1.0. About this document
This aims to be the beginning of a book in progress on the foundations of mathematics and
mathematical physics, with emphasis on geometry.
These are mainly classical subjects, but presented in a new approach, unrelated to any of the
usual teaching programs.
It is especially intended for gifted students who do not only aim to succeed their exams but want
to acquire deep understandings of those subjects.
1.1. What is mathematics ?
To say it brieﬂy, mathematics is the study of possibly inﬁnite systems made up of pure elementary
objects, whose existence is abstract, independent of the exterior world. The only nature of each
component (element, relation. . . ) of these systems the fact of being exact, of the kind “all or nothing”:
two objects are either equal or diﬀerent, are related or unrelated, a given operation gives exactly one
result. . .
The choice of such systems to study is itself a mathematical choice, in the sense that it is given
by conditions of limited complexity (unlike biology for example, which depends on arbitrary choices
secretly accumulated by Nature during billions of years).
So, mathematical science is autonomous with respect to the circumstances of our world or uni-
verse, but can still apply to each of its so many places or particular aspects where such situations of
systems of limited complexity can be met.
Though the nature of these objects is independent of any form of particular sensation or ap-
pearance, the only way to understand them is to represent them in some way. There can be several
formulations or ways to represent a given theory, all being rigourously equivalent but their eﬃciency,
their relevance can be very diﬀerent. Most often, ideas are ﬁrst developed in forms of more or less
visual intuitions or imaginations, then cristallised into formulas and litteral sentences to be written,
communicated or worked under these forms. And one can be freed from a given particular form
of representation only by developing other forms of representation, and by exercising to translate
objects and concepts from one form to the other.
1.2. What is geometry ?
Notion of geometry
Etymologically, the word geometry means “measure of the earth”. Thus it is originally a science
related to concrete reality: the one of the space around us, or the one of planes: the ground when
it is ﬂat, or any ﬂat surface (paper, board. . . ), that one can see and on which one can easily draw.
The Greek geometer Euclid ﬁrst wrote an axiomatic formulation of it in the case of the plane, thus
giving his name to the usual (plane) geometry.
Euclidean geometry is the ﬁrst physical theory, that is, a mathematical theory modelizing the
physical reality, while idealizing it: whereas two points too close to each other can hardly be distin-
guished by physical measures, in mathematics one must split cases: they are either equal, or diﬀerent.
(then, between two diﬀerent points there are other points, and so on: in a segment with ﬁnite length
there is an inﬁnity of points). And one chooses the simplest model: this theory is deﬁned to be the
simplest theory of the plane or space in agreement with usual physical experience. This is also the
unique theory in which is exactly true any simple statement that seems true for this experience, up
to measurement uncertainties.
For a modern mathematician, the name “geometry” is the family name carried by a large number
of mathematical theories that have more or less parenthood together, like the fact it is more or less
possible to visualize them, to draw ﬁgures of them. From a mathematical point of view, they are all
equally “true”, as studies of mathematical realities (systems of “points”).
How can one visualize other geometries ?
But ﬁrst, how can one visualize Euclidean geometry ? It is not the one we directly see, as the
geometry of the visual ﬁeld is the spherical geometry.
Indeed, the sight of a point only informs us on the half-line from the eye, to which this point
belongs; and this half-line can be represented by its intersection point with a sphere centered on the
eye. When one moves the eye and/or the head, the modiﬁcation of what we see corresponds to a
rotation of this sphere. The clearest example of this notion is the one of astronomical objects that
we perceive only by their image on the celestial sphere (that rotates in appearance because of the
rotation of the Earth).
But Euclidean geometry (of the plane or space) is only the one we are most used to conceive.
Indeed, by seeing or imagining a two-dimensional image (on the vision sphere), we assign to each
point a sensation of proximity or distance, and we are familiar with the way in which this perception
is transformed when we turn the object or change the viewpoint with respect to it. It is this capacity,
fruit of a mental construction, which constitutes our understanding of geometry; and our cleverness
to handle it renders its use so easy like the one of an elementary sensation, so that its indirect nature
becomes nearly anecdotical.
But, the intuitive perception of other geometries holds on the same principle: to see a thing
for conceiving another. And to manage to conceive thus other geometries based on a vision which
(unfortunately) remains the same, a good way is to work on this ability to conceive Euclidean space,
by particularizing, modifying or generalizing it. (But then, it appears that this link to vision is
not easily kept, and the conception work becomes more intense and shifting into formulas with a
relatively weaker role of vision).
