Inphysics, theLorentz transformations are a six-parameter family oflineartransformations from acoordinate frame inspacetime to another frame that moves at a constantvelocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the DutchphysicistHendrik Lorentz.
The most common form of the transformation, parametrized by the real constant representing a velocity confined to thex-direction, is expressed as[1][2]where(t,x,y,z) and(t′,x′,y′,z′) are the coordinates of an event in two frames with the spatial origins coinciding att =t′ = 0, where the primed frame is seen from the unprimed frame as moving with speedv along thex-axis, wherec is thespeed of light, andis theLorentz factor. When speedv is much smaller thanc, the Lorentz factor is negligibly different from 1, but asv approachesc, grows without bound. The value ofv must be smaller thanc for the transformation to make sense.
Expressing the speed as a fraction of the speed of light, an equivalent form of the transformation is[3]
Frames of reference can be divided into two groups:inertial (relative motion with constant velocity) andnon-inertial (accelerating, moving in curved paths, rotational motion with constantangular velocity, etc.). The term "Lorentz transformations" only refers to transformations betweeninertial frames, usually in the context of special relativity.
In eachreference frame, an observer can use a local coordinate system (usuallyCartesian coordinates in this context) to measure lengths, and a clock to measure time intervals. Anevent is something that happens at a point in space at an instant of time, or more formally a point inspacetime. The transformations connect the space and time coordinates of anevent as measured by an observer in each frame.[nb 1]
Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed oflight was observed to be independent of thereference frame, and to understand the symmetries of the laws ofelectromagnetism. The transformations later became a cornerstone forspecial relativity.
The Lorentz transformation is alinear transformation. It may include a rotation of space; a rotation-free Lorentz transformation is called aLorentz boost. InMinkowski space—the mathematical model of spacetime in special relativity—the Lorentz transformations preserve thespacetime interval between any two events. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as ahyperbolic rotation of Minkowski space. The more general set of transformations that also includes translations is known as thePoincaré group.
Lorentz (1892–1904) and Larmor (1897–1900), who believed the luminiferous aether hypothesis, also looked for the transformation under whichMaxwell's equations are invariant when transformed from the aether to a moving frame. They extended theFitzGerald–Lorentz contraction hypothesis and found out that the time coordinate has to be modified as well ("local time").Henri Poincaré gave a physical interpretation to local time (to first order inv/c, the relative velocity of the two reference frames normalized to the speed of light) as the consequence of clock synchronization, under the assumption that the speed of light is constant in moving frames.[8] Larmor is credited to have been the first to understand the crucialtime dilation property inherent in his equations.[9]
In 1905, Poincaré was the first to recognize that the transformation has the properties of amathematical group,and he named it after Lorentz.[10]Later in the same yearAlbert Einstein published what is now calledspecial relativity, by deriving the Lorentz transformation under the assumptions of theprinciple of relativity and the constancy of the speed of light in anyinertial reference frame, and by abandoning the mechanistic aether as unnecessary.[11]
Derivation of the group of Lorentz transformations
Anevent is something that happens at a certain point in spacetime, or more generally, the point in spacetime itself. In any inertial frame an event is specified by a time coordinatect and a set ofCartesian coordinatesx,y,z to specify position in space in that frame. Subscripts label individual events.
in all inertial frames for events connected bylight signals. The quantity on the left is called thespacetime interval between eventsa1 = (t1,x1,y1,z1) anda2 = (t2,x2,y2,z2). The interval betweenany two events, not necessarily separated by light signals, is in fact invariant, i.e., independent of the state of relative motion of observers in different inertial frames, as isshown using homogeneity and isotropy of space. The transformation sought after thus must possess the property that:
D2
where(t,x,y,z) are the spacetime coordinates used to define events in one frame, and(t′,x′,y′,z′) are the coordinates in another frame. First one observes that (D2) is satisfied if an arbitrary4-tupleb of numbers are added to eventsa1 anda2. Such transformations are calledspacetime translations and are not dealt with further here. Then one observes that alinear solution preserving the origin of the simpler problem solves the general problem too:
D3
(a solution satisfying the first formula automatically satisfies the second one as well; seePolarization identity). Finding the solution to the simpler problem is just a matter of look-up in the theory ofclassical groups that preservebilinear forms of various signature.[nb 2] First equation in (D3) can be written more compactly as:
D4
where(·, ·) refers to the bilinear form ofsignature(1, 3) onR4 exposed by the right hand side formula in (D3). The alternative notation defined on the right is referred to as therelativistic dot product. Spacetime mathematically viewed asR4 endowed with this bilinear form is known asMinkowski spaceM. The Lorentz transformation is thus an element of the groupO(1, 3), theLorentz group or, for those that prefer the othermetric signature,O(3, 1) (also called the Lorentz group).[nb 3] One has:
D5
which is precisely preservation of the bilinear form (D3) which implies (by linearity ofΛ and bilinearity of the form) that (D2) is satisfied. The elements of the Lorentz group arerotations andboosts and mixes thereof. If the spacetime translations are included, then one obtains theinhomogeneous Lorentz group or thePoincaré group.
