Lorentz-Transformation

The Lorentz transformations, named after Hendrik Antoon Lorentz, join in the special theory of relativity and the Lorentz ether theory between the time and space coordinates, which specify different observers when and where events take place. It is straightforward to uniformly moving observers and to coordinates in which force-free particles just pass through world-lines. For Lorentz transformations, the speed of light remains unchanged, conversely, the constancy of the speed of light was the starting point of Einstein's derivation of the Lorentz transformation.

Affected by the Lorentz transformation are:

• The velocity vector parallel to the local variables
• The electromagnetic field perpendicular to the velocity vector components
• Time.

Lorentz transformation for locations and times

Is a uniformly moving observer is moving with velocity in direction over another observer, so hang the coordinates, which he attributes to an event by the Lorentz transformation

With the coordinates of the other observer for the same event together, if the two reference systems have the same orientation and at the time have a common origin, as in a previous one, triggering event.

For a coordinate- free representation of this transformation is decomposed the distance between two events into components parallel and perpendicular to the velocity vector:

With the abbreviation.

Lorentz transformation of the electromagnetic field

Even at low velocities occur with regard to the electromagnetic field relativistic effects. This fundamental fact is illustrated by a simple thought experiment:

• An observer observing a ( relative to it is not moving) charge, to measure an electric field, however, due to the lack of current flow no magnetic field.

• The observer moves, however, on the charge to or away from it, he is on the one hand be noted that due to the motion of the electric field changes. This means that the observer measures a different E-field at the same distance from the load, but other relative speed to the load. On the other hand, the observer interprets the charge as well as a current moving towards or away from him to him. The observer is thus additionally to the electric field detect a magnetic field.

As well as places and times, therefore, the electromagnetic field components of a Lorentz transformation must be subjected if the reference system of observation is changed. For the electrical and magnetic quantities apply:

In the nonrelativistic approximation, ie for velocities, applies approximately. In this case, does not need to be a distinction between places and times in different reference systems, and applies to the box sizes:

Historical Development

The work of Woldemar Voigt (1887 ), Hendrik Antoon Lorentz (1895, 1899, 1904), Joseph Larmor (1897, 1900) and Henri Poincaré (1905, which the Lorentz transformations gave her name ) showed that the solutions of the equations of electrodynamics by Lorentz transformations are mapped to each other or in other words, that the Lorentz transformations are symmetries of the Maxwell equations.

They tried then to explain the electromagnetic phenomena by a hypothetical ether. The most remarkable feature of this ether, however, it turned out that he had to prove no trace. In its Lorentz ether theory could explain this by the fact that the length scales become shorter when moving in the direction of motion and show that moving clocks a slower passage of time, which he called local time. The stated by Lorentz transformations of lengths and times that are displayed by moving clocks and scales, formed a group and were mathematically consistent. Even if a uniform motion relative to the ether was undetectable in Lorentz's ether theory, Lorentz held fast to the idea of an ether, the distinguished an absolutely dormant, but not detectable system.

Einstein's special theory of relativity superseded Newtonian mechanics and the ether hypothesis. He derived his theory from from the principle of relativity, that can not be distinguished from uniform motion in a vacuum, neglecting gravitational effects alone. In particular, light in a vacuum for any observer the same speed. The time and space coordinates, with which two uniformly moving observers describe events, then hang by a Lorentz transformation with each other, rather than as in Newtonian mechanics by a Galilean transformation.

Derivation

The first derivations based on the wave equation of the elastic theory of light or of electrodynamics. Voigt (1887) could transformation formulas, however, are not reciprocal, derived on the basis of the wave equation for an elastic incompressible transmission medium. It was later shown that the exact Lorentz transformation formulas can be rigorously derived from the electromagnetic wave equation, where the demand for linearity and reciprocity is considered. Since the electromagnetic wave equation can be deduced from the Maxwell equations, the Lorentz transformation in the same method and from the Maxwell's equations can be derived. In solutions of Maxwell's electrodynamics, the first transformation of Lorentz (1892-1904) and of Larmor (1897-1900) was used. In the context of electrodynamics the derivation of the Lorentz transformation can also be done taking into account the potential of a moving charge ( Liénard -Wiechert potential).

The derivations that are commonly presented in modern textbooks, based on the interpretation of the transformations in the sense of special relativity, which relate to this space and time themselves, and are independent of assumptions about the electrodynamics. Einstein ( 1905) himself used this two postulates: the principle of relativity and the principle of the constancy of the speed of light. More general derivations, which go back to Vladimir Ignatowski (1910 ), based on group-theoretical considerations.