For example, one can conceive spaces (Euclidean or not) with dimension greater than 3. They
do not mathematically bear any signiﬁcant novelty in comparison to the line, plane or usual space,
though the total absence of daily experience with them gives newbies the impression of an over-
To conceive these spaces, one can for example say that we just see two or three of these dimensions
at once in the form of a cut or projection, but there are others present, we can change them. . .
Does Euclidean geometry exactly describe physical space ?
No. Not only is time added to the dimensions of physical space in special relativity theory to
make up a 4-dimensional space (space-time) which is not Euclidean and ismore real (close to the
physical reality) than our usual 3-dimensional space, but even if we restrict the consideration to its
3-dimensional appearance, the General Relativity theory teaches us that in reality, this space, as
long as it can still be measured, does not exactly obey Euclidean geometry (independently of any
problem of objects materializing or measurement accuracy). Of course, the error in assuming it
Euclidean is extremely weak, much weaker than other errors much easier to understand and already
often neglected: when we make a map of a region of the Earth (e.g. roughly shaped like a square),
with diameter L, representing it by a region of a plane, the necessary error due to the roundness of
the Earth is (as a ground distance) in the range of order of
where R = 6370 km is the radius of the Earth, which for example gives 2,4 cm for 10 km; the one
due to general relativity, which thus remains in the tangent plane to the Earth, is of
where g = 9, 8 m.s−2 is the acceleration of gravity and c = 3.108 m.s−1 is the speed of light. It
is thus 720 millions of times weaker (precisely, for any given region); in the case of a measurement
of the whole Earth instead of a region, the error is of a few centimeters. So, given a choice of an
“equator” as a perfect circle, its length is lower than its distance to the center times 2π (but the
exact calculation of its distance to the center depends on the distribution of mass of the Earth among
the diﬀerend depths).
If now one considers a vertical plane, another error appears, also stronger than the one from
general relativity: the solid of length L that is turned to compare horizontal and vertical lengths is
deformed by its own wheight (stretched if hanged, pressed if put), with an amplitude of about
where c is the speed of sound in the solid, which is necessarily lower than c according to special
On the other hand, the physics of the very small distances poses such problems that some
theorician physicists seriously consider that the usual conceps of geometry do not hold anymore
at the Planck scale (which is much smaller than the smallest distance for which the behaviour of
particules is presently known).
1.3. What is mathematical physics ?
There are two types of work in mathematical physics: fundamental physics and applied physics.
The theoretical part of applied physics primarily consists in developing the consequences of fun-
damental physics using appropriate approximations (while sometimes using experiments when the
approximations are too diﬃcult to implement on a theoretical level).
The purpose of this work will be to present well-established and largely checked to date funda-
mental physical theories, from the simplest (mathematically simple and close to daily experience) to
the most advanced (mathematically sophisticated and at the base of the explanation of the largest
number of phenomena). But this order of arrangement is not strict. From the point of view of the
“physical reality”, only the last ones would have the status of fundamental physical theories as they
are physically more fundamental and precisely in conformity with reality, whereas the ﬁrst ones are
only results and approximations of them; but the fact is that the teaching order of the fundamental
theories (the order of comprehension in learning them and the intuitive meaning given to the con-
cepts) is more or less the reverse of their dependence order as physical theories. (It is ﬁnally logical
since if a theory A preceded a theory B both physically and pedagogically, one would classify B in
First there is classical mechanics, the one of Galileo and Newton, which deals with forces and
movements; the law of gravitation is formulated there simply. From it one can advance in two
directions, which are the two types of modern physics which are opposed to classical physics: on
the one hand Relativity (special then general), on the other hand, quantum physics and statistical
physics, which are two parent theories closely related together. These theories will be introduced in
more details in the corresponding chapters.
We will not specify the theoretical and experimental justiﬁcations which assess them, because
these justiﬁcations are currently innumerable and among them, those at the historical origin of their
discovery already lost any speciﬁc role.
Such a teaching approach deprived from matters of experimental justiﬁcation is largely satisfac-
tory since these well-established theories checked with an extreme precision, already include in their
application ﬁeld most of the realizable physical experiments. However, many of these experiments
escape in practice this reduction because of the extreme complexity of their theoretical expression
or of the too high diﬀulty of the resolution of the corresponding mathematical problems. This does
not make fundamental physics a dead science, not only because the established theories are visibly
incomplete (general relativity and quantum physics are very hardly compatible for reasons impossible
to popularize), but quantum theory in its complete form compatible with special relativity (quantum
ﬁeld theory) appears, at the fundamental level, of a very high complexity lacking clear mathematical
foundations, and the work of reformulation in the search of a better base remains active.
