Length contraction

The Lorentz contraction or relativistic length contraction is a phenomenon of special relativity. It says that a moving observer a shorter distance between two points in space as a measure of resting. Because the length of an object, the distance between its end points, the length measurement of a moving object results in a shorter length than the same measurement on the stationary object. The effect occurs only in the direction of the relative movement and decreases with increasing relative speed. The Lorentz contraction, together with the time dilation and the relativity of simultaneity is one of the basic phenomena of special relativity theory and plays an important role in the analysis of experiments in particle accelerators.

The size of the effect is calculated by means of the contraction formula

It is

The length contraction was originally introduced in 1892 by Hendrik Antoon Lorentz, up with the speed to explain the Michelson - Morley experiment relative to the hypothetical ether. She received in 1905 by Albert Einstein their modern, relativistic interpretation, in which the speed between the observer and observed object is.

History

The length contraction was originally a qualitative form of George Francis FitzGerald (1889 ) and in quantitative form by Hendrik Antoon Lorentz (1892 ) postulated to explain the negative result of the Michelson -Morley experiment and to save the idea of ​​a stationary ether ( Fitzgerald - Lorentz contraction hypothesis). As an analogy was used already in 1888 by Oliver Heaviside established fact that moving electrostatic fields are deformed ( Heaviside ellipsoid). Because at the time, however, no reason existed to believe that the intermolecular forces behave exactly like electromagnetic forces (or electromagnetic nature are) the Lorentz contraction was classified as an ad hoc hypothesis, which only served to the discovery with the hypothesis of ether to reconcile. Below Joseph Larmor developed in 1897 a theory in which the matter itself of electromagnetic origin. The contraction hypothesis is then no longer be considered as pure ad hoc hypothesis, but would be a consequence of the electromagnetic constitution of matter. A purely electromagnetic explanation of the matter soon turned out to be unworkable: Henri Poincaré showed in 1905 that non- electrical forces another ad hoc hypothesis had to be introduced to ensure the stability of the electrons and to explain the contraction dynamically.

This problem was solved when Albert Einstein in 1905 be required to adopt by reformulation of the concepts of space and time, and without any dynamic ether effects, in the framework of special relativity achieved a simple kinematic derivation. This statement, which was based on the principle of relativity and the principle of the constancy of the speed of light, the effect took his final ad- hoc nature and forms the basis of the modern conception of the Lorentz contraction. This was further out, among others, Hermann Minkowski, who developed a vivid geometric representation of the relativistic effects in space-time.

Explanation

For the understanding of the Lorentz contraction is the careful consideration of the methods for measuring the length of stationary and moving objects is of fundamental importance. By " object" is simply a line meant to rest their endpoints always to each other and always move at the same speed. If the observer does not move relative to the observed object, so they rest in the same inertial frame, then the can " rest or proper length " of the object can be easily determined by direct application of a scale.

However, there is a relative velocity > 0, can be taken as follows: The observer provides a number of watches on, all of which are synchronized, either

• By the exchange of light signals according to the Poincaré Einstein synchronization or
• By "slow clock transport". In this method, a clock is ( to be able to neglect the influence of the time dilation ) sufficiently slowly transferred to each clock of the row and transfers them to its time display.

After synchronization, the object to be measured is moving along these watches series. Each clock records the time at which the right and the left end of the object passes through the respective clock. You wrote down using the values ​​stored in the watches of the time and place of a clock A at which the left end has determined and the location of a clock B at the same time has found the right end. It is clear that the distance AB clock is identical to the length of the moving object.

The definition of the simultaneity of events is thus crucial for the length measurement of moving objects. In classical physics, the simultaneity is absolute, and are thus and always coincide. However, in the theory of relativity makes the constancy of the speed of light in all inertial systems and the related relativity of simultaneity destroy this match. So if observers claim in an inertial system to have the two end points of the object measured simultaneously, observers will claim in all other inertial frames, that these measurements were not made simultaneously, that is a Lorentz transformation from the value to be calculated - see the section derivation. As a result, we have: While the rest length remains unchanged and still is the largest measured length of the body, is at a relative movement between the object and measuring instrument - measured contracted length - with respect to the resting length. This occurs only in the contraction direction of motion is represented by the following relationship (in which the relative speed, and the speed of light ):

Consider, for example, a train and a train station, which move relative to each other at a constant speed. The station lies in the inertial frame S, the train rests in S '. The train system S ' is now a bar are that there has a rest length. From the perspective of the station system S, however, the rod is moved, and it is measured according to the following formula, the contracted length:

The rod is then thrown out of the train at the station and comes to a stop, so that the observer must again determine the length of the rod, taking into account the above measurement specifications. Now it is the railway station system S in which the resting length of the rod is measured by ( the rod has increased in S), whereas the rod from the perspective of the train system S ' is moving and is measured contracted in accordance with the following formula:

As required by the principle of relativity, the same laws of nature must be valid in all inertial systems. Thus, the length contraction falls from symmetrical: Resting the rod in the train, he has measured in drawing system S ' its resting length and is contracted at the station S system. If, however, he is transported to the station, then the station system S its resting length and measured in the train system S ' its contracted length.

