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SOME ARGUMENTS ABOUT TIME
Etienne Klein
DRF/LARSIM
Centre d’Etudes de Saclay
91191 Gif sur Yvette cédex
Language is not neutral. The way we use the word “time” has made it a sort of autonomous entity
that exists on its own, independently of things and processes. A first question is: does this
ontological promotion have anything to do with the physical reality of time?
It so happens that the question of whether or not time is a particular substance is precisely one of
the most important issue facing contemporary theoretical physics. For three-quarters of a century,
this question has been torn between two representations of spacetime which it would like to unify to
allow the four fundamental interactions of nature to be defined at the same time. The first
representation comes from quantum physics, which describes the movements of elementary
particles and considers spacetime to be flat and static. The second is general relativity, which
describes how spacetime is curved by gravitation, and considers it to be flexible and dynamic,
constantly deformed by the movement of the matter and the energy it contains. In other words,
researchers must now elaborate a theory capable of describing “quantum gravity”.
But what exactly is meant by quantum theory of gravity and on what can it be built? Is it a matter
of applying quantum physics procedures to general relativity? Or unifying two approaches, which
would entail modifying standard quantum physics? Or else elaborating a new theory which would
exceed, and include at the same time, quantum physics and general relativity?
These different approaches can be divided into three groups:
1/ all the procedures that apply quantization rules to ordinary general relativity. Two approaches
can be distinguished in this group: “covariant” approaches, which make no attempt to find an a
priori definition of time; and “canonical” approaches (such as loop quantum gravity), which start
from an a priori definition of time in the spacetime of general relativity.
2/ Superstring theory, which is currently the most widely studied approach.
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3/ All the approaches which do not come under the first two, for example Penrose’s twistor
theory or Alain Connes’ noncommutative geometry
These ongoing avenues of research have a point in common: they return to and renew major
philosophical problems which, for several centuries, concerned nature of space and time, and which
now touch upon the nature of spacetime in general.
Let us begin by looking at the debates on space. In the early 18th century, at the time of the
famous controversy between Samuel Clarke, a disciple of Newton, who believed in the substantial
reality of time, and Gottfried Wilhelm Leibniz, for whom neither time nor space had any real
existence outside the objects they connected1. If space is a particular substance, then it can exist
even if nothing else does. It is a stage containing all physical objects. It exists by itself, prior to the
objects, in such a way that these can be said to move within it. If, however, space is relational, it
must be considered that the world is first made up of physical objects, but without jumping to the
immediate conclusion that these are in space. According to Leibniz, space does not precede physical
objects. It is not self-sufficient. It is merely a web of relationships between things, not what today’s
physicists call the “background of phenomena”. Its role appears secondary to that of objects, to
express the relations of contiguity that exist between them.
In fact, the substantialist conception of space posits the existence of two types of entity in the
universe: physical objects and space. For the relational conception, however, there is only one
reality: interrelated physical objects2.
Where does physics stand in the debate? It does not put all its eggs in one basket. It does not
grant space the same status in all its theories. Like Newtonian mechanics, quantum physics posits a
substantialist conception of space, describing it as a manifold possessing determinate spatial
properties (for example, the distance from one point to any other is clearly defined without it being
necessary to consider the material content of the universe). The special theory of relativity does the
1. Leibniz’s arguments were, in particular, based on the principle of sufficient reason and on that of the
identity of indiscernibles. If absolute space exists, why would God have put it here rather than two feet
farther away? And would it not be possible to tell two perfectly identical objects apart simply by their
different positions in space?
2. In his Science of Mechanics, published in 1883, Ernst Mach undertook a radical criticism of the concept
of absolute space, in which he took up some of Leibniz’s arguments, and developed others, drawing on the
physics of his day. In his opinion, considerations relating to the choice of a particular class of reference
frame represented an intrusion of unjustifiable metaphysical considerations in the scientific domain. His
criticism was based on the fact that space had no separate existence. It could only be seen as a relationship
between objects. Mach, however, added new arguments to this classical relationism, encouraged by the
positivist context of the late 19th century, which sought to reduce physical properties to observable quantities
alone. He proclaimed that only experimental reality mattered, advocated that senses formed the basis of all
mental experiences, and wished to rid scientific statements of any metaphysics.
