Vol.2, No.3, 212-219 (2010) Natural Science
http://dx.doi.org/10.4236/ns.2010.23033
What does the “arrow of time” stand for?
Etienne Klein
Laboratoire de recherche sur les sciences de la matiere (larsim)centre d’etudes de saclay 91191 gif sur yvette cedex France;
Received 24 November 2009; revised 28 December 2009; accepted 28 January 2010.
ABSTRACT
One hundred and thirty years after the work of
Ludwig Boltzmann on the interpretation of the
irreversibility of physical phenomena, and one
century after Einstein's formulation of Special
Relativity, we are still not sure what we mean
when we talk of “time” or “arrow of time”. We
shall try to show that one source of this
difficulty is our tendency to confuse, at least
verbally, time and becoming, i.e. the course of
time and the arrow of time, two concepts that
the formalisms of modern physics are careful to
distinguish. The course of time is represented
by a time line that leads us to define time as the
producer of duration. It is customary to place on
this time line a small arrow that, ironically, must
not be confused with the “arrow of time”. This
small arrow is only there to indicate that the
course of time is oriented, has a well-defined
direction, even if this direction is arbitrary. The
arrow of time, on the other hand, indicates the
possibility for physical systems to experience,
over the course of time, changes or transforma-
tions that prevent them from returning to their
initial state forever. Contrary to what the ex-
pression “arrow of time” suggests, it is there-
fore not a property of time itself but a property
of certain physical phenomena whose dynamic
is irreversible. By its very definition, the arrow
of time presupposes the existence of a well-
established course of time within which – in
addition – certain phenomena have their own
temporal orientation. We think that it is worth-
while to emphasize the difference between sev-
eral issues traditionally subsumed under the
label “the problem of the direction of time”. If
the expressions “course of time”, “direction of
time” and “arrow of time” were better defined,
systematically distinguished from one another
and always used in their strictest sense, the
debate about time, irreversibility and becoming
in physics would become clearer.
Keywords: Time, Time’S Arrow; Temporal Asym-
metry; Principle of Causality; Irreversibility
1. INTRODUCTION
Each one of us can make the following observation: we
often talk about time as if it corresponded only to a “be-
coming”, to the stream of changes affecting a thing, a
person, an institution, a physical system. Certainly,
change is truly the phenomenon that best suggests the
idea of time, and one can easily understand why: we
never encounter a specific and directly perceptible reality
that would be the time. We only see around us changing
things, things becoming others, and it is therefore through
the concrete effect of change that the course of time first
appears to us. But to conclude from this, as our natural
language does, that time and becoming are the same is a
step that is too easily made: without proceeding to further
investigation, time is mostly referred to as if it resembled
what it holds. It is said to stop or disappear when nothing
seems to be happening as if all its dynamics depended not
upon time itself but upon its contents.
Such short-cuts serve to answer, before it is even for-
mulated, the question of the relationship between time
and events, as well as the question of its dynamics: is
time an abstract structure into which events are inserted,
that is to say a reality in itself preceding all possible
events, or is it composed of the stream of events itself?
In other words, is it legitimate to differentiate time from
the concrete succession of events? If the answer is yes,
then what makes time go forward? If the answer is no,
do events determine its flow?
Those questions are neither purely academic nor
purely theoretical. They deserve to be thoroughly exam-
ined since concealing them creates a confusion between
time and becoming, which leads to a kind of intellectual
vagueness through repetition: concepts get lost, clouded
and confused with one another. Moreover, this confusion
constantly exposes our conception of temporality to
metaphors that ultimately discourage us from thinking
about time for what it is. By willing too much to harmo-
nize the representation of time with the experience of
becoming, common thinking in fact engenders confusion

E. Klein / Natural Science 2 (2010) 212-219
213
between time and irreversible physical phenomena. At
least for the sake of hypothesis, and without going back
to a Newtonian conception of time, why not conceive of
time as a more fundamental entity than temporal phe-
nomena? A kind of skeletal being whose flesh would be
made of events? The discreet background of any concrete
becoming? Those are the questions we would like to dis-
cuss by examining the kind of answers physics provide…
For this purpose, we have studied the theoretical con-
structions specific to conventional physics (classical
physics, quantum mechanics, special or general relativ-
ity). As we will see, this study shows that through its
operational formalisms, physics distinguishes time from
becoming. It even distinguishes them completely; on one
side, there is the course of time, a primitive entity, on the
other side there is the arrow of time, which is not a
property of time but of the majority of phenomena tak-
ing place in time, specifically irreversible temporal phe-
nomena.
