Changes

Unlike most of physics, the theories of relativity have discontinuities whereby the limit of a physical quantity as a variable (such as mass or velocity) approaches a fixed value is not the same as the physical quantity at the fixed value.  For example, the limit of momentum as mass approaches 0 and velocity approaches the speed of light is not equal to the momentum of (massless) light.<ref>Discontinuities in General Relativity are also well-recognized. See, e.g., [http://www.springerlink.com/content/u47l341u2q555455/]</ref>

Unlike most of physics, the theories of relativity have discontinuities whereby the limit of a physical quantity as a variable (such as mass or velocity) approaches a fixed value is not the same as the physical quantity at the fixed value.  For example, the limit of momentum as mass approaches 0 and velocity approaches the speed of light is not equal to the momentum of (massless) light.<ref>Discontinuities in General Relativity are also well-recognized. See, e.g., [http://www.springerlink.com/content/u47l341u2q555455/]</ref>

More generally, and also unlike most of physics, the theories of relativity consist of complex mathematical equations relying on several hypotheses.  For example, at Hofstra University general relativity is taught as part of an upperclass math course on differential geometry, based on three stated assumptions.<ref>http://people.hofstra.edu/Stefan_Waner/diff_geom/tc.html</ref>  The equations for special relativity assume that it is forever impossible to attain a velocity faster than the speed of light and that all inertial frames of reference are equivalent, hypotheses that can never be fully tested. Relativity rejects Newton's [[action at a distance]], which is basic to Newtonian gravity and [[quantum mechanics]].  The mathematics of relativity assume no exceptions, yet in the time period immediately following the origin of the universe the relativity equations could not possibly have been valid.

More generally, the theories of relativity consist of complex mathematical equations relying on several hypotheses.  For example, at Hofstra University general relativity is taught as part of an upperclass math course on differential geometry, based on three stated assumptions.<ref>http://people.hofstra.edu/Stefan_Waner/diff_geom/tc.html</ref>  The equations for special relativity assume that it is forever impossible to attain a velocity faster than the speed of light and that all inertial frames of reference are equivalent, hypotheses that can never be fully tested. Relativity rejects Newton's [[action at a distance]], which is basic to Newtonian gravity and [[quantum mechanics]].  The mathematics of relativity assume no exceptions, yet in the time period immediately following the origin of the universe the relativity equations could not possibly have been valid.

The "continuous" nature of space and time postulated by relativity is in conflict with the "discrete" nature in [[quantum mechanics]],<ref>For example, Relativity claims that space and time are smooth and continuous, while [[quantum mechanics]] suggests otherwise. [http://www.csmonitor.com/Science/Cool-Astronomy/2010/1025/Is-the-universe-a-big-hologram-This-device-could-find-out.]  Relativity also denies [[action-at-a-distance]], while quantum mechanics suggests otherwise.  Relativity denies any role for chance, while quantum mechanics is heavily dependent on it.</ref> and although theories like [[string theory]] and [[quantum field theory]] have attempted to unify relativity and quantum mechanics, neither has been entirely successful or proven.

The "continuous" nature of space and time postulated by relativity is in conflict with the "discrete" nature in [[quantum mechanics]],<ref>For example, Relativity claims that space and time are smooth and continuous, while [[quantum mechanics]] suggests otherwise. [http://www.csmonitor.com/Science/Cool-Astronomy/2010/1025/Is-the-universe-a-big-hologram-This-device-could-find-out.]  Relativity also denies [[action-at-a-distance]], while quantum mechanics suggests otherwise.  Relativity denies any role for chance, while quantum mechanics is heavily dependent on it.</ref> and although theories like [[string theory]] and [[quantum field theory]] have attempted to unify relativity and quantum mechanics, neither has been entirely successful or proven.

===Length contraction===

===Length contraction===

When two inertial reference frames move past each other in a straight line with constant relative velocity, an observer in one reference frame would observe a metre rule in the other frame to be shorter.

When two inertial reference frames move past each other in a straight line with constant relative velocity, an observer in one reference frame would observe a metre rule in the other frame to be shorter along the direction parallel to the relative motion.

The length, <math>l</math>, of an object as seen by a (relative) stationary observer is given by:

The length, <math>l</math>, of an object as seen by a (relative) stationary observer is given by:

Where  

Where  

:<math>m_0</math> is the "rest mass" or the mass of the object when it is at rest.

:<math>m_0</math> is the "rest mass" or the mass of the object measured by an observer in the same reference frame as the object.

:<math>v</math> is the relative velocity of the object.

:<math>v</math> is the relative velocity of the object.

:<math>c</math> is the speed of light (3x10⁸ ms^-1).

:<math>c</math> is the speed of light (3x10⁸ ms^-1).