The relations between the primed and unprimed spacetime coordinates are theLorentz transformations, each coordinate in one frame is alinear function of all the coordinates in the other frame, and theinverse functions are the inverse transformation. Depending on how the frames move relative to each other, and how they are oriented in space relative to each other, other parameters that describe direction, speed, and orientation enter the transformation equations.
Transformations describing relative motion with constant (uniform) velocity and without rotation of the space coordinate axes are calledLorentz boosts or simplyboosts, and the relative velocity between the frames is the parameter of the transformation. The other basic type of Lorentz transformation is only a rotation in the spatial coordinates. Unlike boosts, these are inertial transformations since there is no relative motion, the frames are simply tilted (and not continuously rotating), and in this case quantities defining the rotation are the parameters of the transformation (e.g.,axis–angle representation, orEuler angles, etc.). A combination of a rotation and a boost is ahomogeneous transformation, which transforms the origin back to the origin.
The full Lorentz groupO(3, 1) also contains special transformations that are neither rotations nor boosts, but ratherreflections in a plane through the origin. Two of these can be singled out;spatial inversion in which the spatial coordinates of all events are reversed in sign andtemporal inversion in which the time coordinate for each event gets its sign reversed.
Boosts should not be conflated with mere displacements in spacetime; in this case, the coordinate systems are simply shifted and there is no relative motion. However, these also count as symmetries forced by special relativity since they leave the spacetime interval invariant. A combination of a rotation with a boost, followed by a shift in spacetime, is aninhomogeneous Lorentz transformation, an element of the Poincaré group, which is also called the inhomogeneous Lorentz group.
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: frameF′ moves at velocityv along thex-axis of frameF. Bottom: frameF moves at velocity −v along thex′-axis of frameF′.[12]
A "stationary" observer in frameF defines events with coordinatest,x,y,z. Another frameF′ moves with velocityv relative toF, and an observer in this "moving" frameF′ defines events using the coordinatest′,x′,y′,z′.
The coordinate axes in each frame are parallel (thex andx′ axes are parallel, they andy′ axes are parallel, and thez andz′ axes are parallel), remain mutually perpendicular, and relative motion is along the coincidentxx′ axes. Att =t′ = 0, the origins of both coordinate systems are the same,(x,y,z) = (x′,y′,z′) = (0, 0, 0). In other words, the times and positions are coincident at this event. If all these hold, then the coordinate systems are said to be instandard configuration, orsynchronized.
If an observer inF records an eventt,x,y,z, then an observer inF′ records thesame event with coordinates[13]
Here,v is theparameter of the transformation, for a given boost it is a constant number, but can take a continuous range of values. In the setup used here, positive relative velocityv > 0 is motion along the positive directions of thexx′ axes, zero relative velocityv = 0 is no relative motion, while negative relative velocityv < 0 is relative motion along the negative directions of thexx′ axes. The magnitude of relative velocityv cannot equal or exceedc, so only subluminal speeds−c <v <c are allowed. The corresponding range ofγ is1 ≤γ < ∞.
The transformations are not defined ifv is outside these limits. At the speed of light (v =c)γ is infinite, andfaster than light (v >c)γ is acomplex number, each of which make the transformations unphysical. The space and time coordinates are measurable quantities and numerically must be real numbers.
As anactive transformation, an observer inF′ notices the coordinates of the event to be "boosted" in the negative directions of thexx′ axes, because of the−v in the transformations. This has the equivalent effect of thecoordinate systemF′ boosted in the positive directions of thexx′ axes, while the event does not change and is simply represented in another coordinate system, apassive transformation.
The inverse relations (t,x,y,z in terms oft′,x′,y′,z′) can be found by algebraically solving the original set of equations. A more efficient way is to use physical principles. HereF′ is the "stationary" frame whileF is the "moving" frame. According to the principle of relativity, there is no privileged frame of reference, so the transformations fromF′ toF must take exactly the same form as the transformations fromF toF′. The only difference isF moves with velocity−v relative toF′ (i.e., the relative velocity has the same magnitude but is oppositely directed). Thus if an observer inF′ notes an eventt′,x′,y′,z′, then an observer inF notes thesame event with coordinates
Inverse Lorentz boost (x direction)
and the value ofγ remains unchanged. This "trick" of simply reversing the direction of relative velocity while preserving its magnitude, and exchanging primed and unprimed variables, always applies to finding the inverse transformation of every boost in any direction.[14][15]
Sometimes it is more convenient to useβ =v/c (lowercasebeta) instead ofv, so thatwhich shows much more clearly the symmetry in the transformation. From the allowed ranges ofv and the definition ofβ, it follows−1 <β < 1. The use ofβ andγ is standard throughout the literature. In the case of three spatial dimensions[ct,x,y,z], where the boost is in thex direction, theeigenstates of the transformation are[1, 1, 0, 0] with eigenvalue,[1, −1, 0, 0] with eigenvalue, and[0, 0, 1, 0] and[0, 0, 0, 1], the latter two with eigenvalue 1.
