suggests
that the real, physical features of the world must be the ones that hold
regardless of the coordinate system being used: the ones that hold in all
allowable coordinate systems or reference frames.9

Hence our best theories are typically formulated in an invariant,
coordinate-independent way. This is the way best suited to getting at
the nature of world, apart from our descriptions of it. Mathematically,
these theories are formulated in terms of invariant geometric objects,
rather than coordinate-dependent numerical ones—in terms of vectors,
for example, rather than their components. And when we consider the
time reversal (or other) symmetry of a theory, we should consider our best,
most fundamental theory, in its best formulation. Then, once we have
the proper transformation in hand, we can evaluate what happens to the
fundamental objects under it, and make our inference to the spacetime
structure undergirding the theory.

Consider a different case in which a theory’s geometric formulation
brings to light the underlying spacetime structure: Newtonian mechanics.
Newton’s �rst law says that every object continues in a state of rest or
uniform motion unless acted on by a net external force. Geometrically, it
says that every object continues on a straight spacetime trajectory unless
acted on by a net force. Stated this way, Newton’s law clearly requires
that there be a genuine distinction between the trajectories of objects
at rest or in uniform motion, and the trajectories of objects that are
accelerating. If there were no objective, frame-independent distinction
between accelerated and unaccelerated trajectories, then the law would
not make any sense. Hence a world governed by Newtonian mechanics
must possess the spacetime structure needed to ground that distinction.
Such a world requires suf�cient structure to distinguish, in an objective,
frame-independent way, straight from curved paths through spacetime
(cf. Maudlin 1994, Chapter 2).

9I have been in�uenced here by Maudlin (1994, Chapter 2), Maudlin (2006).

10



Here is our guiding inference, put in these terms. Formulate our best
fundamental theory in an invariant, geometric, coordinate-independent
way. Then see what spacetime structure we need to assume in order to
do this. The structure we need is the structure we infer the spacetime to
have. Initially, make use of any structure we like in formulating a theory,
including a temporal orientation structure. Just be sure to go back and
check whether we really needed it.

How do we �gure out whether we needed some bit of structure, such as
a temporal orientation structure? Trot out the relevant transformation—
in this case, our time reversal transformation. Apply it to the theory
and see what happens. If the theory looks different after undergoing the
transformation, then it seems to require or presuppose such a structure. If
the theory looks the same after the transformation, then it does not seem
to presuppose this structure; it turns out we did not need it to formulate
the theory after all, and we correspondingly infer that there is none. The
proper time reversal transformation should thus tell us whether we need
a temporal orientation in order to formulate a theory in an invariant way;
if we do, then we have reason to infer that there is one.10

Let us revisit the standard transformation, put in these terms. This
means updating our discussion to spacetime. For the time being, stick
with the assumption of a �at spacetime.

The mirroring action is now this: �ip the fundamental contents of
spacetime across a time slice. Notice this is still captured by the idea of
a movie playing backward, since the mirroring of objects across a time
slice is the restacking of time slices in reverse temporal order. This is the
spacetime equivalent of playing out the time-reversed sequence of states.

To apply this transformation, choose a frame within which to do the
�ipping, and choose a time slice within that frame to be the temporal
origin. This time slice is our “mirror.” Then shift the fundamental objects,
or states, to their temporal mirror-image locations across the time slice
mirror. For example, shift particle and scalar �eld value locations from
(t, x, y, z) to (−t, x, y, z). Passively, �ip the time axis across the time slice
mirror, changing these coordinate locations in the same way.

10Keep in mind the distinction between the coordinate-invariant formulation of a
theory, i.e., its invariance under allowable coordinate changes, and a theory’s invariance
under time reversal.

11



It is worth taking a moment to see how this transformation acts on
other kinds of objects in spacetime. For now, focus on the active version.
Since the standard transformation, understood actively, shifts particles to
their temporal mirror image locations, this means that it will �ip particle
worldlines, taking their mirror images across the time slice mirror. This
is the spacetime analog of particle locations getting mapped to their
temporal mirror image locations in classical three-dimensional space.

It is a bit trickier to see what happens to 4-velocities. The 4-velocity
of a particle at a spacetime point is the tangent to the worldline at that
point.11 In addition to �ipping particle worldlines, the standard trans-
formation also re�ects their 4-velocities across the time slice mirror.
The result is that their spatial components �ip sign and their temporal
components remain the same.