Would we assume the physical objects have some “real nature”, we could not in any case know
what it is from experiment because all our perceptions pass by a translation from the external
phenomena into the conscience we can have of them. Therefore, the best we can do is to study the
physical world as a mathematical system, which lets us free to represent things in forms unrelated
with those under which we usually perceive them. Let us explain that in details:
At least until now, it seems that any claim that an element of imagination would be more than
another resembling the “true nature” of a certain element of physical reality is meaningless, because
there is no means of comparing these natures : all our perceptions of the external world are indirect,
converted by our body into nervous impulses, that the brain reprocesses to distinguish their global
forms. Therefore, of the external physical world, only the structures, i.e. the relations between
things, are accessible to our senses and can be the subject of our discussions, not any ontological
nature. Anyway, how could a physical object have any identity of nature with a mental phenomenon
For example there is no resemblance of nature between the feeling of a color and the qualities of
the objects seen, which are responsible for this color. This quality is due to the absorption coeﬃcients
of the various frequencies of electromagnetic waves by these objects, that determine the composition
in frequencies of the light which the eye receives. Then, the cells of the eye only transmit a weak
part of information from this composition (according to the properties of the molecules in the retina
which do this work).
Also, we can hardly imagine time even just in accordance with the way in which we live it, which
would show the nature of this lively experience of time beyond the mere form of the straight line
which represents it mathematically: in no one moment can we imagine a duration as such, since
one moment does not contain any duration; there are only future durations which do not exist yet,
and past durations, which we perceive in our memory. But already, the memory of a duration is no
duration any more, because it only represents this duration by a certain other instantaneous feeling.
Thus, any claim to describe things “as they are” would be meaningless, or then, it is no more
physics but metaphysics, which can be possibly defended if announced clearly. Possibly, it can be
useful when one works with a provisional theory being questioned to try to replace it with a more
accurate one, that explicits subjacent structures to a given structure (for example, that is relevant to
pass from the moles to the molecules, and from classical thermodynamics to statistical mechanics).
But that only takes place in a relative and nonabsolute way, and is not the general case.
The object of Physics is thus to describe the physical laws, i.e. the expression as mathematical
theories, of the obseved relations between the results of our observations of the physical world, since
it is a general characteristic of mathematics to be focused on the structures, relations between the
objects, and not on their nature. Then, between several mathematically equivalent presentations
(translations) of a theory, that make various choices of forms of imagination to represent such or
such aspect of reality, the best one is the one which makes it possible to grasp this mathematical
theory in the easiest or most eﬀective way.
The daily intuition of the physical world is adapted to the understanding of the problems of
everyday life. But when one wants to come to study the structures of reality which appear in other
contexts, it can be judicious to modify this choice. But in any case, a speech thus judiciously based
on an unusual system of correspondences between real objects and forms of imagination used to
represent them, is of course equally empty of signiﬁcance regarding the true nature of these aspects
or objects of physical reality than any other speech.
In lack of a complete theory of physics, we have partial theories, each of which deals with
the properties of the observations in a certain experimental ﬁeld of reality, i.e. a certain type of
experiments with a certain degree of accuracy of measurements (depending on the parameters of the
experiment). During the exploration of mathematical physics, we happen to consider a succession
of theories corresponding to various experimental ﬁelds, aiming to be increasingly broad and with
theoretical consequences more and more precisely in agreement with experience (or which gives more
and more information to test).
Each one of these theories is ﬁrst by nature a mathematical theory; its physical signiﬁcance (its
title of physical theory) then consists in allotting to each experiment of the considered experimental
ﬁeld a model in the theory, i.e.
a certain subsystem of objects of the theory which represents
mathematically the experimental device used.