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same with spacetime, which it presents as a given making up what is called the “background” of
events.
General relativity, however, advances what seems to be a more relational conception in which
spatial relationships are not considered as existing a priori. There is no prior determination of
spacetime geometry: it obeys a dynamic equation (Einstein’s equation) which makes it dependent
on the quantities of matter and radiation present in the universe. It can thus be seen as a variable of
the theory. Within this context, no radical distinction can be made between space (or rather
spacetime) and physical objects or processes, as spacetime itself obeys physical laws. The notion of
empty spacetime, in the sense of a stage that would exist even if nothing was happening on it, does
not apply, as spacetime itself is physical and obeys laws that make it dependent on its contents, and
is interlinked with the phenomena occurring within it.
I should add that this theory also involves a “metric tensor”, which is used to calculate angles,
lengths and durations1. This tensor is space- and time-dependent and is not cancelled out anywhere.
Consequently, spacetime appears non-empty in that it always contains a non-zero field - namely the
metric tensor - which obeys Einstein’s equation, just as electromagnetic fields obey Maxwell’s
equations. Thus, empty space, in the Newtonian sense, does not exist for general relativity, as space
also comes with its metric.
It nevertheless remains that this conclusion is problematic and has been hotly debated,
particularly by Stephen Hawking and Roger Penrose2.
Now let us consider time. It was long considered separately from space and has also been at the
centre of a controversy between those who claim it is substantial in nature, and those who say it is
relational. Newton, of course, sided with the first group: the absolute time of mechanics, which
passes homogenously, unrelated to anything outside it, is one of the incarnations of time as
substance. In the second group, we find Leibniz again, as well as Ernst Mach.
If time is relational, then we cannot say we evolve in time. Time is simply the reflection of a
dynamic related to phenomena and which cannot be defined in terms of time. In this context, time
emerges from a world that does not contain it. It no longer belongs to the background.
The question of whether or not time is substantial in nature is by no means settled. On the
contrary, it is the issue of the hour, and the physicists working to elaborate a quantum gravity theory
give frequent reminders that their research rekindles this debate. Some of them, including Abhay
Ashtekar, Carlo Rovelli and Lee Smolin, the three founders of loop quantum gravity 3, go so far as
1. This metric tensor is denoted by “g()”.
2. See Stephen Hawking, Roger Penrose, The Nature of Space and Time
3. Apologies for the jargon needed here to describe the principles of this theory. The theory is based on the
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to claim that any physical theory worthy of the name must be formulated without referring to any
pre-existing background or any spacetime frame given a priori.
While not all physicists share this view, they all agree that the problem of unifying general
relativity and quantum physics boils down to the question: does time receive events or does it
emanate from them? Could time derive from one or more concepts that are deeper than itself?
WILL AN EQUATION TELL US WHERE TIME COMES FROM?
The standard model of particle physics posits that time exists, independently of phenomena, it
notes that it goes by, but without specifying its nature or what makes it go by.
The standard model comes up against several conceptual problems. First, at energy levels higher
than those attainable using particle colliders, its underlying principles themselves enter into
collision with one another, with the result that the equations no longer work. This indicates that the
conceptual framework currently in use does not describe the phenomena that occurred at higher
energy levels in the primordial universe. Second, the standard model of particle physics ignores the
gravity. How can it be integrated?
To do so, theoreticians feel free to formulate all kinds of strange hypotheses. According to some,
for example, spacetime could be discrete rather than smooth, or might not really exist, or could have
more than four dimensions, etc.
Could space and time be a sort of foam?