One might wonder why. One might think that the temporal compo-
nent ought to �ip. We are, after all, applying a time reversal transforma-
tion. Though the books say that the spatial components invert, they do
not give much justi�cation for this other than noting the desired result
for 3-velocities: their spatial directions �ip sign, just as they should in a
backward-running movie.

Malament provides a justi�cation for this by pointing to an important
feature of 4-velocities. (See also Arntzenius (2005).) The worldline of a
particle at a time really has two tangents associated with it, one pointing
in each temporal direction. This means that there is no unique 4-velocity
associated with a given point on a worldline. How then do we get the
4-velocity of a particle at a time? We need a temporal orientation on the
spacetime manifold. A temporal orientation picks out, at each spacetime
point, which direction is to the future of that point and which is to the
past. Given a temporal orientation, we can then pick out the unique
future-directed 4-velocity at any point on a worldline—future-directed
according to the background temporal orientation.

I say below why this gives rise to the standard transformation proper-
ties of 4-velocities. The important lesson for us is this. Whereas tangents
to worldlines are de�ned without reference to a temporal orientation,

11The 4-velocity is the geometric object representing instantaneous velocity in a
four-dimensional spacetime. If the worldline is parameterized by the proper time, then
the 4-velocity at a point is a unit vector tangent to the worldine at the point.

12



4-velocities are only de�ned relative to a temporal orientation and a
worldline. Absent a temporal orientation, there simply is no unique 4-
velocity associated with a point on a worldline. In this sense, 4-velocities
are not truly fundamental objects. This will be important when we get to
Malament’s view.

(Why do the spatial components of 4-velocities invert? Assume a back-
ground temporal orientation, and so a unique 4-velocity associated with
any point on a worldline. The standard active transformation mirrors
worldlines across the time slice mirror, with the result that a 4-velocity’s
spatial components get the opposite sign. The reason is that the 4-velocity
must remain tangent to the time-reversed worldline [at the corresponding
time-reversed point], and the time-reversed worldline is the mirror im-
age of the original worldline. The temporal component will not change
because the time-reversed 4-velocity must still point to the future ac-
cording to the �xed background temporal orientation. Since 4-velocities
are de�ned relative to a temporal orientation and a worldline, and since
the temporal orientation remains �xed while worldlines get �ipped, this
is what should happen to 4-velocities. Note that this yields the usual
transformation properties of 3-velocities: since a 3-velocity comprises
the three spatial components of the 4-velocity [the 4-velocity projected
onto the x-y-z hypersurface], to say that the 3-velocity inverts [spatial]
direction under time reversal is just to say that the three spatial compo-
nents of the 4-velocity invert. But now we see a justi�cation for these
transformation properties of 4-velocities, over and above their yielding
the expected results for time-reversed 3-velocities.12)

6. Malament’s transformation

Malament’s time reversal transformation is simple. Here it is in a nutshell:
invert the temporal orientation.

That is it. Do not �ip the contents of spacetime across a time slice.
Do not shift particles and �eld values to their temporal mirror image

12Since 4-forces and 4-accelerations are de�ned independently of a temporal orien-
tation, under standard time reversal the temporal components of these objects invert
and their spatial components do not. It is now easy to see how the electromagnetic �eld
behaves under standard time reversal, as Malament shows. See also Arntzenius (2005).

13



locations. Do not invert particle worldlines. Only �ip the temporal
orientation vector �eld.

In the next section I give reasons for preferring this transformation
to the standard one. Here, I outline the way in which it is quite natural,
details aside.

It is natural for a very simple reason. What is a time reversal transfor-
mation? Just a �ipping of the direction of time! That is all there is to a
transformation that changes how things are with respect to time: change
the direction of time itself. In Malament’s words, “The time reversal
operation is naturally understood as one taking �elds [and other funda-
mental objects] on [the spacetime manifold] M as determined relative to
one temporal orientation to corresponding �elds on M as determined
relative to the other” (2004, 306).