But a theory has also an intuitive meaning, which lies in the choice of the form of representation
best adapted to the comprehension of the theory. In this exploration, many geometrical theories
intervene, not only to account for the behavior of the matter evolving inside the space which is
familiar for us, but also to replace this space itself by other ones (this last case does not often
happen: mainly in Special Relativity, General Relativity and what includes quantum gravity). And,
among our usual intuitions or abilities to perceive the physical world, vision is the one which is clearest
mathematically, and easiest to work on to be able, with it, to conceive many of these geometrical
The diﬃculty of the newbie to question the form of his perceptions also arises in terms of the
principle of reality, whereby “physical reality exists” whether we perceive it or not, in the name of
what he tries to continue to think things in terms of “real” or “concrete” objects such as he usually
However the problem is not to know whether reality contains or not other objects than those
of our experiment or perception, because in any case the physical theories with which we work
present some. The problem is to knowing which ones. The newbie imagines these objects of the
underlying reality, which he cannot perceive directly, as resembling those which are familiar to him.
In principle, he is not always wrong this way. Indeed, insofar as such or such phenomenon that he
needs to explain can be due to mechanisms located in an experimental ﬁeld broader than his daily
experience, but not yet so broader that his intuition ceases being valid, his unability to directly
observe these mechanisms in terms of this same intuition may be a mere matter of circumstance
remediable by convenient apparatus (microscopes, telescopes, etc).
On the other hand, when while going further in experimental ﬁelds we are confronted with a
more fundamental impossibility to reﬁne the measuring instruments to observe the same type of
explanatory real mechanisms, it is time to consider that there may be a good reason for this: namely,
that there is no underlying reality resembling what the newbie would like to believe, because reality
is of another form which can be understood only by diﬀerent means, in diﬀerent terms.
1.5. A few words of Euclidean geometry
Let us evoke here the Euclidean geometries of dimensions 1, 2 and 3. The one of dimension 1
(of the straight line) is mathematically simplest; the two following ones (of the plane and space) are
almost of equivalent complexity, and will be treated here in the same manner.
Our vision is accustomed to understand the plane and space, but is a little broad to speak about
the straight line: which line? horizontal, vertical. . . ? One is tempted to place this line somewhere.
One is already less tempted to situate the plane in space, since the two dimensions of the plane ﬁll
the two dimensions of our ﬁeld of vision, not letting us see the vacuum around it. But whatever can
be our modes of representations and possible relationships with our physical Universe, each one of
these geometries, as a mathematical theory, is limited to consider constructions than can be made
inside its own system (the line, the plane, the space) independently of any consideration of a possible
external space in which this system could be situated. Even if such a consideration can be useful for
intermediate calculations and reasoning or as an intuitive support, it is not regarded as belonging to
the core realities of this theory, and must thus disappear in the objects of study and the ﬁnal results.
This independence of each geometry clearly appears in the example already evoked of the ﬁeld of
vision, which has the shape (or geometry) of a sphere: this sphere certainly has a well-deﬁned center
(the eye) but does not have a radius, it would be wrong to deﬁne it as a particular sphere situated
in space. (It is the set of all half-lines from the eye.)
For this reason, it is wise to provide to the disposal of our intuition some forms of representations
other than vision to ﬁgure out the geometry of the straight line (or geometry of dimension 1). Here
are two. First, the feeling of time. Then, the concept of wire (inﬁnitely thin), as a mechanical object
whose nature is preserved whatever its conﬁguration in space (straight lines, curves, broken lines. . . ).
Now let us present an intuitive introduction to Euclidean geometry.
Consider a wire of ﬁnite length, stopped in two ends (it is cut there, or its extensions are forgotten
as if they did not exist). We deﬁne a curve as the image of a wire, the subset of the plane or space
it can take. Each wire is characterized by its length, which is a positive quantity. For each curve,
the wires that can be laid out on it are all those which have a same length, that one then calls the
length of the curve. Any point of a wire divides it into two parts, whose sum of lengths is equal to
the length of the ﬁrst wire. The length of a curve is also measured by the time taken to follow this
curve at a ﬁxed speed (so are related these three unidimensional intuitions: wires, time and curves).
Marking the wire with a regular graduation to measure the lengths from an end (or more generally
from any given point of the wire), or in an equivalent way, counting the times during its following
at constant speed, provides a marking of each point of the curve by a number or quantity called its
1.6. Experimental foundations of aﬃne plane geometry
We will deﬁne the aﬃne geometry of the plane by the following experiment.
Let us observe from far away a ﬁgure drawn on a plane; we do not know (and will never know)
at which distance this plane is, nor how it is laid out with respect to us (straight or skew; we do not
see its border), nor if we see it from front or from behind (or equivalently, through a mirror). Various
such observers will be always even unaware of how they are positioned in space the ones with respect
to to the others.
We only know what we see, and its apparent size is small (it could hold in the lunar disc for
example): we can take it in photograph, and make geometrical measurements on the photograph.