When the boost velocity is in an arbitrary vector direction with the boost vector, then the transformation from an unprimed spacetime coordinate system to a primed coordinate system is given by[16][17]
where theLorentz factor is. Thedeterminant of the transformation matrix is +1 and itstrace is. The inverse of the transformation is given by reversing the sign of. The quantity is invariant under the transformation: namely.
The Lorentz transformations can also be derived in a way that resembles circular rotations in 3-dimensional space using thehyperbolic functions. For the boost in thex direction, the results are
Lorentz boost (x direction with rapidityζ)
whereζ (lowercasezeta) is a parameter calledrapidity (many other symbols are used, includingθ,ϕ,φ,η,ψ,ξ). Given the strong resemblance to rotations of spatial coordinates in 3-dimensional space in the Cartesianxy,yz, andzx planes, a Lorentz boost can be thought of as ahyperbolic rotation of spacetime coordinates in the xt, yt, and zt Cartesian-time planes of 4-dimensionalMinkowski space. The parameterζ is thehyperbolic angle of rotation, analogous to the ordinary angle for circular rotations. This transformation can be illustrated with aMinkowski diagram.
The hyperbolic functions arise from thedifference between the squares of the time and spatial coordinates in the spacetime interval, rather than a sum. The geometric significance of the hyperbolic functions can be visualized by takingx = 0 orct = 0 in the transformations. Squaring and subtracting the results, one can derive hyperbolic curves of constant coordinate values but varyingζ, which parametrizes the curves according to the identity
Conversely thect andx axes can be constructed for varying coordinates but constantζ. The definitionprovides the link between a constant value of rapidity, and theslope of thect axis in spacetime. A consequence these two hyperbolic formulae is an identity that matches the Lorentz factor
Comparing the Lorentz transformations in terms of the relative velocity and rapidity, or using the above formulae, the connections betweenβ,γ, andζ are
Taking the inverse hyperbolic tangent gives the rapidity
Since−1 <β < 1, it follows−∞ <ζ < ∞. From the relation betweenζ andβ, positive rapidityζ > 0 is motion along the positive directions of thexx′ axes, zero rapidityζ = 0 is no relative motion, while negative rapidityζ < 0 is relative motion along the negative directions of thexx′ axes.
The inverse transformations are obtained by exchanging primed and unprimed quantities to switch the coordinate frames, and negating rapidityζ → −ζ since this is equivalent to negating the relative velocity. Therefore,
Inverse Lorentz boost (x direction with rapidityζ)
The inverse transformations can be similarly visualized by considering the cases whenx′ = 0 andct′ = 0.
So far the Lorentz transformations have been applied toone event. If there are two events, there is a spatial separation and time interval between them. It follows from thelinearity of the Lorentz transformations that two values of space and time coordinates can be chosen, the Lorentz transformations can be applied to each, then subtracted to get the Lorentz transformations of the differences:with inverse relationswhereΔ (uppercasedelta) indicates a difference of quantities; e.g.,Δx =x2 −x1 for two values ofx coordinates, and so on.
These transformations ondifferences rather than spatial points or instants of time are useful for a number of reasons:
in calculations and experiments, it is lengths between two points or time intervals that are measured or of interest (e.g., the length of a moving vehicle, or time duration it takes to travel from one place to another),
the transformations of velocity can be readily derived by making the difference infinitesimally small and dividing the equations, and the process repeated for the transformation of acceleration,
if the coordinate systems are never coincident (i.e., not in standard configuration), and if both observers can agree on an eventt0,x0,y0,z0 inF andt′0,x′0,y′0,z′0 inF′, then they can use that event as the origin, and the spacetime coordinate differences are the differences between their coordinates and this origin, e.g.,Δx =x −x0,Δx′ =x′ −x′0, etc.
A critical requirement of the Lorentz transformations is the invariance of the speed of light, a fact used in their derivation, and contained in the transformations themselves. If inF the equation for a pulse of light along thex direction isx =ct, then inF′ the Lorentz transformations givex′ =ct′, and vice versa, for any−c <v <c.
For relative speeds much less than the speed of light, the Lorentz transformations reduce to theGalilean transformation:[18][19]in accordance with thecorrespondence principle. It is sometimes said that nonrelativistic physics is a physics of "instantaneous action at a distance".[20]
Three counterintuitive, but correct, predictions of the transformations are:
Suppose two events occur along the x axis simultaneously (Δt = 0) inF, but separated by a nonzero displacementΔx. Then inF′, we find that, so the events are no longer simultaneous according to a moving observer.
Suppose there is a clock at rest inF. If a time interval is measured at the same point in that frame, so thatΔx = 0, then the transformations give this interval inF′ byΔt′ =γΔt. Conversely, suppose there is a clock at rest inF′. If an interval is measured at the same point in that frame, so thatΔx′ = 0, then the transformations give this interval inF byΔt =γΔt′. Either way, each observer measures the time interval between ticks of a moving clock to be longer by a factorγ than the time interval between ticks of his own clock.
