text:
Lorentz transformation
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Part of a series on
Spacetime
Special relativity
General relativity
Spacetime concepts[show]
Spacetime manifold
Equivalence principle
Lorentz transformations
Minkowski space
General relativity[show]
Introduction to general relativity
Mathematics of general relativity
Einstein field equations
Classical gravity[show]
Introduction to gravitation
Newton's law of universal gravitation
Relevant mathematics[show]
Four-vector
Derivations of relativity
Spacetime diagrams
Differential geometry
Curved spacetime
Mathematics of general relativity
Spacetime topology
v
t
e
In physics, the Lorentz transformations (or transformation) are coordinate transformations between two coordinate frames that move at constant velocity relative to each other. The transformations are named after the Dutch physicist Hendrik Lorentz.
Frames of reference can be divided into two groups: inertial (relative motion with constant velocity) and non-inertial (accelerating in curved paths, rotational motion with constant angular velocity, etc.). The term "Lorentz transformations" only refers to transformations between inertial frames, usually in the context of special relativity.
In each reference frame, an observer can use a local coordinate system (most exclusively Cartesian coordinates in this context) to measure lengths, and a clock to measure time intervals. An observer is a real or imaginary entity that can take measurements, say humans, or any other living organism—or even robots and computers. An event is something that happens at a point in space at an instant of time, or more formally a point in spacetime. The transformations connect the space and time coordinates of an event as measured by an observer in each frame.[nb 1]
They supersede the Galilean transformation of Newtonian physics, which assumes an absolute space and time (see Galilean relativity). The Galilean transformation is a good approximation only at relative speeds much smaller than the speed of light. Lorentz transformations have a number of unintuitive features that do not appear in Galilean transformations. For example, they reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events, but always such that the speed of light is the same in all inertial reference frames. The invariance of light speed is one of the postulates of special relativity.
Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism. The Lorentz transformation is in accordance with special relativity, but was derived before special relativity.
The Lorentz transformation is a linear transformation. It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space, the mathematical model of spacetime in special relativity, the Lorentz transformations preserve the spacetime interval between any two events. This property is the defining property of a Lorentz transformation. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a hyperbolic rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.
Contents
[hide]
1 History
2 Derivation of the group of Lorentz transformations
3 Generalities
4 Physical formulation of Lorentz boosts
4.1 Coordinate transformation
4.2 Physical implications
4.3 Vector transformations
4.4 Transformation of velocities
4.5 Transformation of other quantities
5 Mathematical formulation
5.1 Homogeneous Lorentz group
5.2 Proper transformations
5.2.1 The Lie group SO+(3,1)
5.2.2 The Lie algebra so(3,1)
5.3 Improper transformations
5.4 Inhomogeneous Lorentz group
6 Tensor formulation
6.1 Contravariant vectors
6.2 Covariant vectors
6.3 Tensors
6.3.1 Transformation of the electromagnetic field
6.4 Spinors
6.4.1 Transformation of general fields
7 See also
8 Footnotes
9 Notes
10 References
10.1 Websites
10.2 Papers
10.3 Books
11 Further reading
12 External links
History[edit]
Main article: History of Lorentz transformations
Many physicists—including Woldemar Voigt, George FitzGerald, Joseph Larmor, and Hendrik Lorentz[1] himself—had been discussing the physics implied by these equations since 1887.[2] Early in 1889, Oliver Heaviside had shown from Maxwell's equations that the electric field surrounding a spherical distribution of charge should cease to have spherical symmetry once the charge is in motion relative to the aether. FitzGerald then conjectured that Heaviside’s distortion result might be applied to a theory of intermolecular forces. Some months later, FitzGerald published the conjecture that bodies in motion are being contracted, in order to explain the baffling outcome of the 1887 aether-wind experiment of Michelson and Morley. In 1892, Lorentz independently presented the same idea in a more detailed manner, which was subsequently called FitzGerald–Lorentz contraction hypothesis.[3] Their explanation was widely known before 1905.[4]
Lorentz (1892–1904) and Larmor (1897–1900), who believed the luminiferous aether hypothesis, also looked for the transformation under which Maxwell's equations are invariant when transformed from the aether to a moving frame. They extended the FitzGerald–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 in v/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.[5] Larmor is credited to have been the first to understand the crucial time dilation property inherent in his equations.[6]
In 1905, Poincaré was the first to recognize that the transformation has the properties of a mathematical group, and named it after Lorentz.[7] Later in the same year Albert Einstein published what is now called special relativity, by deriving the Lorentz transformation under the assumptions of the principle of relativity and the constancy of the speed of light in any inertial reference frame, and by abandoning the mechanistic aether as unnecessary.[8]
Derivation of the group of Lorentz transformations[edit]
Main articles: Derivations of the Lorentz transformations and Lorentz group
An event 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 coordinate ct and a set of Cartesian coordinates x, y, z to specify position in space in that frame. Subscripts label individual events.
From Einstein's second postulate of relativity follows
c 2 ( t 2 − t 1 ) 2 − ( x 2 − x 1 ) 2 − ( y 2 − y 1 ) 2 − ( z 2 − z 1 ) 2 = 0 (lightlike separated events 1, 2) {\displaystyle c^{2}(t_{2}-t_{1})^{2}-(x_{2}-x_{1})^{2}-(y_{2}-y_{1})^{2}-(z_{2}-z_{1})^{2}=0\quad {\text{(lightlike separated events 1, 2)}}}
(D1)
in all inertial frames for events connected by light signals. The quantity on the left is called the spacetime interval between events a1 = (t1, x1, y1, z1) and a2 = (t2, x2, y2, z2). The interval between any 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 is shown using homogeneity and isotropy of space. The transformation sought after thus must possess the property that
c 2 ( t 2 − t 1 ) 2 − ( x 2 − x 1 ) 2 − ( y 2 − y 1 ) 2 − ( z 2 − z 1 ) 2 = c 2 ( t 2 ′ − t 1 ′ ) 2 − ( x 2 ′ − x 1 ′ ) 2 − ( y 2 ′ − y 1 ′ ) 2 − ( z 2 ′ − z 1 ′ ) 2 (all events 1, 2) . {\displaystyle c^{2}(t_{2}-t_{1})^{2}-(x_{2}-x_{1})^{2}-(y_{2}-y_{1})^{2}-(z_{2}-z_{1})^{2}=c^{2}(t_{2}'-t_{1}')^{2}-(x_{2}'-x_{1}')^{2}-(y_{2}'-y_{1}')^{2}-(z_{2}'-z_{1}')^{2}\quad {\text{(all events 1, 2)}}.}
(D2)
where (ct, x, y, z) are the spacetime coordinates used to define events in one frame, and (ct′, x′, y′, z′) are the coordinates in another frame. First one observes that (D2) is satisfied if an arbitrary 4-tuple b of numbers are added to events a1 and a2. Such transformations are called spacetime translations and are not dealt with further here. Then one observes that a linear solution preserving the origin of the simpler problem
c 2 t 2 − x 2 − y 2 − z 2 = c 2 t ′ 2 − x ′ 2 − y ′ 2 − z ′ 2 or c 2 t 1 t 2 − x 1 x 2 − y 1 y 2 − z 1 z 2 = c 2 t 1 ′ t 2 ′ − x 1 ′ x 2 ′ − y 1 ′ y 2 ′ − z 1 ′ z 2 ′ {\displaystyle c^{2}t^{2}-x^{2}-y^{2}-z^{2}=c^{2}t'^{2}-x'^{2}-y'^{2}-z'^{2}\quad {\text{ or }}\quad c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}=c^{2}t'_{1}t'_{2}-x'_{1}x'_{2}-y'_{1}y'_{2}-z'_{1}z'_{2}}
(D3)
solves the general problem too. (A solution satisfying the left formula automatically satisfies the right formula, see polarization identity.) Finding the solution to the simpler problem is just a matter of look-up in the theory of classical groups that preserve bilinear forms of various signature.[nb 2] Equation (D3) can be written more compactly as
( a , a ) = ( a ′ , a ′ ) or a ⋅ a = a ′ ⋅ a ′ , {\displaystyle (a,a)=(a',a')\quad {\text{or}}\quad a\cdot a=a'\cdot a',}
(D4)
where (·, ·) refers to the bilinear form of signature (1, 3) on ℝ4 exposed by the right hand side formula in (D3). The alternative notation defined on the right is referred to as the relativistic dot product. Spacetime mathematically viewed as ℝ4 endowed with this bilinear form is known as Minkowski space M. The Lorentz transformation is thus an element of the group Lorentz group O(1, 3), the Lorentz group or, for those that prefer the other metric signature, O(3, 1) (also called the Lorentz group).[nb 3] One has
( a , a ) = ( Λ a , Λ a ) = ( a ′ , a ′ ) , Λ ∈ O ( 1 , 3 ) , a , a ′ ∈ M , {\displaystyle (a,a)=(\Lambda a,\Lambda a)=(a',a'),\quad \Lambda \in \mathrm {O} (1,3),\quad a,a'\in M,}
(D5)
which is precisely preservation of the bilinear form (D3) which implies (by linearity of Λ and bilinearity of the form) that (D2) is satisfied. The elements of the Lorentz group are rotations and boosts and mixes thereof. If the spacetime translations are included, then one obtains the inhomogeneous Lorentz group or the Poincare group.