The course of time establishes an asymmetry between
past and future: if two events are not simultaneous, then
one of them is earlier than the other one. It also estab-
lishes a difference of position (but not of nature) be-
tween past and future moments: on a timeline, tomorrow
isn’t set where yesterday is – a certain amount of time
separates them definitely. So defined, the course of time
expresses the irreversibility of time itself. As for the ar-
row of time, it represents the fact that some physical
systems evolve in an irreversible way throughout time:
they won’t go back to their previous states. So defined,
the arrow of time expresses the irreversibility of phe-
nomena within the course of time. It is the concrete
manifestation of becoming.
Today, physics has become so spectacularly effective
that it is possible to imagine that the distinction it makes
between time and becoming could be transferred to phi-
losophy, which often aggregates the two notions. Espe-
cially because physics not only distinguishes them but
almost opposes them. In a way, it considers the course of
time as that which never becomes, in the sense that it
never changes its way of renewing the present moment
or, to put it differently, its way of being time. Therefore,
almost ignoring the meaning of words and going against
how we normally think of time when we don’t really
think about it, physics conceives of the existence, within
the course of time itself, of a principle that remains and
never changes: within passing time, there is something
that doesn’t pass, something that time doesn’t affect.
2. TIME AND TEMPORAL PHENOMENA:
DON’T JUDGE BY APPEARANCES
Because of a transference process between concepts, we
don’t distinguish between time and what happens within
it: we automatically mistake the contents for its container.
Because we notice that cyclical phenomena exist around
us, we pretend that the course of time is cyclical. Or, if
our schedules get crammed, if we manufactured objects
at a frantic pace, we declare that it is time itself that’s
speeding up. Our notion of time is always of a time
marked by events, by the phenomena it contains, a full
time excluding any notion of emptiness or abstraction.
It is of course possible to pretend that those are only
figures of speech, frequently used expressions that don’t
hinder our understanding of time in any way. But this
would be to ignore that this shift, this simplification of
language doesn’t exclusively happen in daily life. Some
philosophers have repeated it: as soon as they learned
that scientists have discovered a new category of phe-
nomena, they declare that time itself has therefore been
modified even that it has just been the object of an onto-
logical revolution.
On the subject of chaos theory, which shows that a
system defined by deterministic equations doesn’t nec-
essarily have a predictable evolution, the French phi-
losopher Michel Serres wrote: “Time does not always
flow according to a line nor according to a plan but,
rather, according to an extraordinarily complex mixture,
as though it reflected stopping points, ruptures, deep
wells, chimneys of thunderous acceleration, rending,
gaps – all sown at random, at least in a visible disorder.
Thus, the development of history truly resembles what
chaos theory describes.”
What does this mean? That the existence of chaotic
phenomena necessarily implies that the course of time
itself is chaotic? But chaos theory, as revolutionary as it
first appears, can be explained through classical Newto-
nian mechanics. It even belongs to it completely. Neither
the status nor the representation of time have been modi-
fied by this theory or even questioned by it: time in
chaos theory is the same as Newtonian time. The dis-
covery of a new typology of phenomena doesn’t neces-
sarily require a new conception of time in order to be
characterized.
Similar remarks could be made about a book from
1979 that has had a lasting influence: The New Alliance
by Ilya Prigogine and Isabelle Stengers, which is often
presented as the manifesto for “temps retrouvé” (the
rediscovery of time) supposedly because its authors con-
clude that: “Today’s physics no longer denies time. It
acknowledges the irreversible time of evolutions toward
equilibrium, the rhythmical time of structures whose
impulse feeds on the world that runs through them, the
zigzagging time of evolutions through instabilities and
amplifications of fluctuations, and even the microscopic
time that reflects the indeterminacy of macroscopic
physical evolutions.2” But has physics ever “denied”
time? And what is “rhythmical time”? What about “zig-
1Conversations on Science, Culture and Time, Michel Serres with
Bruno Latour, University of Michigan Press, 1995, p. 57.
2Ilya Prigogine, Isabelle Stengers, La Nouvelle Alliance, Paris, Galli-
mard, 1979, p. 275.