Suppose there is a rod at rest inF aligned along thex axis, with lengthΔx. InF′, the rod moves with velocity−v, so its length must be measured by taking two simultaneous (Δt′ = 0) measurements at opposite ends. Under these conditions, the inverse Lorentz transform shows thatΔx =γΔx′. InF the two measurements are no longer simultaneous, but this does not matter because the rod is at rest inF. So each observer measures the distance between the end points of a moving rod to be shorter by a factor1/γ than the end points of an identical rod at rest in his own frame. Length contraction affects any geometric quantity related to lengths, so from the perspective of a moving observer, areas and volumes will also appear to shrink along the direction of motion.
An observer in frameF observesF′ to move with velocityv, whileF′ observesF to move with velocity−v.The coordinate axes of each frame are still parallel[according to whom?] and orthogonal. The position vector as measured in each frame is split into components parallel and perpendicular to the relative velocity vectorv. Left: Standard configuration.Right: Inverse configuration.
The use of vectors allows positions and velocities to be expressed in arbitrary directions compactly. A single boost in any direction depends on the full relativevelocity vectorv with a magnitude|v| =v that cannot equal or exceedc, so that0 ≤v <c.
Only time and the coordinates parallel to the direction of relative motion change, while those coordinates perpendicular do not. With this in mind, split the spatialposition vectorr as measured inF, andr′ as measured inF′, each into components perpendicular (⊥) and parallel ( ||) tov,then the transformations arewhere· is thedot product. The Lorentz factorγ retains its definition for a boost in any direction, since it depends only on the magnitude of the relative velocity. The definitionβ =v/c with magnitude0 ≤β < 1 is also used by some authors.
Introducing aunit vectorn =v/v =β/β in the direction of relative motion, the relative velocity isv =vn with magnitudev and directionn, andvector projection and rejection give respectively
Accumulating the results gives the full transformations,
Lorentz boost (in directionn with magnitudev)
The projection and rejection also applies tor′. For the inverse transformations, exchanger andr′ to switch observed coordinates, and negate the relative velocityv → −v (or simply the unit vectorn → −n since the magnitudev is always positive) to obtain
Inverse Lorentz boost (in directionn with magnitudev)
The unit vector has the advantage of simplifying equations for a single boost, allows eitherv orβ to be reinstated when convenient, and the rapidity parametrization is immediately obtained by replacingβ andβγ. It is not convenient for multiple boosts.
The vectorial relation between relative velocity and rapidity is[21]and the "rapidity vector" can be defined aseach of which serves as a useful abbreviation in some contexts. The magnitude ofζ is the absolute value of the rapidity scalar confined to0 ≤ζ < ∞, which agrees with the range0 ≤β < 1.
The transformation of velocities provides the definitionrelativistic velocity addition⊕, the ordering of vectors is chosen to reflect the ordering of the addition of velocities; firstv (the velocity ofF′ relative toF) thenu′ (the velocity ofX relative toF′) to obtainu =v ⊕u′ (the velocity ofX relative toF).
Defining the coordinate velocities and Lorentz factor bytaking the differentials in the coordinates and time of the vector transformations, then dividing equations, leads to
The velocitiesu andu′ are the velocity of some massive object. They can also be for a third inertial frame (sayF′′), in which case they must beconstant. Denote either entity byX. ThenX moves with velocityu relative toF, or equivalently with velocityu′ relative toF′, in turnF′ moves with velocityv relative toF. The inverse transformations can be obtained in a similar way, or as with position coordinates exchangeu andu′, and changev to−v.
TheLorentz transformations of acceleration can be similarly obtained by taking differentials in the velocity vectors, and dividing these by the time differential.
In general, given four quantitiesA andZ = (Zx,Zy,Zz) and their Lorentz-boosted counterpartsA′ andZ′ = (Z′x,Z′y,Z′z), a relation of the formimplies the quantities transform under Lorentz transformations similar to the transformation of spacetime coordinates;
The decomposition ofZ (andZ′) into components perpendicular and parallel tov is exactly the same as for the position vector, as is the process of obtaining the inverse transformations (exchange(A,Z) and(A′,Z′) to switch observed quantities, and reverse the direction of relative motion by the substitutionn ↦ −n).