Generalities[edit]
The relations between the primed and unprimed spacetime coordinates are the Lorentz transformations, each coordinate in one frame is a linear function of all the coordinates in the other frame, and the inverse 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 called boosts, and the relative velocity between the frames is the parameter of the transformation. The other basic type of Lorentz transformations is rotations in the spatial coordinates only, these are also inertial frames 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, or Euler angles, etc.). A combination of a rotation and boost is a homogeneous transformation, which transforms the origin back to the origin.
The full Lorentz group O(3, 1) also contains special transformations that are neither rotations nor boosts, but rather reflections 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 and temporal 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 an inhomogeneous Lorentz transformation, an element of the Poincaré group, which is also called the inhomogeneous Lorentz group.
Physical formulation of Lorentz boosts[edit]
Further information: Derivations of the Lorentz transformations
Coordinate transformation[edit]
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 v along the x-axis of frame F.
Bottom: frame F moves at velocity −v along the x′-axis of frame F′.[9]
A "stationary" observer in frame F defines events with coordinates t, x, y, z. Another frame F′ moves with velocity v relative to F, and an observer in this "moving" frame F′ defines events using the coordinates t′, x′, y′, z′.
The coordinate axes in each frame are parallel (the x and x′ axes are parallel, the y and y′ axes are parallel, and the z and z′ axes are parallel), remain mutually perpendicular, and relative motion is along the coincident xx′ axes. At t = 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 in standard configuration, or synchronized.
If an observer in F records an event t, x, y, z, then an observer in F′ records the same event with coordinates[10]
Lorentz boost (x direction)
t ′ = γ ( t − v x c 2 ) x ′ = γ ( x − v t ) y ′ = y z ′ = z {\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\\y'&=y\\z'&=z\end{aligned}}}
where v is the relative velocity between frames in the x-direction, c is the speed of light, and
γ = 1 1 − v 2 c 2 {\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}
(lowercase gamma) is the Lorentz factor.
Here, v is the parameter 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 velocity v > 0 is motion along the positive directions of the xx′ axes, zero relative velocity v = 0 is no relative motion, while negative relative velocity v < 0 is relative motion along the negative directions of the xx′ axes. The magnitude of relative velocity v cannot equal or exceed c, so only subluminal speeds −c < v < c are allowed. The corresponding range of γ is 1 ≤ γ < ∞.
The transformations are not defined if v is outside these limits. At the speed of light (v = c) γ is infinite, and faster than light (v > c) γ is a complex number, each of which make the transformations unphysical. The space and time coordinates are measurable quantities and numerically must be real numbers.
As an active transformation, an observer in F′ notices the coordinates of the event to be "boosted" in the negative directions of the xx′ axes, because of the −v in the transformations. This has the equivalent effect of the coordinate system F′ boosted in the positive directions of the xx′ axes, while the event does not change and is simply represented in another coordinate system, a passive transformation.
The inverse relations (t, x, y, z in terms of t′, x′, y′, z′) can be found by algebraically solving the original set of equations. A more efficient way is to use physical principles. Here F′ is the "stationary" frame while F is the "moving" frame. According to the principle of relativity, there is no privileged frame of reference, so the transformations from F′ to F must take exactly the same form as the transformations from F to F′. The only difference is F moves with velocity −v relative to F′ (i.e., the relative velocity has the same magnitude but is oppositely directed). Thus if an observer in F′ notes an event t′, x′, y′, z′, then an observer in F notes the same event with coordinates
Inverse Lorentz boost (x direction)
t = γ ( t ′ + v x ′ c 2 ) x = γ ( x ′ + v t ′ ) y = y ′ z = z ′ , {\displaystyle {\begin{aligned}t&=\gamma \left(t'+{\frac {vx'}{c^{2}}}\right)\\x&=\gamma \left(x'+vt'\right)\\y&=y'\\z&=z',\end{aligned}}}
and the value of γ 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.
Sometimes it is more convenient to use β = v/c (lowercase beta) instead of v, so that
c t ′ = γ ( c t − β x ) , x ′ = γ ( x − β c t ) , {\displaystyle {\begin{aligned}ct'&=\gamma \left(ct-\beta x\right)\,,\\x'&=\gamma \left(x-\beta ct\right)\,,\\\end{aligned}}}
which shows much more clearly the symmetry in the transformation. From the allowed ranges of v and the definition of β, it follows −1 < β < 1. The use of β and γ is standard throughout the literature.
The Lorentz transformations can also be derived in a way that resembles circular rotations in 3d space using the hyperbolic functions. For the boost in the x direction, the results are
Lorentz boost (x direction with rapidity ζ)
c t ′ = c t cosh  ζ − x sinh  ζ x ′ = x cosh  ζ − c t sinh  ζ y ′ = y z ′ = z {\displaystyle {\begin{aligned}ct'&=ct\cosh \zeta -x\sinh \zeta \\x'&=x\cosh \zeta -ct\sinh \zeta \\y'&=y\\z'&=z\end{aligned}}}
where ζ (lowercase zeta) is a parameter called rapidity (many other symbols are used, including θ, ϕ, φ, η, ψ, ξ). Given the strong resemblance to rotations of spatial coordinates in 3d space in the Cartesian xy, yz, and zx planes, a Lorentz boost can be thought of as a hyperbolic rotation of spacetime coordinates in the xt, yt, and zt Cartesian-time planes of 4d Minkowski space. The parameter ζ is the hyperbolic angle of rotation, analogous to the ordinary angle for circular rotations. This transformation can be illustrated with a Minkowski diagram.
The hyperbolic functions arise from the difference 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 taking x = 0 or ct = 0 in the transformations. Squaring and subtracting the results, one can derive hyperbolic curves of constant coordinate values but varying ζ, which parametrizes the curves according to the identity
cosh 2  ζ − sinh 2  ζ = 1 . {\displaystyle \cosh ^{2}\zeta -\sinh ^{2}\zeta =1\,.}
Conversely the ct and x axes can be constructed for varying coordinates but constant ζ. The definition
tanh  ζ = sinh  ζ cosh  ζ , {\displaystyle \tanh \zeta ={\frac {\sinh \zeta }{\cosh \zeta }}\,,}
provides the link between a constant value of rapidity, and the slope of the ct axis in spacetime. A consequence these two hyperbolic formulae is an identity that matches the Lorentz factor
cosh  ζ = 1 1 − tanh 2  ζ . {\displaystyle \cosh \zeta ={\frac {1}{\sqrt {1-\tanh ^{2}\zeta }}}\,.}
Comparing the Lorentz transformations in terms of the relative velocity and rapidity, or using the above formulae, the connections between β, γ, and ζ are
β = tanh  ζ , γ = cosh  ζ , β γ = sinh  ζ . {\displaystyle {\begin{aligned}\beta &=\tanh \zeta \,,\\\gamma &=\cosh \zeta \,,\\\beta \gamma &=\sinh \zeta \,.\end{aligned}}}
Taking the inverse hyperbolic tangent gives the rapidity
ζ = tanh − 1  β . {\displaystyle \zeta =\tanh ^{-1}\beta \,.}
Since −1 < β < 1, it follows −∞ < ζ < ∞. From the relation between ζ and β, positive rapidity ζ > 0 is motion along the positive directions of the xx′ axes, zero rapidity ζ = 0 is no relative motion, while negative rapidity ζ < 0 is relative motion along the negative directions of the xx′ axes.
