# Lorentz contraction

The Lorentz contraction or relativistic length contraction is a phenomenon of the special theory of relativity . The measured distance between two points in space depends on the relative movement of the measuring and measured system. If the points whose distance is to be measured are at rest in the measuring system, the measurement gives the maximum value - the so-called rest length. The faster the measuring and the measured system move relative to one another, the smaller the measured distance becomes. Since the length of an object is 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 only occurs in the direction of relative movement and increases with increasing relative speed. The Lorentz contraction, like the time dilation and the relativity of simultaneity, is one of the fundamental phenomena of the special theory of relativity and plays an important role in the evaluation of experiments in particle accelerators .

## The formula in general terms

The size of the effect along the direction of movement of an object is calculated using the contraction formula:

${\ displaystyle L = {\ frac {L_ {0}} {\ gamma}}}$.

Where:

${\ displaystyle L}$the contracted length, i.e. the length measured in an inertial system of an object moving relative to this inertial system,
${\ displaystyle L_ {0}}$ the rest length, i.e. the length of the same object measured in the inertial system in which the object is at rest, and
${\ displaystyle \ gamma = {\ frac {1} {\ sqrt {1-v ^ {2} / c ^ {2}}}}}$the Lorentz factor with the speed of light and the speed of the object to be measured.${\ displaystyle c}$${\ displaystyle v}$

The formula for the Lorentz contraction can thus be written as:

${\ displaystyle L = L_ {0} {\ sqrt {1-v ^ {2} / c ^ {2}}}}$.
${\ displaystyle L_ {0} = L {\ frac {1} {\ sqrt {1-v ^ {2} / c ^ {2}}}}}$.

The length contraction was originally introduced in 1892 by Hendrik Antoon Lorentz to explain the Michelson-Morley experiment with the velocity relative to the hypothetical ether . In 1905 it received its modern, relativistic interpretation from Albert Einstein , in which the speed is between the observer and the observed object. ${\ displaystyle v}$${\ displaystyle v}$

## history

The length contraction was originally formulated in qualitative form by George Francis FitzGerald (1889) and in quantitative form by Hendrik Antoon Lorentz (1892) (initially postulated) in order to explain the negative outcome of the Michelson-Morley experiment and the idea of ​​a stationary ether to save ( Fitzgerald-Lorentz contraction hypothesis ).

The fact, already established by Oliver Heaviside in 1888 , that moving electrostatic fields are deformed ( Heaviside ellipsoid ) served as an analogy . However, since at that time there was no reason to assume that the intermolecular forces behave in the same way as electromagnetic forces (or are of an electromagnetic nature), the Lorentz contraction was classified as an ad hoc hypothesis , which only served to reconcile the discovery with the hypothesis of the To bring ether into harmony. In the following, Joseph Larmor developed a theory in 1897 in which matter itself is of electromagnetic origin. The contraction hypothesis is then no longer to be regarded as a pure ad hoc hypothesis, but would be a consequence of the electromagnetic constitution of matter. A purely electromagnetic explanation of matter soon turned out to be impracticable: Henri Poincaré showed in 1905 that another ad hoc hypothesis had to be introduced with non-electrical forces in order to guarantee the stability of the electrons and to dynamically explain the contraction.

This problem was solved when Albert Einstein in 1905 succeeded in reformulating the concepts of space and time, and without having to assume any dynamic etheric effects, within the framework of the special theory of relativity, a simple kinematic derivation. This explanation, which was based on the principle of relativity and the principle of the constancy of the speed of light, finally took the effect of its ad hoc character and forms the basis of the modern conception of the Lorentz contraction. This was continued by Hermann Minkowski , among others , who developed a clear geometric representation of the relativistic effects in space-time .

## Explanation

To understand Lorentz contraction, careful consideration of the methods for measuring the length of stationary and moving objects is of fundamental importance. “Object” simply means a route whose end points always rest in relation to one another or always move at the same speed. If the observer does not move relative to the observed object, i.e. if they rest in the same inertial system, then the "rest or proper length " of the object can be determined simply by directly applying a scale. ${\ displaystyle L_ {0}}$

However, if there is a relative speed> 0, the following procedure can be used: The observer sets up a series of clocks, which are all synchronized, either

• by exchanging light signals according to the Poincaré-Einstein synchronization or
• through "slow watch transport". With this method, a clock is transported sufficiently slowly (so that the influence of time dilation can be neglected) to each individual clock in the series and transfers its time display to it.