A first promising avenue of research was proposed by the mathematician Alain Connes, who
elaborated “noncommutative” geometries. Physicists previously preferred to consider space and
time as “smooth” entities, which could thus be represented as continuous quantities: there would be
space everywhere, and there would always be time, with no possible gaps, making it possible to
consider lengths and durations, no matter how small, without ever reaching a limit.
But why not imagine that space itself could be “discrete”, not continuous, with a structure like
that of a network in which the finite, non-zero mesh would represent the shortest possible distance?
Alain Connes and his noncommutative geometry makes it possible to consider discrete structures
without breaking fundamental symmetries. In order to build such atypical geometries, the usual
canonical quantisation of general relativity in a Hamiltonian formulation, with the other three fundamental
interactions initially left aside. One of its predictions is that space must have a discrete structure, as opposed
to the spacetime continuum of general relativity: areas and surfaces and quantised. This theory competes
with superstring theory, at least as far as gravity is concerned. See Carlo Rovelli’s work What is Time? What
is Space?
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spatial coordinates, which are ordinary numbers, must be replaced by “algebraic operators” which
display the property of not commuting with each other. These algebraic operators are not “neutral”
in the sense that they verify certain relationships that define spatial properties at a very small scale.
Consequently, there is always “space”, or more precisely a spatial structure which, on very close
examination, does not exhibit the ordinary properties of space.
This new construction owes its strength to the fact it produces all the usual properties of space but
on a larger scale. It allows to consider that space as we know it actually emerges from an underlying
structure that is very different from it. For example, the smooth aspect of space - its apparent
continuity - could be regarded as a foam floating above a discrete network of points. It is like
looking at a television screen. If we watch with our nose pressed up against it, we can only see dots
in three different colours, and no picture as such; the picture only emerges gradually, along with all
its colours, as we move away from the screen. Similarly, space with all its continuous properties,
may only have appeared once the universe had exceeded a certain size.
Couldn’t a similar concept work with time, on the grounds that it is tied to space by the theory of
relativity? Couldn’t it be discrete deep down? We must avoid jumping to conclusions here based
only on our common sense. Calculations may one day disprove our experience of time, and their
consequences might seem absurd at first glance.
Could spacetime be an application of causality?
It is traditionally considered that an event represented by a point in spacetime is a primary datum,
and that the relationships between two events are only secondary data: only the event itself is
considered real, while causal relationships are only ever accessory. Wouldn’t it be possible,
however, to reverse the situation and consider that causal relationships are genuine fundamental
elements, that can then be used as a basis for defining events?
In the 1980s, Roger Penrose opened up another avenue by proposing a conception of spacetime
based on what he called the “causal structure of the universe”, in which spacetime is built from
causality, rather than being the arena in which it (causality) is expressed. According to general
relativity, spacetime geometry dictates the propagation of light, which can only follow certain paths
called geodesics of light1. In order for two events to have a causal relationship, a particle must have
been propagated from one to the other. No particle, however, can travel faster than light. That being
the case, knowing the geodesics of light allows us to determine what event(s) may have been caused
1. The term “geodesic” refers to the shortest path between one point and another. In curved space,
geodesics are generally not straight lines. The geodesics of light, those along which photons travel, are a
particular form of geodesic, of zero length.
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by a given other event. The answer is all the events linked to the event in question by a signal
travelling at or below the speed of light. Spacetime geometry therefore contains information relating
to the causal relationships between events - information that makes up the “causal structure of the
universe”.
Knowing this structure makes it possible to determine whether or not one particular region of the
universe can transmit information to another, and thus learn which region can have a causal
influence on another. It forms a kind of spacetime web that indicates all the paths along which
causal relationships can propagate
According to Roger Penrose, this causal structure of the universe is its key property. What is
essential is no longer all the events likely to occur within spacetime, but rather all the possible paths
travelled by rays of light capable of interconnecting events. This predominance of light leads
Penrose to a complete reversal of perspective. Instead of considering that spacetime geometry
determines causal relationships, he suggests that it is causal relationships that determine spacetime
geometry. His argument is simple: most of the information we need to define spacetime geometry
is integrally frozen once we know how light travels there.