The inverse transformations are obtained by exchanging primed and unprimed quantities to switch the coordinate frames, and negating rapidity ζ → −ζ since this is equivalent to negating the relative velocity. Therefore,
Inverse Lorentz boost (x direction with rapidity ζ)
c t = c t ′ cosh  ζ + x ′ sinh  ζ x = x ′ cosh  ζ + c t ′ sinh  ζ y = y ′ z = z ′ {\displaystyle {\begin{aligned}ct&=ct'\cosh \zeta +x'\sinh \zeta \\x&=x'\cosh \zeta +ct'\sinh \zeta \\y&=y'\\z&=z'\end{aligned}}}
The inverse transformations can be similarly visualized by considering the cases when x′ = 0 and ct′ = 0.
So far the Lorentz transformations have been applied to one event. If there are two events, there is a spatial separation and time interval between them. It follows from the linearity 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;
Δ t ′ = γ ( Δ t − v Δ x c 2 ) , Δ x ′ = γ ( Δ x − v Δ t ) , {\displaystyle {\begin{aligned}\Delta t'&=\gamma \left(\Delta t-{\frac {v\Delta x}{c^{2}}}\right)\,,\\\Delta x'&=\gamma \left(\Delta x-v\Delta t\right)\,,\end{aligned}}}
with inverse relations
Δ t = γ ( Δ t ′ + v Δ x ′ c 2 ) , Δ x = γ ( Δ x ′ + v Δ t ′ ) . {\displaystyle {\begin{aligned}\Delta t&=\gamma \left(\Delta t'+{\frac {v\Delta x'}{c^{2}}}\right)\,,\\\Delta x&=\gamma \left(\Delta x'+v\Delta t'\right)\,.\end{aligned}}}
where Δ (uppercase delta) indicates a difference of quantities; e.g., Δx = x2 − x1 for two values of x coordinates, and so on.
These transformations on differences 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 event t0, x0, y0, z0 in F and t0′, x0′, y0′, z0′ in F′, 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′ − x0′, etc.
Physical implications[edit]
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 in F the equation for a pulse of light along the x direction is x = ct, then in F′ the Lorentz transformations give x′ = ct′, and vice versa, for any −c < v < c.
For relative speeds much less than the speed of light, the Lorentz transformations reduce to the Galilean transformation
t ′ ≈ t x ′ ≈ x − v t {\displaystyle {\begin{aligned}t'&\approx t\\x'&\approx x-vt\end{aligned}}}
in accordance with the correspondence principle. It is sometimes said that nonrelativistic physics is a physics of "instantaneous action at a distance".[11]
Three unintuitive, but correct, predictions of the transformations are:
Time dilation
Suppose there is a clock at rest in F. If a time interval (say a "tick") is measured at the same point so that Δx = 0, then the transformations give this tick in F′ by Δt′ = γΔt. Conversely, suppose there is a clock at rest in F′. If a tick is measured at the same point so that Δx′ = 0, then the transformations give this tick in F by Δt = γΔt′. Either way, the boosted observer measures longer time intervals than the observer in the other frame.
Relativity of simultaneity
Suppose two events occur simultaneously (Δt = 0) along the x axis, but separated by a nonzero displacement Δx. Then in F′, we find that Δ t ′ = γ − v Δ x c 2 {\displaystyle \Delta t'=\gamma {\frac {-v\Delta x}{c^{2}}}} , so the events are no longer simultaneous according to a moving observer.
Length contraction
Suppose there is a rod at rest in F aligned along the x axis, with length Δx. In F′, 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 that Δx = γΔx′. In F the two measurements are no longer simultaneous, but this does not matter because the rod is at rest in F. We conclude that the boosted observer measures a shorter length, by a factor of γ, than the observer in the rest frame of the rod. 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.
Vector transformations[edit]
Further information: Euclidean vector and vector projection
An observer in frame F observes F′ to move with velocity v, while F′ observes F to move with velocity −v. The coordinate axes of each frame are still parallel and orthogonal. The position vector as measured in each frame is split into components parallel and perpendicular to the relative velocity vector v. 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 relative velocity vector v with a magnitude |v| = v that cannot equal or exceed c, so that 0 ≤ 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 spatial position vector r as measured in F, and r′ as measured in F′, each into components perpendicular (⊥) and parallel ( ‖ ) to v,
r = r ⊥ + r ∥ , r ′ = r ⊥ ′ + r ∥ ′ , {\displaystyle \mathbf {r} =\mathbf {r} _{\perp }+\mathbf {r} _{\|}\,,\quad \mathbf {r} '=\mathbf {r} _{\perp }'+\mathbf {r} _{\|}'\,,}
then the transformations are
t ′ = γ ( t − r ∥ ⋅ v c 2 ) r ∥ ′ = γ ( r ∥ − v t ) r ⊥ ′ = r ⊥ {\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {\mathbf {r} _{\parallel }\cdot \mathbf {v} }{c^{2}}}\right)\\\mathbf {r} _{\|}'&=\gamma (\mathbf {r} _{\|}-\mathbf {v} t)\\\mathbf {r} _{\perp }'&=\mathbf {r} _{\perp }\end{aligned}}}
where · is the dot product. The Lorentz factor γ retains its definition for a boost in any direction, since it depends only on the magnitude of the relative velocity. The definition β = v/c with magnitude 0 ≤ β < 1 is also used by some authors.
Introducing a unit vector n = v/v = β/β in the direction of relative motion, the relative velocity is v = vn with magnitude v and direction n, and vector projection and rejection give respectively
r ∥ = ( r ⋅ n ) n , r ⊥ = r − ( r ⋅ n ) n {\displaystyle \mathbf {r} _{\parallel }=(\mathbf {r} \cdot \mathbf {n} )\mathbf {n} \,,\quad \mathbf {r} _{\perp }=\mathbf {r} -(\mathbf {r} \cdot \mathbf {n} )\mathbf {n} }
Accumulating the results gives the full transformations,
Lorentz boost (in direction n with magnitude v)
t ′ = γ ( t − v n ⋅ r c 2 ) , r ′ = r + ( γ − 1 ) ( r ⋅ n ) n − γ t v n . {\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {v\mathbf {n} \cdot \mathbf {r} }{c^{2}}}\right)\,,\\\mathbf {r} '&=\mathbf {r} +(\gamma -1)(\mathbf {r} \cdot \mathbf {n} )\mathbf {n} -\gamma tv\mathbf {n} \,.\end{aligned}}}
The projection and rejection also applies to r′. For the inverse transformations, exchange r and r′ to switch observed coordinates, and negate the relative velocity v → −v (or simply the unit vector n → −n since the magnitude v is always positive) to obtain
Inverse Lorentz boost (in direction n with magnitude v)
t = γ ( t ′ + r ′ ⋅ v n c 2 ) , r = r ′ + ( γ − 1 ) ( r ′ ⋅ n ) n + γ t ′ v n , {\displaystyle {\begin{aligned}t&=\gamma \left(t'+{\frac {\mathbf {r} '\cdot v\mathbf {n} }{c^{2}}}\right)\,,\\\mathbf {r} &=\mathbf {r} '+(\gamma -1)(\mathbf {r} '\cdot \mathbf {n} )\mathbf {n} +\gamma t'v\mathbf {n} \,,\end{aligned}}}
The unit vector has the advantage of simplifying equations for a single boost, allows either v or β to be reinstated when convenient, and the rapidity parametrization is immediately obtained by replacing β and βγ. It is not convenient for multiple boosts.
The vectorial relation between relative velocity and rapidity is[12]
β = β n = n tanh  ζ , {\displaystyle {\boldsymbol {\beta }}=\beta \mathbf {n} =\mathbf {n} \tanh \zeta \,,}
and the "rapidity vector" can be defined as
ζ = ζ n = n tanh − 1  β , {\displaystyle {\boldsymbol {\zeta }}=\zeta \mathbf {n} =\mathbf {n} \tanh ^{-1}\beta \,,}
each of which serves as a useful abbreviation in some contexts. The magnitude of ζ is the absolute value of the rapidity scalar confined to 0 ≤ ζ < ∞, which agrees with the range 0 ≤ β < 1.
Transformation of velocities[edit]
Further information: differential of a function and velocity addition formula
The transformation of velocities provides the definition relativistic velocity addition ⊕, the ordering of vectors is chosen to reflect the ordering of the addition of velocities; first v (the velocity of F′ relative to F) then u′ (the velocity of X relative to F′) to obtain u = v ⊕ u′ (the velocity of X relative to F).