After synchronization has taken place, the object to be measured moves along this row of clocks. Each clock records the time at which the right and left ends of the object pass the respective clock. You wrote down from the stored values 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 which the same has been the right end. It is clear that the clock spacing AB is identical to the length of the moving object. ${\ displaystyle L}$

Symmetry of the Lorentz contraction: there are three structurally identical blue and red rods that move past each other. If the left ends of A and D occupy the same position on the x-axis, it now follows in S that the simultaneous position of the left end of A and the right end of C is significantly further apart than the simultaneous position of the left end of D and the right end of F. On the other hand, in S 'the simultaneous position of the left end of D and the right end of F is significantly further apart than the simultaneous position of the left end of A and the right end of C.

The definition of the simultaneity of events is therefore of decisive importance for the length measurement of moving objects. In classical physics simultaneity is absolute, and consequently will and always coincide. However, in the theory of relativity, the constancy of the speed of light in all inertial systems and the related relativity of simultaneity nullify this agreement. So if observers in one inertial system claim to have measured the two end points of the object at the same time, observers in all other inertial systems will claim that these measurements were not made at the same time, namely around a value to be calculated from the Lorentz transformation - see the section Derivation . As a result: While the rest length remains unchanged and is always the largest measured length of the body, a relative movement between the object and the measuring instrument - with respect to the rest length - measures a contracted length. This contraction, which only occurs in the direction of movement , is represented by the following relationship (where the Lorentz factor is): ${\ displaystyle L}$${\ displaystyle L_ {0}}$${\ displaystyle \ gamma}$

${\ displaystyle L = L_ {0} / \ gamma.}$

For example, consider a train and a station moving at a constant speed relative to one another . The station rests in the inertial system S, the train rests in S '. In the traction system S 'there should now be a rod that has a rest length of there. From the point of view of the station system S, however, the rod is moved and the contracted length is measured according to the following formula : ${\ displaystyle v = 0 {,} 8c}$${\ displaystyle L_ {0} ^ {'} = 30 \ \ mathrm {cm}}$${\ displaystyle L}$

${\ displaystyle L = L_ {0} ^ {'} / \ gamma = 18 \ \ mathrm {cm}}$

The rod is now thrown from the train and comes to a standstill at the station, so that the observers have to determine the length of the rod again, taking into account the above measurement regulations. Now it is the station system S, in which the rest length of the rod is measured by (the rod has become larger in S), whereas the rod is moved from the point of view of the train system S 'and is measured contracted according to the following formula: ${\ displaystyle L_ {0} = 30 \ \ mathrm {cm}}$

${\ displaystyle L '= L_ {0} / \ gamma = 18 \ \ mathrm {cm}}$

As required by the relativity principle , the same laws of nature must apply in all inertial systems. The length contraction is therefore symmetrical: If the rod is at rest in the train, it has its rest length in the train system S 'and is measured contracted in the station system S. If, on the other hand, it is transported to the station, then its rest length is measured in the station system S and its contracted length is measured in the train system S '.

## Derivation

### Lorentz transformation

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

{\ displaystyle {\ begin {aligned} x ^ {'} & = \ gamma \ left (x-vt \ right), \\ t ^ {'} & = \ gamma \ left (t-vx / c ^ {2 } \ right). \ end {aligned}}}

### Moved length was measured

In the inertial system S, and denote the measured end points for an object of length moving there . Since the object is moved here, according to the above measurement procedure and its length was measured by the simultaneous determination of the endpoints already so . The end points in S ', where the object is at rest, are now to be determined by the Lorentz transformation. A transformation of the time coordinates would result in a difference, which, however, is irrelevant since the object in the target system S 'always rests at the same location and the time of the measurements is irrelevant there. Consequently, the transformation of the spatial coordinates is sufficient in this measurement situation: ${\ displaystyle x_ {1}}$${\ displaystyle x_ {2}}$${\ displaystyle L}$${\ displaystyle L}$${\ displaystyle t_ {1} = t_ {2}}$

