|
|
| Line 1: |
Line 1: |
| {{Special relativity}}
| | The considered consuming green coffee bean extract to start out losing fat may appear extremely new yet it has already assisted a wonderful deal of individuals go back to their awesome fat plus more. Losing weight is never a easy thing to strive for. Besides the undeniable fact which it usually require a great deal of effort plus discipline, 1 is not absolutely sure if the several goods being presented available is efficient o-r not. The question today is, can this extract be an perfect method to lose excess weight? Can it be safe? How swiftly can you actually see results? Keep reading and discover.<br><br>It is always important to check the elements of the [http://greencoffeeweightlossplan.com green coffee bean extract] Extract supplement we are wanting to test. Make sure it's all-natural and 100% pure before you even think of spending any money. You are able to even have a talk with your Doctor before taking any fat loss supplements plus find out when there are any acknowledged side effects or additional issues which you should be aware of.<br><br>Always make certain the supplements we choose are made in the USA or Europe. I may tell we which I have usually intended to purchase a product, considering the advertising found on the bottle was appealing, branded, and had inside it what I desired or felt I needed. But countless brand producers try to hide the fact that their treatments are made overseas in Asia. So in the event you don't consider the box or bottle, carefully, we won't see which inside print almost too tiny to see... yes, there it is... "Made inside China," where the standards for quality are often low or non-existent.<br><br>So, how does it function? The active ingredient, because the advertisers selected to refer to the central plank to their pitch, is chlorogenic acid or acids. Some extract processors and pill manufacturers main on really several - or 1 - of these acids plus put it inside a bottle. The acids abound in the unroasted coffee bean. Roast them and the fat-burning ingredients are destroyed. Yes. Fat burning is the trick.<br><br>The biggest community of scientists has done a study on the benefits of utilizing green coffee extract for fat loss. They gave an extract of green coffee to a sizable group of individuals. They told to the group to not change anything about their lifestyle. This really is how they were able to isolate the benefits of the green coffee extract.<br><br>This causes the beans to turn brown plus it also destroys the chlorogenic acid. A board certified nutritionist plus holistic doctor by the name of Dr. Lindsey Duncan was a guest speaker on Dr. Oz's show, that was aired on April 2012.<br><br>Our nation has gone wild for supplements, from capsules labeled "all-natural migraine relief" to medications marketed because "super-sized stress reduction." If it's sold with a label containing words like "holistic health" plus "natural," you seem to think it's safe and powerful. The latest pumped-up pill: Green coffee bean extract for fat loss.<br><br>Indeed, coffee comes inside all different forms. There is no proper or wrong choice; it is simply a matter of individual opinion. When choosing a coffee roast a common guideline of thumb is that as the sunlight rises inside the morning it was time for a light roast coffee and because the sun sets, it is time for a dark roast coffee. |
| In [[physics]], '''length contraction''' is the phenomenon of a decrease in [[length]] measured by an observer of objects which are traveling at any non-zero velocity relative to the observer. This contraction (more formally called '''Lorentz contraction''' or '''Lorentz–FitzGerald contraction''' after [[Hendrik Lorentz]] and [[George FitzGerald]]) is usually only noticeable at a substantial fraction of the [[speed of light]]. Length contraction is only in the direction parallel to the direction in which the observed body is travelling. This effect is negligible at everyday speeds, and can be ignored for all regular purposes. Only at greater speeds does it become relevant. At a speed of 13,400,000 m/s (30 million mph, 0.0447{{math|''c''}}), the contracted length is 99.9% of the length at rest; at a speed of 42,300,000 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 dominant, as can be seen from the formula:
| |
| | |
| :<math>L=\frac{L_{0}}{\gamma(v)}=L_{0}\sqrt{1-v^{2}/c^{2}}</math>
| |
| | |
| where
| |
| :{{math|''L''<sub>0</sub>}} is the [[proper length]] (the length of the object in its rest frame),
| |
| :{{math|''L''}} is the length observed by an observer in relative motion with respect to the object,
| |
| :{{math|''v''}} is the relative velocity between the observer and the moving object,
| |
| :{{math|''c''}} is the [[speed of light]],
| |
| | |
| and the ''[[Lorentz factor]]'', {{math|''γ(v)''}}, is defined as
| |
| | |
| :<math>\gamma (v) \equiv \frac{1}{\sqrt{1-v^2/c^2}} \ </math>.
