ues. It is not clear from the 1904 paper [3] that Lorentz was aware of the group structure he had discovered. Indeed in order to see such a structure one needs to change variables in a way which is somewhat involved. The right change of parameters could come only from a new physical principle that was the mission of Albert Einstein to clarify, in his celebrated 1905 paper On the electrodynamics of moving bodies [4], and that of Minkowski to interpret geometrically. In previous years, starting from 1895, in an attempt to explain the puzzle provided by the Michelson Morley experiment, Lorentz had proposed that moving bodies contract in the direction of motion and, to the present time, this relativistic effect is named the Lorentz contraction. He also realized that the transition from one reference frame to another could be simplified by using a new time variable which he called local time [5]. Such local time depended on two variables, the first is what Lorentz regarded as the universal time t , but was simply the time of one of the two considered frames. The second variable entering the formula for the local time was the space-location under consideration. In 1900, Henri Poincaré declared that Lorentz’s local time was a wonderful invention and illustrated it by showing that clocks in moving frames are synchronized by exchanging light signals that are assumed to travel at the same speed in both directions, namely when they travel against and when they travel with the motion of the frame.

1.3 The Principle of Special Relativity

It should be clear to the reader of the previous pages that all the tiles of the puzzle were, by the end of 1904, ready and just waited a clear logical mind such as that of Einstein to be assembled together in a meaningful picture. On one side Michelson Morley experiment had shown that light travels always at the same speed, independently from the state of motion of its observer. Secondly Lorentz had shown that the most important Laws of Nature, apart from Newton’s law of gravitational attraction, namely those codified in Maxwell equations, are covariant not with respect to the transformations of the Galilei group, rather with respect to another set of transformations, those that bear his name. Albert Einstein (see Fig. 1.5) transformed these two facts into the axioms of his new Theory of Special Relativity:

(a)The speed of light c is constant and the same in all inertial reference frames.

(b)All the Laws of Nature should, like Maxwell equations, have a form, in inertial reference frames, that is covariant with respect to Lorentz transformations.

Said differently, the correct transformations from one inertial frame to another are those of Lorentz and rather than searching for complicated interpretations of Lorentz covariance of electrodynamics, one should rather concentrate on mechanics and change the laws of Newtonian mechanics so that they become Lorentz covariant.

Einstein showed that these principles implied a critical revision of the concept of contemporaneity. Namely events that happen at the same time for one inertial observer may happen at different times for another observer in relative motion with

Fig. 1.5 Albert Einstein (1879–1955) is the most famous of all physicists of the XX century and he is the principal actor in the story told in the present book. He was born in Ulm, Germany, and died in Princeton in the USA. His citizenship changed three times. Born German he became Swiss, than German again and finally American citizen. He was awarded the Nobel Prize in 1921 for his discovery of the law of the photoelectric effect. This discovery is contained in one of his three fundamental papers of 1905, dealing respectively with the photoelectric phenomenon, the Brownian motion of molecules and the third on Special Relativity. His major achievement, namely the Theory of General Relativity was published in 1915 after a decade of studies. We do not dwell here on Einstein’s biography, since many books have been published on the subject. Moreover his thoughts and ideas will be constantly recalled throughout the development of the present book and many citations will occur

respect to the first. Secondly using various arguments he showed that the Principle of Special Relativity implied the equivalence of mass and energy, according to the celebrated formula E = mc2. The meaning of this equivalence is that, even when at rest, a particle of mass m has an energy, which through interaction with other particles or radiation can be extracted or exchanged. For instance a massive particle can decay by means of the emission of a light particle endowed with high kinetic energy and this kinetic energy is subtracted from the rest energy of the decaying particle. The remnant of the decay has necessarily a lower mass than its predecessor.

The essential implication of Einstein new approach to the formulation of natural laws was the suppression of the ancestral separation of time from space and the fusion of the former with the latter into a newly born stage for physical processes, named space-time. Intuitively this latter is a continuous space, whose points, named the events are labeled by four parameters, the first of which t , defines when the event occurred, while the last three x, y, z define where it happened. It was the historical mission of Hermann Minkowski (see Fig. 1.6) to make this intuitive idea mathematical sound and construct explicitly the geometrical arena of special relativity. In terms of Minkowski space the formulation of special relativistic theories becomes extremely simple and Einstein ideas become algorithmic.

Fig. 1.6 Hermann Minkowski (1864–1909) was born in Lithuania, belonging at that time to the Russian Empire. His family was Jewish, partly of Lithuanian, partly of Polish descent. His higher education, however, was German and took place in the historical University of Königsberg, where Immanuel Kant had taught and developed his philosophical ideas one century before. Having become a refined mathematician, whose scientific interests centered on the theory of quadratic forms, Minkowski received prestigious international recognition, including a Prize from the French Academy of Sciences and taught in various Universities of Germanic language, Bonn, Göttingen, Könisberg and Zürich. In the Swiss Polytechnic of Zürich he happened to be one among the teachers of Albert Einstein. Since 1902 he was appointed professor in Göttingen and became one of the closest friends and collaborators of David Hilbert. It was just in 1907, two years after the 1905 paper by Einstein and two years before his premature death that he had the brilliant idea of interpreting Special Relativity in terms of a continuous geometrical space that joined space and time together and was endowed with the metric which bears his name and is invariant under Lorentz transformations