Derivation

Lorentz transformation

The relationship between resting length and moving length can be, depending on the measurement situation, derived by means of the Lorentz transformation.

Moving length was measured

In the inertial system S, respectively, and the measured endpoints perform on an object moving length. Since the object is moved here, and its length was measured already by simultaneous determination of the end points, so according to the above measurement specification. There will now in the end points S ', in which the object rests, is determined by the Lorentz transformation. A transformation of the time coordinate would result in a difference, but this is irrelevant because the object ' rests in the target system S in the same place and time of measurements there does not matter. Consequently, in this situation, measuring the transformation of the space coordinates is sufficient:

Since and, we have:

The resting length in S ' is thus greater than the turbulent length in S, the latter is therefore contracted with respect to the resting length to:

According to the principle of relativity must also reversed in S ' objects at rest be subject from the perspective of S a contraction. If in the above formulas, the signs and underscores replaced symmetrically, there is actually:

So

Resting length was measured

However, if an object given the rest in S, then there its resting length is measured. When used to calculate the moving length in S ', is to pay attention to simultaneity of the measurement of the end points in the target system, since the position of the endpoints is constantly changing due to the movement there. The Lorentz transformation is:

With and results in the following non- simultaneous differences:

To determine the simultaneous positions of the two end points has the distance that has been traveled from the second endpoint in time, be removed from the non- simultaneous removal. The result is

Conversely, the above calculation method for a in S ' object at rest the symmetric result:

Time dilation

The length contraction can also be derived by means of time dilation. According to this effect, the transition rate of an individual is "moving" clock that displays their invariant proper time less than that of two synchronized " resting" clocks which display the time. Time dilation has been experimentally confirmed many times and is represented by the relationship:

A rod with length at rest in and a clock at rest in moving along each other. The respective travel times of the clock by a rod end to the other are given with in and so are the periods and lengths. By inserting the Zeitdilatationsformel the ratio of the lengths obtained by:

Consequently, the measured length in with results

So the effect displays that in the moving clock a lower travel time due to the time dilation is in interpreted as caused by the contraction of the moving rod. If now rest in the clock and the bar would rest in, would the above approach, the symmetric result:

Minkowski diagram

Now, the Lorentz transformation corresponding geometric rotation in a four-dimensional space-time, and the following from their effects as the Lorentz contraction can therefore be illustrated with the aid of a Minkowski diagram.

Is a resting rod given in S ', so are its endpoints on the ct' - axis and the axis parallel to it. In S ' yields the ( parallel to the x' axis ) simultaneous positions of the endpoints O and B, ie, a resting length of OB. In contrast, the S ( parallel to the x- axis) the simultaneous positions of the endpoints O and A are given, which is a contracted length of OA.

If, however, a stationary rod in S given, are its endpoints on the ct- axis and the axis parallel to it. In S there are the ( parallel to the x axis ) simultaneous positions of the endpoints O and D, ie, a resting length of OD. In contrast, in S ' the ( parallel to the x' axis ) simultaneous positions of the endpoints O and C are given, ie a contracted length of OC.

Experimental confirmations

A direct experimental confirmation of the Lorentz contraction is difficult because the effect would be detectable only upon an almost moving at the speed of light particles. However, their spatial dimension as particles is negligible. In addition, they can only be detected by an observer who is not like the observed object in the same inertial frame. For a comoving observer is the same contraction subject as object to be observed, and thus both can look dormant due to the relativity principle as in the same inertial frame (see for example the Trouton - Rankine experiment or experiments of Rayleigh and Brace ). For the moving observer 's own contraction is therefore non-existent.

However, there are a number of indirect confirmations of the Lorentz contraction, wherein the evaluation was made, by definition, from the viewpoint of a non- inertial entrained out.