| |
| | |
| In this equation it is assumed that the object is parallel with its 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 viewing 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.
| |
| | |
| ==History==
| |
| {{Main|History of special relativity}}
| |
| Length contraction was postulated by [[George FitzGerald]] (1889) and [[Hendrik Antoon Lorentz]] (1892) to explain the negative outcome of the [[Michelson-Morley experiment]] and to rescue the hypothesis of the stationary aether ([[Lorentz ether theory#Length contraction|Lorentz–FitzGerald contraction hypothesis]]).<ref>{{Citation|author=FitzGerald, George Francis|year=1889|title=[[s:The Ether and the Earth's Atmosphere|The Ether and the Earth's Atmosphere]]|journal=Science|volume=13|pages=390|doi=10.1126/science.ns-13.328.390|pmid=17819387|issue=328|bibcode = 1889Sci....13..390F }}</ref><ref>{{Citation|last=Lorentz|first=Hendrik Antoon|year=1892|title=[[s:Translation:The Relative Motion of the Earth and the Aether|The Relative Motion of the Earth and the Aether]]|journal=Zittingsverlag Akad. V. Wet.|volume=1|pages=74–79}}</ref>
| |
| Although both FitzGerald and Lorentz alluded to the fact that electrostatic fields in motion were deformed ("Heaviside-Ellipsoid" after [[Oliver Heaviside]], who derived this deformation from electromagnetic theory in 1888), it was considered an [[ad hoc hypothesis]], because at this time there was no sufficient reason to assume that intermolecular forces behave the same way as electromagnetic ones. In 1897 [[Joseph Larmor]] developed a model in which all forces are considered to be of electromagnetic origin, and length contraction appeared to be a direct consequence of this model. Yet it was shown by [[Henri Poincaré]] (1905) that electromagnetic forces alone cannot explain the electron's stability. So he had to introduce another ad hoc hypothesis: non-electric binding forces ([[Electromagnetic mass#Poincaré stresses and 4/3 problem|Poincaré stresses]]) that ensure the electron's stability, give a dynamical explanation for length contraction, and thus hide the motion of the stationary aether.<ref name=pais>{{Citation
| |
| |author=Pais, Abraham
| |
| |authorlink=Abraham Pais
| |
| |year=1982
| |
| |title= Subtle is the Lord: The Science and the Life of Albert Einstein
| |
| |location= New York
| |
| |publisher=Oxford University Press
| |
| |isbn=0-19-520438-7}}</ref>
| |
| | |
| Eventually, [[Albert Einstein]] (1905) was the first<ref name=pais /> who completely removed the ad hoc character from the contraction hypothesis, by demonstrating that this contraction was not due to motion through a supposed aether, but was due to the change in the notions of space, time and simultaneity brought about by [[special relativity]].<ref>{{Citation
| |
| |doi=10.1002/andp.19053221004
| |
| |author=Einstein, Albert
| |
| |year=1905a
| |
| |title=Zur Elektrodynamik bewegter Körper
| |
| |journal=Annalen der Physik
| |
| |volume=322
| |
| |issue=10
| |
| |pages=891–921|bibcode = 1905AnP...322..891E
| |
| |url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_891-921.pdf}}. See also: [http://www.fourmilab.ch/etexts/einstein/specrel/ English translation].</ref> Einstein's view was further elaborated by [[Hermann Minkowski]] and others, who demonstrated the geometrical meaning of all relativistic effects in [[spacetime]].<ref>{{Citation|author=Minkowski, Hermann|year=1909|title=[[s:de:Raum und Zeit (Minkowski)|Raum und Zeit]]|journal=Physikalische Zeitschrift|volume=10|pages=75–88}}
| |
| :*Various English translations on Wikisource: [[s:Space and Time|Space and Time]]</ref>
| |
| | |
| ==Basis in relativity==
| |
| [[File:Lorentzkontraktion.png|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.]]