The mapping from the real ﬁgure to the photograph or ﬁgure seen is called an aﬃne transformation.
In these conditions, which are the concepts of geometry (properties or measurements of a ﬁgure
– this will further be called the structures) whose appearance from any point of view of this type is
always faithful to reality, up to measurement uncertainties (neglecting the eﬀects of distortion related
to the size of the object with respect to its distance) ? We call aﬃne plane geometry the geometry of
the plane* reduced to the study of the concepts which appear accurately in this experiment. It thus
diﬀers from the Euclidean geometry to which we are used, by the fact that its language is reduced:
when two observers placed from diﬀerent points of view communicate their observations of the same
ﬁgure by phone, while having only the right to express themselves in this language, they will always
agree. And they can even introduce new terms, as long as they can deﬁne them entirely by the only
means of this language, they will still always agree.
Thus, all the concepts of aﬃne geometry are concepts which also exist in Euclidean geometry.
But as certain usual concepts of Euclidean geometry are no more valid, we need then, to be still able
to work, to use instead some less usual concepts which remain in aﬃne geometry, that can correspond
to these Euclidean concepts for particular observers.
The same object of aﬃne geometry admits multiple possible interpretations (or measurements)
in the language of Euclidean geometry (each observer has his own interpretation deﬁned by what
he sees), and no measurement or property which only uses the aﬃne language can express how to
distinguish “which is the true one”. Anyone can be regarded as true from the corresponding point
The concept of straight line belongs to aﬃne geometry. All concepts that can be deﬁned from
concepts of aﬃne geometry, also belong to aﬃne geometry. For example, the following concepts
belong to aﬃne geometry: alignment of points, half-lines, segments, parallels and parallelograms,
lengths ratios of parallel segments, ratios of surfaces, barycentres, translations and homotheties,
parabolas, as well as ellipses and hyperbolas with their centers.
In fact, the concept of straight line is suﬃcient to deﬁne all the other concepts of aﬃne geometry,
i.e. which do not depend on the observer in the experiment above. Therefore, any transformation
which maps straight lines into straight lines is an aﬃne transformation.
The following concepts do not belong to aﬃne geometry, as observers in diﬀerent places can
disagree on them: distances, lengths of curves or segments, angles, orthogonality, circles, rotations,
scalar product, axes and focuses of conics.
For example, if one sees a circle, someone else will not see a circle but only an ellipse. This can
serve as a deﬁnition of the concept of ellipse: an ellipse is a ﬁgure obtained by an aﬃne transformation
* Here is why one can speak about the (inﬁnite) plane although the ﬁgure is supposed to hold in
the lunar disc: it is enough to start from much smaller ﬁgures than the lunar disc (one thus needs a
telescope to observe it), from what the lunar disc will seem almost inﬁnitely large like a plane: we
can freely continue there the ﬁgure. . .
from a circle. In the same way, isometries as deﬁned from Euclidean geometry are aﬃne transforma-
tions which in aﬃne geometry are no more distinguishable in larger sets of aﬃne transformations of
We saw in Euclidean geometry the conﬁgurations of wires in the plane, which are certain maps
of a wire into the plane. Here, it is no more necessary to distinguish them inside some larger set of
maps from a wire in the plane, for example the one of parameterized curves (the wire is replaced by
an elastic; if we keep the intuition of time to imagine the wire, the speed of travel along the curve
can then vary in any way, diﬀerent from one point of view to another).
2. The strangeness of Relativity theory
This introductory chapter to relativity and its strangeness aims to analyze the methodology on
which usual introductory courses to special relativity traditionally rests, and to justify the adoption of
a new approach better reﬂecting the use of this theory in a broader context of theoretical physics. It
is particularly addressed to those having seen the usual presentation, but should be at least partially
understandable by all.
It is possible to directly come to the core of the subject by skipping this and going to chapter 3.
2.1. Its name and some other exterior aspects
Relativity is in two stages called “special” and “general”. Special relativity theory is a theory of
space and time that replaces the one previously stated by Galileo and Newton, by reducing it to the
status of an approximation valid when relative speeds of the studied objects are negligible compared
to the speed of light. (We will imply sometimes the adjective “special”, as we will not study general
relativity in this text).
A consequence of relativity is that no signal, no information can be transmitted faster than the
speed of light in the vacuum (which is also the one of radio waves), no more than it can arrive before
it left, for reasons of non-contradiction of reality.