${\ displaystyle x '_ {1} = \ gamma \ left (x_ {1} -vt_ {1} \ right) \ quad \ mathrm {and} \ quad x' _ {2} = \ gamma \ left (x_ { 2} -vt_ {2} \ right).}$

There and is, it results: ${\ displaystyle L = x_ {2} -x_ {1} \}$${\ displaystyle L_ {0} ^ {'} = x_ {2} ^ {'} - x_ {1} ^ {'}}$

(1) ${\ displaystyle L_ {0} ^ {'} = L \ gamma.}$

The rest length in S 'is greater than the moved length in S, the latter is thus contracted with respect to the rest length by:

(2) ${\ displaystyle L = L_ {0} ^ {'} / \ gamma.}$

Conversely, according to the principle of relativity, objects at rest in S must also be subject to a contraction from the point of view of S '. If the signs and dashes are symmetrically replaced in the above formulas, the result is actually:

(3) ${\ displaystyle L_ {0} = L '\ gamma,}$

so

(4) ${\ displaystyle L '= L_ {0} / \ gamma.}$

### Rest length was measured

However, if there is an object that is at rest in S, then its rest length is measured there. If the moved length in S 'is calculated from this, attention must be paid to the simultaneity of the measurement of the end points in the target system, since the position of the end points changes constantly there due to the movement. So the Lorentz transformation is:

{\ displaystyle {\ begin {aligned} 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 {aligned}}.}

With and the following non-simultaneous differences arise: ${\ displaystyle t_ {1} = t_ {2} \}$${\ displaystyle L_ {0} = x_ {2} -x_ {1}}$

{\ displaystyle {\ begin {aligned} \ Delta x '& = \ gamma L_ {0} \\\ Delta t' & = \ gamma vL_ {0} / c ^ {2} \ end {aligned}}}

In order to determine the simultaneous positions of the two end points, the distance that has been covered by the second end point in the time must be subtracted from the non-simultaneous distance . It turns out ${\ displaystyle v}$${\ displaystyle \ Delta t '}$${\ displaystyle \ Delta x '}$

{\ displaystyle {\ begin {aligned} L '& = \ Delta x'-v \ Delta t' \\ & = \ gamma L_ {0} - \ gamma v ^ {2} L_ {0} / c ^ {2 } \\ & = L_ {0} / \ gamma \ end {aligned}}}

Conversely, the above calculation method gives the symmetrical result for an object at rest in S ':

${\ displaystyle L = L_ {0} ^ {'} / \ gamma}$.

### Time dilation

The length contraction can also be derived from time dilation . According to this effect, the rate of a single “moving” watch, which shows its invariant proper time, is lower than that of two synchronized “resting” watches which show the time . The time dilation has been confirmed experimentally many times and is represented by the relationship: ${\ displaystyle T_ {0}}$${\ displaystyle T}$

${\ displaystyle T = T_ {0} \ gamma}$.

A rod with rest length in and a clock at rest in S 'move along each other. The respective travel times of the clock from one end of the rod to the other are given in S and S ', i.e. the lengths covered are and . By inserting the time dilation formula, the ratio of the lengths results with: ${\ displaystyle L_ {0}}$${\ displaystyle S}$${\ displaystyle T = L_ {0} / v}$${\ displaystyle T '_ {0} = L' / v}$${\ displaystyle L_ {0} = Tv}$${\ displaystyle L '= T' _ {0} v}$

${\ displaystyle {\ frac {L '} {L_ {0}}} = {\ frac {T' _ {0} v} {Tv}} = 1 / \ gamma}$

As a result, the measured length results in S 'with

${\ displaystyle L '= L_ {0} / \ gamma}$.

So the effect that the moving clock shows a shorter travel time due to the time dilation is interpreted in S 'as being caused by the contraction of the moving rod. If the clock was now in S and the rod in S ', the above procedure would give the symmetrical result: ${\ displaystyle S}$

${\ displaystyle L = L '_ {0} / \ gamma}$.

## Minkowski diagram

Minkowski diagram and length contraction. In S all events on parallels to the x-axis are simultaneous, and in S 'all events on parallels to the x'-axis are simultaneous.

The Lorentz transformation corresponds geometrically to a rotation in four-dimensional space - time , and the effects resulting from it, such as the Lorentz contraction, can consequently be clearly illustrated with the aid of a Minkowski diagram.