Defining the coordinate velocities and Lorentz factor by
u = d r d t , u ′ = d r ′ d t ′ , γ v = 1 1 − v ⋅ v c 2 {\displaystyle \mathbf {u} ={\frac {d\mathbf {r} }{dt}}\,,\quad \mathbf {u} '={\frac {d\mathbf {r} '}{dt'}}\,,\quad \gamma _{\mathbf {v} }={\frac {1}{\sqrt {1-{\dfrac {\mathbf {v} \cdot \mathbf {v} }{c^{2}}}}}}}
taking the differentials in the coordinates and time of the vector transformations, then dividing equations, leads to
u ′ = 1 1 − v ⋅ u c 2 [ u γ v − v + 1 c 2 γ v γ v + 1 ( u ⋅ v ) v ] {\displaystyle \mathbf {u} '={\frac {1}{1-{\frac {\mathbf {v} \cdot \mathbf {u} }{c^{2}}}}}\left[{\frac {\mathbf {u} }{\gamma _{\mathbf {v} }}}-\mathbf {v} +{\frac {1}{c^{2}}}{\frac {\gamma _{\mathbf {v} }}{\gamma _{\mathbf {v} }+1}}\left(\mathbf {u} \cdot \mathbf {v} \right)\mathbf {v} \right]}
The velocities u and u′ are the velocity of some massive object. They can also be for a third inertial frame (say F′′), in which case they must be constant. Denote either entity by X. Then X moves with velocity u relative to F, or equivalently with velocity u′ relative to F′, in turn F′ moves with velocity v relative to F. The inverse transformations can be obtained in a similar way, or as with position coordinates exchange u and u′, and change v to −v.
The transformation of velocity is useful in stellar aberration, the Fizeau experiment, and the relativistic Doppler effect.
The Lorentz transformations of acceleration can be similarly obtained by taking differentials in the velocity vectors, and dividing these by the time differential.

Transformation of other quantities[edit]
In general, given four quantities A and Z = (Zx, Zy, Zz) and their Lorentz-boosted counterparts A′ and Z′ = (Z′x, Z′y, Z′z), a relation of the form
A 2 − Z ⋅ Z = A ′ 2 − Z ′ ⋅ Z ′ {\displaystyle A^{2}-\mathbf {Z} \cdot \mathbf {Z} ={A'}^{2}-\mathbf {Z} '\cdot \mathbf {Z} '}
implies the quantities transform under Lorentz transformations similar to the transformation of spacetime coordinates;
A ′ = γ ( A − v n ⋅ Z c ) , Z ′ = Z + ( γ − 1 ) ( Z ⋅ n ) n − γ A v n c . {\displaystyle {\begin{aligned}A'&=\gamma \left(A-{\frac {v\mathbf {n} \cdot \mathbf {Z} }{c}}\right)\,,\\\mathbf {Z} '&=\mathbf {Z} +(\gamma -1)(\mathbf {Z} \cdot \mathbf {n} )\mathbf {n} -{\frac {\gamma Av\mathbf {n} }{c}}\,.\end{aligned}}}
The decomposition of Z (and Z′) into components perpendicular and parallel to v 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 substitution n ↦ −n).
The quantities (A, Z) collectively make up a four vector, where A is the "timelike component", and Z the "spacelike component". Examples of A and Z are the following:
Four vector
A
Z
Position four vector Time (multiplied by c), ct Position vector, r
Four momentum Energy (divided by c), E/c Momentum, p
Four-wave vector angular frequency (divided by c), ω/c wave vector, k
Four spin (No name), st Spin, s
Four current Charge density (multiplied by c), ρc Current density, j
Electromagnetic four potential Electric potential (divided by c), φ/c Magnetic potential, A
For a given object (e.g., particle, fluid, field, material), if A or Z correspond to properties specific to the object like its charge 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 energy E 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 a rest 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 in relativistic quantum mechanics spin s 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 quantity st, however a boosted observer will perceive a nonzero timelike component and an altered spin.[13]
Not all quantities are invariant in the form as shown above, for example orbital angular momentum L does not have a timelike quantity, and neither does the electric field E nor the magnetic field B. The definition of angular momentum is L = r × p, and in a boosted frame the altered angular momentum is L′ = r′ × p′. Applying this definition using the transformations of coordinates and momentum leads to the transformation of angular momentum. It turns out L transforms with another vector quantity N = (E/c2)r − tp related to boosts, see relativistic angular momentum for details. For the case of the E and B fields, the transformations cannot be obtained as directly using vector algebra. The Lorentz force is the definition of these fields, and in F it is F = q(E + v × B) while in F′ it is F′ = 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.
Mathematical formulation[edit]
Main article: Lorentz group
Further information: Matrix (mathematics), matrix product, linear algebra, and rotation formalisms in three dimensions
Throughout, italic non-bold capital letters are 4×4 matrices, while non-italic bold letters are 3×3 matrices.
Homogeneous Lorentz group[edit]
Writing the coordinates in column vectors and the Minkowski metric η as a square matrix
X ′ = [ c t ′ x ′ y ′ z ′ ] , η = [ − 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] , X = [ c t x y z ] {\displaystyle X'={\begin{bmatrix}c\,t'\\x'\\y'\\z'\end{bmatrix}}\,,\quad \eta ={\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\,,\quad X={\begin{bmatrix}c\,t\\x\\y\\z\end{bmatrix}}}
the spacetime interval takes the form (T denotes transpose)
X ⋅ X = X T η X = X ′ T η X ′ {\displaystyle X\cdot X=X^{\mathrm {T} }\eta X={X'}^{\mathrm {T} }\eta {X'}}
and is invariant under a Lorentz transformation
X ′ = Λ X {\displaystyle X'=\Lambda X}
where Λ is a square matrix which can depend on parameters.
The set of all Lorentz transformations Λ in this article is denoted L {\displaystyle {\mathcal {L}}} . This set together with matrix multiplication forms a group, in this context known as the Lorentz group. Also, the above expression X·X is a quadratic form of signature (3,1) on spacetime, and the group of transformations which leaves this quadratic form invariant is the indefinite orthogonal group O(3,1), a Lie group. In other words, the Lorentz group is O(3,1). As presented in this article, any Lie groups mentioned are matrix Lie groups. In this context the operation of composition amounts to matrix multiplication.
From the invariance of the spacetime interval it follows
η = Λ T η Λ {\displaystyle \eta =\Lambda ^{\mathrm {T} }\eta \Lambda }
and this matrix equation contains the general conditions on the Lorentz transformation to ensure invariance of the spacetime interval. Taking the determinant of the equation using the product rule[nb 4] gives immediately
[ det ( Λ ) ] 2 = 1 ⇒ det ( Λ ) = ± 1 {\displaystyle [\det(\Lambda )]^{2}=1\quad \Rightarrow \quad \det(\Lambda )=\pm 1}
Writing the Minkowski metric as a block matrix, and the Lorentz transformation in the most general form,
η = [ − 1 0 0 I ] , Λ = [ Γ − a T − b M ] , {\displaystyle \eta ={\begin{bmatrix}-1&0\\0&\mathbf {I} \end{bmatrix}}\,,\quad \Lambda ={\begin{bmatrix}\Gamma &-\mathbf {a} ^{\mathrm {T} }\\-\mathbf {b} &\mathbf {M} \end{bmatrix}}\,,}
carrying out the block matrix multiplications obtains general conditions on Γ, a, b, M to ensure relativistic invariance. Not much information can be directly extracted from all the conditions, however one of the results
Γ 2 = 1 + b T b {\displaystyle \Gamma ^{2}=1+\mathbf {b} ^{\mathrm {T} }\mathbf {b} }
is useful; bTb ≥ 0 always so it follows that
Γ 2 ≥ 1 ⇒ Γ ≤ − 1 , Γ ≥ 1 {\displaystyle \Gamma ^{2}\geq 1\quad \Rightarrow \quad \Gamma \leq -1\,,\quad \Gamma \geq 1}
The negative inequality may be unexpected, because Γ multiplies the time coordinate and this has an effect on time symmetry. If the positive equality holds, then Γ is the Lorentz factor.