This yields the following procedure to �gure out whether a theory re-
quires a temporal orientation. Take the theory in its coordinate-invariant
formulation. Flip the temporal orientation (this is the step of applying the
time reversal transformation). See whether the theory remains the same
afterward. In other words, invert the direction of time, and see whether
things still behave in accord with the theory. If they do, then the theory
does not pick out one temporal direction as opposed to the other; it says
the same thing regardless of the direction of time. In that case, it does
not presuppose that there is a real distinction between the two directions.
If things do look different afterward, then the theory does require the
distinction, for it says different things depending on the direction of time.
In this case, the theory does presuppose a real distinction between the
two temporal directions, in only one of which things can lawfully evolve.
And a temporal orientation is the spacetime structure that supports such
a distinction.

As a word of caution, let me emphasize that more needs to be done in
order to show that the procedure will work as a general rule. It may not
always be clear what role the temporal orientation is playing in a given
theory’s invariant formulation. Malament works this out for the particular
case of classical electromagnetism, but it remains to be seen whether his
procedure will work for any theory we take to be time reversal invariant.13

13In classical electromagnetism, the key is Malament’s realization that we can charac-

14



7. Two Views

The standard time reversal transformation mirrors the fundamental ob-
jects in spacetime across a time, keeping the background temporal ori-
entation (if there is one) �xed. A theory is time reversal invariant if this
does not alter the theory: a solution always gets mapped to a solution.
Malament’s transformation leaves the fundamental objects in spacetime
alone and �ips the temporal orientation. A theory is time reversal invari-
ant if a solution always remains a solution after we invert the temporal
orientation.

In this section, I compare Malament’s view with the standard view
on six points, concluding that Malament’s transformation is the one we
should use in deciding whether a given theory is time reversal invariant.
Before doing that, let me �rst note some features of it that may seem
counterintuitive.

Malament’s approach seems to require that we treat the temporal
orientation as a real physical �eld. As Malament discusses it, the temporal
orientation takes on a life of its own as a fundamental vector �eld on the
spacetime manifold. Indeed, there is no such thing as a 4-velocity, let
alone a time reversal transformation, without it.

One might balk at this understanding of the temporal orientation.
It is certainly an odd-sounding sort of �eld. Should we posit a physical
�eld corresponding to any noninvariance of a theory? If our theories are
non-invariant under Galilean transformations, should we infer an “abso-
lute rest vector �eld” on the spacetime? An even odder-sounding �eld!
Further, for a temporally symmetric theory like Newtonian mechanics,
we want to say that there simply is no temporal orientation �eld. How
then can we de�ne the time reversal invariance of a theory in terms of
this (non-existent) �eld?

My own view is that this is the right way to understand these coordinate-
independent objects in general, and the temporal orientation in particular.

terize the electromagnetic �eld without reference to a temporal orientation, as a linear
map from tangent lines to 4-forces (or 4-accelerations). The standard characterization
of the electromagnetic �eld in terms of the tensor �eld Fab does make reference to a
temporal orientation; this, too, falls out of Malament’s formulation, once we introduce
a background temporal orientation.

15



We need these abstract-seeming objects in our (best formulations of) fun-
damental physics, and I see no reason to distinguish between a temporal
orientation, on the one hand, and an electromagnetic �eld—or, for that
matter, a metric or other spacetime structure—on the other, as far as their
being physically real goes (North, 2009). Nevertheless, one does not have
to adopt this understanding of the temporal orientation in order to accept
Malament’s basic notion of time reversal. Just apply your preferred view
of the abstract geometric objects used in physics, and the physical objects
they represent, to this case. And when we apply our time reversal test,
we do not have to (absurdly) posit the existence of a nonexistent �eld.
Rather, we are considering what would happen to the theory if there were
such a �eld. If the theory would be the same regardless, then we infer
that there is no such �eld. If the theory would be different, then we infer
that this is because there is such a �eld.

Second, on Malament’s approach, there is no sense to be made of
4-velocities absent a temporal orientation. This might seem odd as well.
But it is correct: the de�nition of a 4-velocity just does not allow for a
unique such object without a temporal orientation.14 And note that this
does not mean that we are left without any well-de�ned 4-velocities in a
nontemporally oriented spacetime. It is open to conventionally choose a
temporal orientation if the spacetime does not come equipped with one.
(This does mean that we cannot de�ne 4-velocities in a nontemporally
orientable spacetime, but such a spacetime is so strange anyway that this
need not worry us here.)15

Third, Malament’s view goes against what I presented as the intuitive
understanding of time reversal. Malament’s time reversal transforma-
tion keeps every fundamental object in spacetime at the same location.

