Wikiprojekt:SKFiz/brudnopis/Skrócenie Lorentza

Skrócenie Lorentza – zjawisko fizyczne polegające na pozornej zmianie długości obiektu podróżującego z dowolną niezerową prędkością w stosunku do obserwatora. Skrócenie Lorentza ma mijesce tylko wzdłuż kierunku poruszania się ciała i jest na ogół obserwowalne dla prędkości bliskich prędkości światła. Przykładowo dla prędkości 13.400.000 m/s (48.240.000 km/h, .0447 prędkości światła), długość obserwowana to 99.9% długości spoczynkowej; dla prędkości 42.300.000 m/s (152.280.000 km/h, 0.141 prędkości światła), długość obserwowana to 99% długości spoczynkowej. Zmiana długości w efekcie skrócenia Lorentza opisana jest wzorem:

$L={\frac {L_{0}}{\gamma (v)}}=L_{0}{\sqrt {1-v^{2}/c^{2}}}$

gdzie

$L_{0}$ to długość spoczynkowa obiektu,

$L$ to długość zmierzona przez obserwatora względem którego porusza się obiekt mierzony,

$v\,$ to prędkość obiektu względem obserwatora,

$c\,$ to prędkość światła,

a Czynnik Lorentzowski jest zdefiniowany jako:

$\gamma (v)\equiv {\frac {1}{\sqrt {1-v^{2}/c^{2}}}}\$.

W powyższym równaniu zakłada się, że długość jest mierzona wzdłuż kierunku ruchu poprzez odjęcie od siebie jednocześnie niezmierzonych położenia obu końców obiektu.

}}</ref>

In an inertial reference frame S', $x_{1}^{'}$ and $x_{2}^{'}$ shall denote the endpoints for an object of length $L_{0}^{'}$ at rest in this system. The coordinates in S' are connected to those in S by the Lorentz transformations as follows:

$x_{1}^{'}={\frac {x_{1}-vt_{1}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$    and    $x_{2}^{'}={\frac {x_{2}-vt_{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}.$

As this object is moving in S, its length $L$ has to be measured according to the above convention by determining the simultaneous positions of its endpoints, so we have to put $t_{1}=t_{2}\,$. Because $L=x_{2}-x_{1}\,$ and $L_{0}^{'}=x_{2}^{'}-x_{1}^{'}$, we obtain

(1) $L_{0}^{'}={\frac {L}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}.$

Thus the length as measured in S is given by

(2) $L=L_{0}^{'}\cdot {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}.$

According to the relativity principle, objects that are at rest in S have to be contracted in S' as well. For this case the Lorentz transformation is as follows:

$x_{1}={\frac {x_{1}^{'}+vt_{1}^{'}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$     and    $x_{2}={\frac {x_{2}^{'}+vt_{2}^{'}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}.$

By the requirement of simultaneity $t_{1}^{'}=t_{2}^{'}\,$ and by putting $L_{0}=x_{2}-x_{1}\$ and $L^{'}=x_{2}^{'}-x_{1}^{'}$, we actually obtain:

(3) $L_{0}={\frac {L^{'}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}.$

Thus its length as measured in S' is given by:

(4) $L^{'}=L_{0}\cdot {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}.$

So (1), (3) give the proper length when the contracted length is known, and (2), (4) give the contracted length when the proper length is known.

Geometrical representation

Szablon:See also

Minkowski diagram and length contraction. 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.

The Lorentz transformation geometrically corresponds to a rotation in four-dimensional spacetime, and it can be illustrated by a Minkowski diagram: 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.

Left: a rotated cuboid in three-dimensional euclidean space E³. 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^1,2, 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^1,2 at right, and in the sense of E³ at left).

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 E³ (see left half figure at the right). This is the Euclidean analog of boosting a cuboid in E^1,2. In the latter case, however, we can interpret the boosted cuboid as the world slab of a moving plate.

In special relativity, Poincaré transformations are a class of affine transformations which can be characterized as the transformations between alternative Cartesian coordinate charts on Minkowski spacetime corresponding to alternative states of inertial motion (and different choices of an origin). Lorentz transformations are Poincaré transformations which are linear transformations (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 rotations 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:

Experimental verifications

Since the occurrence of length contraction depends on the inertial frame chosen, it can only be measured by an observer not at rest in the same inertial frame, i.e., it exists only in a non-co-moving frame. This is because the effect vanishes after a Lorentz transformation into the rest frame of the object, where a co-moving observer can judge himself and the object as at rest in the same inertial frame in accordance with the relativity principle (as it was demonstrated by the Trouton-Rankine experiment). In addition, even in a non-co-moving frame, direct experimental confirmations of Lorentz 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 in fact the negative result of a famous experiment, that required the introduction of Lorentz 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 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.

Other indirect confirmations are: Heavy ions 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 Lorentz contraction is considered. Another confirmation is the increased ionization ability of electrically charged particles in motion. According to pre-relativistic physics the ability should decrease at high speed, however, the Lorentz contraction of the Coulomb field leads to an increase of the electrical field strength normal to the line of motion, which leads to the actually observed increase of the ionization ability. Lorentz contraction is also necessary to understand the function of free-electron lasers. 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 Lorentz contraction and the rel. Doppler effect, the extremely small wavelength of undulator radiation can be explained. Another example is the observed lifetime of muons in motion and thus their range of action, which is much higher than that of muons at low velocities. In the proper frame of the atmosphere, this is explained by the time dilation of the moving muons. However, in the proper frame of the muons their lifetime is unchanged, but the atmosphere is contracted so that even their small range is sufficient to reach the surface of earth.

Reality of Lorentz contraction

Another issue that is sometimes discussed concerns the question whether this contraction is "real" or "apparent". However, this problem only stems from terminology, as our common language attributes different meanings to both of them. Yet, whatever terminology is chosen, in physics the measurement and the consequences of length contraction with respect to any reference frame are clearly and unambiguously defined in the way stated above.

In 1911 Vladimir Varićak asserted that length contraction is "real" according to Lorentz, while it is "apparent or subjective" according to Einstein. Einstein replied:

The author unjustifiably stated a difference of Lorentz's view and that of mine concerning the physical facts. The question as to whether the Lorentz 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.

{{Cytat}} Brakujące pola: treść. Przestarzałe pola: 1 oraz 2.

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 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.

Visual effects

Length contraction refers to measurements of position made at simultaneous times according to a coordinate system. This could naively lead to a thinking 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, it is important to realize that 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. In 1959 Roger Penrose and James Terrell published papers demonstrating that length contraction instead actually shows up as elongation or even a rotation in a photographic image. This kind of visual rotation effect is called Penrose-Terrell rotation.

See also

• Time dilation
• Ehrenfest paradox
• Ladder paradox
• Lorentz transformation
• Relativity of simultaneity
• Kennedy–Thorndike experiment
• Trouton–Rankine experiment
• Michelson–Morley experiment