The determinant and inequality provide four ways to classify Lorentz transformations (herein LTs for brevity). Any particular LT has only one determinant sign and only one inequality. There are four sets which include every possible pair given by the intersections ("n"-shaped symbol meaning "and") of these classifying sets.
Intersection, ∩
Antichronous (or non-orthochronous) LTs
L ↓ = { Λ : Γ ≤ − 1 } {\displaystyle {\mathcal {L}}^{\downarrow }=\{\Lambda \,:\,\Gamma \leq -1\}}
Orthochronous LTs
L ↑ = { Λ : Γ ≥ 1 } {\displaystyle {\mathcal {L}}^{\uparrow }=\{\Lambda \,:\,\Gamma \geq 1\}}
Proper LTs
L + = { Λ : det ( Λ ) = + 1 } {\displaystyle {\mathcal {L}}_{+}=\{\Lambda \,:\,\det(\Lambda )=+1\}}
Proper antichronous LTs
L + ↓ = L + ∩ L ↓ {\displaystyle {\mathcal {L}}_{+}^{\downarrow }={\mathcal {L}}_{+}\cap {\mathcal {L}}^{\downarrow }}
Proper orthochronous LTs
L + ↑ = L + ∩ L ↑ {\displaystyle {\mathcal {L}}_{+}^{\uparrow }={\mathcal {L}}_{+}\cap {\mathcal {L}}^{\uparrow }}
Improper LTs
L − = { Λ : det ( Λ ) = − 1 } {\displaystyle {\mathcal {L}}_{-}=\{\Lambda \,:\,\det(\Lambda )=-1\}}
Improper antichronous LTs
L − ↓ = L − ∩ L ↓ {\displaystyle {\mathcal {L}}_{-}^{\downarrow }={\mathcal {L}}_{-}\cap {\mathcal {L}}^{\downarrow }}
Improper orthochronous LTs
L − ↑ = L − ∩ L ↑ {\displaystyle {\mathcal {L}}_{-}^{\uparrow }={\mathcal {L}}_{-}\cap {\mathcal {L}}^{\uparrow }}
where "+" and "−" indicate the determinant sign, while "↑" for ≥ and "↓" for ≤ denote the inequalities.
The full Lorentz group splits into the union ("u"-shaped symbol meaning "or") of four disjoint sets
L = L + ↑ ∪ L − ↑ ∪ L + ↓ ∪ L − ↓ {\displaystyle {\mathcal {L}}={\mathcal {L}}_{+}^{\uparrow }\cup {\mathcal {L}}_{-}^{\uparrow }\cup {\mathcal {L}}_{+}^{\downarrow }\cup {\mathcal {L}}_{-}^{\downarrow }}
A subgroup of a group must be closed under the same operation of the group (here matrix multiplication). In other words, for two Lorentz transformations Λ and L from a particular set, the composite Lorentz transformations ΛL and LΛ must be in the same set as Λ and L. This will not always be the case; it can be shown that the composition of any two Lorentz transformations always has the positive determinant and positive inequality, a proper orthochronous transformation. The sets L + ↑ {\displaystyle {\mathcal {L}}_{+}^{\uparrow }} , L + {\displaystyle {\mathcal {L}}_{+}} , L ↑ {\displaystyle {\mathcal {L}}^{\uparrow }} , and L 0 = L + ↑ ∪ L − ↓ {\displaystyle {\mathcal {L}}_{0}={\mathcal {L}}_{+}^{\uparrow }\cup {\mathcal {L}}_{-}^{\downarrow }} all form subgroups. The other sets involving the improper and/or antichronous properties (i.e. L + ↓ {\displaystyle {\mathcal {L}}_{+}^{\downarrow }} , L − ↓ {\displaystyle {\mathcal {L}}_{-}^{\downarrow }} , L − ↑ {\displaystyle {\mathcal {L}}_{-}^{\uparrow }} ) do not form subgroups, because the composite transformation always has a positive determinant or inequality, whereas the original separate transformations will have negative determinants and/or inequalities.
Proper transformations[edit]
The Lorentz boost is
X ′ = B ( v ) X {\displaystyle X'=B(\mathbf {v} )X}
where the boost matrix is
B ( v ) = [ γ − γ β n x − γ β n y − γ β n z − γ β n x 1 + ( γ − 1 ) n x 2 ( γ − 1 ) n x n y ( γ − 1 ) n x n z − γ β n y ( γ − 1 ) n y n x 1 + ( γ − 1 ) n y 2 ( γ − 1 ) n y n z − γ β n z ( γ − 1 ) n z n x ( γ − 1 ) n z n y 1 + ( γ − 1 ) n z 2 ] . {\displaystyle B(\mathbf {v} )={\begin{bmatrix}\gamma &-\gamma \beta n_{x}&-\gamma \beta n_{y}&-\gamma \beta n_{z}\\-\gamma \beta n_{x}&1+(\gamma -1)n_{x}^{2}&(\gamma -1)n_{x}n_{y}&(\gamma -1)n_{x}n_{z}\\-\gamma \beta n_{y}&(\gamma -1)n_{y}n_{x}&1+(\gamma -1)n_{y}^{2}&(\gamma -1)n_{y}n_{z}\\-\gamma \beta n_{z}&(\gamma -1)n_{z}n_{x}&(\gamma -1)n_{z}n_{y}&1+(\gamma -1)n_{z}^{2}\\\end{bmatrix}}\,.}
The boosts along the Cartesian directions can be readily obtained, for example the unit vector in the x direction has components nx = 1 and ny = nz = 0.
The matrices make one or more successive transformations easier to handle, rather than rotely iterating the transformations to obtain the result of more than one transformation. If a frame F′ is boosted with velocity u relative to frame F, and another frame F′′ is boosted with velocity v relative to F′, the separate boosts are
X ″ = B ( v ) X ′ , X ′ = B ( u ) X {\displaystyle X''=B(\mathbf {v} )X'\,,\quad X'=B(\mathbf {u} )X}
and the composition of the two boosts connects the coordinates in F′′ and F,
X ″ = B ( v ) B ( u ) X . {\displaystyle X''=B(\mathbf {v} )B(\mathbf {u} )X\,.}
Successive transformations act on the left. If u and v are collinear (parallel or antiparallel along the same line of relative motion), the boost matrices commute: B(v)B(u) = B(u)B(v) and this composite transformation happens to be another boost.
If u and v 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) and B(u)B(v) are not equal. Also, each of these compositions is not a single boost, but still a Lorentz transformation as each boost still preserves invariance of 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 of R(ρ)B(w) or B(w)R(ρ). The w and w are composite velocities, while ρ and ρ are rotation parameters (e.g. axis-angle variables, Euler angles, etc.). The rotation in block matrix form is simply
R ( ρ ) = [ 1 0 0 R ( ρ ) ] , {\displaystyle \quad R({\boldsymbol {\rho }})={\begin{bmatrix}1&0\\0&\mathbf {R} ({\boldsymbol {\rho }})\end{bmatrix}}\,,}
where R(ρ) is a 3d rotation matrix, which rotates any 3d vector in one sense (active transformation), or equivalently the coordinate frame in the opposite sense (passive transformation). It is not simple to connect w and ρ (or w and ρ) to the original boost parameters u and v. In a composition of boosts, the R matrix is named the Wigner rotation, and gives rise to the Thomas precession. These articles give the explicit formulae for the composite transformation matrices, including expressions for w, ρ, w, ρ.
In this article the axis-angle representation is used for ρ. The rotation is about an axis in the direction of a unit vector e, through angle θ (positive anticlockwise, negative clockwise, according to the right-hand rule). The "axis-angle vector"
θ = θ e {\displaystyle {\boldsymbol {\theta }}=\theta \mathbf {e} }
will serve as a useful abbreviation.
Spatial rotations alone are also Lorentz transformations 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), and R(θ)−1 = R(−θ) (rotation in the opposite sense about the same axis)
identity transformation for no relative motion/rotation: B(0) = R(0) = I
unit determinant: det(B) = det(R) = +1. This property makes them proper transformations.
matrix symmetry: B is symmetric (equals transpose), while R is nonsymmetric but orthogonal (transpose equals inverse, RT = R−1).
The most general proper Lorentz transformation Λ(v, θ) includes a boost and rotation together, and is a nonsymmetric matrix. As special cases, Λ(0, θ) = R(θ) and Λ(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 writes Λ(ζ, θ) and B(ζ).