14We could de�ne a 4-velocity, or alternatively a worldline, in a nonstandard way,
as an intrinsically temporally directed object. I am open to this idea, though it is
both intuitively odd and highly nonstandard. See Arntzenius and Greaves (2007) for
discussion.

15There are spacetimes of general relativity that are not globally temporally orientable.
Our own spacetime locally has a Lorentz signature metric: locally, it is temporally
orientable. This does not guarantee global temporal orientability. But as John Earman
(2002, 257) argues, so long as we have reason to think the fundamental laws hold in all
regions of our universe, we have good reason to infer that local temporal orientability
yields global temporal orientability.

16



Nothing gets shifted at all: we only �ip the temporal orientation vector
�eld. This means that particle positions and worldlines remain put, in-
stead of shifting to their temporal mirror-image locations as they would
in a reverse-running movie; clockwise motions are not time-reversed
into counterclockwise motions, for example.16 In reply, I think we need
only point to the naturalness of this sense of time reversal, discussed in
the last section, and to the arguments coming up in this section. The
backward-running movie idea is a relic left over from classical theories
set in three-dimensional space. (Think of the backward-running movie
as the inverted sequence of movie frames, i.e., the inverted history of
objects’ spatial locations at the same time.)

Now to compare the views on six points. Together, these points make
the case for Malament’s transformation over the standard view, though
the last two alone should suf�ce.

7.1. Justi�cation

One problem with the standard view is its lack of a clear motivation.
Physics books typically do not offer much justi�cation other than to point
to desired results for classical theories, as we saw above for their view on
4-velocities. Malament shows that there is a justi�cation available, once
we see that 4-velocities are de�ned relative to a temporal orientation. But
this raises a question: Why not invert the temporal orientation as well?
The temporal orientation can be thought of as a fundamental object,
just as worldlines are. Then why does it not get �ipped across the time
slice mirror too?17 The books do not say anything about this. They just
assume that the temporal orientation remains �xed. (They are not explicit
about this; Malament brings the assumption to light.)

Malament, though, has a clear motivation for his view. To �gure out
whether a theory requires a temporal orientation, compare the theory with

16Malament suggests that this falls of his view given a reference frame (in his sense),
though I am not sure that this preserves the initial idea. It seems to me that one of
the main bene�ts of Malament’s view (the applicability to arbitrary curved spacetimes)
indicates that we will have to give up the classical movie-playing picture.

17This might suggest a third option: �ip particle worldlines and the temporal orien-
tation. But this transformation will always give us our original states right back. The
time-reversed solution is just the original solution, looked at from a different perspective.

17



what it says after �ipping the temporal orientation. This understanding
of time reversal yields a clear justi�cation for objects’ transformation
properties. Take 4-velocities. Their transformation properties will differ
from those on the standard view: since 4-velocities are de�ned relative
to a temporal orientation and a worldline, when we invert the temporal
orientation, keeping everything else (worldlines included) �xed, all the
components will �ip, temporal and spatial. More generally, when we
�ip the temporal orientation—which is all that this transformation does,
remember—all the components of objects de�ned relative to a temporal
orientation will �ip sign; the components of objects not so de�ned will
not. Thus, fundamental objects (other than the temporal orientation
itself) are left alone by the transformation; those de�ned relative to a
temporal orientation completely invert. Worldlines are invariant, as
are their tangents, whereas 4-velocities completely invert. Indeed, only
objects de�ned relative to a temporal orientation change at all under this
transformation.18

So the reason 4-velocities completely invert on Malament’s view is the
same as the reason that only their spatial components invert on the stan-
dard view: 4-velocities are de�ned relative to a temporal orientation and
a worldline. The difference in transformation properties comes from the
difference in the basic action of the time reversal operation. Malament’s
transformation �ips the temporal orientation and leaves worldlines intact;
the standard view �ips worldlines and leaves the temporal orientation
intact. Yet Malament has a clear justi�cation for the transformation prop-
erties of the different objects in spacetime, based on how these objects
are de�ned and what the time reversal operation does. The standard view
has a justi�cation for the transformation properties of some objects, like
4-velocities, but this only reveals the lack of a justi�cation for other as-
pects of the view, such as its leaving the background temporal orientation

184-forces and 4-accelerations will not change any components. As a result, classical
electromagnetism is time reversal invariant on this transformation, as it is on the standard
one. Malament locates a conception of the electromagnetic �eld that differs from
the standard Maxwell-Faraday tensor (as a map from tangents to 4-forces), which is
independent of a temporal orientation. Given this and the above characterization of
4-velocities, it is easy to see how the standard electromagnetic �eld tensor transforms
under time reversal.