| |
| | |
| 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 > 0, then one can proceed as follows:
| |
| | |
| 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 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 after 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 [[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|author=Edwin F. Taylor, John Archibald Wheeler|title=Spacetime Physics: Introduction to Special Relativity|year=1992|publisher=W. H. Freeman|location=New York|ISBN=0-7167-2327-1}}</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, yet the length of the same object as measured in another inertial frame is shorter than the proper length. This contraction only occurs in the line of motion, and can be represented by the following relation (where <math>v</math> is the relative velocity and <math>c</math> the speed of light)
| |
| | |
| :<math>L=L_{0}/\gamma.</math>
| |
| | |
| == Symmetry ==
| |
| {{multiple image
| |
| |align=right
| |
| |width=250
| |
| |direction=vertical
| |
| |background color = white
| |
| |image1=Minkowski diagram - length contraction.svg
| |
| |caption1=In S all events parallel to the axis of x are simultaneous, while in S' all events parallel to the axis of x' are simultaneous.
| |
| |image2=LoedRodTransport.svg
| |
| |caption2=A rod is transported from S to S'
| |
| |header = [[Minkowski diagram]]
| |
| }}
| |
| | |
| The principle of relativity (according to which the laws of nature must assume the same form in all inertial reference frames) requires that length contraction is symmetrical: If a rod rests in inertial frame S, it has its proper length in S and its length is contracted in S'. However, if a rod rests in S', it has its proper length in S' and its length is contracted in S. This can be vividly illustrated using symmetric [[Minkowski diagram]]s (or Loedel diagrams), because the Lorentz transformation geometrically corresponds to a rotation in four-dimensional [[spacetime]].<ref>{{cite book |author=Albert Shadowitz |title=Special relativity |isbn=0-486-65743-4 |publisher=Courier Dover Publications |edition=Reprint of 1968 edition |year=1988|pages=20–22}}</ref><ref name=sartori>{{cite book |author=Leo Sartori |title=Understanding Relativity: a simplified approach to Einstein's theories |isbn=0-520-20029-2 |publisher=University of California Press |year=1996|pages=151ff}}</ref>
| |
| | |
| ''First image'': If a rod at rest in S' is given, then its endpoints are located upon the ct' axis and the axis parallel to it. In this frame the simultaneous (parallel to the axis of x') positions of the endpoints are O and B, thus the ''proper'' length is given by OB. But in S the simultaneous (parallel to the axis of x) positions are O and A, thus the ''contracted'' length is given by OA.
| |
| | |
| On the other hand, if another rod is at rest in S, then its endpoints are located upon the ct axis and the axis parallel to it. In this frame the simultaneous (parallel to the axis of x) positions of the endpoints are O and D, thus the ''proper'' length is given by OD. But in S' the simultaneous (parallel to the axis of x') positions are O and C, thus the ''contracted'' length is given by OC.
| |
| | |
| ''Second image'': A train at rest in S and a station at rest in S' with relative velocity of <math>v = 0{.}8c</math> are given. In S a rod with proper length <math>L_0=\mathrm{AB}=30\ \mathrm{cm}</math> is located, so its contracted length <math>L'</math> in S' is given by:
| |
| | |
| :<math>L'=\mathrm{AC}=L_{0}/\gamma=18\ \mathrm{cm}.</math>
| |
| | |
| Then the rod will be thrown out of the train in S and will come to rest at the station in S'. Its length has to be measured again according to the methods given above, and now the proper length <math>L'_0 = \mathrm{EF} =30\ \mathrm{cm}</math> will be measured in S' (the rod has become larger in that system), while in S the rod is in motion and therefore its length is contracted (the rod has become smaller in that system):
| |
| | |
| :<math>L=\mathrm{DE}=L'_{0}/\gamma=18\ \mathrm{cm}.</math>
| |
| | |
| ==Experimental verifications==
| |
| {{See also|Tests of special relativity}}
| |
| | |
| Any observer co-moving with the observed object cannot measure the object's contraction, because he can judge himself and the object as at rest in the same inertial frame in accordance with the principle of relativity (as it was demonstrated by the [[Trouton-Rankine experiment]]). So Length contraction cannot be measured in the object's rest frame, but only in a frame in which the observed object is in motion. In addition, even in such a non-co-moving frame, ''direct'' experimental confirmations of Length contraction are hard to achieve, because at the current state of technology, objects of considerable extension cannot be accelerated to relativistic speeds. And the only objects traveling with the speed required are atomic particles, yet whose spatial extensions are too small to allow a direct measurement of contraction.