Roger Penrose calls all rays of light together the “twistor space”: each ray, which corresponds to
a geodesic of light in spacetime, is represented by a simple point in twistor space and, reciprocally,
each point in spacetime can be reconsidered as all the rays of light passing through it, in other
words as a manifold in twistor space.1 This gives rise to a relationship of correspondence between
twistor space and spacetime, a relationship which leads us to consider that the second
is....secondary, i.e. that it derives from the first. This is just a step away from saying that twistor
space is a more fundamental entity than spacetime and that it should be considered as the basis for
reformulating the laws of physics. Roger Penrose did not hesitate to take that step.
Twistor theory2 developed rapidly over the twenty years that followed Penrose’s initial proposal.
It came as a surprise to nearly all physicists that many equations could be reformulated in twistor
space.
The successes of this approach are only partial, but they have convinced many theoretical
physicists that the concept of causality operates at levels underlying spacetime itself. Since then,
research aimed at understanding the nature of space and time have all used a combination of three
1. Mathematicians know that complex numbers can be represented in a plane (the complex plane) or, if a
point at infinity is added, on a sphere known as the Riemann sphere. This sphere can revolve around its own
axis and thus become a twistor. In spacetime, rays of light are geodesics. In twistor space, each point in
spacetime, in other words, each event, is represented by a Riemann sphere, which corresponds to all the rays
of light passing through it.
2. Roger Penrose himself provides a fairly accessible presentation of this theory in a book that he wrote
with Stephen Hawking entitled The Nature of Space..., op. cit.
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fundamental ideas: spacetime is emerging; its most fundamental description is discrete; and
causality is a crucial part of this description.
Does spacetime have hidden dimensions?
Superstring theory took up and added to the Kaluza-Klein hypothesis by introducing ten-
dimensional spacetime, six of which are curled up. The six extra dimensions are assumed to be
spatial rather than temporal, so that this theory should not have any impact on the way we represent
time, which continues to be purely one-dimensional1. Furthermore, unlike twistor theory and
noncommutative geometries, spacetime in superstring theory does not derive from an entity deeper
than itself. It is posited a priori, before any other thing, even if the theory strangely suggests that it
could change, or even disappear locally in a black hole. In a word, we cannot reasonably expect this
theory, or rather this research programme, to disclose the profound nature of spacetime in the very
near future. It might, however, at least tell us how many dimensions spacetime has.
Physics happens to produce theoretical or experimental results that illuminate or even
modify the philosophical answers that we bring to philosophical questions. One of these
philosophical questions that collide with physics is the following: Is time identical to becoming?
Physics does tell us something about that. I say “does tell us,” which is a strange way of talking
about physics since physics does not speak. But what I would like to do in this paper is try to
express what the equations of physics would say about time if they could speak.
1 Is Time Identical to Change? Roman Opalka’s Lesson
We notice time because of change and in most situations, time and change appear entangled
to the point that they seem the same thing. But that entanglement does not imply that time is
change. In fact, situations exist in which they can be explicitly separated. Let us look at the work of
Roman Opalka, who, every day since 1965, has been painting a series of integers on canvases then
1. If at least one of the extra dimensions were to be temporal, it would imply contemplating the existence
of several times at once, which does not seem easy. A further difficulty lies in the fact that these extra
temporal dimensions would be curled up to form loops, within which the principle of causality could not
apply.
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photographing himself after each work session. The succession of numbers materializes the
irreversible course of time; each number drawn is new (and each moment is completely new), but it
is always obtained in the same way––that is, by adding a unit to the preceding number. As for the
photographs that the artist takes of himself periodically, they show a series of physical changes over
time, that is to say, the irreversibility of his own becoming. On the one hand, the course of time is
represented by the succession of numbers; on the other, becoming is represented by a series of
photographs of the same being (Opalka himself) changing and becoming older. This dual
representation is enough to demonstrate that these two kinds of irreversibility can be distinguished
and that their difference can even be made visible.