18



�xed.

7.2. Active versus passive time reversal

In a classical space, there is a passive time reversal transformation cor-
responding to the standard active one, as we saw above. This is less
straightforward in a four-dimensional spacetime. In fact, now the stan-
dard view seems to lose the correspondence between active and passive
time reversal entirely. Recall that the standard active transformation
mirrors the material contents of spacetime across a time slice. What is
the corresponding passive transformation?

The passive transformation should be just a change in time coordinate
from t to −t. Yet this would yield different time-reversed 4-velocities,
inverting their temporal and not their spatial components. Take any 4-
velocity Vµ, say with original components (1, 2, 3, 4) in a given coordinate
system. Under standard active time reversal, the components become
(1,−2,−3,−4), whereas under this passive transformation, they become
(−1, 2, 3, 4).

This is weird. Usually there is a passive transformation corresponding
to any given active one. And, intuitively, it should not matter whether
we change the coordinate-dependent representations of invariant objects
by shifting the objects themselves or by moving the coordinate system
around them. Not only is this intuitively odd, but it seems to undercut
our inference to a direction of time. Since the active and passive transfor-
mations have different effects on objects in spacetime, whether a theory
is deemed time reversal invariant or not could depend on whether we
use the active or passive transformation as our test. Our inference to a
temporal orientation would then similarly depend on this.

Since the books assert that 4-velocities’ spatial components �ip sign
under time reversal, it seems the standard transformation must be under-
stood actively. The passive transformation simply will not yield the result
the books claim for time-reversed 4-velocities. We could try to construct
a passive transformation to yield the same results as the active one. But
a passive transformation is de�ned as a change in coordinate-dependent
representations of coordinate-independent objects due to a change in coor-
dinate system. To de�ne a passive transformation in terms of an active one

19



is to give up on the distinction between active and passive transformations
altogether.19 Unlike in classical space, then, here we cannot understand
the standard transformation (passively) as an inversion of the time axis.
Instead, both the chosen coordinate system and the background temporal
orientation must remain intact.

Here is another way of defending the standard view of 4-velocities,
and by means of what is perhaps a passive version of the standard transfor-
mation.20 Consider a transformation that �ips the time coordinate, not
the temporal orientation or worldlines, and also says that the proper time
goes with the time coordinate. Once we �ip the temporal coordinate, the
parameterization of a worldline by the proper time will also �ip. The
result (assuming that a 4-velocity is de�ned relative to a parameteriza-
tion by proper time) is that a 4-velocity’s spatial components �ip sign,
and the temporal component does not, just as the standard view says.
We can regard this as a kind of passive transformation corresponding to
the standard active one, in this case changing not just t to −t but also
the parameterization. (I say “kind of” since this does not just change
the coordinate system, and I am not sure whether this should count as
active or passive.) On this view, too, 4-velocities are not fundamental
objects: they are de�ned relative to a parameterization. This is another
way of defending the standard view, and with something that looks like a
passive transformation—it does not shift around fundamental objects in
spacetime—albeit not the usual one that only changes t to −t. Nonethe-
less, this will also face the two key problems below.

Malament’s time reversal is also an active transformation: invert the
temporal orientation, understood as a real vector �eld, represented by an
abstract geometric object, on the spacetime manifold. Here, too, it seems
there simply is no corresponding passive transformation.21 No passive
transformation will get the same transformation properties for objects in

19Earman (2002) de�nes time reversal as a change in coordinate system plus a �ipping
of the temporal orientation. This too mixes the notion of passive with active time
reversal in such a way as to lose the distinction.

20This was pointed out by Hartry Field (in conversation) and is advocated by Earman
(2002, Section 3).

21But see Leeds (2006) for an argument that Malament’s transformation is �rst and
foremost a passive transformation, with an active version.