E. Klein / Natural Science 2 (2010) 212-219
214
zagging time”? And what kind of time would not be
“irreversible”?
To postulate that time is equivalent to what it contains,
or even that it is generated by phenomena is to concede
explicitly that a multiplicity of time exists: there would
then be the same number of times than the number of
temporalities. In this sense, time is labelled with several
adjectives that specify the type of phenomena associated
with it: there is a psychological time, a geological time, a
astrophysical time, a subjective time, one or several his-
torical times, even a “rhythmical time” or a “zigzagging
time”, because those phenomena inhabit time in a way
that varies according to each… Such posture makes be-
lieve that multiple kinds of time can coexist. But could
there seriously be a time for rocks and one for atoms, a
time for stars and one for galaxies?
Ilya Prigogine found it useful to add another element
to the list: the “entropic time”, which is supposed to re-
flect the irreversible evolution of a system by quantify-
ing the scope of changes that have affected it. This
“time” is proportional to the variation of a system’s en-
tropy during a given irreversible process.3 What does
this mean?
As we know, the entropy of a system is a quantity that
characterizes the system’s ability to experience sponta-
neous changes: the bigger the entropy, the smaller the
system’s ability to transform itself. The second law of
thermodynamics states that a closed system’s entropy
can only increase throughout time. This means that by
changing, the system necessarily loses some of its ability
to change even more. A closed system spontaneously
tends toward a state of maximum entropy, in which any
spontaneous transformation will become impossible.
Let’s take an example: the total entropy of a sugar cube
and of a cup of coffee being inferior to the entropy of a
cup of coffee with sugar, a sugar cube dropped into a
cup of coffee has no choice but to be dissolved into the
coffee. This phenomenon is irreversible: the sugar cube
dissolved in the coffee cup will never return to its origi-
nal square shape, nor will it recover its whiteness, and
the coffee won’t ever get its bitterness back. As such, the
second law of thermodynamics is in accordance with the
fact that physical phenomena seem to go in a truly spe-
cific direction.
Another way of defining entropy is to say that it
measures the energy quality available within a system.
The energy is conserved and can therefore only be
transformed. But during this transformation, its quality
lessens and it becomes less and less usable. Good quality
energy is organized energy with little entropy, like the
entropy of a waterfall whose overall falling movement
can activate a water turbine. At the bottom of the fall,
water molecules have lost the vertical organization they
had during the fall due to the action of gravity. Their
energy is now of lower quality: it can’t be used so easily.
That being said, let’s go back to Ilya Prigogine who,
by using the entropic variation, links the course of
physical time to processes that happen within it. In fact,
he creates a kind of time that resembles the phenomena
it contains and clearly depends on the system. Somehow,
he “mixes up” the length of time during which the sys-
tem changes and the intensity of these changes: if the
system doesn’t change, its entropic time doesn’t increase;
if it changes, its entropic time shifts from physical time
and this shift depends on the production of entropy. The
specificity of entropic time is that it stops as soon as the
system’s entropy becomes constant while physical time
keeps on passing. Entropic time is similar to Bergsonian
time: it only passes when it creates novelty. This time is
only valid for the changes it brings and stops as soon as
the system can’t evolve anymore4. Besides, the course of
entropic time has a clearly defined motor: it is “pulled”
or “pushed” by the irreversibility of temporal processes
that occur within it.
But Prigogine is far from having created a new time
since his definition of entropic time explicitly relies on
physical time. Entropy variation being the manifestation
of any irreversible change, it can certainly be used to
characterize a system’s inner dynamics through its rela-
tionship to the duration of that system’s change. But is it
therefore judicious to measure a system’s irreversible
evolution using a variable called “time” if this variable
has actually been set to proceed outside of…time? “En-
tropic time” appears as a misleading expression in the
sense that it implies that entropy could be generated by
time even though it is a quantity linked to what is hap-
pening throughout time.
This kind of confusion between time and temporal
phenomena seems implicit, evident, but is it even rele-
vant or justified? Can such assimilation of time with
temporal developments be found among the formalisms
of physics?
