Special Relativity theory made intuitive

Short abstract:
Starting from a philosophical introduction on theoretical physics and its understanding, we will express Special Relativity theory by a work on geometrical intuition. This gives a powerful method to solve problems without ever using any Lorentz transformation formula.

A few more comments:
The traditional presentation of Special Relativity can roughly be qualified as a way to let one get used to some formulas that can produce the right results in a computational (but possibly tedious) way, and then wait until one gets really familiar with their properties, to finally grasp the meaning of the theory and go further. Instead, we propose here a way to make the effort necessary to really understand the core of the theory more directly and deeply like theorician physicists do, according to the mathematical terms in which Special Relativity is involved as a base for further developments of mathematical physics.

(It was first written in French in 2002 with further parts written in 2003, then this translation into English of some beginning of the text was done in august 2004).

Download the document (.pdf, 210 kb, 18 pages)

What is special with this presentation

This presentation:

- deeply explains the meaning of concepts. This requires a certain amount of conceptual work and it is expressed in terms different from those of traditional presentations of Relativity. (Its goal is not thus to make you succeed in your exams of the Special theory of Relativity, which most probably still follows the traditional approach).
- In particular, this reformulation from a new base makes it possible to reduce the "shock" of the paradoxes usually associated with this theory. (These paradoxes not only come from the natural intrinsic novelty of this theory that differs from our daily intuition, but also more particularly with the traditional approach in which all concepts used were unfortunately defined based on the very daily intuition and inspiration, so unnecessarily feels like a contradiction since it it must then continuously deny these the misleading intuition which results from it and might tire the thought unnecessarily).
- Is nevertheless bold (probably even more), but in a positive way (by the audacity to consider other intuitive ways of thought, in fact an expression of the very intuition of the theorists, finally fully explained) rather than negative (rejection of the intuition).
- Is mainly expressed in common language, uses hardly any mathematical formulas (in particular, the Lorentz transformation formulas are not used since they are not useful to this approach) but is nevertheless mathematically accurate and is a full expression of the theory (it is geometry !). The main formula used is the mathematical theorem :

If v²=-c² then cos(a/v) = ch(a/c) and v sin(a/v) = c sh(a/c)

(which can be justified from calculations on complex numbers or on formal power series)

Contents of this .pdf

1. Introduction

1.0. About this document
1.1. What is mathematics ?
1.2. What is geometry ?
1.3. What is mathematical physics ?
1.5. A few words of Euclidean geometry
1.6. Experimental foundations of affine plane geometry

2. The strangeness of Relativity theory

2.1. Its name and some other exterior aspects
2.2. The origin of its paradoxes: the Galilean intuition
2.3. The methodological trap of its current teaching
2.4. For a new approach of the theory

3. New presentation of special relativity theory

3.1. The core logic of the theory
3.2. Link with the experience
3.3. Deformed study of the Euclidean plane geometry

Further parts, not translated yet (see the French version):

1.4. What is mathematical logics ?
3.4. Visions of the relativity of simultaneity with one space dimension
3.5. Relativistic transformations of images
4. The mechanics of equilibrium, classical and relativistic mechanics (including E=mc²), phase space
5. Foundations of statistical mechanics
6. Introduction to quantum physics (including the EPR paradox)

A few excerpts and abstracts of the ideas

(Again, see the document for the details)

What is mathematical physics ?

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 come through a translation from the external phenomena into the consciousness 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 new ways unrelated with those of our usual direct perception.
Then, between several mathematically equivalent presentations (translations) of a theory, based on various choices of forms of imagination to represent aspects of reality, the best one is the one which makes it possible to grasp this mathematical theory in the easiest or most effective way.
Anyway, a speech judiciously based on an unusual system of correspondences between "real" things and forms of imagination used to represent them, is of course as empty of significance regarding the "true nature" of these aspects or elements of physical reality, as any other speech.

The core logic of Special Relativity theory

Special Relativity theory can be summed up into the following sentence:

Imagine a 4-dimensional world in which things are fixed, in equilibrium

This rises three questions, corresponding to the three independent parts of the understanding of the theory.

Philosophical aspect (link with experience)

Since the world is fixed and 4-dimensional, how can we see it as evolving and 3-dimensional ?
Answer: we see it as moving in our view because in fact, we (our conciousness) are moving in it, at a constant speed v which is the speed of physical (geometrical) time flowing with respect to our time (v²=-c² where c is the speed of light).

Geometrical aspect

What is the geometry of this space ? In fact, the dimension is not important. The geometry is not the one of an Euclidean 4-dimensional space (which mathematically exists) but is the pseudo-Euclidean geometry, which is similar to the Euclidean one with few differences. This difference can already be studied for 2-dimensional geometry.
The geometrical figures of the pseudo-Euclidean plane can be drawn preserving the underlying affine geometry, which is the same in both geometries.
For General Relativity (not studied here), the affine geometry is lost.

Physical aspect

(document chapter 4 not translated yet)
What is the physics of equilibrium ? We are somehow used to it, but what are its laws mathematically ?
(It is expressed depending on a geometrical space, and the replacement of the usual 3-dimensional Euclidean space by the 4-dimensional pseudo-Euclidean space of Special Relativity makes no problem). Its two-dimensional version can be easily (and "miraculously") expressed, but its general expression needs (in my opinion) the tool of tensor calculus. if we want to express not only the force vector but also momentums (do you seriously think that the usual presentation of torsors, without tensor calculus, is nice ? - Ok, the usual presentation of tensor calculus is even worse but I plan to make it clean in the future).
Which is the dictionnary translating the language of equilibrium into the one of relativistic mechanics ?
And how do the particles look like in terms of equilibrium physics ?

Here is the dictionary (but my English is not perfect so I can make some mistakes in names):
The potential energy in equilibrium physics is translated by the action in relativistic physics; the differential of the restriction of this potential to the solid moves (isometries at the first order near identity) of some external support gives in equilibrium mechanics the torsor of the force received from this support. The force vector comes by still restricting this to translations; in the language of relativistic mechanics it is called energy-momentum quadrivector ; when this world apparently splits into time and the 3-dimensional space, its time component is called energy, and its space component (projection parallel to the time direction) is called momentum vector.
The relativistic particles are like elactics, with the property that the tension (norm of the force vector) is constant and called the mass of the particle. This phenomenon of constant tension of 1-dimensional objects also exists in larger dimension: the surface of water also has a constant surface tension. Such constant tensions (=negative pressures) unaffected by movement, are associated with a density of energy.

In dimension 3 there is the pressure of a gas (except that it is affected by movements), and in dimension 4 it is the cosmological constant.

(Some more details in the french version)
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