The question is: Does physics also distinguish between time and becoming?
The first point is to notice that physical time - let us say Newtonian time to begin with - does
not have the properties that our way of speaking about time attributes implicitly to it. Let us take an
example. We often say that time flows like a river, which suggests that time has a certain speed,
because the flow of a river does have a speed (by the way, in everyday language, time is constantly
granted the property of speed). But speed is the derivative of a certain quantity in relationship to
time. The speed of time is then obtained by our determining the rate of the variation of time in
relation to itself, and this operation has of course no meaning. If one really wants to define the
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speed of time, one still can say that it is equal to one hour per hour and that won’t help you very
much. The metaphor of the river turns the simplest arguments into traps.1
The second point is an observation. We talk about time as if it corresponded only to a
“becoming,” in other words to the stream of changes affecting an object, a person, an institution, a
physical system. Change is truly the phenomenon that best suggests the idea of time, and one can
easily understand why: In life, 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 argument that time and becoming are the same is a step that is too easily made
without further investigation. We should be careful before saying that, for the following reason:
time is mostly referred to as if it looked like what it holds, in the sense that common thinking
engenders confusion between time and temporal phenomena. For example, time is said to stop or
disappear when nothing seems to be happening, as if its dynamics only depended upon its contents.
But are we right when we say such a thing? We should not answer too quickly, because a crucial
question has to be first examined: Is time an abstract structure into which events are inserted, that is
to say a reality in itself preceding all possible events and as such different from becoming ? Or is it
composed of the stream of events itself?
These are the questions I would like to discuss by examining the kind of answers physics provides.
2 Time and Becoming: Physics Sees Double
Physical theories are mostly composed of equations. What would the equations say about
time and becoming if they could speak? For this purpose, we have to study the structure of physical
theories (that is classical physics, quantum mechanics, special or general relativity). This study
shows that the formalisms of physics do distinguish time from becoming. On one side, there is the
1 On this point, see Klein (2005)
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course of time, a primitive entity, on the other side there is the arrow of time, which is not a
property of time but a property of the majority of phenomena taking place in time, specifically
irreversible phenomena.
The course of time does establish an asymmetry between earlier and later. If two events are
not simultaneous, then one of them is earlier than the other one. So defined, the course of time
expresses the irreversibility of time itself; it is not possible to reverse the earlier/later order of the
events.1 As for the arrow 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 phenomena within the course of time and not the
irreversibility of time itself.
Today, physics has become so effective (and the discovery of the Higgs boson in 2012 is a
new demonstration of that) that it is possible to imagine that the distinction it makes between time
and becoming could be transferred to philosophy, which often aggregates the two notions. In other
words, the distinction it establishes between time and becoming represents a “negative
philosophical discovery” because it modifies the terms in which the philosophical question of
becoming is stated.
The course of time is usually represented by a line, a timeline on which a little arrow is
usually drawn, an arrow that is not the arrow of time in the sense introduced above: it is there to
indicate that time has a dynamics oriented in a single direction, and that time travel is indeed
impossible. It is impossible to come back or to go through the same instant twice.
By the way, we have to notice that the depiction of time as a line is incomplete because it
omits indicating how this line is built. Because the present does not bring another present by itself,
1 This issue is more complicated in the Special Theory of Relativity, but this irreversibility
holds for events that are time-like separated. See Klein (2005).
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there has to be something, an “engine” of time, to do this “work.” This little engine does build the
course of time, in the sense that it continuously renews the present. 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 we have to consider that the true mystery of time exists in the hidden dynamic
that builds the timeline.
As for the arrow of time, contrary to what the expression 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 arrow, wrongly called the “arrow of time.”
The problem of the arrow of time can be 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: the future
is the direction toward which the entropy of the system increases. But in fact, the question asked
does not concern the arrow of time because 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), which has to be distinguished
from the first one.