3. TIME AND BECOMING: PHYSICS
SEES DOUBLE
Physics formalisms are mostly composed of equations,
which condense fundamental relationships, reflect es-
sential properties, and reveal things more profound than
our human discourse can express however subtle it
might be. But equations don’t speak, at least in the
common sense of the verb. So what can we say about
time and becoming that the equations would say if they
3If dS is the variation of entropy for a system and P(t) the production o
f
entropy per unit of physical time, then of course: dS = P(t)dt. The idea
on which the concept of entropic time is based consists in building a
“time” t* for the variation of entropy during the time interval dt to be
p
roportionnal to dt*: dS = P(t)dt =a.dt*, where a is a constant. We
therefore have: ∫dt* = (1/a)∫P(t)dt.
4« Time is invention or it is nothing at all », Henri Bergson, Creative
E
volution, Courier Dover Publication,1998, p. 341.

E. Klein / Natural Science 2 (2010) 212-219
215
could speak?
Here is the short answer to this question: modern
physics has been constructed through formalisms that
make a distinction between time and becoming, or more
precisely between the course of time and the arrow of
time. Since its birth, that is since the apparition of New-
tonian mechanics, physics has conceived a time that
doesn’t need something to happen somewhere to pass.
Physics’ operational effectiveness, its experimental suc-
cesses have become so impressive that it seems right to
believe that the distinction it establishes between time
and becoming represents a “negative philosophical dis-
covery” of the highest importance since it alters the
terms in which the philosophical question of becoming is
stated.
We are now going to explain the comment above.
What does course of time and arrow of time mean?
The course of time only reflects the passing of time.
In a way, it is time itself. It is represented by a line, a
timeline on which a little arrow is usually drawn, an
arrow that is not the arrow known in physics as the ar-
row of time. It is there to indicate that time has a single
direction, and that time travel is indeed impossible; our
presence on the timeline at a specific moment conditions
our subsequent positions on that line. It is also impossi-
ble to come back, nor to go through the same moment
twice.
We have to notice that this depiction of time as a line
is fundamentally incomplete because it omits indicating
how this line is built. Since the present does not bring
another present by itself, there has to be something, an
“engine” of time, to do this “work”. This little engine is
responsible of the course of time, in the sense that it
continuously renews the present: without it, the newness
of each instant could not arise. Where does this engine
come from? Is it a property of time itself or a property of
the arrangement of things in time? Is it linked to a global
property of the Universe or to our consciousness? The
answers to these questions have still to be elucidated, so
that we have to consider that the mystery of time exists
less in the line through which we represent it than in the
hidden dynamic that builds this line.
The course of time has a structure, which by itself
guarantees that each event is necessarily definitive. As
soon as the event occurs, this fact can’t be erased by
anything. If the imprint that the event might have left
can be erased, this erasure will change the close or dis-
tant future, but not the past. More generally, any present
action can only have consequences in the future, never in
the past. The course of time thus maintains a constant
possibility of distinguishing the past from the future.
As for the arrow of time, what is it if not the little ar-
row placed on a timeline? Contrary to what the expres-
sion might suggest, it is not related to time itself but to
what happens within it. It is not an attribute of time, but a
potential property of physical phenomena; most of what
exists at our scale is transformed irreversibly throughout
time and can’t return to its original state. The dynamics
of those physical phenomena is then marked with an ar-
row, wrongly called the “arrow of time” (since this arrow
is linked to the dynamics, not to the time itself).
The problem of the arrow of time is often summarized
by the following question: why do we remember the past
and not the future? The answer usually given is that the
only way of distinguishing between past and future is by
means of the second law of thermodynamics. But in fact,
the question asked doesn’t concern the arrow of time
since the invocation of the course of time is enough to
answer it: if we do not remember the future, it is because
we have not yet been present in… the future! Asking
“Why are we in a different state in the future than in the
past?” is quite another question (whose answer can be,
this time, the second law of thermodynamics) that has to
be distinguished from the first one.
This example of confusion shows that it is worthwhile
to emphasize the difference between several issues tradi-
tionally subsumed under the label “the problem of the
direction of time”. The most invoked concepts are the
concepts of irreversibility and of time-reversal invari-
ance. Time-reversal invariance is a property of physical
laws: a law is time-reversal invariant when it is ex-
pressed by a differential equation which is invariant un-
der the transformation t → -t. By contrast, irreversibility
is a property of processes: a process is irreversible if it is
always observed in the same temporal order, and never
in the inverse one. The problem of the arrow of time
consists in finding out how irreversible processes can be
explained by means of time-reversal invariant laws (as
we see, this problem is conceptually different from the
explanation of the origin of the course of time, and also
from the description of the engine of time).
