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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