This example of confusion shows that it is worthwhile to emphasize the difference between
several issues traditionally labelled “the problem of the direction of time.” The most invoked
concepts are the concepts of irreversibility and of time-reversal invariance. Time-reversal
invariance is a property of physical laws: a law is time-reversal invariant when it is expressed by a
differential equation which is invariant under the transformation t → -t. By contrast, irreversibility
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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.
Since Newton, the principle of causality has always constrained from the outside the
representation of the course of time in physics. This principle has generally been summarized by
our saying that every event has a cause that precedes it, but this formulation has to be refined
because the concept of cause appears to be unclear in quantum physics. The principle of causality
has now a statement which does not refer to the idea of cause: it says that recorded history cannot
be changed, in the sense that any event that has occurred cannot be eliminated from the past.
The principle of causality sets an absolute temporal order between several types of events,
even if none can be presented as the cause of another, and it thus imposes a “directionality” to time.
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 cannot influence its cause retroactively). In special
relativity, causality 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 equation.1 In particle physics, causality made it
possible to predict the existence of antimatter, and it is now formally expressed by CPT (charge
conjugation, parity transformation, time reversal) invariance to which the dynamics of physical
phenomena must respond. What does CPT invariance represent? It represents the fact that physical
1 In 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 automatically
respected.
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laws ruling our universe are perfectly identical to the rules of a universe where matter and
antimatter would interchange their roles, if observed in a mirror, and where time would go
backward.
But one thing has to be emphasized: In every physical theory, once the principle of causality
is taken into account, the course of time then becomes irreversible in the sense that an instant cannot
occur twice. This argument leads to the question of knowing whether the course of time is
irreversible by itself or whether it is due to the fact that it contains events causally linked to each
other. But the key point is that this irreversibility of time can never be compensated for or erased by
the reversibility of any movement or dynamical process; as fast as one can possibly return from
Paris after being to London, time has irreversibly passed during the trip and one is therefore a bit
older (which would not necessarily make you look any different). More generally, the absence of
any arrow of time doesn’t stop time from passing.
When it exists, the arrow of time appears in addition: it “fills up” the irreversible course of time
with irreversible phenomena. We shall later see that physicists have identified possible explanations
for the arrow of time: all of them presuppose the existence of a set course of time within which
time-oriented phenomena take place.
While time passes, it doesn’t change its way of being time. Thus it escapes becoming. 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. The notion of “the course of time” therefore
precedes the notion of becoming, as in the work of Roman Opalka.
3 Where Does the Arrow of Time Come From?
When a phenomenon is irreversible, that is to say when an arrow of time appears, what is its
origin?
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The arrow of time was not part of fundamental formalisms of physics from the start, neither
in classical mechanics, nor in quantum physics nor the theory of relativity. Therefore how do we
understand it?
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, they would be the same if the course of time was going in the
opposite direction. If they allow a certain process to occur when time goes in one direction,
they allow it to happen when it goes in the opposite direction; the initial and final states
could be interchanged (for example, according to the Newtonian equations for gravitation,
planets could rotate around the Sun in directions opposite to what they are). 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 phenomena? 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
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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 (see Price 2002).
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 inter-relations. I will just mention
them briefly (see Zeh 1989 and Savitt 1995):
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 temporally asymmetrical process.
The violation of CP symmetry during certain phenomena governed by the weak interaction.
Some unstable particles––for example, neutral kaons–– do not behave exactly like their anti-
particles. More specifically, they don’t disintegrate into other particles at the same pace as
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 microscopic 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 argument can appear
contradictory because 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 contracting, 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 (1994) and Roger
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Penrose (1989), think that this arrow of time could be the arrow mastering all the others, but
not all physicists share this position.
We have gone far enough to be able to make two remarks.
