# Special theory of relativity

Special Relativity (SRT) is a generalization of classical mechanics. In contrast to General Relativity, Special Relativity deals with processes observed in so called inertial frames (frames of observation without the influence of acceleration or gravity). It is based on two main observations from different experiments.

1. The speed of light is constant for all (inertial) observers, regardless of their velocities relative to each other.
2. The laws of physics are identical in all inertial reference frames.

It is not difficult to demonstrate that it is impossible to reconcile these conditions with a Newtonian mechanics, in which the coordinates are formed in a three-dimensional space. In SRT, each object moving in its own inertial frame has its own time, which constitutes a fourth coordinate describing its state.

The three most prominent SRT effects are time dilation, length contraction and the equivalence of mass and energy.

## Mathematics

The central equation of special relativity is[1]:

γ=$\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$, where:
• γ is the factor relating relativistic time, mass, and momentum to non-relativistic time, mass, and momentum.
• v is the speed of the object in question
• c is the speed of light

### Classical mechanics

When v, the speed of the object in question, is low relative to c, γ approaches $\frac{1}{\sqrt{1-0}} = 1$, causing the Einstein-Lorentz relativity equations to be equivalent to Newton's equations. This is why classical mechanics, governed by Isaac Newton's laws of motion, works for particles at low mass and low speed. However, at higher velocities, γ diverges, causing relativity to be essential.

Electromagnetism, including for light and gamma radiation, where the quanta (photons) travel at light speed have no rest mass, is always relativistic.

### Universal speed limit

At the other extreme, when v approaches c, γ approaches $\frac{1}{\sqrt{1-1}} = \frac{1}{0} = \infty$. Since infinite γ means infinite mass, no object can ever reach the speed of light. (Some theorists have postulated hypothetical tachyons, which would always travel faster than the speed of light[2]. No evidence has been found for them. If they existed, special relativity would mean that they cannot ever travel slower than the speed of light.)

## Experimental Proofs

1. Michelson-Moreley experiment
3. GPS clocks (general relativity)
4. Lifetimes of fast traveling particles

## History

In the beginning of the last century the two assumption mentioned before where found experimentally in the form of the Maxwell equations, which describe electromagnetic waves. The general idea at that point was that any wave is carried by some medium, called the luminiferous aether. However, as the earth is moving through space, there should be a difference in the interferometrically determined light wavelength, when measured at noon, at evening and at night at the same geographic location. However the Michelson-Morley experiment did reject the aether hypothesis. With this rejection it became an inevitable fact that equations descibing physics, which where fitting to a three dimensional space with a single timeframe for all observers exist. The mathematical framework and physical significance was developed by Henri Poincaré and Hendrik Lorentz. Albert Einstein gave an alternate derivation in terms of postulates. Many other scientists contributed modifications of this theory. In particular, the Irish Physicist Fitzgerald proposed that the failure of the Michelson Morley experiment was as a result of a length contraction effect. This idea was taken up by Hendrik Lorentz and shown by others to be a useful mechanism by which theory could be forced into conformance with experimental results. However, in 2005, Michael Strauss a computer engineer invalidated much of Special Relativity theory by showing clear contradictions in the theory. Special Relativity

## Spin as a relativistic effect

The relativistic extension of Quantum Mechanics, described by the Dirac Equation allows, due to the symmetry of the equation in 4-space, an additional quantum number to exist, called spin. electron spin was known from chemistry before relativity arose.

Some consequences of the SRT are:

1. It is impossible ever to transmit information faster than the speed of light.[3]
2. The laws of physics are identical, without any variation, in every location throughout the universe.
3. The laws of physics are identical, without any variation, no matter how fast something is traveling (in an inertial reference frame).

Special relativity alters Isaac Newton's laws of motion by assuming that the speed of light will be the same for all observers, despite their relative velocities and the source of the light. (Therefore, if A sends a beam of light to B, and both measure the speed, it will be the same for both, no matter what the relative velocity of A and B. In Newtonian/Galilean mechanics, If A sends a physical object at a particular velocity towards B, and nothing slows it, the velocity of the object relative to B depends on the velocities of the object and of B relative to A.)

In the framework of special relativity, several thought experiments can be constructed, which lead to apparent paradoxes. The most striking one is the twin paradox. If you take twins, one on earth, and one in traveling to the next star with high speed and back, their biological age will not be the same, even though you could redefine the system of the traveling twin to be resting. This paradox is resolved because the second twin is not in an inertial frame - he has accelerated, most significantly at his turn-around point. This points out, that while neglected in special relativity, acceleration has a non-trivial role. This is considered in the General theory of relativity.

## References

1. http://www2.slac.stanford.edu/vvc/theory/relativity.html
2. http://www.math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html
3. This assumption is commonly restated in this manner. For example, a discussion of hypothetical tachyons talks "about using tachyons to transmit information faster than the speed of light, in violation of Special Relativity."[1] However, there is some question whether the Theory of Special Relativity really restricts faster-than-light communication of information.