20



spacetime that Malament’s time reversal does. Think of inverting the time
axis. Whereas Malament’s transformation �ips all the components of 4-
velocities, this passive transformation �ips only the temporal component.
In general, geometric objects transform differently under Malament’s
time reversal depending on whether they are de�ned relative to a temporal
orientation or not, and there is no passive transformation, no mere change
in coordinate system, that will get this result.22

Thus, both views face the problem of losing the correspondence
between active and passive time reversal. On this, the views are tied. I am
inclined to think this is not such a problem for either view. Once we start
investigating theories in their coordinate-invariant formulations, it seems
their symmetry properties ought to be linked to transformations that
do their job independently of an arbitrarily chosen coordinate system.
Since the theories themselves are written in a way that is invariant under
coordinate transformations, it is reasonable to expect that the proper time
reversal (or other) transformation should be too.

7.3. Time translation invariance

Standard time reversal presupposes that theories are time translation
invariant.23 The standard transformation mirrors objects across a chosen
temporal origin. If a theory is not time translation invariant, then whether
it is deemed time reversal invariant will differ depending on the choice
of temporal origin. Indeed, the standard picture of time reversal is a bit
unclear if a theory did depend on the time. During what time sequence
should we compare an allowable process with its time reverse?

Although the theories we take seriously for our world all appear to be
time translation invariant, it is a bene�t of Malament’s view that it does

22Keep in mind that �ipping the temporal orientation is different from inverting
the time coordinate axis. (1) The temporal orientation is fundamental coordinate-
independent object:, represented by a vector �eld on the spacetime; the time axis is a
coordinate axis we lay down on the spacetime in order to represent invariant objects.
(2) Inverting the time axis has a different effect on 4-velocities from inverting the
temporal orientation. The standard view does keep both the time axis and the temporal
orientation �xed, but not because these must go together. It is because the view inverts
worldlines and not the temporal orientation.

23I thank Matt Kotzen for pointing this out to me.

21



not presuppose this. Since Malament’s transformation does not require
any choice of temporal origin, it will deem a theory time reversal invariant,
or not, regardless of whether it possesses this additional symmetry.

7.4. Indeterministic theories

Malament’s test for time reversal can be easily adapted to indeterministic
theories. Formulate the theory in a coordinate-independent way. Then
take any (conditional) probability the theory assigns to a history of states.
Flip the temporal orientation. See whether the probabilities remain the
same. If they do, then the theory is time reversal invariant, and we did
not need a temporal orientation to formulate it in the �rst place.

Although the standard transformation can be made to work for in-
deterministic theories, the story is more involved. I refer the reader to
Arntzenius (1997) for that story.

7.5. Curved spacetimes

Malament’s transformation applies to background spacetimes of arbitrary
curvature. This means it is suitable for the spacetimes of general relativity.
The only requirement is that the spacetime be temporally orientable.

The standard transformation will not generalize in this way. It �ips
fundamental objects across a time slice, (actively) shifting them to space-
time locations at an equal temporal distance away from the time slice
mirror. This requires a choice of reference frame in which to do the
�ipping and a choice of time slice mirror within that frame. The mir-
roring action must shift objects’ locations to an equal temporal distance
away from this time slice, while retaining relative distances along the
mirror-image, time-reversed worldlines.

All of this requires that we can measure an equal distance in space from
a given time. And it is assumed that we can do this along a straight line.
But the notion of distances along a straight line and of equal distances
from a time straightforwardly make sense only in a �at spacetime. The
standard picture requires a background spacetime with enough time-
re�ection symmetries that we can mirror objects across a time slice in this
way. This suggests that it presupposes a spacetime with the full structure
of a Minkowski spacetime.

22



We could try using distances along a spacetime’s geodesics instead
of distances along straight lines; geodesics are the curved-spacetime
analogs of straight lines in a �at spacetime. But this will not work as
a general rule. Even if the geodesics are orthogonal to our time slice
mirror (the spacelike hypersurface we choose as our temporal origin),
they may not be orthogonal to a later time slice. In general, geodesics
have all sorts of strange properties that do not �t nicely with the standard
idea of time reversal as a mirroring across a time. Geodesics can cross,
curve, not remain parallel, they can intersect each other more than once.
This means, for instance, that two distinct spacetime points could be
mapped to the same point under time reversal, and we would not want
that to happen: two distinct points that are simultaneous in a frame
should remain simultaneous after time reversal, not get mapped to the
same spacetime location. In short, it is crucial to the standard view that
two straight parallel lines do not intersect, and taking temporal mirror
image locations along geodesics will not guarantee this, for geodesics can
intersect.