| |
| | |
| However, there are ''indirect'' confirmations of this effect in a non-co-moving frame:
| |
| *It was the negative result of a famous experiment, that required the introduction of length contraction: the [[Michelson-Morley experiment]] (and later also the [[Kennedy–Thorndike experiment]]). In special relativity its explanation is as follows: In its rest frame the interferometer can be regarded as at rest in accordance with the relativity principle, so the propagation time of light is the same in all directions. Although in a frame in which the interferometer is in motion, the transverse beam must traverse a longer, diagonal path with respect to the non-moving frame thus making its travel time longer, the factor by which the longitudinal beam would be delayed by taking times L/(c-v) & L/(c+v) for the forward and reverse trips respectively is even longer. Therefore, in the longitudinal direction the interferometer is supposed to be contracted, in order to restore the equality of both travel times in accordance with the negative experimental result(s). Thus the two-way speed of light remains constant and the round trip propagation time along perpendicular arms of the interferometer is independent of its motion & orientation.
| |
| [[File:MyonLoed.svg|thumb|250px|Muon-atmosphere-scenario]]
| |
| * The range of action of [[muon]]s at high velocities is much higher than that of slower ones. The atmosphere has its proper length in the Earth frame, while the increased muon range is explained by their longer lifetimes due to time dilation (see [[Time dilation of moving particles]]). However, in the muon frame their lifetime is unchanged, but the atmosphere is contracted so that even their small range is sufficient to reach the surface of earth.<ref name=sexl />
| |
| *Heavy [[ion]]s that are spherical when at rest should assume the form of "pancakes" or flat disks when traveling nearly at the speed of light. And in fact, the results obtained from particle collisions can only be explained, when the increased nucleon density due to length contraction is considered.<ref>{{cite web|author=Brookhaven National Laboratory|url=http://www.bnl.gov/rhic/physics.asp|title=The Physics of RHIC|accessdate=2013}}</ref><ref>{{cite web|author=Manuel Calderon de la Barca Sanchez|url=http://nuclear.ucdavis.edu/~calderon/Research/physicsResearch.html|title=Relativistic heavy ion collisions|accessdate=2013}}</ref><ref>{{Cite journal|author=Hands, Simon|title=The phase diagram of QCD|year=2001|journal=Contemporary Physics|volume=42|issue=4|pages=209-225|doi=10.1080/00107510110063843|arxiv=physics/0105022}}</ref>
| |
| * The [[ionization]] ability of electrically charged particles with large relative velocities is higher than expected. In pre-relativistic physics the ability should decrease at high velocities, because the time in which ionizing particles in motion can interact with the electrons of other atoms or molecules is diminished. Though in relativity, the higher-than-expected ionization ability can be explained by length contraction of the [[Coulomb's law|Coulomb field]] in frames in which the ionizing particles are moving, which increases their electrical field strength normal to the line of motion.<ref>{{Citation|author=Williams, E. J.|title=The Loss of Energy by β -Particles and Its Distribution between Different Kinds of Collisions|year=1931|journal=Proceedings of the Royal Society of London. Series A|volume=130|issue=813|pages=328-346|doi=10.1098/rspa.1931.0008}}</ref><ref name=sexl>{{Citation|author=Sexl, Roman & Schmidt, Herbert K.|title=Raum-Zeit-Relativität|year=1979|publisher=Vieweg|location=Braunschweig|isbn=3-528-17236-3}}</ref>
| |
| * In [[free-electron laser]]s, relativistic electrons were injected into an [[undulator]], so that [[synchrotron radiation]] is generated. In the proper frame of the electrons, the undulator is contracted which leads to an increased radiation frequency. Additionally, to find out the frequency as measured in the laboratory frame, one has to apply the [[relativistic Doppler effect]]. So, only with the aid of length contraction and the relativistic Doppler effect, the extremely small wavelength of undulator radiation can be explained.<ref>{{cite web|author=DESY photon science|url=http://photon-science.desy.de/research/studentsteaching/primers/synchrotron_radiation/index_eng.html|title=What is SR, how is it generated and what are its properties?|accessdate=2013}}</ref><ref>{{cite web|author=DESY photon science|url=http://flash.desy.de/sites2009/site_vuvfel/content/e395/e2188/FLASH-Broschrefrs_web.pdf|title=FLASH The Free-Electron Laser in Hamburg (PDF 7,8 MB)|accessdate=2013}}</ref>
| |
| | |
| ==Reality of length contraction==
| |
| [[File:EinsteinContraction.svg|thumb|300px|Minkowski diagram of Einstein's 1911 [[thought experiment]] on length contraction. Two rods of rest length <math>A'B'=A''B''=L_0</math> are moving with 0.6c in opposite direction, resulting in <math>A^\ast B^\ast <L_0</math>.]]