The first one is that attempts to explain the arrow of time resort to arguments that all differ
from the restrictions imposed on the course of time by the principle of causality. (I 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 shows 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 this or that physical theory, but they
are not incorporated into any formalism. There is indeed no operating physical theory that
integrates becoming from the start (for example through the use of irreversible fundamental
equations). We can not exclude that this conclusion may change in the future thanks to the building
of a new kind of physical formalism, but for the moment, it seems that becoming can only be
accounted for in physics through the interpretation of theories that do not include it among their
principles. So interpretations of the arrow of time’s origins end up mixing physics and philosophy.
Thus, they can be subject to disagreement and are indeed very ardently disputed. Some physicists
think this is only a fake problem: on the pretext that no arrow of time appears in physics’
fundamental equations, they believe that becoming is only pure appearance and is closely related to
how our limited senses make us perceive the world. Others conclude that because actual physics
can’t explicitly account for becoming, it is either wrong, or incomplete.
17
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 do
not include the arrow of time. Becoming was not integrated directly into its principles, but physics
has always referred to the course of time. One can regret that physics has not integrated becoming
from the start––or, better, suggest how physics could make room for becoming in its formalisms––
but it cannot be blamed for forgetting to integrate the course of time because it did not forget it.
Although “on paper” it is possible to change the sign of time in a physical equation, this
does not imply that the course of time can be “physically” reversed.
4 Should We Adapt Our Vocabulary to What We Know?
What can these considerations teach us? They teach us that a more carefully chosen
vocabulary and a more rigorous conceptualization 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.
They allow us to better think about the question of time in general.
The principle of causality, for example, could benefit from being renamed “antecedence principle”
or “principle of chronological protection,”, as Stephen Hawking (1975) proposed. Similarly, when
we refer to a physical process, the quite awkward expression “time reversal” could be replaced by
the expression “movement reversal” because the intention is not to create a time machine but to
reverse the speed of the physical entities concerned. When a phenomenon’s dynamics is reversible,
the direction of time is indeed arbitrary, but, once it has been chosen, it cannot simply be reversed.
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 derives from a convention.
To change this convention and declare that an electron’s charge is positive and a proton’s negative
would not change anything to the laws of physics or the universe. Beginning with a conventional
choice makes it possible to design physical laws that are unconventional.
18
To claim that the course of time does not exist according to physics under the pretence that
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 do not change if
each charge’s sign is reversed.
5 An Open Question: What Makes Time Flow?
The nature of the “engine of time” that makes us feel the flow of time has not yet been
elucidated, but a a great deal of theoretical work is now being devoted to this problem. Different
avenues are being explored. In fact, there have been three major theories of time’s flow. The first 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, today’s
physical laws, or, if it has been missed there, then in future physical 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 and have the same degree
of existence within 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, which can be considered as a variation of the previous one, time would
only be a psychological 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 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.
19
Contrary to the first and the second theories, the third one 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 non-existent, to be created
later as the now advances. Thus physics should grant time’s flow a well-defined place in its
formalisms. (See, for example, Elitzur and Dolev 2005).
It is not my purpose here to discuss these theories in detail or to argue for or against any one
of them. I merely wished to stress that the common semantic carelessness 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,
systematically distinguished from one another and always used in their strictest sense, the debate
about time, irreversibility, and becoming in physics would become clearer.
6 Conclusion
I have shown that the formalisms of modern physics do clearly distinguish the course of
time and the arrow of time. The course of time is represented by a timeline that leads us to define
time as the producer of duration. As I have pointed out, it is customary to place on this timeline 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 and has a well-defined direction, even if this
direction is arbitrary.
The arrow of time, however, indicates the possibility for physical systems to experience,
over the course of time, changes or transformations that prevent them from returning to their initial
state forever. Contrary to what the expression “arrow of time” suggests, it is therefore 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.
20
Today, physics has become so effective that the distinction it establishes between time and
becoming could be transferred to philosophy, which often aggregates the two notions. We could
even state that it represents a “negative philosophical discovery” because it modifies the terms in
which the philosophical question of becoming is stated.
21
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