The Lie group SO+(3,1)[edit]
The set of transformations
{ B ( ζ ) , R ( θ ) , Λ ( ζ , θ ) } {\displaystyle \{B({\boldsymbol {\zeta }}),R({\boldsymbol {\theta }}),\Lambda ({\boldsymbol {\zeta }},{\boldsymbol {\theta }})\}}
with matrix multiplication as the operation of composition forms a group, called the "restricted Lorentz group", and is the special indefinite orthogonal group SO+(3,1). (The plus sign indicates positive unit determinant).
For simplicity, look at the infinitesimal Lorentz boost in the x 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 the Taylor expansion of the boost matrix to first order about ζ = 0,
B x = I + ζ ∂ B x ∂ ζ | ζ = 0 + ⋯ {\displaystyle B_{x}=I+\zeta \left.{\frac {\partial B_{x}}{\partial \zeta }}\right|_{\zeta =0}+\cdots }
where the higher order terms not shown are negligible because ζ is small, and Bx is simply the boost matrix in the x direction. The derivative 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 at ζ = 0,
∂ B x ∂ ζ | ζ = 0 = − K x . {\displaystyle \left.{\frac {\partial B_{x}}{\partial \zeta }}\right|_{\zeta =0}=-K_{x}\,.}
For now, Kx is defined by this result (its significance will be explained shortly). In the limit of an infinite number of infinitely small steps, the finite boost transformation in the form of a matrix exponential is obtained
B x = lim N → ∞ ( I − ζ N K x ) N = e − ζ K x {\displaystyle B_{x}=\lim _{N\rightarrow \infty }\left(I-{\frac {\zeta }{N}}K_{x}\right)^{N}=e^{-\zeta K_{x}}}
where the limit definition of the exponential has been used (see also characterizations of the exponential function). More generally[nb 5]
B ( ζ ) = e − ζ ⋅ K , R ( θ ) = e θ ⋅ J . {\displaystyle B({\boldsymbol {\zeta }})=e^{-{\boldsymbol {\zeta }}\cdot \mathbf {K} }\,,\quad R({\boldsymbol {\theta }})=e^{{\boldsymbol {\theta }}\cdot \mathbf {J} }\,.}
The axis-angle vector θ and rapidity vector ζ are altogether six continuous variables which make up the group parameters (in this particular representation), and the generators of the group are K = (Kx, Ky, Kz) and J = (Jx, Jy, Jz), each vectors of matrices with the explicit forms[nb 6]
K x = [ 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ] , K y = [ 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 ] , K z = [ 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 ] {\displaystyle K_{x}={\begin{bmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0\\\end{bmatrix}}\,,\quad K_{y}={\begin{bmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0\end{bmatrix}}\,,\quad K_{z}={\begin{bmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\end{bmatrix}}}
J x = [ 0 0 0 0 0 0 0 0 0 0 0 − 1 0 0 1 0 ] , J y = [ 0 0 0 0 0 0 0 1 0 0 0 0 0 − 1 0 0 ] , J z = [ 0 0 0 0 0 0 − 1 0 0 1 0 0 0 0 0 0 ] {\displaystyle J_{x}={\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&0&-1\\0&0&1&0\\\end{bmatrix}}\,,\quad J_{y}={\begin{bmatrix}0&0&0&0\\0&0&0&1\\0&0&0&0\\0&-1&0&0\end{bmatrix}}\,,\quad J_{z}={\begin{bmatrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\end{bmatrix}}}
These are all defined in an analogous way to Kx 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 the rotation generators which correspond to angular momentum, and K are the boost generators which correspond to the motion of the system in spacetime. The derivative of any smooth curve C(t) with C(0) = I in the group depending on some group parameter t with respect to that group parameter, evaluated at t = 0, serves as a definition of a corresponding group generator G, 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 map G smoothly back into the group via t → exp(tG) for all t; this curve will yield G again when differentiated at t = 0.
Expanding the exponentials in their Taylor series obtains
B ( ζ ) = I − sinh  ζ ( n ⋅ K ) + ( cosh  ζ − 1 ) ( n ⋅ K ) 2 {\displaystyle B({\boldsymbol {\zeta }})=I-\sinh \zeta (\mathbf {n} \cdot \mathbf {K} )+(\cosh \zeta -1)(\mathbf {n} \cdot \mathbf {K} )^{2}}
R ( θ ) = I + sin  θ ( e ⋅ J ) + ( 1 − cos  θ ) ( e ⋅ J ) 2 . {\displaystyle R({\boldsymbol {\theta }})=I+\sin \theta (\mathbf {e} \cdot \mathbf {J} )+(1-\cos \theta )(\mathbf {e} \cdot \mathbf {J} )^{2}\,.}
which 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 the infinitesimal level the product
Λ = ( I − ζ ⋅ K + ⋯ ) ( I + θ ⋅ J + ⋯ ) = ( I + θ ⋅ J + ⋯ ) ( I − ζ ⋅ K + ⋯ ) = I − ζ ⋅ K + θ ⋅ J + ⋯ {\displaystyle {\begin{aligned}\Lambda &=(I-{\boldsymbol {\zeta }}\cdot \mathbf {K} +\cdots )(I+{\boldsymbol {\theta }}\cdot \mathbf {J} +\cdots )\\&=(I+{\boldsymbol {\theta }}\cdot \mathbf {J} +\cdots )(I-{\boldsymbol {\zeta }}\cdot \mathbf {K} +\cdots )\\&=I-{\boldsymbol {\zeta }}\cdot \mathbf {K} +{\boldsymbol {\theta }}\cdot \mathbf {J} +\cdots \end{aligned}}}
is 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
Λ ( ζ , θ ) = e − ζ ⋅ K + θ ⋅ J . {\displaystyle \Lambda ({\boldsymbol {\zeta }},{\boldsymbol {\theta }})=e^{-{\boldsymbol {\zeta }}\cdot \mathbf {K} +{\boldsymbol {\theta }}\cdot \mathbf {J} }.}
The converse is also true, but the decomposition of a finite general Lorentz transformation into such factors is nontrivial. In particular,
e − ζ ⋅ K + θ ⋅ J ≠ e − ζ ⋅ K e θ ⋅ J , {\displaystyle e^{-{\boldsymbol {\zeta }}\cdot \mathbf {K} +{\boldsymbol {\theta }}\cdot \mathbf {J} }\neq e^{-{\boldsymbol {\zeta }}\cdot \mathbf {K} }e^{{\boldsymbol {\theta }}\cdot \mathbf {J} },}
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 rotation in principle (this usually does not yield an intelligible expression in terms of generators J and K), see Wigner 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 the Baker–Campbell–Hausdorff formula applies.
The Lie algebra so(3,1)[edit]
Lorentz generators can be added together, or multiplied by real numbers, to obtain more Lorentz generators. In other words, the set of all Lorentz generators
V = { ζ ⋅ K + θ ⋅ J } {\displaystyle V=\{{\boldsymbol {\zeta }}\cdot \mathbf {K} +{\boldsymbol {\theta }}\cdot \mathbf {J} \}}
together with the operations of ordinary matrix addition and multiplication of a matrix by a number, forms a vector space over the real numbers.[nb 7] The generators Jx, Jy, Jz, Kx, Ky, Kz form a basis set of V, and the components of the axis-angle and rapidity vectors, θx, θy, θz, ζx, ζy, ζz, are the coordinates of a Lorentz generator with respect to this basis.[nb 8]
Three of the commutation relations of the Lorentz generators are
[ J x , J y ] = J z , [ K x , K y ] = − J z , [ J x , K y ] = K z , {\displaystyle [J_{x},J_{y}]=J_{z}\,,\quad [K_{x},K_{y}]=-J_{z}\,,\quad [J_{x},K_{y}]=K_{z}\,,}
where the bracket [A, B] = AB − BA is known as the commutator, and the other relations can be found by taking cyclic permutations of x, y, z components (i.e. change x to y, y to z, and z to x, repeat).
These commutation relations, and the vector space of generators, fulfill the definition of the Lie algebra s o ( 3 , 1 ) {\displaystyle {\mathfrak {so}}(3,1)} . In summary, a Lie algebra is defined as a vector space V over a field of numbers, and with a binary operation [ , ] (called a Lie bracket in this context) on the elements of the vector space, satisfying the axioms of bilinearity, alternatization, and the Jacobi identity. Here the operation [ , ] is the commutator which satisfies all of these axioms, the vector space is the set of Lorentz generators V 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.