If a rod at rest is given in S ', its endpoints are on the ct' axis and the axis parallel to it. The simultaneous positions of the end points with O and B (parallel to the x'-axis) result in S ', i.e. a rest length of OB. In contrast, the simultaneous positions of the end points with O and A (parallel to the x-axis) are given in S, i.e. a contracted length of OA.

If, on the other hand, a rod at rest is given in S, its endpoints are on the ct axis and the axis parallel to it. In S, the simultaneous positions of the endpoints with O and D (parallel to the x-axis) result, i.e. a rest length of OD. In contrast, the simultaneous positions of the end points with O and C (parallel to the x 'axis) are given in S', i.e. a contracted length of OC.

## Experimental confirmations

A direct experimental confirmation of the Lorentz contraction is difficult, since the effect would only be detectable with particles moving approximately at the speed of light. However, their spatial dimension as particles is negligible. In addition, it can only be detected by an observer who is not in the same inertial system as the observed object. Because a moving observer is subject to the same contraction as the object to be observed, and thus both can consider themselves to be at rest in the same inertial system due to the principle of relativity (see for example the Trouton-Rankine experiment or the experiments by Rayleigh and Brace ). For the observer who is moving with it, his own contraction is consequently nonexistent.

However, there are a number of indirect confirmations of the Lorentz contraction, whereby the assessment was made, by definition, from the standpoint of a non-moving inertial system.

• It was the negative outcome of the Michelson-Morley experiment that made the introduction of the Lorentz contraction necessary. In the context of the SRT, its explanation is as follows: For a moving observer, the interferometer is at rest and the result is negative due to the principle of relativity. But from the point of view of a non-moving observer (in classical physics this corresponds to the point of view of an observer resting in the ether), the interferometer must be contracted in the direction of movement in order to bring the negative result into agreement with Maxwell's equations and the principle of constancy of the speed of light.
• It follows from the Lorentz contraction that in the state of rest spherical heavy ions at relativistic speeds in the direction of movement must take the form of flat disks or pancakes. And it actually turns out that the results obtained with particle collisions can only be explained taking into account the high nucleon density caused by the Lorentz contraction or the high frequencies in the electromagnetic fields. This fact means that the effects of the Lorentz contraction have to be taken into account in the design of the experiments.
• A further confirmation is the increase in the ionization capacity of electrically charged particles with increasing speed. According to classical physics, this capacity would have to decrease, but the Lorentz contraction of the Coulomb field leads to an increase in the electric field strength perpendicular to the direction of movement with increasing speed , which leads to the actually observed increase in ionization capacity.
• Another example are muons in the earth's atmosphere, which arise at a distance of approx. 10 km from the earth's surface. If the half-life of resting and moving muons were to match, they could only travel approx. 600 m even at almost the speed of light - they still reach the earth's surface. In the rest system of the atmosphere, this phenomenon is explained by the time dilation of moving particles , which increases the lifespan and thus the range of the muons accordingly. In the muon rest system, the range is unchanged at 600 m, but the atmosphere is agitated and consequently contracted, so that even the short range is sufficient to reach the surface.
• Likewise, the length contraction together with the relativistic Doppler effect is in agreement with the extremely small wavelength of the undulator radiation of a free-electron laser . Relativistic electrons are injected into an undulator and synchrotron radiation is generated. In the rest system of the particles, the undulator moves almost at the speed of light and is contracted, which leads to an increased frequency. The relativistic Doppler effect must now be applied to this frequency to determine the frequency in the laboratory system.

Linked to this is the question of whether the length contraction is real or apparent . But this affects more the choice of words, because in the theory of relativity the relationship between rest length and contracted length is operationally unequivocally defined, and can and is used in physics as just mentioned. Einstein himself also rejected Vladimir Varičak's assertion in 1911 , according to which, according to Lorentz, the contraction is "real", but according to Einstein it is only "apparent, subjective" (emphasis placed in the original):

“The author wrongly stated a difference between Lorentz 's conception and mine with reference to physical facts . The question of whether the Lorentz abbreviation really exists or not is misleading. For it does not “really” exist insofar as it does not exist for an observer who is moving along with it; but it "really" exists, i. H. in such a way that it could in principle be proven by physical means for an observer who is not moving. "

- Albert Einstein

## Optical perception

As explained above, to measure the length contraction of moving objects it is necessary that clocks or measuring instruments are located at the location of the object to be measured or at its end points. Another question is what a moving object looks like from a greater distance - for example on a photograph or a camera film . The result is that the Lorentz contraction cannot be recognized as such in a photo, since the contraction effect is also accompanied by optical effects that lead to a distortion of the image. Instead of a compressed object, the observer sees the original object rotated, the apparent angle of rotation depending on the speed of the body.