We can construct a later time slice by following the spacetime’s
geodesics, making sure that any time slice along the way remains or-
thogonal to them. This could also yield time-reversed locations at equal
temporal distances away from the time slice mirror as the original loca-
tions are. Typically, though, the hypersurface at the later time will itself
be curved and crossed over itself, since the geodesics we use to construct
it can do this. But then the hypersurface will no longer reasonably be
a time slice! There simply is no guarantee of getting �at hypersurfaces
over time—genuine time slices—by constructing them in this way. (We
could �nd time slices by means of Killing vector �elds orthogonal to
the geodesics, but this is a special case. We cannot in general construct
hypersurfaces through an arbitrary curved spacetime so that the geodesics
in any direction will be orthogonal to the time slices.)24

24As Malament pointed out to me (in correspondence), there are non-Minkowski
spacetimes that allow for a limited sense of standard time reversal, namely those ad-
mitting one well-de�ned family of time-re�ection maps, by means of a hypersurface-
orthogonal timelike Killing vector �eld. Hypersurface-orthogonality means that (at least
locally) we can foliate the spacetime with a family of spacelike hypersurfaces everywhere
orthogonal to the Killing vector �eld. Even so, only in a Minkowski spacetime will there

23



The standard transformation therefore presupposes a �at background
spacetime (modulo the above). Moreover, it is de�ned relative to a frame
of reference and a time slice within it. Objects then have their trans-
formation properties relative to that frame; the spatial components of
4-velocities �ip sign within the frame in which we choose to do the �ip-
ping, for example. This is to be expected from a transformation that
tries as much as possible to preserve the classical picture of time reversal.
Consider two objects moving with constant relative velocity in a classical
spacetime: the time-reversed velocities clearly depend on the frame in
which we choose to carry out the time reversal.

Malament’s transformation generalizes to the (temporally orientable)
arbitrary curved spacetimes of general relativity, while yielding the same
results as the standard view for spacetimes that have the above symmetries.
Further, Malament’s is an entirely frame-independent notion of time
reversal. Objects completely invert, in any frame, if they are de�ned
relative to a temporal orientation, or remain invariant, in any frame, if
they are not so de�ned. We are always free to arbitrarily choose a frame
in order to employ the standard transformation. In discussing the view
above, we assumed that such a choice had been made. But it is certainly
cleaner if we do not have to do this.

7.6. The inference to a direction of time

Malament’s transformation yields a better test for a temporal orientation,
the very thing we want from our time reversal transformation. There are
two ways to see this.

The �rst is its naturalness. Malament’s test goes like this. Take our
best fundamental theory and take a solution to it. Invert the temporal
orientation and see whether it remains a solution. If it always does, then

be many hypersurface-orthogonal timelike Killing �elds, allowing for many families of
time re�ection maps, which we can apply to arbitrary motions. (Malament offers this
precisi�cation as a proposition: Let (M, gab) be a (not necessarily temporally orientable)
spacetime. Suppose that for all points p in M, and all timelike vectors ξa at p, ξa can
be extended to a hypersurface-orthogonal Killing �eld on M. Further suppose that
(M, gab) is geodesically complete. Then (M, gab) is isometric to Minkowski spacetime:
locally, the timelike Killing �elds means the spacetime metric is �at, with vanishing
Riemann curvature.)

24



the theory did not require the temporal orientation to begin with. It says
the same thing whether time goes in one way or the other. If a solution
does become a nonsolution, then this is indication that the temporal
orientation matters to the theory. After all, this is the only thing that
changed under the transformation.25

Second, Malament’s time reversal transformation, unlike the stan-
dard one, does not invert spacetime handedness. To see this, think of a
spatiotemporally handed object, say in a three-dimensional spacetime.26
Suppose the object comprises three arrows, one pointing up along the
temporal axis (call this component arrow t), the other two in the x-y plane
(call these x and y). This is a triple of linearly independent vectors, one
timelike, the other two spacelike. Suppose this is a fundamental object,
not de�ned relative to any other fundamental objects in the spacetime.