The quantities(A,Z) collectively make up afour-vector, whereA is the "timelike component", andZ the "spacelike component". Examples ofA andZ are the following:
For a given object (e.g., particle, fluid, field, material), ifA orZ correspond to properties specific to the object like itscharge density,mass density,spin, etc., its properties can be fixed in the rest frame of that object. Then the Lorentz transformations give the corresponding properties in a frame moving relative to the object with constant velocity. This breaks some notions taken for granted in non-relativistic physics. For example, the energyE of an object is a scalar in non-relativistic mechanics, but not in relativistic mechanics because energy changes under Lorentz transformations; its value is different for various inertial frames. In the rest frame of an object, it has arest energy and zero momentum. In a boosted frame its energy is different and it appears to have a momentum. Similarly, in non-relativistic quantum mechanics the spin of a particle is a constant vector, but inrelativistic quantum mechanics spins depends on relative motion. In the rest frame of the particle, the spin pseudovector can be fixed to be its ordinary non-relativistic spin with a zero timelike quantityst, however a boosted observer will perceive a nonzero timelike component and an altered spin.[22]
Not all quantities are invariant in the form as shown above, for example orbitalangular momentumL does not have a timelike quantity, and neither does theelectric fieldE nor themagnetic fieldB. The definition of angular momentum isL =r ×p, and in a boosted frame the altered angular momentum isL′ =r′ ×p′. Applying this definition using the transformations of coordinates and momentum leads to the transformation of angular momentum. It turns outL transforms with another vector quantityN = (E/c2)r −tp related to boosts, seeRelativistic angular momentum for details. For the case of theE andB fields, the transformations cannot be obtained as directly using vector algebra. TheLorentz force is the definition of these fields, and inF it isF =q(E +v ×B) while inF′ it isF′ =q(E′ +v′ ×B′). A method of deriving the EM field transformations in an efficient way which also illustrates the unit of the electromagnetic field uses tensor algebra,given below.
Writing the coordinates in column vectors and theMinkowski metricη as a square matrixthe spacetime interval takes the form (superscriptT denotestranspose)and isinvariant under a Lorentz transformationwhereΛ is a square matrix which can depend on parameters.
Theset of all Lorentz transformations in this article is denoted. This set together with matrix multiplication forms agroup, in this context known as theLorentz group. Also, the above expressionX·X is aquadratic form of signature (3,1) on spacetime, and the group of transformations which leaves this quadratic form invariant is theindefinite orthogonal group O(3,1), aLie group. In other words, the Lorentz group is O(3,1). As presented in this article, any Lie groups mentioned arematrix Lie groups. In this context the operation of composition amounts tomatrix multiplication.
From the invariance of the spacetime interval it followsand this matrix equation contains the general conditions on the Lorentz transformation to ensure invariance of the spacetime interval. Taking thedeterminant of the equation using the product rule[nb 4] gives immediately
Writing the Minkowski metric as a block matrix, and the Lorentz transformation in the most general form,carrying out the block matrix multiplications obtains general conditions onΓ,a,b,M to ensure relativistic invariance. Not much information can be directly extracted from all the conditions, however one of the resultsis useful;bTb ≥ 0 always so it follows that
The negative inequality may be unexpected, becauseΓ multiplies the time coordinate and this has an effect ontime symmetry. If the positive equality holds, thenΓ is the Lorentz factor.
The determinant and inequality provide four ways to classifyLorentzTransformations (hereinLTs for brevity). Any particular LT has only one determinant signand only one inequality. There are four sets which include every possible pair given by theintersections ("n"-shaped symbol meaning "and") of these classifying sets.
Intersection, ∩
Antichronous (or non-orthochronous) LTs
Orthochronous LTs
Proper LTs
Proper antichronous LTs
Proper orthochronous LTs
Improper LTs
Improper antichronous LTs
Improper orthochronous LTs
where "+" and "−" indicate the determinant sign, while "↑" for ≥ and "↓" for ≤ denote the inequalities.
The full Lorentz group splits into theunion ("u"-shaped symbol meaning "or") of fourdisjoint sets
Asubgroup of a group must beclosed under the same operation of the group (here matrix multiplication). In other words, for two Lorentz transformationsΛ andL from a particular subgroup, the composite Lorentz transformationsΛL andLΛ must be in the same subgroup asΛ andL. This is not always the case: the composition of two antichronous Lorentz transformations is orthochronous, and the composition of two improper Lorentz transformations is proper. In other words, while the sets,,, and all form subgroups, the sets containing improper and/or antichronous transformations without enough proper orthochronous transformations (e.g.,,) do not form subgroups.
If a Lorentz covariant 4-vector is measured in one inertial frame with result, and the same measurement made in another inertial frame (with the same orientation and origin) gives result, the two results will be related bywhere the boost matrix represents the rotation-free Lorentz transformation between the unprimed and primed frames and is the velocity of the primed frame as seen from the unprimed frame. The matrix is given by[23]
where is the magnitude of the velocity and is the Lorentz factor. This formula represents a passive transformation, as it describes how the coordinates of the measured quantity changes from the unprimed frame to the primed frame. The active transformation is given by.
If a frameF′ is boosted with velocityu relative to frameF, and another frameF′′ is boosted with velocityv relative toF′, the separate boosts areand the composition of the two boosts connects the coordinates inF′′ andF,Successive transformations act on the left. Ifu andv arecollinear (parallel or antiparallel along the same line of relative motion), the boost matricescommute:B(v)B(u) =B(u)B(v). This composite transformation happens to be another boost,B(w), wherew is collinear withu andv.