| |
| In 1911 [[Vladimir Varićak]] asserted that length contraction is "real" according to Lorentz, while it is "apparent or subjective" according to Einstein.<ref>{{Citation |author=Miller, A.I.| year=1981 | title= Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) | chapter=Varičak and Einstein| place= Reading |publisher=Addison–Wesley |pages=249–253 | isbn=0-201-04679-2}}</ref> Einstein replied:
| |
| | |
| {{Quote|text=The author unjustifiably stated a difference of Lorentz's view and that of mine ''concerning the physical facts''. The question as to whether length contraction ''really'' exists or not is misleading. It doesn't "really" exist, in so far as it doesn't exist for a comoving observer; though it "really" exists, ''i.e.'' in such a way that it could be demonstrated in principle by physical means by a non-comoving observer.<ref name=einst1911>{{cite journal|author=Einstein, Albert|title=Zum Ehrenfestschen Paradoxon. Eine Bemerkung zu V. Variĉaks Aufsatz|journal=Physikalische Zeitschrift|volume=12|pages=509–510|year=1911}}; Original: Der Verfasser hat mit Unrecht einen Unterschied der ''Lorentz''schen Auffassung von der meinigen ''mit Bezug auf die physikalischen Tatsachen'' statuiert. Die Frage, ob die ''Lorentz''-Verkürzung ''wirklich'' besteht oder nicht, ist irreführend. Sie besteht nämlich nicht „wirklich“, insofern sie für einen mitbewegten Beobachter nicht existiert; sie besteht aber „wirklich“, d. h. in solcher Weise, daß sie prinzipiell durch physikalische Mittel nachgewiesen werden könnte, für einen nicht mitbewegten Beobachter.</ref>|sign=Albert Einstein, 1911}}
| |
| | |
| Einstein also argued in that paper, that length contraction is not simply the product of ''arbitrary'' definitions concerning the way clock regulations and length measurements are performed. He presented the following thought experiment: Let A'B' and A"B" be the endpoints of two rods of same proper length. Let them move in opposite directions with same speed with respect to a resting coordinate x-axis. Endpoints A'A" meet at point A*, and B'B" meet at point B*, both points being marked on that axis. Einstein pointed out that length A*B* is shorter than A'B' or A"B", which can also be demonstrated by one of the rods when brought to rest with respect to that axis.<ref name=einst1911 />
| |
| | |
| ==Paradoxes==
| |
| Due to superficial application of the contraction formula some paradoxes can occur. For examples see the [[Ladder paradox]] or [[Bell's spaceship paradox]]. However, those paradoxes can simply be solved by a correct application of relativity of simultaneity. Another famous paradox is the [[Ehrenfest paradox]], which proves that the concept of [[Rigid body|rigid bodies]] is not compatible with relativity, and reduces the applicability of [[Born rigidity]]. It also shows that for a co-rotating observer the geometry is in fact [[non-euclidean Geometry|non-euclidean]].