The following considerations clarify how coordinates are related, the inertial observer (observer are fixed to an inertial frame ) used to name the time and place of events. The observer should be exemplary Anna and Bert here. Anna's coordinate system is given by Bert and painted by the variables. If it were rectangular coordinates.

To keep the formula image of the latter derivations is below, as is customary, the unit of length traveled by the distance light travels in one second, defined. Then, time and length have the same unit, and the dimensionless speed of light is the speed Moreover, the unity of the speed of light. Investigations in other systems of units do not bring deeper insights.

Linearity

For all uniformly moving observers free particles just pass through world-lines. Therefore, the transformation must map straight lines to straight lines. Mathematically, this indicates that the transformation is linear inhomogeneous.

If the two match observer in the choice of the time zero and of the spatial origin, then the desired transformation is linear and homogeneous.

Bert is moving relative to Anna with speed. The coordinate systems are oriented such that and lie on a straight line in one direction. Then you can be limited to the coordinates.

The required Lorentz transformation ( LT) is then

The unknowns are now to be determined.

Light cone

A light pulse, the Anna unleashing currently at the site is described by. Since the speed of light is absolute, must apply for Bert. The equations require the Plus and the equations with the minus sign. It follows and or

This applies to all the LTs, regardless of the relative velocity of the observer.

Relative velocity

Anna describes Bert's through exercise, Bert his own course through. The LT of Annas to Bert's coordinate system must convert these expressions into each other. It follows then, that

It only remains to determine the pre-factor. From the coordinates he can not depend on, otherwise the LT would not be linear. Thus remains only a function of the relative speed. One writes. Since the LT is not to depend on the direction of even applies.

Prefactor

To determine the pre-factor, one introduces another inertial observer Clara with the coordinates and the relative velocity in relation to Bert. The LT of Berts to Clara's coordinates must have the same form as the above because of the principle of relativity, ie

It was abbreviated.

It now combines the two transformations, ie expects the coordinates of Anna at the Clara. It is enough to compute one of the two coordinates:

Sits next to Clara Anna, is, and the double primed coordinate are equal to the unprimed. The factor then disappears, and the pre-factor must be equal to 1. Ways and then must

. apply With the abbreviation is then

In the usual units, the Lorentz transformations read

Addition theorem

Sits next to Anna Clara not, can be calculated from the above expression that associates with the coordinates of Anna, the addition theorem of velocities read ( in units ):

Invariant

It can be shown by inserting the LT that

Must apply. The term is therefore an invariant of the LT, ie constant in all LTs associated with coordinate systems. In three space dimensions is the invariant. The generalization of the LT in three space dimensions is therefore trivial.

Invariance of the transverse coordinates

Wherein relative movement in the x-direction defines a scale which is set in the y- direction, a strip parallel to the x -axis. The observer in the primed and unprimed system, the width of the strip to any selected time points, ie completely independent of the time compare. Unlike scales in the x- direction in which it comes to Lorentz contraction, the relativity of simultaneity is not reflected here. Since the inertial systems are equivalent, the strips must have the same width, ie.

Alternative derivation

Time dilation

With an argument of Macdonald, you can win the transformation formulas with little effort from the time dilation. On a light front moving in the positive x - direction, the Differenzkoordinate has everywhere the same value as well. One considers a front that goes through the event E, and at some point ( before or after) to the moving coordinate origin O ' is true, which is actually slower than light. Because of the steady state values ​​, the difference coordinate each other as at point O ' at E in the same relation. At this applies to, and after the Dilatationsformel being. Therefore applies to the difference coordinates

Analog has on a light front moving in the negative x - direction, the Summenkoordinate everywhere the same value as well. Even such a front passes through E ( with same coordinates as above) and O ' (at a time other than above). In the equation analogous to the previous sums instead of differences now be formed, therefore it is

Addition and subtraction of the two equations as a function of results.

Empirical derivation

Howard Percy Robertson and others have shown that the Lorentz transformation may also be derived empirically. For this it is necessary to provide general transformation formulas between different inertial frames with experimentally determinable parameters. For example, assuming that a single inertial system exists, in which the velocity of light is constant and isotropic, the transformation formulas are relatively moved to the system:

Corresponds to temporal changes, longitudinal and transverse changes in length. corresponds to the currently selected clock synchronization convention, however, has no physical effect. The ratio of longitudinal and transverse changes in length is determined from the Michelson -Morley experiment, the ratio of temporal changes and longitudinal length changes by the Kennedy - Thorndike experiment, and finally changes over time with the Ives - Stilwell experiment. This could be determined and with great accuracy, which transferred above transformation in the Lorentz transformation.