Ifu andv are not collinear but in different directions, the situation is considerably more complicated. Lorentz boosts along different directions do not commute:B(v)B(u) andB(u)B(v) are not equal. Although each of these compositions isnot a single boost, each composition is still a Lorentz transformation as it preserves the spacetime interval. It turns out the composition of any two Lorentz boosts is equivalent to a boost followed or preceded by a rotation on the spatial coordinates, in the form ofR(ρ)B(w) orB(w)R(ρ). Thew andw arecomposite velocities, whileρ andρ are rotation parameters (e.g.axis-angle variables,Euler angles, etc.). The rotation inblock matrix form is simplywhereR(ρ) is a3 × 3rotation matrix, which rotates any 3-dimensional vector in one sense (active transformation), or equivalently the coordinate frame in the opposite sense (passive transformation). It isnot simple to connectw andρ (orw andρ) to the original boost parametersu andv. In a composition of boosts, theR matrix is named theWigner rotation, and gives rise to theThomas precession. These articles give the explicit formulae for the composite transformation matrices, including expressions forw,ρ,w,ρ.
In this article theaxis-angle representation is used forρ. The rotation is about an axis in the direction of aunit vectore, through angleθ (positive anticlockwise, negative clockwise, according to theright-hand rule). The "axis-angle vector"will serve as a useful abbreviation.
Spatial rotations alone are also Lorentz transformations since they leave the spacetime interval invariant. Like boosts, successive rotations about different axes do not commute. Unlike boosts, the composition of any two rotations is equivalent to a single rotation. Some other similarities and differences between the boost and rotation matrices include:
inverses:B(v)−1 =B(−v) (relative motion in the opposite direction), andR(θ)−1 =R(−θ) (rotation in the opposite sense about the same axis)
The most general proper Lorentz transformationΛ(v,θ) includes a boost and rotation together, and is a nonsymmetric matrix. As special cases,Λ(0,θ) =R(θ) andΛ(v,0) =B(v). An explicit form of the general Lorentz transformation is cumbersome to write down and will not be given here. Nevertheless, closed form expressions for the transformation matrices will be given below using group theoretical arguments. It will be easier to use the rapidity parametrization for boosts, in which case one writesΛ(ζ,θ) andB(ζ).
The set of transformationswith matrix multiplication as the operation of composition forms a group, called the "restricted Lorentz group", and is thespecial indefinite orthogonal group SO+(3,1). (The plus sign indicates that it preserves the orientation of the temporal dimension).
For simplicity, look at the infinitesimal Lorentz boost in thex direction (examining a boost in any other direction, or rotation about any axis, follows an identical procedure). The infinitesimal boost is a small boost away from the identity, obtained by theTaylor expansion of the boost matrix to first order aboutζ = 0,where the higher order terms not shown are negligible becauseζ is small, andBx is simply the boost matrix in thex direction. Thederivative of the matrix is the matrix of derivatives (of the entries, with respect to the same variable), and it is understood the derivatives are found first then evaluated atζ = 0,
The axis-angle vectorθ and rapidity vectorζ are altogether six continuous variables which make up the group parameters (in this particular representation), and the generators of the group areK = (Kx,Ky,Kz) andJ = (Jx,Jy,Jz), each vectors of matrices with the explicit forms[nb 6]
These are all defined in an analogous way toKx above, although the minus signs in the boost generators are conventional. Physically, the generators of the Lorentz group correspond to important symmetries in spacetime:J are therotation generators which correspond toangular momentum, andK are theboost generators which correspond to the motion of the system in spacetime. The derivative of any smooth curveC(t) withC(0) =I in the group depending on some group parametert with respect to that group parameter, evaluated att = 0, serves as a definition of a corresponding group generatorG, and this reflects an infinitesimal transformation away from the identity. The smooth curve can always be taken as an exponential as the exponential will always mapG smoothly back into the group viat → exp(tG) for allt; this curve will yieldG again when differentiated att = 0.
Expanding the exponentials in their Taylor series obtainswhich compactly reproduce the boost and rotation matrices as given in the previous section.
It has been stated that the general proper Lorentz transformation is a product of a boost and rotation. At theinfinitesimal level the productis commutative because only linear terms are required (products like(θ·J)(ζ·K) and(ζ·K)(θ·J) count as higher order terms and are negligible). Taking the limit as before leads to the finite transformation in the form of an exponential
The converse is also true, but the decomposition of a finite general Lorentz transformation into such factors is nontrivial. In particular,because the generators do not commute. For a description of how to find the factors of a general Lorentz transformation in terms of a boost and a rotationin principle (this usually does not yield an intelligible expression in terms of generatorsJ andK), seeWigner rotation. If, on the other hand,the decomposition is given in terms of the generators, and one wants to find the product in terms of the generators, then theBaker–Campbell–Hausdorff formula applies.
Lorentz generators can be added together, or multiplied by real numbers, to obtain more Lorentz generators. In other words, theset of all Lorentz generatorstogether with the operations of ordinarymatrix addition andmultiplication of a matrix by a number, forms avector space over the real numbers.[nb 7] The generatorsJx,Jy,Jz,Kx,Ky,Kz form abasis set ofV, and the components of the axis-angle and rapidity vectors,θx,θy,θz,ζx,ζy,ζz, are thecoordinates of a Lorentz generator with respect to this basis.[nb 8]
Three of thecommutation relations of the Lorentz generators arewhere the bracket[A,B] =AB −BA is known as thecommutator, and the other relations can be found by takingcyclic permutations ofx,y,z components (i.e. changex toy,y toz, andz tox, repeat).
