lorentz transformation for frames in standard configuration

8/13/2019 Lorentz Transformation for Frames in Standard Configuration

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Lorentz transformation for frames in standard configuration

Consider two observers Oand O, each using their ownCartesian coordinate systemto

measure space and time intervals. Ouses (t,x,y,z) and O uses (t,x,y,z). Assumefurther that the coordinate systems are oriented so that, in 3 dimensions, thex-axis and

thex-axis are collinear,they-axis is parallel to they-axis, and thez-axis parallel to

thez-axis. The relative velocity between the two observers is valong the commonx-

axis; O measures O to move at velocity v along the coincident xx axes, while O

measures Oto move at velocity valong the coincident xxaxes. Also assume that the

origins of both coordinate systems are the same, that is, coincident times and positions.

If all these hold, then the coordinate systems are said to be in standard configuration.

Theinverseof a Lorentz transformation relates the coordinates the other way round;

from the coordinates O measures (t,x,y,z) to the coordinates Omeasures (t,x,y,z),

so t,x,y,zare in terms of t,x,y,z. The mathematical form is nearly identical to the

original transformation; the only difference is the negation of the uniform relative

velocity (from vto v), and exchange of primed and unprimed quantities, because Omoves at velocity vrelative to O, and equivalently, Omoves at velocity vrelative to

O. This symmetry makes it effortless to find the inverse transformation (carrying out

the exchange and negation saves a lot of rote algebra), although more fundamentally;

it highlights that all physical laws should remain unchanged under a Lorentz

transformation.[8]

Below, the Lorentz transformations are called "boosts" in the stated directions.
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Boost in the x-direction

The spacetime coordinates of an event, as measured by each observer in their inertial

reference frame (in standard configuration) are shown in the speech bubbles.

Top: frame F moves at velocity valong thex-axis of frame F.

Bottom: frame Fmoves at velocity valong thex-axis of frame F.[9]

These are the simplest forms. The Lorentz transformation for frames in standard

configuration can be shown to be (see for example[10]and[11]):

where:
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vis the relative velocity between frames in thex-direction, cis thespeed of light,

is theLorentz factor(Greeklowercasegamma),

is the velocity coefficient (Greeklowercasebeta), again for thex-direction.

The use of and is standard throughout the literature.[12]For the remainder of the

articlethey will be also used throughout unless otherwise stated. Since the above is

a linear system of equations (more technically a linear transformation), they can be

written inmatrixform:

According to the principle of relativity, there is no privileged frame of reference, so

the inverse transformations frameF to frameFmust be given by simply negating v:

where the value of remains unchanged.
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Boost in the yor zdirections

The above collection of equations apply only for a boost in thex-direction. The

standard configuration works equally well in theyorzdirections instead ofx, and so

the results are similar.

For they-direction:

summarized by

where vand so are now in they-direction.

For thez-direction:

summarized by


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where vand so are now in thez-direction.

The Lorentz or boost matrix is usually denoted by (Greekcapitallambda). Above

the transformations have been applied to thefour-positionX,

The Lorentz transform for a boost in one of the above directions can be compactly

written as a single matrix equation:

Boost in any direction

Boost in an arbitrary direction.

Vector form

Further information:Euclidean vectorandvector projection

For a boost in an arbitrary direction with velocity v, that is, Oobserves Oto move in

direction vin theFcoordinate frame, while Oobserves Oto move in direction vin

the Fcoordinate frame, it is convenient to decompose the spatial vector rinto

components perpendicular and parallel to v:
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so that

where denotes thedot product(see alsoorthogonalityfor more information). Then,

only time and the component rin the direction of v;

are "warped" by the Lorentz factor:

.

The parallel and perpendicular components can be eliminated, by substituting

into r:

Since rand vare parallel we have

where geometrically and algebraically:

v/vis a dimensionlessunit vectorpointing in the same direction as r, r= (r v)/vis theprojectionof rinto the direction of v,

substituting for rand factoring vgives
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This method, of eliminating parallel and perpendicular components, can be applied to

any Lorentz transformation written in parallel-perpendicular form.