To better grasp this distinction between the course of
time, the arrow of time and the engine of time, we
should remember that since Newton, the principle of
causality has always constrained from the outside the
representation of the course of time in physics. This
principle is generally summarized by saying that every
event has a cause that precedes it, but a point should be
clarified here.
Even if it might not seem so, the notion of cause, in
the strict sense of the term, gave a lot of trouble to
physicists who ended up abandoning it. After playing an
essential role in 17th and 18th century physics, its impor-
tance declined with the emergence of probabilities in
statistical physics. In the 20th century, quantum physics
definitely wiped it out. Indeed, quantum physics uses
probabilities in a way that precludes referring to a cause,
in the strict sense of the term, when talking about quan-
tum processes. Therefore, the notion of cause slowly
disappeared from scientific theories in favour of the no-
tion of physical laws, or was absorbed by the dynamics
of systems.

E. Klein / Natural Science 2 (2010) 212-219
216
The fact remains that the principle of causality, even
freed of the notion of cause, has enabled the develop-
ment of modern physics’ theories, and deeply structures
those currently being elaborated, such as superstring
theory. It sets an absolute and compulsory order between
several types of phenomena, even if none can be pre-
sented as the cause of another, and thus imposes a direc-
tion to time. In short, the principle is reduced to a classi-
fication method for events that fall under its influence, a
rule that organizes them according to a systematically
constraining order.
In practice, the different formalisms of physics adapt
the principle of causality to themselves by giving it a
form that depends on how events and phenomena are
represented. Its consequences are always constraining. In
Newtonian physics, causality implies that time is linear
and non cyclical (which is enough to guarantee that an
effect can’t influence its cause retroactively). In special
relativity, it posits that a particle can’t travel faster than
the speed of light (which is enough to render travelling to
the past impossible). In non-relativistic quantum physics,
causality is guaranteed by the structure of Schrödinger’s
equation5. In particle physics, causality made it possible
to predict the existence of antimatter, and it is now for-
mally expressed by CPT invariance to which the dynam-
ics of physical phenomena must respond. What does CPT
invariance represent? The fact that physical laws ruling
our universe are perfectly identical to the rules of a uni-
verse in which matter and antimatter would interchange
their roles, observed in a mirror, and where time would
go backward (CPT invariance imposes that the mass and
lifetime of particles must be strictly equal to the mass and
lifetime of their antiparticles).
Putting aside their most technical aspects, these dec-
linations and implications of the principle of causality
are quite clear, so clear that they tend to hide a funda-
mental conceptual problem. In fact, the notion of causal-
ity can’t be thought of, or even defined, outside of events
that embody it. How it relates to the course of time then
becomes partly ambiguous: if the principle of causality
constrains the course of time, this means that the latter is
indirectly “contaminated” by causally related phenom-
ena that occur within it. In other words, the principle of
causality (partly) aggregates the course of time and
temporal phenomena despite the distinction established
between them.
But we would like to emphasize one thing: in every
physical theory, once the course of time is dependent
upon the principle of causality, it then becomes com-
pletely irreversible, in the sense that an instant can’t oc-
cur twice. This irreversibility can never be compensated
or erased by the reversibility of any movement or dy-
namical process; as fast as one can possibly return from
Paris after being to Genève, time has irreversibly passed
during the trip and one is therefore a bit older. More
generally, the absence of the arrow of time doesn’t stop
the hours from passing.
The course of time possesses a direction that is quite
different from the arrow of time. When it exists, the ar-
row of time appears in addition, “filling up” the irre-
versible course of time with irreversible phenomena. We
shall later see that physicists have identified possible
explanations for the irreversibility of phenomena. All of
them presuppose the existence of a set course of time
within which time-oriented phenomena take place.
While time passes, the course of time in itself doesn’t
change (all successive instants are equal in the sense that
they have the same status with respect to physical laws).
It doesn’t change throughout time its way of being time.
Thus it escapes becoming (the variable t, designing the
time, doesn’t vary according to time). It is the arrow of
time that constitutes the true expression of becoming. It
manifests itself within the course of time, which it
doesn’t affect in any way but which it overruns with
mostly irreversible phenomena. In some respects, the
notion of “the course of time” therefore precedes the
notion of becoming.