This speed is a universal constant c ≈ 3.108 m.s−1, which means precisely that what sends a
signal at this speed, in the form of radio waves for example, to a point located at a distance of
3.108 m where it is immediately sent back, will have to wait two seconds after the ﬁrst emission
before receiving the return signal.
One should not be mistaken by the name of “relativity” which badly reﬂects the meaning of this
theory: some authors of the theory regretted it thereafter, thinking that the name of “chronogeom-
etry” for example would have ﬁt better, but it was too late. The caricatural slogan “all is relative”
had already taken possession of popular imagination, but its meaning was neither new nor correctly
representative of the theory. Indeed, on the one hand the Principle of relativity was already contained
in the Galilean theory, the only diﬀerence being that the change of reference frame aﬀects a larger
number of parameters (for example time) which describe given physical objects, in relativity theory
compared to galilean theory (and this principle is thus recovered after electromagnetism seemed to
contradict it). In addition, this slogan is likely to cause for example psychological inhibitions against
the acceptance of the objective and measurable character of the rotational movement of an object
(gyroscope, Foucault pendulum. . . ).
Special relativity was discovered by diﬀerent researchers: Poincar´
e, Lorentz, Minkowski, Einstein.
General relativity was discovered by Einstein and Hilbert; it includes special relativity and gravitation
(which it interprets as not being a force but the eﬀect of the curvature of space-time. . . ).
Between the two we can place electromagnetism, for its level of complexity (general relativity
does not rigourously depend on it but uses about the same mathematical tools). General relativity
cohabits naturally with electromagnetism : the electromagnetic ﬁeld and the gravitational ﬁeld
interact in a coherent way while remaining distinct objects.
One can notice a mathematical tool which crosses a large number of physical theories: tensor
calculus. To summarize, this formalism consists of systems of general operations of “multiplications”
between vectors that can belong to diﬀerent vector spaces. It is expressed in all its extent in quantum
ﬁeld theory where any tensorial expression that one can write from certain objects (given by the types
of available particles) is realized by the corresponding Feynman diagram.
Its absence from the usual courses of electromagnetism is in fact an awkwardness: certain calcu-
lations (in classical mechanics and electromagnetism) use vectorial formulas or other rules of trans-
formations of expressions which seem then of very mysterious origin or require complicated demon-
strations, but would become simple and natural once translated into tensorial expressions. What
makes this situation last is that it does not seem to currently exist in the literature a suﬃciently
clear introductory presentation of tensor calculus so that one can seriously consider introducing it
on this level of teaching.
Historically, electromagnetism was developed before special relativity, and currently electromag-
netism is still taught ﬁrst in the ﬁrst years of university. By observing it implies that the speed of
light is constant, it later serves as a justiﬁcation to develop special relativity. But this approach
inherited from history, perpetuated by inertia, lack strongly of elegance (without relativity, the elec-
tromagnetic ﬁeld is presented in the form of two objects, electric ﬁeld and magnetic ﬁeld, obeying
the four Maxwell’s equations which seem quite unnatural, whereas one or two tensorial equations on
only one object is enough in the context of relativity). It would be more elegant and eﬀective to start
with special relativity and tensor calculus, if only suitable approaches of these theories were ﬁndable
in the literature.
Electromagnetism within the relativistic framework is also called classical electrodynamics (in
opposition to quantum electrodynamics which treats this force in quantum physics). One can conceive
this theory as an extension of electrostatics, in the following way.
There is a formal similarity between the theories of electrostatics and magnetic induction (in
three-dimensional space) which correspond by replacing the clarges of dimension zero (densities)
by charges of dimension 1 (currents). Once noticed in terms of tensor calculus, it is enough to
reexpress this same theory, formally independent of the choice of a particular dimension, to the case
of relativistic space-time, to obtain classical electrodynamics. The magnetic ﬁeld and the electric
ﬁeld are only the two components of the same ﬁeld split relatively to the chosen inertial observer.
Classical electrodynamics presents the strange defect to be unable to regard the electrons as
inﬁnitely small material points in all exactitude without absurd consequences, but only according to
approximations, giving up the idea to deﬁne their positions with a precision ﬁner than a distance
of the order of the classical radius∗ of the electron re = 2, 82.10−15 m. However, this size is quite
lower than the famous “uncertainty” of quantum physics (within which this problem is solved but in
a very complicated way).