The exponential map (Lie theory) from the Lie algebra to the Lie group,
e x p : s o ( 3 , 1 ) → S O ( 3 , 1 ) , {\displaystyle \mathrm {exp} \,:\,{\mathfrak {so}}(3,1)\rightarrow \mathrm {SO} (3,1),}
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. It the case of the Lorentz group, the exponential map is just the matrix exponential. Globally, the exponential map is not one-to-one, but in the case of the Lorentz group, it is surjective (onto). Hence any group element can be expressed as an exponential of an element of the Lie algebra.
Improper transformations[edit]
Lorentz transformations also include parity inversion
P = [ 1 0 0 − I ] {\displaystyle P={\begin{bmatrix}1&0\\0&-\mathbf {I} \end{bmatrix}}}
which negates all the spatial coordinates only, and time reversal
T = [ − 1 0 0 I ] {\displaystyle T={\begin{bmatrix}-1&0\\0&\mathbf {I} \end{bmatrix}}}
which negates the time coordinate only, because these transformations leave the spacetime interval invariant. Here I is the 3d identity matrix. These are both symmetric, they are their own inverses (see involution (mathematics)), and each have determinant −1. This latter property makes them improper transformations.
If Λ is a proper orthochronous Lorentz transformation, then TΛ is improper antichronous, PΛ is improper orthochronous, and TPΛ = PTΛ is proper antichronous.
Inhomogeneous Lorentz group[edit]
Two other spacetime symmetries have not been accounted for. For the spacetime interval to be invariant, it can be shown[14] that it is necessary and sufficient for the coordinate transformation to be of the form
X ′ = Λ X + C {\displaystyle X'=\Lambda X+C}
where C is a constant column containing translations in time and space. If C ≠ 0, this is an inhomogeneous Lorentz transformation or Poincaré transformation.[15][16] If C = 0, this is a homogeneous Lorentz transformation. Poincaré transformations are not dealt further in this article.
Tensor formulation[edit]
Main article: Representation theory of the Lorentz group
For the notation used, see Ricci calculus.
Contravariant vectors[edit]
Writing the general matrix transformation of coordinates as the matrix equation
[ x ′ 0 x ′ 1 x ′ 2 x ′ 3 ] = [ Λ 0 0 Λ 0 1 Λ 0 2 Λ 0 3 Λ 1 0 Λ 1 1 Λ 1 2 Λ 1 3 Λ 2 0 Λ 2 1 Λ 2 2 Λ 2 3 Λ 3 0 Λ 3 1 Λ 3 2 Λ 3 3 ] [ x 0 x 1 x 2 x 3 ] {\displaystyle {\begin{bmatrix}{x'}^{0}\\{x'}^{1}\\{x'}^{2}\\{x'}^{3}\end{bmatrix}}={\begin{bmatrix}{\Lambda ^{0}}_{0}&{\Lambda ^{0}}_{1}&{\Lambda ^{0}}_{2}&{\Lambda ^{0}}_{3}\\{\Lambda ^{1}}_{0}&{\Lambda ^{1}}_{1}&{\Lambda ^{1}}_{2}&{\Lambda ^{1}}_{3}\\{\Lambda ^{2}}_{0}&{\Lambda ^{2}}_{1}&{\Lambda ^{2}}_{2}&{\Lambda ^{2}}_{3}\\{\Lambda ^{3}}_{0}&{\Lambda ^{3}}_{1}&{\Lambda ^{3}}_{2}&{\Lambda ^{3}}_{3}\\\end{bmatrix}}{\begin{bmatrix}x^{0}\\x^{1}\\x^{2}\\x^{3}\end{bmatrix}}}
allows the transformation of other physical quantities that cannot be expressed as four-vectors; e.g., tensors or spinors of any order in 4d spacetime, to be defined. In the corresponding tensor index notation, the above matrix expression is
x ′ ν = Λ ν μ x μ , {\displaystyle {x^{\prime }}^{\nu }={\Lambda ^{\nu }}_{\mu }x^{\mu },}
where lower and upper indices label covariant and contravariant components respectively,[17] and the summation convention is applied. It is a standard convention to use Greek indices that take the value 0 for time components, and 1, 2, 3 for space components, while Latin indices simply take the values 1, 2, 3, for spatial components. Note that the first index (reading left to right) corresponds in the matrix notation to a row index. The second index corresponds to the column index.
The transformation matrix is universal for all four-vectors, not just 4-dimensional spacetime coordinates. If A is any four-vector, then in tensor index notation
A ′ ν = Λ ν μ A μ . {\displaystyle {A^{\prime }}^{\nu }={\Lambda ^{\nu }}_{\mu }A^{\mu }\,.}
Alternatively, one writes
A ν ′ = Λ ν ′ μ A μ . {\displaystyle A^{\nu '}={\Lambda ^{\nu '}}_{\mu }A^{\mu }\,.}
in which the primed indices denote the indices of A in the primed frame. This notation cuts risk of exhausting the Greek alphabet roughly in half.
For a general n-component object one may write
X ′ α = Π ( Λ ) α β X β , {\displaystyle {X'}^{\alpha }={\Pi (\Lambda )^{\alpha }}_{\beta }X^{\beta }\,,}
where Π is the appropriate representation of the Lorentz group, an n×n matrix for every Λ. In this case, the indices should not be thought of as spacetime indices (sometimes called Lorentz indices), and they run from 1 to n. E.g., if X is a bispinor, then the indices are called Dirac indices.
Covariant vectors[edit]
There are also vector quantities with covariant indices. They are generally obtained from their corresponding objects with contravariant indices by the operation of lowering an index; e.g.,
x ν = η μ ν x μ , {\displaystyle x_{\nu }=\eta _{\mu \nu }x^{\mu },}
where η is the metric 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 by
x μ = η ν μ x ν , {\displaystyle x^{\mu }=\eta ^{\nu \mu }x_{\nu },}
where, when viewed as matrices, ημν is the inverse of ημν. As it happens, ημν = ημν. This is referred to as raising an index. To transform a covariant vector Aμ, first raise its index, then transform it according to the same rule as for contravariant 4-vectors, then finally lower the index;
A ′ ν = η ρ ν Λ ρ σ η μ σ A μ . {\displaystyle {A'}_{\nu }=\eta _{\rho \nu }{\Lambda ^{\rho }}_{\sigma }\eta ^{\mu \sigma }A_{\mu }.}
But
η ρ ν Λ ρ σ η μ σ = ( Λ − 1 ) μ ν , {\displaystyle \eta _{\rho \nu }{\Lambda ^{\rho }}_{\sigma }\eta ^{\mu \sigma }={\left(\Lambda ^{-1}\right)^{\mu }}_{\nu },}
I. e., it is the (μ, ν)-component of the inverse Lorentz transformation. One defines (as a matter of notation),
Λ ν μ ≡ ( Λ − 1 ) μ ν , {\displaystyle {\Lambda _{\nu }}^{\mu }\equiv {\left(\Lambda ^{-1}\right)^{\mu }}_{\nu },}
and may in this notation write
A ′ ν = Λ ν μ A μ . {\displaystyle {A'}_{\nu }={\Lambda _{\nu }}^{\mu }A_{\mu }.}
Now for a subtlety. The implied summation on the right hand side of
A ′ ν = Λ ν μ A μ = ( Λ − 1 ) μ ν A μ {\displaystyle {A'}_{\nu }={\Lambda _{\nu }}^{\mu }A_{\mu }={\left(\Lambda ^{-1}\right)^{\mu }}_{\nu }A_{\mu }}
is running over a row index of the matrix representing Λ−1. Thus, in terms of matrices, this transformation should be thought of as the inverse transpose of Λ acting on the column vector Aμ. That is, in pure matrix notation,
A ′ = ( Λ − 1 ) T A . {\displaystyle A'=\left(\Lambda ^{-1}\right)^{\mathrm {T} }A.}
This means exactly that covariant vectors (thought of as column matrices) transform according to the dual representation of the standard representation of the Lorentz group. This notion generalizes to general representations, simply replace Λ with Π(Λ).