This is also what the second law of thermodynamics
suggests: the assertion that the entropy of isolated sys-
tems going through a spontaneous transformation only
increases with time implicitly presupposes that the
transformation in question follows the direction of a time
that leads us from our “past” toward our “future”, and
not in the opposite direction. It presupposes that the
course of time has first been defined.
It is possible to illustrate the distinction between time
and becoming, to make it visible. Let’s look at the work
of Roman Opalka who, every day since 1965, has been
painting a series of integers on canvases then photo-
graphing himself after each work session. The succes-
sion of numbers materializes the irreversible course of
time, which happens even if nothing happens; if each
number drawn is new (and each moment is completely
new), it is always obtained by adding a unit to the pre-
ceding number. As for the photographs the artist takes of
himself regularly under unchanging conditions (on a
white background, with a white shirt, under a white light,
with the same facial expression), they show a series of
physical changes over time, that is to say the irreversi-
bility of his own becoming. On the one hand, the course
of time is represented by the succession of numbers and
the accumulation of canvases; on the other, becoming is
represented by a series of photographs of the same being
changing. This dual representation is enough to prove, if
not to demonstrate, that these two kinds of irreversibility,
which always appear entangled to the point that they
seem whole, can in fact be separated. But since they’re
so often combined and have simultaneous effects, it is
5In quantum physics, the Hamiltonian is the mathematical operator that
describes a physical system’s evolution throughout time. Schrödinger’s
equation makes this operator into the infinitesimal generator of time
translations. The principle of causality is therefore respected.

E. Klein / Natural Science 2 (2010) 212-219
217
hard for us not to confuse them even though they are
always just superposed.
4. WHERE DOES THE ARROW OF TIME
COME FROM?
Why are some temporal phenomena irreversible or oth-
ers not? When a phenomena is irreversible, that is to say
when an arrow of time appears, what is its origin?
The arrow of time wasn’t part of physics fundamental
formalisms from the start, neither in classical mechanics,
nor in quantum physics and the theory of relativity.
Therefore how do we understand it? Where could it have
come from?
The question appeared only a century and half ago,
when physicists started to ask themselves if physical
phenomena could “go in both directions”: can a dynamic
process capable of changing a system from a state A to a
state B make it change from a state B to a state A? This
question was born of the conjunction of two apparently
contradictory observations:
1) Daily, we can observe around us many physical
processes for which corresponding reverse processes
have never been observed or are exceptional. Therefore
these are, by definition, irreversible phenomena.
2) Yet none of the dynamics laws that govern these
processes contain temporal asymmetries, that is to say
that they would be the same if the course of time was
going in the opposition direction. If they allow a certain
process to occur when time goes into one direction, they
allow it to happen when it goes into the opposite direc-
tion: the initial and final states could be interchanged.
Such equations are called “T-invariant equations”: if a
system can go from state A to state B, it should be able
to go from state B to state A (in that case, the system
isn’t concerned with the arrow of time).
Therefore, why are there some irreversible phenom-
ena? Why is there an arrow of time, that is to say an
asymmetry in the dynamics of certain phenomena that
we observe, even though the equations of physics have
no room for it?
In view of what we have stated above, these questions
can’t be answered by explaining “the direction of time”,
by setting out the reasons why it flows in one direction
rather than another, or even less by explaining why we
don’t remember the future. The issue is solely related to
the asymmetry of physical processes within time and not
to the asymmetry of time itself. It is an asymmetry of the
“contents” of time, not an asymmetry of the container
itself.
To try to solve this riddle, physicists advance four
categories of argument that can delimit the origins of the
arrow of time, and they also study their possible in-
ter-relations. We will present them briefly:
 The second law of thermodynamics, or the increase
of the entropy of isolated systems. In Boltzmann’s
interpretation, which underlies this principle, there is
no arrow of time at the microscopic level, but on a
macroscopic level, one can get the impression that
one exists.
 The process of measurement in quantum physics,
which has been the subject of intense debate for
eighty years. Generally, it is understood as a tempo-
rally asymmetrical process.
