Editing Length contraction (section)

==Basis in relativity==
[[File:Observer in special relativity.svg|thumb|In special relativity, the observer measures events against an infinite latticework of synchronized clocks.]]

First it is necessary to carefully consider the methods for measuring the lengths of resting and moving objects.<ref name=born /> Here, "object" simply means a distance with endpoints that are always mutually at rest, ''i.e.'', that are at rest in the same [[inertial frame of reference]]. If the relative velocity between an observer (or his measuring instruments) and the observed object is zero, then the [[proper length]] <math>L_0</math> of the object can simply be determined by directly superposing a measuring rod. However, if the relative velocity is greater than zero, then one can proceed as follows:

[[File:Lorentzkontraktion.svg|thumb|300px|right|''Length contraction'': Three blue rods are at rest in S, and three red rods in S'. At the instant when the left ends of A and D attain the same position on the axis of x, the lengths of the rods shall be compared. In S the simultaneous positions of the left side of A and the right side of C are more distant than those of D and F, while in S' the simultaneous positions of the left side of D and the right side of F are more distant than those of A and C.]]
The observer installs a row of clocks that either are synchronized a) by exchanging light signals according to the [[Einstein synchronization|Poincaré–Einstein synchronization]], or b) by "slow clock transport", that is, one clock is transported along the row of clocks in the limit of [[zero of a function|vanishing]] transport velocity. Now, when the synchronization process is finished, the object is moved along the clock row and every clock stores the exact time when the left or the right end of the object passes by. After that, the observer only has to look at the position of a clock A that stored the time when the left end of the object was passing by, and a clock B at which the right end of the object was passing by ''at the same time''. It's clear that distance AB is equal to length <math>L</math> of the moving object.<ref name=born /> Using this method, the definition of [[Relativity of simultaneity|simultaneity]] is crucial for measuring the length of moving objects.

Another method is to use a clock indicating its [[proper time]] <math>T_0</math>, which is traveling from one endpoint of the rod to the other in time <math>T</math> as measured by clocks in the rod's rest frame. The length of the rod can be computed by multiplying its travel time by its velocity, thus <math>L_{0} = T\cdot v</math> in the rod's rest frame or <math>L = T_{0}\cdot v</math> in the clock's rest frame.<ref>{{cite book|author1=Edwin F. Taylor| author2=John Archibald Wheeler| title=Spacetime Physics: Introduction to Special Relativity| year=1992| publisher=W. H. Freeman| location=New York| isbn=0-7167-2327-1| url=https://archive.org/details/spacetimephysics00edwi_0}}</ref>

In Newtonian mechanics, simultaneity and time duration are absolute and therefore both methods lead to the equality of <math>L</math> and <math>L_0</math>. Yet in relativity theory the constancy of light velocity in all inertial frames in connection with [[relativity of simultaneity]] and [[time dilation]] destroys this equality. In the first method an observer in one frame claims to have measured the object's endpoints simultaneously, but the observers in all other inertial frames will argue that the object's endpoints were ''not'' measured simultaneously. In the second method, times <math>T</math> and <math>T_0</math> are not equal due to time dilation, resulting in different lengths too.

The deviation between the measurements in all inertial frames is given by the formulas for [[Lorentz transformation]] and time dilation (see [[#Derivation|Derivation]]). It turns out that the proper length remains unchanged and always denotes the greatest length of an object, and the length of the same object measured in another inertial reference frame is shorter than the proper length. This contraction only occurs along the line of motion, and can be represented by the relation
:<math>L = \frac{1}{\gamma(v)}L_0</math>

where
*<math>L</math> is the length observed by an observer in motion relative to the object
*<math>L_0
</math> is the proper length (the length of the object in its rest frame)
*<math>\gamma(v)</math> is the ''[[Lorentz factor]]'',  defined as <math display="block">\gamma (v) \equiv \frac{1}{\sqrt{1 - v^2/c^2}} </math> where
**<math>v</math> is the relative velocity between the observer and the moving object
**<math>c</math> is the speed of light

Replacing the Lorentz factor in the original formula leads to the relation
:<math>L = L_0\sqrt{1 - v^2/c^2}</math>

In this equation both <math>L</math> and <math>L_0
</math> are measured parallel to the object's line of movement. For the observer in relative movement, the length of the object is measured by subtracting the simultaneously measured distances of both ends of the object. For more general conversions, see the [[Lorentz transformations]]. An observer at rest observing an object travelling very close to the speed of light would observe the length of the object in the direction of motion as very near zero.

Then, at a speed of {{val|13,400,000|u=m/s}} (30 million mph, 0.0447{{math|''c''}}) contracted length is 99.9% of the length at rest; at a speed of {{val|42,300,000|u=m/s}} (95 million mph, 0.141{{math|''c''}}), the length is still 99%. As the magnitude of the velocity approaches the speed of light, the effect becomes prominent.