The theory of relativity was formulated by Albert Einestein in the beginning of the 19th century. It applies to systems travelling at speeds close to the speed of light. It is a four dimensional theory that includes, in addition to the three cartesian coordinates x, y and z, the element of time (t) as a fourth variable. The postulates of the special theory of relativity are that the speed of light is constant in all inertial reference systems and that all laws of physics are valid in all inertial reference systems.
In classical physics, length, time and mass have constant values when measured in different inertial reference systems. At speeds close to the speed of light the measurement of these quantities are affected by the choice of the reference frame in which the measurement is done.
In the theory of special relativity, time measurement in different inertial systems gives different values for the time measured. The time measured in one inertial system is related to that measured in another inertial reference system by the following: relation:
t' = t*/SQRT(1-(v/c)(v/c)) where t' is the time measured in the frame that is moving at a speed very close to the speed of light, and t is the time that is measured in the frame that is at rest. Where SQRT means the square root.
Analogously the distance that is measured in two different reference systems, one is at rest and the other travelling at a speed very close to the speed of light c, is not the same but is different by the factor 1/SQRT(1-(v/c)(v/c)).
This factor is the same as that one that relates measurements of time in different inertial systems one at rest and the other travelling at a speed v close to the speed of light c.
The equation that relates the measurements of length in two inertial systems, one is at rest and the other travelling at a speed of v close to the speed of light is as follows:
L' = l/SQRT(1-(v/c)(v/c)) where l' is the length measured at the travelling reference system and l is the length measured at the reference system at rest.
The third physical variable that its measurement is affected by applying the theory of special relativity is the mass of a body. Measurement of the mass depends on the reference system whether it is at rest or it is moving at a speed of v close to the speed of light. Travelling at speeds close to the speed of light leads to enlargement of the mass by the factor 1/SQRT(1-(v/c)(v/c)) where SQRT is the square root. At v=c the mass becomes infinite in its value. The relation between the measured mass in the two different reference systems is given according to the following equation:
M'=m/SQRT(1-(v/c)(v/c)) where SQRT denotes the square root and m' is the mass measured at the travelling reference system and m is the mass meaured at the reference frame at rest.
It is noteworthy to mention that the theory of special relativity is valid only for inertial systems. Namely it is valid only for reference systems that travel at constant speed close to the speed of light. This is in contrast to the theory of general relativity which is applicable to non-inertial systems. Namely, reference systems that are accelerated and are travelling at speeds close to the speed of light.
The theory of special relativity has a mass-energy relationship which predicts the conversion of mass to energy and vice-versa. This discovery lead to the development of the nuclear energy based on the transformation of nuclear mass into energy by fusion or fission processes.
The total energy of a relativistic particle is related to its momentum by the following equation:
E*E=PC*PC+ m*m*C*C*C*C where E is the total energy of the particle and P is its momentum and C is the speed of light.
This relativistic equation leads to the development of the relativistic quantum mechanics known as the Dirac equation and Klein-Gordon equation.