Matrix forms

These equations can be expressed inblock matrixform as

where Iis the 33identity matrixand = v/c is the relative velocity vector (in units

of c) as acolumn vectorincartesianandtensor index notationit is:

T= vT/c is thetransposearow vector:

and is themagnitudeof :

More explicitly stated:
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The transformation can be written in the same form as before,

which has the structure:[13]

and the components deduced from above are:

where ijis theKronecker delta,and by convention:Latin lettersfor indices take the

values 1, 2, 3, for spatial components of a 4-vector (Greekindices take values 0, 1, 2,

3 for time and space components).

Note that this transformation is only the "boost," i.e., a transformation between two

frames whosex,y, andzaxis are parallel and whose spacetime origins coincide. The

most general proper Lorentz transformation also contains a rotation of the three axes,

because the composition of two boosts is not a pure boost but is a boost followed by a
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rotation. The rotation gives rise to Thomas precession. The boost is given by a

symmetric matrix, but the general Lorentz transformation matrix need not be

symmetric.

Composition of two boosts

The composition of two Lorentz boosts B(u) and B(v) of velocities uand vis given

by:[14][15]

,

where

B(v) is the 4 4 matrix that uses the components of v, i.e. v1, v2, v3in the entriesof the matrix, or rather the components of v/c in the representation that is used

above,

is thevelocity-addition, Gyr[u,v] (capital G) is the rotation arising from the composition. If the 3 3

matrix form of the rotation applied to spatial coordinates is given by gyr[u,v],

then the 4 4 matrix rotation applied to 4-coordinates is given by:[14]

gyr (lower case g) is thegyrovector spaceabstraction of thegyroscopic Thomasprecession, defined as an operator on a velocity win terms of velocity addition:

for all w.
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The composition of two Lorentz transformations L(u, U) and L(v, V) which include

rotations Uand Vis given by:[16]

Visualizing the transformations in Minkowski space

Main article:Minkowski space

Lorentz transformations can be depicted on the Minkowski light cone spacetime

diagram.

The momentarily co-moving inertial frames along the world line of a rapidly

accelerating observer (center). The vertical direction indicates time, while the

horizontal indicates distance, the dashed line is thespacetimetrajectory ("world line")

of the observer. The small dots are specific events in spacetime. If one imagines these

events to be the flashing of a light, then the events that pass the two diagonal lines in

the bottom half of the image (the pastlight coneof the observer in the origin) are the

events visible to the observer. The slope of the world line (deviation from being

vertical) gives the relative velocity to the observer. Note how the momentarily co-

moving inertial frame changes when the observer accelerates.
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Particle travelling at constant velocity (straightworldlinecoincident with time t axis).

Acceleratingparticle (curvedworldline).

Lorentz transformations on theMinkowskilight conespacetime diagram,for one space

and one time dimension.

The yellow axes are the rest frame of an observer, the blue axes correspond to the frame

of a moving observer
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The red lines areworld lines,a continuous sequence of events: straight for an object

travelling at constant velocity, curved for an object accelerating. Worldlines of light

form the boundary of the light cone.

The purplehyperbolaeindicate this is ahyperbolic rotation,the hyperbolic angle is

called rapidity (see below). The greater the relative speed between the reference

frames, the more "warped" the axes become. The relative velocity cannot exceed c.

The black arrow is adisplacementfour-vectorbetween two events (not necessarily on

the same world line), showing that in a Lorentz boost; time dilation (fewer time

intervals in moving frame) and length contraction (shorter lengths in moving frame)

occur. The axes in the moving frame are orthogonal (even though they do not look so).

Rapidity

The Lorentz transformation can be cast into another useful form by defining a

parameter called therapidity(an instance ofhyperbolic angle)such that

and thus

Equivalently:

Then the Lorentz transformation in standard configuration is:
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Hyperbolic expressions

From the above expressions for eand e

and therefore,

Hyperbolic rotation of coordinates

Substituting these expressions into the matrix form of the transformation, it is evident

that

Thus, the Lorentz transformation can be seen as ahyperbolic rotationof coordinates in

Minkowski space,where the parameter represents the hyperbolic angle of rotation,

often referred to as rapidity. This transformation is sometimes illustrated with a

Minkowski diagram,as displayed above.
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This 4-by-4 boost matrix can thus be written compactly as aMatrix exponential,

where the simpler Lie-algebraic hyperbolic rotation generator Kx is called a boost

generator.

