Annex 8 - Special Theory of Relativity

The special theory of relativity grew out of electrodynamics and optics which it simplified their theoretical structure. It has also reduced the number of independent hypothesis at the base of the theories. Moreover the changes brought up by the special theory of relativity are only important when the velocities involved are close to the speed of light.

The classical laws said that the kinetic energy of a body of mass m was: mv²/2 (1)
The special theory of relativity changed the expression to: mc²/sqrt(1- v²/c²) (2).
This expression becomes infinite when the speed is equal to the velocity of light (v = c), a velocity that cannot be reached.
If we develop (2) in a series we get: mc² + mv²/2 + 3/8 mv²v²/c² + … (3)
When v/c is small, then the third term of the expression (3) is negligible and the second term is the classical expression (1).

Before Einstein introduced the theory of relativity, physics was based on two independent conservation laws:
- The law of the conservation of energy.
- The law of conservation of mass.
- Relativity united them in one law.

Relativity requires that the law of conservation of energy hold true with reference to a coordinate system K as well as with the coordinate system K' that is in an uniform movement of translation by reference to K (or to every Galilean system of coordinates).

The fundamental equations of the electrodynamics of Maxwell say that a body moving with the velocity v, which absorbs an amount of radiation energy Eo without changing its velocity has its energy increased by: Eo/sqrt(1 - v²/c²) (4)

Taking into consideration the equations (1), (2) and (3) we find that the required energy of the body becomes: (m + Eo/c²)*c²/sqrt(1 - v²/²) (5)

As a result the body has the same energy that a body of mass (m + Eo/c²) (6) moving with the velocity v.

We can also say that if a body takes up an amount of energy Eo, then its inertial mass increases by Eo/c² and this shows that the inertial mass of a body is not constant, but varies according to the change of energy in the body. Moreover the inertial mass of a body can be seen as a measure of its energy. The law of conservation of the mass of a system becomes the same as the law of conservation energy. It is valid only if the system does not absorb or emit energy.

If we write the equation (50 as follow; (mc² + Eo)/sqrt(1 - v²/²) (7) we see that the term mc² is the energy of the body before it absorbed the energy Eo.

Einstein introduced the notion of a four-dimensional space but it was his professor at the Zurich Polytechnics to describe it in a mathematical way. We are all aware, and we understand, the classical three-dimensional space in which we live. In this continuum the position of a point P at rest can be described by means of its three coordinates -x, y, z- valid in the system of reference chosen. A four-dimensional space is much more difficult to understand, and even more to describe. However from a purely mathematical point of view this point P can be described in this four-dimensional continuum by four numbers: x, y and z as before and, in addition, t for the time. Before Einstein introduced his theory of relativity we did not need a four-dimensional space because time was said to be absolute and independent. In classical physics, time is the same to all observers watching an event. The last equation of the Galilean transformation put it clearly in evidence (t = t').

In relativity time is not independent and the fourth of the Lorenz transformation shows it as follow: t' = (t - v/c²*x)/sqrt(1 - v²/c²)

From this equation one sees that the time difference t' of two events relative to K' does not become 0 even when the t of the same two events in the system of reference K is equal to 0. This means simply that the pure "space-distance2 of two events in K results in "time-distance" of the same two events in K'.

Minkowski showed that the four-dimensional space-time continuum of the relativity theory is directly related to the three-dimensional continuum of the Euclidean geometrical space. However to it we must replace the time coordinate t by the imaginary value: sqrt(-1)*ct. In these conditions the laws of the special theory of relativity assume mathematical forms with the time coordinate playing the same role as the three space coordinates. We can also say that the four coordinates of special relativity correspond exactly to the three space coordinates in Euclidean geometry.