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in any inertial reference frame. The magnitude of the 4-momentum, in any inertial frame, equals the rest mass <math>m_0</math> of the particle (in units where <math>c=1</math>).
in any inertial reference frame. The magnitude of the 4-momentum, in any inertial frame, equals the rest mass <math>m_0</math> of the particle (in units where <math>c=1</math>).
== Paradoxes ==
The predictions of the theory of relativity throw up a number of apparent paradoxes and anomalies relating to the effects of time dilatation and length contraction. Whilst these paradoxes are consistent with the theory, they are contrary to everyday human experience and therefore can seem like impossibilities.
=== The Twin Paradox ===
The twin paradox is usually stated as a thought experiment involving two twins, one of whom is sent on a long journey in a spacecraft travelling at close to the speed of light, whilst the other remains on Earth. Time dilatation means that the travelling twin, on his return to Earth, is younger that the twin who has remained at home. However because neither twin is in a special position - each being in an inertial frame of reference - the reverse must also be true, and so the twin remaining on Earth must be younger. Hence each twin is younger than the other - a paradox.
The problem can be resolved in two ways. One is to examine the effects of General Relativity: to come back to Earth, the travelling twin must undergo acceleration in order to reverse his course, causing temporal effects which make him permanently the younger. Alternatively, it can be explained entirely using Special Relativity and noting that the twins are not in symmetrical situations: the one on earth has remained in a single inertial frame of reference, whilst the travelling twin has travelled in two<ref>http://mentock.home.mindspring.com/twins.htm</ref>.
=== The Ehrenfest Paradox ===
The Ehrenfest Paradox considers a rigid wheel or disc rotating a bout its axis at high speed (somewhat like a bicycle wheel spinning freely on its axle). The rim of the wheel travels at close to the speed of light and therefore undergoes length contraction, whereas the radius (the spokes, for the bicycle wheel) does not. Hence the circumference is no longer equal to 2<big><math>\pi</math></big>r, which is paradoxical.
The apparent paradox was finally resolved in 1975 by the Norwegian scientist [[Øyvind Grøn]]<ref>http://www.physicsforums.com/showthread.php?t=224955</ref>.
=== Force Perpendicular to Direction of Motion ===
An object moving at close to the speed of light has an increased mass, and thus a force applied to it would cause less acceleration then if applied to the same body at a lower speed. This paradoxically means that the acceleration caused is different if the force is applied in the direction of travel, or orthogonally to it{{Citation needed|date=December 2011}}.
== Variable Speed of Light ==
== Variable Speed of Light ==