Lorentz - invariant

One size does not change under Lorentz transformations, ie Lorentz invariant. Fixed properties of a physical system which are thus observed by all inertial frames of equal value, one must, if they are not easily reproduced by a always the same numerical value can be expressed by Lorentz invariants. For example, one can form only a single Lorentz - invariant of a four-vector, its norm. With two four-vectors except her two standards and their dot product is Lorentz invariant. A tensor second stage has a lorentzinvariante track, etc. Such physical quantities are the mass (mc is the norm of the energy-momentum vector ), the space- time interval between two events ( norm of the difference of the four-vectors of the two world points ), amount of angular momentum ( angular momentum vector of the standard ), etc. Further lorentzinvariante physical quantities include the speed v = c, the electrical charge, etc.

Poincaré and Lorentz group

The Poincaré group is the set of linear inhomogeneous transformations,

Which leave the distance of two four-vectors invariant. The sub-group of the homogeneous transformation is the Lorentz group, that is the group of the linear transformations on the length of the square

Let each vector of invariant. If we write the length as a square matrix product

Of the column vector (which we note in the text as a line) with the matrix

And the transposed column, row, it must apply to every Lorentz transformed vector

This is precisely the case when the Lorentz transformation equation

Met.

All solutions of this equation, which does not turn the time direction and spatial orientation, are of the form

Here, and rotations

These rotations form the subgroup SO (3) of the Lorentz group. The matrix

Causes the above mentioned Lorentz transformation at a speed

The factor occurring here is called Lorentz factor.

The transformations

Hot Lorentz boost. They transform the coordinates of the moving observer moving with speed in the direction resulting from the rotation of the direction.

Lorentz transformations that change the sign of the time coordinate, the direction of time, not

Provide the subset of orthochronous Lorentz transformation. The Lorentz transformations with

Provide the subset of the actual Lorentz transformation. For the orientation-preserving Lorentz transformations

The time-and orientation-preserving Lorentz transformations

Form the actual orthochronous Lorentz group. It is connected: any actual orthochronous Lorentz transformation can be obtained by continuous change of the six parameters for the three axis of rotation and the angle of rotation and three are converted to the relative speed of the two reference systems of the identity mapping.

Time and space reflection

The non- contiguous with the Lorentz transformations are obtained by mirroring the time or the space reflection

Or both are multiplied by the Lorentz transformation associated with the. The Lorentz group has four connected components.

Velocity addition

The following applies to the speed of light. Executed one after the other Lorentz boosts in the same direction with speed and yield a Lorentz boost to the overall rate

It can be seen immediately thereto, that the speed of light does not change under Lorentz transformations. Is approximately the speed of light, that is, as is also the speed of light.

The above addition formula results from the transformation (see above)

Let's put in a how the factors of the speed dependent, and square we shall apply

This is easy to resolve by the overall speed and shows how to combine when moving in the same direction speeds.

Executed one after the other Lorentz boosts in different directions usually found no Lorentz boosts: the set of Lorentz boosts is not a subgroup of the Lorentz transformations.

Overlay group

The following considerations, that the group of linear transformations of two-dimensional complex vector space whose determinant has the special value, called the special linear group is the simply connected covering of the actual orthochronous Lorentz transformations. The subset of the special two-dimensional unitary transformation, SU (2) superimposed on the group of turns,

Each Hermitian - matrix is of the form:

Since it is reversible uniquely identified by the four real parameters, and since sums and real multiples hermitscher matrices are Hermitian again and belong to the sums and multiples of four-vectors, it is a member of a four-dimensional vector space.

The determinant

Is the length of the square of the four-vector

Multiplying from the left by an arbitrary, complex - adjoint matrix and right with their, so the result is again Hermitian and can be used as writing, which depends linearly on. Is of the special linear group of complex matrices whose determinants have the special value, so does the length of the square and match, so it is a Lorentz transformation. Belongs to each of such virtue

A Lorentz transformation from. Specifically, one of each pair of complex matrices of exactly a Lorentz transformation from the part of which is related to the steadily. This part of the Lorentz group is a representation of the group.

The group is the product variety and simply connected. The group of actual orthochronous Lorentz transformations, however, is not simply connected: rotations about a fixed axis with angles that grow up, form a closed circuit in the rotation group. One can not continuously change in other rotations these transformations, so that this circle shrinks to a point.

Credentials

• Charles Kittel, Walter D. Knight, Malvin A. Ruderman: Mechanics ( Berkeley Physics Course Vol = 1. ). Vieweg, Braunschweig, 1973, ISBN 3-528-08351-4, p 232: Chap. 11
• Norbert Dragon: geometry of the theory of relativity. (PDF file, 2.37 MB )