Transformation of other physical quantities[edit]

For the notation used, seeRicci calculus.

The transformation matrix is universal for all four-vectors, not just 4-dimensional

spacetime coordinates. If Zis any four-vector, then:[13]

or intensor index notation:

in which the primed indices denote indices of Zin the primed frame.

More generally, the transformation of anytensorquantity Tis given by:[17]

where is theinverse matrixof
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Special relativity

The crucial insight of Einstein's clock-setting method is the idea that time is relative.

In essence, each observer's frame of reference is associated with a unique set of clocks,

the result being that time as measured for a location passes at different rates for

different observers.[18]This was a direct result of the Lorentz transformations and is

called time dilation. We can also clearly see from the Lorentz "local time"

transformation that the concept of the relativity of simultaneity and of the relativity of

length contraction are also consequences of that clock-setting hypothesis.[19]

Transformation of the electromagnetic field

For the transformation rules, seeclassical electromagnetism and special relativity.

Lorentz transformations can also be used to prove thatmagneticandelectricfields are

simply different aspects of the same force the electromagnetic force, as a

consequence of relative motion betweenelectric chargesand observers.[20]The fact that

the electromagnetic field shows relativistic effects becomes clear by carrying out a

simple thought experiment:[21]

Consider an observer measuring a charge at rest in a reference frame F. Theobserver will detect a static electric field. As the charge is stationary in this

frame, there is no electric current, so the observer will not observe any magnetic

field.

Consider another observer in frame F moving at relative velocity v(relative toF and the charge). Thisobserver will see a different electric field because the

charge is moving at velocity v in their rest frame. Further, in frame F the

moving charge constitutes an electric current, and thus the observer in frame F

will also see a magnetic field.
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This shows that the Lorentz transformation also applies to electromagnetic field

quantities when changing the frame of reference, given below in vector form.

The correspondence principle

For relative speeds much less than the speed of light, the Lorentz transformations

reduce to theGalilean transformationin accordance with thecorrespondence principle.

The correspondence limit is usually stated mathematically as: as v 0, c . In

words: as velocity approaches 0, the speed of light (seems to) approach infinity. Hence,

it is sometimes said that nonrelativistic physics is a physics of "instantaneous action at

a distance".[18]

Spacetime interval

In a given coordinate systemx, if twoeventsAandBare separated by

thespacetime intervalbetween them is given by

This can be written in another form using the Minkowski metric.In this coordinate

system,

Then, we can write
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or, using theEinstein summation convention,

Now suppose that we make a coordinate transformationxx . Then, the interval in

this coordinate system is given by

or

It is a result ofspecial relativitythat the interval is aninvariant.That is,s2=s 2. For

this to hold, it can be shown that it is necessary (but not sufficient) for the coordinate

transformation to be of the form

Here, Cis a constant vector and a constant matrix, where we require that

Such a transformation is called a Poincar transformation or an inhomogeneous

Lorentz transformation. The Carepresents a spacetime translation. When Ca= 0, the
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transformation is called an homogeneous Lorentz transformation, or simply aLorentz

transformation.

Taking thedeterminantof

gives us

The cases are:

Proper Lorentz transformations have det() = +1, and form a subgroupcalled thespecial orthogonal groupSO(1,3).

Improper Lorentz transformationsare det() = 1, which do not form asubgroup, as the product of any two improper Lorentz transformations will be a

proper Lorentz transformation.

From the above definition of it can be shown that (00)2 1, so either 00 1 or

00

1, called orthochronous and non-orthochronous respectively. An important

subgroup of the proper Lorentz transformations are the proper orthochronous

Lorentz transformationswhich consist purely of boosts and rotations. Any Lorentz

transform can be written as a proper orthochronous, together with one or both of the

two discrete transformations;space inversionPandtime reversalT, whose non-zero

elements are:
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The set of Poincar transformations satisfies the properties of a group and is called the

Poincar group.Under theErlangen program,Minkowski spacecan be viewed as the

geometry defined by the Poincar group, which combines Lorentz transformations

with translations. In a similar way, the set of all Lorentz transformations forms a group,

called theLorentz group.

A quantity invariant under Lorentz transformations is known as aLorentz scalar.
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lorentz transformation for frames in standard configuration

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