 The violation of CP symmetry during certain phe-
nomena governed by the weak interaction: some un-
stable particles, for example neutral kaons, don’t be-
have exactly like their anti-particles. More specifi-
cally, they don’t disintegrate into other particles at
the same pace than their antiparticles. This means
that they disintegrate according to a temporally
asymmetrical law. The fundamental reason for this
temporal asymmetry, which remains hard to interpret,
is not completely understood. It raises the question of
the existence of an “arrow of time” at the micro-
scopic level ;
 The expansion of the universe, which would make it
impossible for any system to return to its initial state
because the universe itself is evolving. This can ap-
pear contradictory since the equations of general
relativity are temporally symmetrical, but in reality
their cosmological solutions, which are supposed to
govern the evolution of the universe, are not. The
universe they describe is either expanding, or con-
tracting, as represented by the existence of an arrow
of cosmic time related to the conditions at the limits
of the universe. Some theorists, including Stephen
Hawking and Roger Penrose, think that this arrow of
time could be the arrow mastering all the others, but
not all physicists share this position.
We’ve gone far enough to be able to make two re-
marks.
The first one is that the attempts to explain the arrow
of time resort to arguments that all differ from the re-
strictions imposed on the course of time by the principle
of causality. (We mentioned them earlier: linear time,
the impossibility of going beyond the speed of light, the
existence of anti-matter, CPT invariance). In conclusion,
the course of time is accounted for in ways that never
coincide with ways in which the arrow of time is justified.
This indicates – or even demonstrates – that the course
of time and the arrow of time are two distinct things in
contemporary physics; the irreversibility of phenomena
doesn’t come from the irreversibility of time and vice
versa.
The second remark is that none of the explanations
given for the arrow of time is likely to constitute a real
theory. They are closer to an interpretation of such or
such physical theories, but are not incorporated into any
formalism. There is indeed no operating physical theory
that integrates becoming from the start (through the use

E. Klein / Natural Science 2 (2010) 212-219
218
of irreversible fundamental equations). Consequently,
becoming can only be accounted for in physics through
the reading of theories that don’t include it among their
principles. So interpretations of the arrow of time’s ori-
gins end up mixing physics and philosophy. Thus, they
can be subject to disagreement and are indeed very ar-
dently disputed. These disagreements are not without
similarities to the debate between supporters of Par-
menides and supporters of Heraclitus. Some physicists
think this is only a fake problem: on the pretext that no
arrow of time appears in physics’ fundamental equations,
they believe, like Parmenides, that becoming is only
pure appearance and is closely related to how our limited
senses make us perceive the world. Others, following the
Heraclitian tradition, consider that because actual phys-
ics can’t explicitly account for becoming, it is either
wrong, or incomplete.
These two positions can be defended as long as there
is agreement on the meaning of words. And also as long
as no one is claiming that physics has negated time just
because its formalisms don’t include the arrow of time.
For if becoming wasn’t integrated directly into its prin-
ciples, physics has always referred to the course of time.
One can regret that physics hasn’t integrated becoming
from the start – or better suggest how physics could
make room for becoming in its formalisms -, but it can’t
be blamed for forgetting to integrate the course of time.
Although, on paper, it is possible to change the sign
of time in a physical equation, this doesn’t imply that the
course of time can be physically reversed. Only the di-
rection of phenomena can be physically reversed, not the
direction of the course of time.
However let us be clear: it is not excluded that the
flow of time and the arrow of time come from the same
source, more profound than they both; that they are
by-products of underlying phenomena that a “new phys-
ics” might reveal. Moreover, some progress has recently
been made in that direction, in characterizing causality
independently of any concept of time and deriving both
time and becoming from an ordering relation on sets of
events taken as primitives. It may thus appear that cau-
sality cannot be understood as a feature of the world that
would exist independently of any phenomenon. Causal-
ity would be intrinsically shaped by the phenomena. As
the principle of causality underlies our representation of
time in physics, this would give some formal foundation
to a close connection between time and becoming. But
for the time being, it would be wiser to formally distin-
guish them in order to make the arguments clearer.
This is not always the case. For example, in the cos-
mological context, it is often told that the problem of the
arrow of time owes its origin to the intuitive asymmetry
between past and future. According to what we have
shown, the expression “course of time” would be there
more appropriate. Because of this confusion, many au-
thors argue that the direction past-to-future is related to
the direction of the gradient of the entropy function of
the universe, forgetting that the definition of the entropy
of the universe is a very controversial matter: entropy is
a thermodynamic magnitude that is typically associated
with subsystems of the universe, and not with the uni-
verse as a whole. To evade this difficulty, some authors6
have tried to define a course of time for the universe
only on the basis of the geometrical properties of
space-time, independently of any entropic consideration.