2.2. The origin of its paradoxes : the Galilean intuition
Seen from the top of a tower, a car can move away any far, indeﬁnitely; it seems to reduce
and to slow down as it approaches the horizon, although actually nothing special happens to it at
its approach. It could never reach this horizon (except because of the roundness of the earth), nor
even less go further. These facts do not astonish anybody. Why can then one be astonished by
the following facts which however are but similar to them: a traveller can accelerate as long as he
wants without anything happening to him, while someone else who would claim himself motionless
would thus see him only go to speeds arbitrarily close to the speed of light, without ever reaching
it, but while having with this approach some distortions, among which a deceleration of time and a
contraction of lengths in the direction of his speed (which in fact does not even visually appear as a
contraction but as a mere rotation; see details later) ?
This astonishment, which expresses a legitimate need to understand, is due to our natural ten-
dency to think the problem in terms of what (unless you have a better suggestion) we will call “the
galilean intuition”, that we will now explain.
∗ Comparable with the size of the atomic nucleus, it is the double of the radius of the sphere
outside which the energy of the electric ﬁeld calculated classically reaches the energy of mass mc2 of
the electron: seen at a lower scale than that, if the electron were a point, by withdrawing from it this
energy and thus this mass which is in the surrounding ﬁeld, the remaining mass would be negative,
defying the laws of the mechanics. . . but in fact, the mass energy of the magnetic ﬁeld due to the
spin of the electron is still more signiﬁcant near the scale of this classical radius and would thus give
a larger radius.
We perceive the world in the form of images received by the eyes and interpreted instantaneously
by the brain in three-dimensional images. There is thus a continuous succession of images, or in other
words an image in evolution. In the same way in the theoretical activity, one imagines ﬁgures or
images which follow one another or evolve during time.
The galilean intuition is then the fact of using this temporal dimension of our imagination, this
faculty to let evolve in our imagination the images during time, to represent the temporal dimension
of the physical universe we try to understand.
This mode of representation was adapted to the Galilean theory, which authorized the instanta-
neous distant interactions and in which the speed of light was inﬁnite. Indeed, the image which an
observer receives from the world at one time corresponded then to the image that another observer
far away from the ﬁrst (and that could be moving relatively to him) also receives at a certain time
(thus at the “same” time), by a simple geometrical transformation (an displacement). It is thus a
3-dimensional image which the theorist can imagine at a certain time, and to which he can give an
But that becomes false in relativity, owing to the fact that the speed of light in the vacuum is
ﬁnite and the maximum possible for information. Therefore, galilean intuition is no more adapted
to conceive this theory.
2.3. The methodological trap of its current teaching
The majority of special relativity courses, even recent ones, that can be found on the market, still
more or less use the same approach faithful to the history of its discovery, far from any attempt of
renewing considered impossible or without object, as this method was repeated more or less just the
same since nearly a century: in the name of the sense of experimental principles and realities, they
start by regarding as a core reality the type of appearances of the world perceived by the miserable
points in the universe which we are. Being thus locked up in the galilean intuition, they then can
only undertake to grasp therein also at any costs the reality of the physical space-time.
As the correspondence between physical reality and Galilean intuition that observation naturally
gives is not satisfactory, they thus deﬁne an artiﬁcial one more satisfactory for the mind, called
“inertial frame”, by means of a virtual, complex and practically unrealizable experimental device.
This concept of inertial frame which was inherited from galilean theory, will ﬁnally be a mathematical
tool equivalent to the concept of an orthogonal frame in Euclidean geometry (or sometimes to the
data of only one base vector of these frames, namely the time vector): useful notion to the study of
geometry (but not absolutely necessary), and diverted here from its basic meaning into the call for
an unsuited intuition. (Indeed, it declares instantaneous independent events, giving to “to see” in
imagination a “present” distant event whose light will arrive only later, as if one could see a light
signal coming before it reaches us. . . ) (In galilean theory, one could also consider non-inertial reference
frames without too many complications, which is no more the case in relativity: the accelerated
material systems that can also be considered by the theory, and which are meant to deﬁne these
other reference frames and did it well in galilean theory, are a worse point of view for the Galilean
intuition in relativity; such a use of this intuition would mathematically correspond to the use of
curvilinear frames of reference.)