Lorentz contraction: optical perception as rotation

The graphic opposite is intended to explain this effect: The observed body is shown in simplified form as a cube in a top view, the stationary observer is represented by an eye. The blue side of the cube is perpendicular to the observer's line of sight for simplicity. If the body is at rest, the observer only sees the side that faces him (shown in blue in the picture). On the other hand, if the body is in motion (perpendicular to the observer's line of sight for the sake of simplicity), the light rays emanating from the red side can also reach the observer's eye. While the red side is invisible with a resting body, with a moving body more and more of it becomes visible with increasing speed. The visible image of a body is determined by the rays of light that reach the eye at the same time. While the light beam from the furthest point on the red side passes the points further ahead on its way to the observer, the body has already moved a little further. Thus, all light rays from closer points that arrive at the viewer at the same point in time arrive offset in the direction of movement of the body. At the same time, the blue side appears compressed because it experiences a Lorentz contraction. Overall, the observer has the same visual impression that a rotated body would produce. The apparent angle of rotation is only dependent on the relative speed of the body perpendicular to the line of sight of the observer: ${\ displaystyle \ varphi}$${\ displaystyle v}$

${\ displaystyle \ sin {\ varphi} = {\ frac {v} {c}}.}$

Superficial application of the contraction formula can lead to apparent paradoxes of the Lorentz contraction. Examples are various paradoxes of length contraction , which can easily be resolved if the measurement rules and the related relativity of simultaneity are taken into account. The relationships are a little more complicated when accelerations such as Bell's spaceship paradox are involved. The change in the inertial system that took place in the process leads to a change in the assessment of the simultaneity of events, and the stresses that arise in the materials used must also be taken into account. The same applies to the rotation of bodies, where it can be demonstrated using Ehrenfest's paradox that no rigid bodies can exist in the SRT. For Einstein this connection was an important step in the development of the general theory of relativity , since for a co-rotating observer the space assumes a non-Euclidean geometry , partly because of the Lorentz contraction .

## Individual evidence

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6. a b Max Born: Moving standards and clocks . In: Einstein's Theory of Relativity . Springer, Berlin / Heidelberg / New York 2003, ISBN 3-540-00470-X , pp. 212-214.
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13. ^ EJ Williams: The Loss of Energy by β-Particles and Its Distribution between Different Kinds of Collisions . In: Proceedings of the Royal Society of London. Series A . 130, No. 813, 1931, pp. 328-346. doi : 10.1098 / rspa.1931.0008 .
14. a b R. Sexl, HK Schmidt: Space-Time-Relativity . Vieweg, Braunschweig 1979, ISBN 3-528-17236-3 .
15. DESY photon science: What is SR, how is it generated and what are its properties? (No longer available online.) Archived from the original on June 3, 2016 ; accessed in 2013 .
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17. See e.g. B. Physics FAQ: "People sometimes argue over whether the Lorentz-Fitzgerald contraction is 'real' or not ... here's a short answer: the contraction can be measured, but the measurement is frame dependent. Whether that makes it 'real' or not has more to do with your choice of words than the physics. "Translation:" Some sometimes debate whether the Lorentz-Fitzgerald contraction is 'real' or not ... here is the short answer : The contraction can be measured, but the measurement depends on the frame of reference. Whether this leads to it being 'real' or not has more to do with our choice of words than with physics. "
18. A. Einstein: To the Ehrenfest paradox. A note on V. Variĉak's paper . In: Physikalische Zeitschrift . 12, 1911, pp. 509-510.
19. ^ Articles on the visualization of length contraction from the University of Tübingen
20. Norbert Dragon and Nicolai Mokros: Relativistic flight through Stonehenge. ( Memento of August 21, 2009 in the Internet Archive ).