• It was the negative result of Michelson - Morley experiment, which required the implementation of the Lorentz contraction. As part of the SRT whose declaration looks like this: For a moving observer, the interferometer is at rest and the result is negative due to the principle of relativity. But from the perspective of a non- co-moving observer (equivalent in classical physics point of view of a resting in the ether observer ), the interferometer must be contracted in the direction of movement to bring the negative result with Maxwell 's equations and the principle of constancy of light speed in accordance.
• It follows from the Lorentz contraction that idle spherical heavy ions at relativistic velocities in the direction of movement in the form of flat discs or pancakes ( " pancakes " ) must accept. And indeed shows that the results obtained in particle collisions can only be explained taking into account the Lorentz contraction caused by the high nucleon density or the high frequencies in the electromagnetic fields. This fact means that the effects of the Lorentz contraction must be considered during the design of the experiments.
• Another confirmation is the increase in Ionisierungsvermögens electrically charged particles with increasing speed. According to classical physics would have to remove such property, but the Lorentz contraction of the Coulomb field leads to an increase with increasing speed of the electric field perpendicular to the direction of motion, which leads to the actually observed increase in Ionisierungsvermögens.
• Another example are muons in the atmosphere that arise at a distance of about 10 km from the Earth's surface. If the half-life match of stationary and moving muons, they could travel only about 600 m, even at almost the speed of light - yet they reach the earth's surface. In the rest frame of the atmosphere, this phenomenon can be explained with the time dilation of moving particles, the lifetime and thus the range of muons extended accordingly by the. In the rest system of the muons, although the range is unchanged at 600 m, but the atmosphere is moved, and consequently contracted, so that even the low range sufficient to reach the surface to.
• Likewise, the length contraction, together with the relativistic Doppler effect in accordance with the wavelength of the ultra-low undulator of a free-electron laser. Here relativistic electrons are injected into an undulator, thereby producing synchrotron radiation. In the rest frame of the particle, the undulator moves nearly the speed of light and is contracted, resulting in an increased frequency. Now in this frequency the relativistic Doppler effect must, in order to determine the frequency in the laboratory, are applied.

Linked to this is the question whether the length contraction is "real" or "apparently" is. But this is largely on word choice, as in the theory of relativity, the ratio of resting length and contracted length is operationally defined unambiguously and can and will be used as just running profitably in physics. Even Einstein himself had in 1911 a replica of the statement Vladimir Varičaks back, which according to the Lorentz contraction " actually " However, according to Einstein only "apparently, subjective" is ( emphasis in original ):

"The author has unjustly statuiert a difference Lorentz's conception of the mine with respect to the physical facts. The question of whether the Lorentz shortening really exists or not is misleading. It is namely not "real ," as it does not exist for a moving observer; but it is "real", ie in such a way that they could in principle be detected by physical means, for a non -moving observer. "

Optical perception

As explained above, it is necessary to measure the length contraction of moving objects that watches and measuring instruments on the location of the object to be measured or at its end points are. Another question is how a moving object from a distance looks like - for example, in a photograph or film of a camera. It follows that in a photo, the Lorentz contraction is not recognizable as such, since the contraction effect to occur even visual effects that lead to a distortion of the image. Instead of a compressed object, the observer sees the original object is rotated, wherein the rotational angle of the apparent velocity of the body dependent.

The adjacent diagram to explain this effect: the body considered is simplified as a cube illustrated in plan view, the resting observer through an eye. The blue side of the cube is the simplicity perpendicular to the line of sight of the observer. If the body is at rest, so the observer sees only the page that shows to him (pictured in blue). If the body is on the other hand ( for simplicity perpendicular to the line of sight of the observer ) so too can the light rays emanating from the red side to reach the eye of the observer in motion. While the red side is invisible in a body at rest in a moving body with increasing speed is always more of it visible. The visible image of a body is determined by the light rays reaching the eye at the same time. While the light beam from the rearmost point of the red side on the way to the observer passes by the more forward points, the body has already moved a piece. Thus all the light rays come closer -lying points in the viewer arrive at the same time, in the direction of movement of the body moved to. At the same time the blue screen appears squashed, as it experiences a Lorentz contraction. Overall, for the observer the same visual impression which would produce a rotated body. The apparent angle of rotation is in this case depends only on the vertical velocity of the body relative to the line of sight of the observer:

Apparent paradoxes

A superficial application of the contraction formula may lead to apparent paradoxes of the Lorentz contraction. Examples are various paradoxes of length contraction, which can be easily resolve on closer consideration of the measurement rules and related to the relativity of simultaneity. Somewhat more complicated are the relationships, if accelerations as in Bell's spaceship paradox in play. In this case, the exchange was carried out of the inertial system leads to a change in the evaluation of the simultaneity of events, and also the arising stresses in the materials used must be taken into account. Similarly, for the rotation of bodies, where it can be demonstrated using the Ehrenfest 's paradox, that can exist no rigid body in the SRT. For Einstein, this relationship was an important step in the development of general relativity, as a co-rotating observer takes for the space partly because of the Lorentz contraction of a non-Euclidean geometry.