These commutation relations, and the vector space of generators, fulfill the definition of theLie algebra. In summary, a Lie algebra is defined as avector spaceV over afield of numbers, and with abinary operation[ , ] (called aLie bracket in this context) on the elements of the vector space, satisfying the axioms ofbilinearity,alternatization, and theJacobi identity. Here the operation[ , ] is the commutator which satisfies all of these axioms, the vector space is the set of Lorentz generatorsV as given previously, and the field is the set of real numbers.
Linking terminology used in mathematics and physics: A group generator is any element of the Lie algebra. A group parameter is a component of a coordinate vector representing an arbitrary element of the Lie algebra with respect to some basis. A basis, then, is a set of generators being a basis of the Lie algebra in the usual vector space sense.
Theexponential map from the Lie algebra to the Lie group,provides a one-to-one correspondence between small enough neighborhoods of the origin of the Lie algebra and neighborhoods of the identity element of the Lie group. In the case of the Lorentz group, the exponential map is just thematrix exponential. Globally, the exponential map is not one-to-one, but in the case of the Lorentz group, it issurjective (onto). Hence any group element in the connected component of the identity can be expressed as an exponential of an element of the Lie algebra.
Lorentz transformations also includeparity inversionwhich negates all the spatial coordinates only, andtime reversalwhich negates the time coordinate only, because these transformations leave the spacetime interval invariant. HereI is the3 × 3identity matrix. These are both symmetric, they are their own inverses (seeInvolution (mathematics)), and each have determinant −1. This latter property makes them improper transformations.
IfΛ is a proper orthochronous Lorentz transformation, thenTΛ is improper antichronous,PΛ is improper orthochronous, andTPΛ =PTΛ is proper antichronous.
Two other spacetime symmetries have not been accounted for. In order for the spacetime interval to be invariant, it can be shown[24] that it is necessary and sufficient for the coordinate transformation to be of the formwhereC is a constant column containing translations in time and space. IfC ≠ 0, this is aninhomogeneous Lorentz transformation orPoincaré transformation.[25][26] IfC = 0, this is ahomogeneous Lorentz transformation. Poincaré transformations are not dealt further in this article.
Writing the general matrix transformation of coordinates as the matrix equationallows the transformation of other physical quantities that cannot be expressed as four-vectors; e.g.,tensors orspinors of any order in 4-dimensional spacetime, to be defined. In the correspondingtensor index notation, the above matrix expression is
where lower and upper indices labelcovariant and contravariant components respectively,[27] and thesummation convention is applied. It is a standard convention to useGreek indices that take the value 0 for time components, and 1, 2, 3 for space components, whileLatin indices simply take the values 1, 2, 3, for spatial components (the opposite for Landau and Lifshitz). Note that the first index (reading left to right) corresponds in the matrix notation to arow index. The second index corresponds to the column index.
The transformation matrix is universal for allfour-vectors, not just 4-dimensional spacetime coordinates. IfA is any four-vector, then intensor index notation
Alternatively, one writes in which the primed indices denote the indices ofA in the primed frame. For a generaln-component object one may write whereΠ is the appropriaterepresentation of the Lorentz group, ann ×n matrix for everyΛ. In this case, the indices shouldnot be thought of as spacetime indices (sometimes called Lorentz indices), and they run from1 ton. E.g., ifX is abispinor, then the indices are calledDirac indices.
There are also vector quantities with covariant indices. They are generally obtained from their corresponding objects with contravariant indices by the operation oflowering an index; e.g.,whereη is themetric tensor. (The linked article also provides more information about what the operation of raising and lowering indices really is mathematically.) The inverse of this transformation is given bywhere, when viewed as matrices,ημν is the inverse ofημν. As it happens,ημν =ημν. This is referred to asraising an index. To transform a covariant vectorAμ, first raise its index, then transform it according to the same rule as for contravariant4-vectors, then finally lower the index;
But
That is, it is the(μ,ν)-component of theinverse Lorentz transformation. One defines (as a matter of notation),and may in this notation write
Now for a subtlety. The implied summation on the right hand side ofis running overa row index of the matrix representingΛ−1. Thus, in terms of matrices, this transformation should be thought of as theinverse transpose ofΛ acting on the column vectorAμ. That is, in pure matrix notation,
This means exactly that covariant vectors (thought of as column matrices) transform according to thedual representation of the standard representation of the Lorentz group. This notion generalizes to general representations, simply replaceΛ withΠ(Λ).
IfA andB are linear operators on vector spacesU andV, then a linear operatorA ⊗B may be defined on thetensor product ofU andV, denotedU ⊗V according to[28]
(T1)
From this it is immediately clear that ifu andv are a four-vectors inV, thenu ⊗v ∈T2V ≡V ⊗V transforms as
(T2)
The second step uses the bilinearity of the tensor product and the last step defines a 2-tensor on component form, or rather, it just renames the tensoru ⊗v.