| |
| | |
| ==Visual effects==
| |
| {{Main|Terrell rotation}}
| |
| Length contraction refers to measurements of position made at simultaneous times according to a coordinate system. This could suggest that if one could take a picture of a fast moving object, that the image would show the object contracted in the direction of motion. However, such visual effects are completely different measurements, as such a photograph is taken from a distance, while length contraction can only directly be measured at the exact location of the object's endpoints. It was shown by several authors such as [[Roger Penrose]] and James Terrell that moving objects do not appear length contracted on a photograph.<ref>{{cite journal |author=Kraus, U. |year=2000 |title=Brightness and color of rapidly moving objects: The visual appearance of a large sphere revisited |journal=American Journal of Physics |volume=68 |issue=1 |pages=56-60 |doi=10.1119/1.19373|url = http://www.tempolimit-lichtgeschwindigkeit.de/sphere/sphere.pdf }}</ref> For instance, for a small angular diameter, a moving sphere remains circular and is rotated.<ref>{{cite book|last=Penrose|first=Roger|title=The Road to Reality|year=2005|publisher=Vintage Books|location=London|isbn=978-0-09-944068-0|pages=430–431}}</ref> This kind of visual rotation effect is called Penrose-Terrell rotation.<ref>[http://math.ucr.edu/home/baez/physics/Relativity/SR/penrose.html Can You See the Lorentz-Fitzgerald Contraction? Or: Penrose-Terrell Rotation]</ref>
| |
| | |
| ==Derivation==
| |
| === Lorentz transformation ===
| |
| Length contraction can be derived from the Lorentz transformation in several ways:
| |
| | |
| :<math>\begin{align}
| |
| x' & =\gamma\left(x-vt\right),\\
| |
| t' & =\gamma\left(t-vx/c^{2}\right).
| |
| \end{align}</math>
| |
| | |
| ==== Moving length is known ====
| |
| In an inertial reference frame S, <math>x_{1}</math> and <math>x_{2}</math> shall denote the endpoints of an object in motion in this frame. There, its length <math>L</math> was measured according to the above convention by determining the simultaneous positions of its endpoints at <math>t_{1}=t_{2}\,</math>. Now, the proper length of this object in S' shall be calculated by using the Lorentz transformation. Transforming the time coordinates from S into S' results in different times, but this is not problematic, as the object is at rest in S' where it does not matter when the endpoints are measured. Therefore, the transformation of the spatial coordinates suffices, which gives:<ref name=born>{{Citation|author=Born, Max|authorlink=Max Born|title = Einstein's Theory of Relativity|publisher=Dover Publications|year =1964|isbn=0-486-60769-0 }}</ref>
| |
| | |
| :<math>x'_{1}=\gamma\left(x_{1}-vt_{1}\right)\quad\mathrm{and}\quad x'_{2}=\gamma\left(x_{2}-vt_{2}\right).</math>
| |
| | |
| Since <math>t_{1}=t_{2}\,</math>, and by setting <math>L=x_{2}-x_{1}\,</math> and <math>L_{0}^{'}=x_{2}^{'}-x_{1}^{'}</math>, the proper length in S' is given by
| |
| | |
| :<math>L_{0}^{'}=L\cdot\gamma. \qquad \qquad \text{(1)},</math>
| |
| | |
| with respect to which the measured length in S is contracted by
| |
| | |
| :<math>L=L_{0}^{'}/\gamma. \qquad \qquad \text{(2)}</math>
| |
| | |
| According to the relativity principle, objects that are at rest in S have to be contracted in S' as well. By exchanging the above signs and primes symmetrically, it follows:
| |
| | |
| :<math>L_{0}=L'\cdot\gamma. \qquad \qquad \text{(3)}</math>
| |
| | |
| Thus the contracted length as measured in S' is given by:
| |
| | |
| :<math>L'=L_{0}/\gamma.\qquad \qquad \text{(4)}</math>
| |
| | |
| ==== Proper length is known ====
| |
| Conversely, if the object rests in S and its proper length is known, the simultaneity of the measurements at the object's endpoints has to be considered in another frame S', as the object constantly changes its position there. Therefore, both spatial and temporal coordinates must be transformed: <ref>{{Cite book|author=Bernard Schutz|title=A First Course in General Relativity|chapter=Lorentz contraction|pages=18|publisher=Cambridge University Press|year=2009|isbn=0521887054}} ({{Google books quote|id=V1CGLi58W7wC|text=lorentz contraction|page=18|GBS, p. 18.}})</ref>
| |
| | |
| :<math>\begin{align}
| |
| x_{1}^{'} & =\gamma\left(x_{1}-vt_{1}\right) & \quad\mathrm{and}\quad & & x_{2}^{'} & =\gamma\left(x_{2}-vt_{2}\right)\\
| |
| t_{1}^{'} & =\gamma\left(t_{1}-vx_{1}/c^{2}\right) & \quad\mathrm{and}\quad & & t_{2}^{'} & =\gamma\left(t_{2}-vx_{2}/c^{2}\right).