Naturally unable to articulate this intuition in accordance with the theory to be conceived,
they declare that the theory can be understood only by the rigour of the formulas and proofs, while
rejecting any intuitive inspiration. But at the same time, in the name of the meaning of realities, they
hopelessly go on clinging to this intuition. Because formulas can’t exist in a pure state without losing
any signiﬁcance, they can only be relations between parameters which correspond to reality, which
is deﬁned as the object of our senses and thus of our intuition, yes but which ones and according to
which correspondance ? It is this unfortunate choice of an arbitrarily ﬁxed mode of representation,
neither faithful to the experiment in its relation to reality, nor adapted to the internal understanding
of the theory to conceive, which determines the appearance of the “fundamental formulas” and their
Then, fault of being able to give an intuitive physical meaning to a formula, they let it the
authority that comes from the lengthy proofs for the fact that there cannot exist any other theory of
space-time or mechanics (compatible with some principles expressed in experimental terms), however
twisted it may be, in which it would not be rigorously true. Thus, in the name of the meaning of
physical realities, they prove the unicity of the theory without giving the means to feel or really
understand it, while almost forgetting to justify its existence (i.e. its non-contradiction), satisﬁed for
that to note that the universe exists. (From where comes the proliferation of pseudo-revolutionary
physicists who claim to refute relativity by simple thought experiments, presenting it as an absurd
and poor interpretation of the famous experiences).
2.4. For a new approach of the theory
We will approach the relativity theory by an intuitive presentation which consists in using,
instead of the Galilean intuition, a purely geometrical intuition.
Here is the reason. We perceive dimensions of physical reality in the form of three sensations :
visual images (of dimension 2), depth and time; they naturally provide three forms of imagination for
the theorist. We recognized that the three dimensions of space have physically the same nature and
are in continuity between them in spite of their diﬀerence of sensitive form (vision and depth). This
identity of nature and this continuity appear intuitive when by a rotation of the object the depth is
exchanged with one of both visual dimensions. Then, let us generalize this reasoning by supposing
that the time dimension itself also has physically the same nature as the three others, and let us
make this fact intuitive by also representing time by a visual dimension of imagination (in fact this
identity of nature is not complete but this approach is no less relevant, and the diﬀerence will be
One could be tempted to reproach this method the absence of rigorous justiﬁcations founded
on reality (statements of physical principles resulting from experimental observations) to build and
prove the theory (or to state it more clearly, the fact that it is not based on the Galilean intuition).
But its aim is to give a clear vision of things without embarrassing oneself at the beginning by
their experimental appearances. The coherence and the consequences of this vision will naturally
show that it describes a possible world compatible with the famous physical principles. The readers
interested by the proof that there is no other possible world, apart from special relativity and the
galilean theory, are thus invited to refer for this to the traditional courses on this subject. (We can
also notice that indeed, general relativity precisely describes another world closer to the truth. . . )
One could also reproach it to not being simpler, and even be more diﬃcult to understand than
the traditional method. Already, people accustomed to the latter (or those who having started to
learn it would like to have it explained better to them), are likely to be reluctant to restart everything
from scratch by changing the approach completely. But perhaps even some, according to their way
of thinking, could really better manage with it. Indeed, it had somehow the advantage of making it
possible to formulate the theory and solve problems without really doing the work to understand it
and come into it: “it suﬃces” to write the proofs and to apply the formulas. However, this advantage
which one can ﬁnd until the formulation of the theory is likely to turn into a handicap later, when
in front of a concrete problem one is lost in front of these formulas which one does not always know
how to apply or handle, or whose use would requires pages of calculation.
Lastly, one could be tempted to think that it is not complete, omitting the devoted formulas
considered necessary to the exact resolution of the practical problems (by often confusing the prac-
ticality of a problem with the arbitrary use of inertial frames and the associated vocabulary and
imagination in its formulation). But ﬁnally, it is not necessary to develop so much speciﬁc formulas
as if it were a completely new theory to our basic knowledge, because the essential part of relativity
is a geometry indeed very similar to the Euclidean geometry (as it will be explained later): it is thus
enough, in order to face the majority of problems, to reuse the various tools of study of Euclidean
geometry, sustained by the corresponding intuition, and to specify how they are modiﬁed in the new
But in fact, the viewpoint that the following presentation aims to express, is not really new, as
appears in the many further developments of theoretical physics that followed the historical discovery
of relativity. Indeed, we can observe that the heavy formulas (“Lorentz transformations” and what
follows) allegedly constitutive of relativity, while remaining true, do not explicitly appear as such any
more in the chapters of general relativity, classical electrodynamics or other theories based on special
relativity. Instead, one uses a simple object (a “metric” called “pseudo-Euclidean”) that testiﬁes the