These observations generalize in an obvious way to more factors, and using the fact that a general tensor on a vector spaceV can be written as a sum of a coefficient (component!) times tensor products of basis vectors and basis covectors, one arrives at the transformation law for anytensor quantityT. It is given by[29]
(T3)
whereΛχ′ψ is defined above. This form can generally be reduced to the form for generaln-component objects given above with a single matrix (Π(Λ)) operating on column vectors. This latter form is sometimes preferred; e.g., for the electromagnetic field tensor.
Lorentz transformations can also be used to illustrate that themagnetic fieldB andelectric fieldE are simply different aspects of the same force — theelectromagnetic force, as a consequence of relative motion betweenelectric charges and observers.[30] The fact that the electromagnetic field shows relativistic effects becomes clear by carrying out a simple thought experiment.[31]
An observer measures a charge at rest in frameF. The observer will detect a static electric field. As the charge is stationary in this frame, there is no electric current, so the observer does not observe any magnetic field.
The other observer in frameF′ moves at velocityv relative toF and the charge.This observer sees a different electric field because the charge moves at velocity−v in their rest frame. The motion of the charge corresponds to anelectric current, and thus the observer in frameF′ also sees a magnetic field.
The electric and magnetic fields transform differently from space and time, but exactly the same way as relativistic angular momentum and the boost vector.
The electromagnetic field strength tensor is given byin with signature(+, −, −, −). In relativity, the factorc may be absorbed into the tensor components to eliminate its explicit appearance in expressions.[32] Consider a Lorentz boost in thex-direction. It is given by[33]where the signature is(−, +, +, +) and the field tensor is displayed side by side for easiest possible reference in the manipulations below.
Here,β = (β, 0, 0) is used. These results can be summarized byand are independent of the metric signature. For SI units, substituteE →E/c.Misner, Thorne & Wheeler (1973) refer to this last form as the3 + 1 view as opposed to thegeometric view represented by the tensor expressionand make a strong point of the ease with which results that are difficult to achieve using the3 + 1 view can be obtained and understood. Only objects that have well defined Lorentz transformation properties (in fact underany smooth coordinate transformation) are geometric objects. In the geometric view, the electromagnetic field is a six-dimensional geometric object inspacetime as opposed to two interdependent, but separate, 3-vector fields inspace andtime. The fieldsE (alone) andB (alone) do not have well defined Lorentz transformation properties. The mathematical underpinnings are equations(T1) and(T2) that immediately yield(T3). One should note that the primed and unprimed tensors refer to thesame event in spacetime. Thus the complete equation with spacetime dependence is
Length contraction has an effect oncharge densityρ andcurrent densityJ, and time dilation has an effect on the rate of flow of charge (current), so charge and current distributions must transform in a related way under a boost. It turns out they transform exactly like the space-time and energy-momentum four-vectors,or, in the simpler geometric view,
Charge density transforms as the time component of a four-vector. It is a rotational scalar. The current density is a 3-vector.
Equation(T1) hold unmodified for any representation of the Lorentz group, including thebispinor representation. In(T2) one simply replaces all occurrences ofΛ by the bispinor representationΠ(Λ),
(T4)
The above equation could, for instance, be the transformation of a state inFock space describing two free electrons.
^One can imagine that in each inertial frame there are observers positioned throughout space, each with a synchronized clock and at rest in the particular inertial frame. These observers then report to a central office, where all reports are collected. When one speaks of aparticular observer, one refers to someone having, at least in principle, a copy of this report. See, e.g.,Sard (1970).
^The separate requirements of the three equations lead to three different groups. The second equation is satisfied for spacetime translations in addition to Lorentz transformations leading to thePoincaré group or theinhomogeneous Lorentz group. The first equation (or the second restricted to lightlike separation) leads to a yet larger group, theconformal group of spacetime.
^The groupsO(3, 1) andO(1, 3) are isomorphic. It is widely believed that the choice between the two metric signatures has no physical relevance, even though some objects related toO(3, 1) andO(1, 3) respectively, e.g., theClifford algebras corresponding to the different signatures of the bilinear form associated to the two groups, are non-isomorphic.
^For two square matricesA andB,det(AB) = det(A)det(B)
^Until now the term "vector" has exclusively referred to "Euclidean vector", examples are positionr, velocityv, etc. The term "vector" applies much more broadly than Euclidean vectors, row or column vectors, etc., seeLinear algebra andVector space for details. The generators of a Lie group also form a vector space over afield of numbers (e.g.real numbers,complex numbers), since alinear combination of the generators is also a generator. They just live in a different space to the position vectors in ordinary 3-dimensional space.
^In ordinary 3-dimensionalposition space, the position vectorr =xex +yey +zez is expressed as a linear combination of the Cartesian unit vectorsex,ey,ez which form a basis, and the Cartesian coordinatesx, y, z are coordinates with respect to this basis.
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Derivation of the Lorentz transformations. This web page contains a more detailed derivation of the Lorentz transformation with special emphasis on group properties.
The Paradox of Special Relativity. This webpage poses a problem, the solution of which is the Lorentz transformation, which is presented graphically in its next page.