| |
| \end{align}</math>
| |
| | |
| With <math>t_{1}=t_{2}</math> and <math>L_{0}=x_{2}-x_{1}</math> this results in non-simultaneous differences:
| |
| | |
| :<math>\begin{align}
| |
| \Delta x' & =\gamma L_{0}\\
| |
| \Delta t' & =\gamma vL_{0}/c^{2}
| |
| \end{align}</math>
| |
| | |
| In order to obtain the simultaneous positions of both endpoints, the distance traveled by the second endpoint with <math>v</math> during <math>\Delta t'</math> must be subtracted from <math>\Delta x'</math>:
| |
| | |
| :<math>\begin{align}
| |
| L' & =\Delta x'-v\Delta t'\\
| |
| & =\gamma L_{0}-\gamma v^{2}L_{0}/c^{2}\\
| |
| & =L_{0}/\gamma
| |
| \end{align}</math>
| |
| | |
| So the moving length in S' is contracted. Likewise, the preceding calculation gives a symmetric result for an object at rest in S':
| |
| | |
| :<math>L=L^{'}_{0}/\gamma</math>.
| |
| | |
| === Time dilation ===
| |
| | |
| Length contraction can also be derived from [[time dilation]],<ref>{{Citation|author=[[David Halliday (physicist)|David Halliday]], [[Robert Resnick]], [[Jearl Walker]]|title = Fundamentals of Physics, Chapters 33-37|publisher=John Wiley & Son|year =2010|isbn=0470547944|pages=1032f }}</ref> according to which the rate of a single "moving" clock (indicating its [[proper time]] <math>T_0</math>) is lower with respect to two synchronized "resting" clocks (indicating <math>T</math>). Time dilation was experimentally confirmed multiple times, and is represented by the relation:
| |
| | |
| :<math>T=T_{0}\cdot\gamma</math>.
| |
| | |
| Suppose a rod of proper length <math>L_0</math> at rest in <math>S</math> and a clock at rest in <math>S'</math> are moving along each other. The respective travel times of the clock between the rod's endpoints are given by <math>T=L_{0}/v</math> in <math>S</math> and <math>T'_{0}=L'/v</math> in <math>S'</math>, thus <math>L_{0}=Tv</math> and <math>L'=T'_{0}v</math>. By inserting the time dilation formula, the ratio between those lengths is:
| |
| | |
| :<math>\frac{L'}{L_{0}}=\frac{T'_{0}v}{Tv}=1/\gamma</math>.
| |
| | |
| Therefore, the length measured in <math>S'</math> is given by
| |
| | |
| :<math>L'=L_{0}/\gamma</math>.
| |
| | |
| So the effect that the moving clock indicates a lower travel time in <math>S</math> due to time dilation, is interpreted in <math>S'</math> as due to length contraction of the moving rod. Likewise, if the clock were at rest in <math>S</math> and the rod in <math>S'</math>, the above procedure would give
| |
| | |
| :<math>L=L'_{0}/\gamma</math>.
| |
| | |
| === Geometrical considerations ===
| |
| | |
| [[File:Slabs.svg|thumb|250px|Cuboids in Euclidean and Minkowski spacetime]]
| |
| | |
| Additional geometrical considerations show, that length contraction can be regarded as a ''trigonometric'' phenomenon, with analogy to parallel slices through a [[cuboid]] before and after a ''rotation'' in '''E'''<sup>3</sup> (see left half figure at the right). This is the Euclidean analog of ''boosting'' a cuboid in '''E'''<sup>1,2</sup>. In the latter case, however, we can interpret the boosted cuboid as the ''world slab'' of a moving plate.
