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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.

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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.

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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.

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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.

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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.

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