Unfortunately, they have called this direction past-to-future
the “global arrow of time”. The expression “global
course of time” would have suited better.
5. WHAT MAKES THE TIME FLOW?
The nature of the “engine of time” that makes us feel the
flow of time has not been elucidated but a lot of theo-
retical work is now being devoted to this problem. Dif-
ferent avenues are being explored. In fact, there have
been three major theories of time’s flow. The first, and
most popular among physicists, is that the flow is an
illusion, the product of the faulty river metaphor. The
second is that it is not an illusion but rather is subjective
being deeply ingrained due to the nature of our minds.
The third is that it is objective, a feature of the
mind-independent reality that is to be found in, say, to-
day scientific laws, or, if it has been missed there, then
in future scientific laws.
The first theory, rooted in the theory of relativity,
represents space-time as a fixed whole and suggests that
the flow of time is a pure illusion: the entire universe
just is, with no special meaning attached to the present
time. All past and future times are equally present, have
the same degree of existence along time, just as different
locations coexist along space. According to this view,
there is nothing special about the “now”. Incidentally, in
the special theory of relativity, there is an uncountable
infinity of nows, and the standard symmetries assure that
none of them can have special significance.
In the second theory, time would only be a psycho-
logical feature linked to the very complex structure of
our brain; in the space-time region we are observing, we
have the feeling that time passes “from the bottom to the
top” of space-time, but in reality the space-time is a rigid
block without any internal dynamics. We observers
would unfold the thread of time ourselves. In other
words, we would be the “engine” of time.
The third theory considers that time’s apparent flow is
real, that it corresponds to a true physical reality. At any
moment in time an observer perceives a “now”; future
events are not only unknown but objectively
6See for example: Mario Castagnino, Olimpia Lombardi, Luis Lara,
The Global Arrow of Time as a Geometrical Property of the Universe,
Foundations of Physics, Vol. 33, N° 6, June 2003; G. Matthews,
“Time’s arrow and the structure of space-time”, Philosophy of Science
46, 82-97 (1979).

E. Klein / Natural Science 2 (2010) 212-219
219
non-existent, to be created later as the now advances.
Thus physics should grant time’s flow a well defined
place in its formalisms7.
It is not our purpose here to discuss these theories in
detail or to argue for or against any one of them. We
merely wished to stress that the common semantic care-
lessness when it comes to the expressions “course of
time”, “direction of time” and “arrow of time” makes the
arguments of all parties more confusing than they really
should be. If these expressions were better defined, sys-
tematically distinguished from one another and always
used in their strictest sense, the debate about time, irre-
versibility and becoming in physics would become
clearer.
6. CONCLUSIONS
What can these considerations teach us? That a more
carefully chosen vocabulary and a more rigorous con-
ceptualization would give us a chance to show how the
different theories formalize the course of time, interpret
the arrow of time, and relate time and becoming. It
would allow us to better think about the question of
time.
Therefore, the principle of causality could benefit
from being renamed “antecedence principle” or “princi-
ple of chronological protection”, as Stephen Hawking
proposed it. Similarly, when referring to a physical
process, the quite awkward expression “time reversal”
could be replaced by the expression “movement rever-
sal”, since the intention is not to create a time machine
but to reverse the speed of the physical entities con-
cerned. When a phenomenon’s dynamics are reversible,
the course of time (or the direction of time) is indeed
arbitrary, but once it has been chosen, it can’t simply be
reverted.
Finally, the situation is the same with the course of
time as with electrical charges. Saying that the electron
carries a negative charge and the proton a positive one
results from a convention. To change this convention
and declare that an electron’s charge is positive and a
proton’s negative wouldn’t change anything to the laws
of physics or the universe. Beginning with a conven-
tional choice, it is therefore possible to design physical
laws that are unconventional.
To claim that the course of time doesn’t exist accord-
ing to physics under the pretence the laws of physics are
time-reversal invariant (so that the direction of time is
arbitrary) is equivalent to saying that electrical charges
have no reality because physical laws don’t change if
each charge’s sign is reversed.
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7See for example “
B
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d
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