| |
| | |
| ''Image'': Left: a ''rotated cuboid'' in three-dimensional euclidean space '''E'''<sup>3</sup>. The cross section is ''longer'' in the direction of the rotation than it was before the rotation. Right: the ''world slab'' of a moving thin plate in Minkowski spacetime (with one spatial dimension suppressed) '''E'''<sup>1,2</sup>, which is a ''boosted cuboid''. The cross section is ''thinner'' in the direction of the boost than it was before the boost. In both cases, the transverse directions are unaffected and the three planes meeting at each corner of the cuboids are ''mutually orthogonal'' (in the sense of '''E'''<sup>1,2</sup> at right, and in the sense of '''E'''<sup>3</sup> at left).
| |
| | |
| In special relativity, [[Poincaré group|Poincaré transformations]] are a class of [[affine transformation]]s which can be characterized as the transformations between alternative [[Cartesian coordinates|Cartesian coordinate charts]] on [[Minkowski spacetime]] corresponding to alternative states of [[inertial frame|inertial motion]] (and different choices of an [[origin (mathematics)|origin]]). Lorentz transformations are Poincaré transformations which are [[linear transformation]]s (preserve the origin). Lorentz transformations play the same role in Minkowski geometry (the [[Lorentz group]] forms the ''isotropy group'' of the self-isometries of the spacetime) which are played by [[rotation]]s in euclidean geometry. Indeed, special relativity largely comes down to studying a kind of noneuclidean [[trigonometry]] in Minkowski spacetime, as suggested by the following table:
| |
| | |
| {| class="wikitable" style="text-align:center"
| |
| |+'''Three plane trigonometries'''
| |
| |-
| |
| ! Trigonometry
| |
| ! Circular !! Parabolic !! Hyperbolic
| |
| |-
| |
| ! Kleinian Geometry
| |
| | euclidean plane || Galilean plane || Minkowski plane
| |
| |-
| |
| ! Symbol
| |
| | '''E'''<sup>2</sup> || '''E'''<sup>0,1</sup> || '''E'''<sup>1,1</sup>
| |
| |-
| |
| ! Quadratic form
| |
| | positive definite || degenerate || non-degenerate but indefinite
| |
| |-
| |
| ! Isometry group
| |
| | '''E'''(2) || '''E'''(0,1) || '''E'''(1,1)
| |
| |-
| |
| ! Isotropy group
| |
| | '''SO'''(2) || '''SO'''(0,1) || '''SO'''(1,1)
| |
| |-
| |
| ! type of isotropy
| |
| | rotations || shears || boosts
| |
| |-
| |
| ! [[Cayley algebra]]
| |
| | complex numbers || dual numbers || split-complex numbers
| |
| |-
| |
| ! ε<sup>2</sup>
| |
| | -1 || 0 || 1
| |
| |-
| |
| ! Spacetime interpretation
| |
| | none || Newtonian spacetime || Minkowski spacetime
| |
| |-
| |
| ! slope
| |
| | tan φ = m || tanp φ = u || tanh φ = v
| |
| |-
| |
| ! "cosine"
| |
| | cos φ = (1+m<sup>2</sup>)<sup>-1/2</sup> || cosp φ = 1 || cosh φ = (1-v<sup>2</sup>)<sup>-1/2</sup>
| |
| |-
| |
| ! "sine"
| |
| | sin φ = m (1+m<sup>2</sup>)<sup>-1/2</sup> || sinp φ = u || sinh φ = v (1-v<sup>2</sup>)<sup>-1/2</sup>
| |
| |-
| |
| ! "secant"
| |
| | sec φ = (1+m<sup>2</sup>)<sup>1/2</sup> || secp φ = 1 || sech φ = (1-v<sup>2</sup>)<sup>1/2</sup>
| |
| |-
| |
| ! "cosecant"
| |
| | csc φ = m<sup>−1</sup> (1+m<sup>2</sup>)<sup>1/2</sup> || cscp φ = u<sup>−1</sup> || csch φ = v<sup>−1</sup> (1-v<sup>2</sup>)<sup>1/2</sup>
| |
| |}
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| {{Relativity}}
| |
| | |
| [[Category:Special relativity]]
| |
| [[Category:History of physics]]
| |
| [[Category:Length]]
| |