Changes

These are the transformations from one coordinate system <math> (x, y, z, t) </math> to another system, <math> (x', y', z', t') </math>, moving past this one at speed u, and with <math>x </math> and <math> x' </math> axes colinear.

These are the transformations from one coordinate system <math> (x, y, z, t) </math> to another system, <math> (x', y', z', t') </math>, moving past this one at speed u, and with <math>x </math> and <math> x' </math> axes colinear.

The Lorentz factor, <math>\gamma </math> occurs frequently all over special relativity and is<ref>http://www2.slac.stanford.edu/vvc/theory/relativity.html</ref>

The [[Lorentz factor]], <math>\gamma </math> occurs frequently all over special relativity and is<ref>http://www2.slac.stanford.edu/vvc/theory/relativity.html</ref>

:<math>\gamma = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}}</math>, where:

:<math>\gamma = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}}</math>, where:

* <math> \gamma </math> is the Lorentz factor and relates time, mass, and momentum in different [[inertial reference frame|inertial frames of reference]].

 

* <math>u</math> is the difference in speed between the [[Inertial frames of reference|frames of reference]]

* <math>c </math> is the [[speed of light]]

* <math>c</math> is the [[speed of light]]

=== Classical mechanics ===

=== Classical mechanics ===

When <math>v</math>, the speed of the object in question, is low relative to <math>c</math>, γ approaches <math>\frac{1}{\sqrt{1-0}} = 1</math>, 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 [[Subatomic particle|particles]] at low mass and low speed.  However, at higher velocities, γ diverges, causing relativity to be essential.

When <math>u</math>, the speed of the object in question, is low relative to <math>c</math>, <math>\gamma </math> approaches <math>\frac{1}{\sqrt{1-0}} = 1</math>, 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 [[Subatomic particle|particles]] at low mass and low speed.  However, at higher velocities, <math>\gamma</math> diverges, causing relativity to be essential.

Electromagnetism, including for [[light]] and gamma radiation, where the quanta ([[photon]]s) travel at [[Speed of light|light speed]] have no rest mass, is always relativistic.

Electromagnetism, including for [[light]] and gamma radiation, where the quanta ([[photon]]s) travel at [[Speed of light|light speed]] have no rest mass, is always relativistic.

===Universal speed limit ===

===Universal speed limit ===

At the other extreme, when <math>v</math> approaches <math>c</math>, γ approaches <math>\frac{1}{\sqrt{1-1}} = \frac{1}{0} = \infty</math>. 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.<ref>http://www.math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html</ref>  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.)

 

At the other extreme, when <math>u</math> approaches <math>c</math>, <math>\gamma </math> approaches <math>\frac{1}{\sqrt{1-1}} = \frac{1}{0} = \infty</math>. Since infinite <math>\gamma </math> means an object would have infinite [[kinetic energy]], no object with mass can ever travel at the speed of light. Relativity does not however forbid particles from traveling faster then <math>c </math>, rather that it is impossible to cross the speed of light barrier. Some theorists have postulated hypothetical [[tachyons]], which would always travel faster than the speed of light.<ref>http://www.math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html</ref>  No evidence has been found for them.

== Experimental Proofs ==

== Experimental Proofs ==