1.3.1 Minkowski Space

The basis of Minkowski’s construction is the realization that the two pillars of Special Relativity, i.e. constancy of light velocity and Lorentz covariance are just two sides of the same medal. Let us introduce a four-dimensional vector space MMink whose elements are m-tuplets of real numbers named the events:

MMink xμ = x0 , x1, x2, . . . , xm−1 (1.3.1)

=ct

where c denotes the speed of light and t the coordinate time; in this way x0 denotes the when and xi (i = 1, . . . , D − 1) the where of a physical event. The statement that MMink is a vector-space implies that events can be summed and subtracted:

These are the same properties with which three-dimensional space is endowed in classical Newtonian mechanics and in Euclidian geometry which provides its mathematical basis.

A Euclidian m-dimensional space Em Rm admits a global notion of distance between any two points based on the existence of a scalar product. The latter is a quadratic bilinear symmetric form on Em:

Typically the scalar product in a Euclidian space is given by the sum of squares of the vector components:

i=1

but any symmetric, non-degenerate matrix Mij with all positive eigenvalues could be used to define , , respecting the same axioms (1.3.4) and (1.3.5, 1.3.6, 1.3.7):

The properties of Mij we spelled out in words correspond to the following formulae:

Given the bilinear form , , the absolute distance between any two points x, y Em can be defined as follows:

and by construction is positive definite and obeys the triangular inequality:

Once time and space are unified into Minkowski space-time, one can introduce a bilinear scalar product (, ) which satisfies the axioms (1.3.4), and non-degeneracy (1.3.5), yet the Principles of Special Relativity require that we remove positive definiteness and we rather choose a different quadratic form. In Sect. 3.6.1 we will tackle the rigorous mathematical definition of signatures of quadratic forms which was clarified in the XIX century by J.J. Sylvester. In a nutshell the signature of a quadratic form defined as in (1.3.9) consists of the signs of the eigenvalues λi of the matrix Mij . When the scalar product is positive definite all the signs are plus:

Minkowski understood that all the Principles of Special Relativity are encoded in the choice of another signature, the Lorentzian signature:

D−1 times

Explicitly the Lorentzian scalar product of Minkowski space can be defined as follows. Identifying the number m with the space-time dimensions, namely with D = 1 + # of space directions, consider the following diagonal matrix:

which is named the flat Minkowski metric. Then for any pair of events xμ, yμ their scalar product is:

The essential novelty attached to Lorentzian signature is that now the square norm of vectors belonging to Minkowski space can be of three types:

1.time-like vectors (x, x) > 0.

2.space-like vectors (x, x) < 0.

3.null-like vectors (x, x) = 0

and, as it will be clear from the mathematical definition of the Lorentz group discussed in the next section, the time, space or null-like character of a vector does not depend on the chosen inertial reference frame. Indeed the Lorentz group is precisely defined as that group of linear substitutions which leaves the Lorentz product (1.3.21) invariant.

Consider now the motion of a particle in Minkowski space-time. A generic motion is described by a world-line of the form:

where τ is some real parameter. Just as in Classical Newtonian Physics we assume that:

Principle 1.3.1 A particle subject to the action of no force travels on a straight line with constant velocity. This means that for such a free particle the world-line is of the form:

where uμ is a constant vector named the D-velocity.

The second principle which encodes the whole of Special Relativity is the following:

Principle 1.3.2 The D-velocity of a physical particle is always either time-like or null-like. It is never space-like. We have two possibilities:

Massive particles When the rest-mass is larger than zero, namely m > 0, the D- velocity is time-like and (u, u) = 1.

Massless particles When the rest-mass vanishes, namely m = 0, the D-velocity is null-like and (u, u) = 0.

This principle states that no physical signal can travel faster than light and establishes that all massless particles travel at the speed of light in whatever inertial reference frame. Indeed let us analyze the implications of (u, u) = 0. In order to satisfy the null-like constraint the four vector uμ must be of the form:

where u is any (D − 1)-component vector. Combining Principles 1.3.1 and 1.3.2, we obtain:

This means that the considered particle travels with a (D − 1)-velocity given by:

On the other hand combining Principles 1.3.1 and 1.3.2 for the case of a massive particle we get:

u0 = ± 1 + |u|2; ui = ui (1.3.28)

where u is once again any (D − 1)-component vector. This implies:

which means that the considered particle travels with the following (D − 1)- velocity:

So any massive particle necessarily travels slower than light. In particular we can satisfy the time-like constraint by setting u = 0, in which case the massive particle is at rest, having vanishing D − 1-velocity. The relation (1.3.30) can be easily inverted finding:

Special relativistic mechanics is now easily constructed. Defining the D-

the space-part of this D-vector takes the form:

− |v|2

1 c2

and it coincides with the Newtonian momentum mv when the velocity of the considered particle is much smaller than the speed of light v c. On the other hand the time component of the momentum D-vector is the following:

− |v|2

1 c2