Now apply the standard time reversal transformation. This trans-
formation (actively) mirrors fundamental objects across a space of one
dimension fewer than the space the objects are in (and which is orthog-
onal to the time axis). Suppose we choose the x-y plane to be the time
slice mirror, and mirror the object across this plane.

What happens when we do this? We invert the object’s spacetime
handedness! (Note that it is the spacetime handedness, not spatial hand-
edness, that is inverted.) After time reversal, the two spacelike arrows in
the x-y plane remain where they are, and the timelike arrow points in
the opposite temporal direction. Before time reversal, when we rotated
x into y, we got t pointing up: this was a right-handed object (right-
spatiotemporal-handed). After time reversal, we can no longer rotate x
into y and get t pointing up: this is a left-handed object.

Any mirroring across a hypersurface of one dimension fewer than the
25Callender (2000) gives a spatial analogy. Imagine a cube-shaped fundamental

particle that emits photons from one of its faces during an experiment. Repeat the
experiment, with the only difference being that the cube is spatially mirror-re�ected. If
the photons are now emitted in the same spatial direction rather than the mirror-image
one, it seems reasonable to conclude that space itself has a preferred direction, “pulling”
the photons that way.

26A handed object cannot be mapped to its mirror image via ordinary rotations and
translations within the space (a right-handed glove in three-dimensional space, the letter
L written on a chalkboard). In any space, there are only two possible handednesses:
objects either have the same or opposite handedness from one another.

25



space—which is what the standard time reversal operation is—will wind
up inverting spacetime handedness as well. Actively: �ip the fundamental
objects across a time, and we change the spacetime handedness of any
objects that are spacetime handed. Passively: �ip the time axis about
the origin, and we change the spacetime handedness of the coordinate
system.27

This means that if a theory is not invariant under standard time rever-
sal, we can reasonably infer that it requires some asymmetric spacetime
structure allowing things to evolve in only one of two ways. But this
could be a spacetime-handedness structure rather than an asymmetric
temporal structure. If a theory changes under this transformation, the
culprit could be either a direction of time or a spacetime-handedness
structure (a spacetime-handedness orientation �eld, say). So this is a bad
test, if what we are ultimately after is evidence of a temporal orientation.

Under Malament’s transformation, no fundamental spacetime object,
including the one above, will undergo any change that could �ddle with
its spacetime handedness (if it has such a handedness). True, after time
reversal, part of the object will point to the past rather than the future.
But its overall spacetime-handedness will not change.28 The reason is
that this transformation leaves everything—every fundamental object in
spacetime, other than the direction of time itself—intact.29

Malament’s time reversal transformation yields a good test for a di-
rection of time precisely because it only �ips the temporal orientation.
If things look different afterward, then the only possible culprit is the

27If the passive and active transformations corresponded, then �ipping spacetime
handedness of coordinates would be equivalent to �ipping objects’ spacetime handed-
ness.

28The spacetime handedness of non-fundamental objects might change, but that will
not affect the time reversal invariance of our fundamental theories.

29You might think that the spacetime handedness must change once the temporal
orientation �ips. Keep in mind that a handedness is just an orientation or an ordering,
where permutations change the handedness (see note 26). Suppose an ordered triple
(1, 2, 3) is right-handed according to the spatiotemporal orientation. After inverting
the temporal orientation, this object should have the same ordering, that is, the same
spacetime handedness. Malament (2004, Sections 5-6) discusses this from the four-
dimensional geometric perspective, which makes this clear. See Arntzenius and Greaves
(2007) for more discussion

26



temporal orientation. Since the direction of time is the only thing altered
in the time-reversed scenario, it is natural to conclude that it plays a real
role in the theory. If this is our best, most fundamental theory, then this
gives us reason to infer that the spacetime itself is temporally oriented.

8. Conclusion

We should use Malament’s time reversal transformation to test whether a
given theory is time reversal invariant, and to infer whether the theory’s
spacetime is temporally oriented. Unlike the standard transformation,
Malament’s time reversal does not mix in considerations of spacetime
handedness or other symmetries, and it generalizes to the arbitrary curved
spacetimes of general relativity. The only drawback is that it fails to
respect our initial intuition about time reversal. But this is insuf�cient
reason for clinging to the standard view, especially when, on further
re�ection, it is Malament’s transformation that captures what we had in
mind all along: to see how a theory fares when oriented with respect to
one direction of time as opposed to the other.

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