Tensors[edit]
If A and B are linear operators on vector spaces U and V, then a linear operator A ⊗ B may be defined on the tensor product of U and V, denoted U ⊗ V according to[18]
( A ⊗ B ) ( u ⊗ v ) = A u ⊗ B v , u ∈ U , v ∈ V , u ⊗ v ∈ U ⊗ V . {\displaystyle (A\otimes B)(u\otimes v)=Au\otimes Bv,\qquad u\in U,v\in V,u\otimes v\in U\otimes V.} (T1)
From this it is immediately clear that if u and v are a four-vectors in V, then u ⊗ v ∈ T2V ≡ V ⊗ V transforms as
u ⊗ v → Λ u ⊗ Λ v = Λ μ ν u ν ⊗ Λ ρ σ v σ = Λ μ ν Λ ρ σ u ν ⊗ v σ ≡ Λ μ ν Λ ρ σ w ν σ . {\displaystyle u\otimes v\rightarrow \Lambda u\otimes \Lambda v={\Lambda ^{\mu }}_{\nu }u^{\nu }\otimes {\Lambda ^{\rho }}_{\sigma }v^{\sigma }={\Lambda ^{\mu }}_{\nu }{\Lambda ^{\rho }}_{\sigma }u^{\nu }\otimes v^{\sigma }\equiv {\Lambda ^{\mu }}_{\nu }{\Lambda ^{\rho }}_{\sigma }w^{\nu \sigma }.} (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 tensor u ⊗ v.
These observations generalize in an obvious way to more factors, and using the fact that a general tensor on a vector space V 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 any tensor quantity T. It is given by[19]
T θ ′ ι ′ ⋯ κ ′ α ′ β ′ ⋯ ζ ′ = Λ α ′ μ Λ β ′ ν ⋯ Λ ζ ′ ρ Λ θ ′ σ Λ ι ′ υ ⋯ Λ κ ′ ζ T σ υ ⋯ ζ μ ν ⋯ ρ , {\displaystyle T_{\theta '\iota '\cdots \kappa '}^{\alpha '\beta '\cdots \zeta '}={\Lambda ^{\alpha '}}_{\mu }{\Lambda ^{\beta '}}_{\nu }\cdots {\Lambda ^{\zeta '}}_{\rho }{\Lambda _{\theta '}}^{\sigma }{\Lambda _{\iota '}}^{\upsilon }\cdots {\Lambda _{\kappa '}}^{\zeta }T_{\sigma \upsilon \cdots \zeta }^{\mu \nu \cdots \rho },} (T3)
where Λχ′ψ is defined above. This form can generally be reduced to the form for general n-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.
Transformation of the electromagnetic field[edit]
Lorentz boost of an electric charge, the charge is at rest in one frame or the other.
Main article: Electromagnetic tensor
Further information: classical electromagnetism and special relativity
Lorentz transformations can also be used to illustrate that the magnetic field B and electric field E are simply different aspects of the same force — the electromagnetic force, as a consequence of relative motion between electric charges and observers.[20] The fact that the electromagnetic field shows relativistic effects becomes clear by carrying out a simple thought experiment.[21]
An observer measures a charge at rest in frame F. 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 frame F′ moves at velocity v relative to F 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 an electric current, and thus the observer in frame F′ 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 by
F μ ν = [ 0 − 1 c E x − 1 c E y − 1 c E z 1 c E x 0 − B z B y 1 c E y B z 0 − B x 1 c E z − B y B x 0 ] (SI units, signature ( + , − , − , − ) ) . {\displaystyle F^{\mu \nu }={\begin{bmatrix}0&-{\frac {1}{c}}E_{x}&-{\frac {1}{c}}E_{y}&-{\frac {1}{c}}E_{z}\\{\frac {1}{c}}E_{x}&0&-B_{z}&B_{y}\\{\frac {1}{c}}E_{y}&B_{z}&0&-B_{x}\\{\frac {1}{c}}E_{z}&-B_{y}&B_{x}&0\end{bmatrix}}{\text{(SI units, signature }}(+,-,-,-){\text{)}}.}
in SI units. In relativity, the Gaussian system of units is often preferred over SI units, even in texts whose main choice of units is SI units, because in it the electric field E and the magnetic induction B have the same units making the appearance of the electromagnetic field tensor more natural.[22] Consider a Lorentz boost in the x-direction. It is given by[23]
Λ μ ν = [ γ − γ β 0 0 − γ β γ 0 0 0 0 1 0 0 0 0 1 ] , F μ ν = [ 0 E x E y E z − E x 0 B z − B y − E y − B z 0 B x − E z B y − B x 0 ] (Gaussian units, signature ( − , + , + , + ) ) , {\displaystyle {\Lambda ^{\mu }}_{\nu }={\begin{bmatrix}\gamma &-\gamma \beta &0&0\\-\gamma \beta &\gamma &0&0\\0&0&1&0\\0&0&0&1\\\end{bmatrix}},\qquad F^{\mu \nu }={\begin{bmatrix}0&E_{x}&E_{y}&E_{z}\\-E_{x}&0&B_{z}&-B_{y}\\-E_{y}&-B_{z}&0&B_{x}\\-E_{z}&B_{y}&-B_{x}&0\end{bmatrix}}{\text{(Gaussian units, signature }}(-,+,+,+){\text{)}},}
where the field tensor is displayed side by side for easiest possible reference in the manipulations below.
The general transformation law (T3) becomes
F μ ′ ν ′ = Λ μ ′ μ Λ ν ′ ν F μ ν . {\displaystyle F^{\mu '\nu '}={\Lambda ^{\mu '}}_{\mu }{\Lambda ^{\nu '}}_{\nu }F^{\mu \nu }.}
For the magnetic field one obtains
B x ′ = F 2 ′ 3 ′ = Λ 2 μ Λ 3 ν F μ ν = Λ 2 2 Λ 3 3 F 23 = 1 × 1 × B x = B x , B y ′ = F 3 ′ 1 ′ = Λ 3 μ Λ 1 ν F μ ν = Λ 3 3 Λ 1 ν F 3 ν = Λ 3 3 Λ 1 0 F 30 + Λ 3 3 Λ 1 1 F 31 = 1 × ( − β γ ) ( − E z ) + 1 × γ B y = γ B y + β γ E z = γ ( B − β × E ) y B z ′ = F 1 ′ 2 ′ = Λ 1 μ Λ 2 ν F μ ν = Λ 1 μ Λ 2 2 F μ 2 = Λ 1 0 Λ 2 2 F 02 + Λ 1 1 Λ 2 2 F 12 = ( − γ β ) × 1 × E y + γ × 1 × B z = γ B z − β γ E y = γ ( B − β × E ) z {\displaystyle {\begin{aligned}B_{x'}&=F^{2'3'}={\Lambda ^{2}}_{\mu }{\Lambda ^{3}}_{\nu }F^{\mu \nu }={\Lambda ^{2}}_{2}{\Lambda ^{3}}_{3}F^{23}=1\times 1\times B_{x}\\&=B_{x},\\B_{y'}&=F^{3'1'}={\Lambda ^{3}}_{\mu }{\Lambda ^{1}}_{\nu }F^{\mu \nu }={\Lambda ^{3}}_{3}{\Lambda ^{1}}_{\nu }F^{3\nu }={\Lambda ^{3}}_{3}{\Lambda ^{1}}_{0}F^{30}+{\Lambda ^{3}}_{3}{\Lambda ^{1}}_{1}F^{31}\\&=1\times (-\beta \gamma )(-E_{z})+1\times \gamma B_{y}=\gamma B_{y}+\beta \gamma E_{z}\\&=\gamma \left(\mathbf {B} -{\boldsymbol {\beta }}\times \mathbf {E} \right)_{y}\\B_{z'}&=F^{1'2'}={\Lambda ^{1}}_{\mu }{\Lambda ^{2}}_{\nu }F^{\mu \nu }={\Lambda ^{1}}_{\mu }{\Lambda ^{2}}_{2}F^{\mu 2}={\Lambda ^{1}}_{0}{\Lambda ^{2}}_{2}F^{02}+{\Lambda ^{1}}_{1}{\Lambda ^{2}}_{2}F^{12}\\&=(-\gamma \beta )\times 1\times E_{y}+\gamma \times 1\times B_{z}=\gamma B_{z}-\beta \gamma E_{y}\\&=\gamma \left(\mathbf {B} -{\boldsymbol {\beta }}\times \mathbf {E} \right)_{z}\end{aligned}}}
For the electric field results
E x ′ = F 0 ′ 1 ′ = Λ 0 μ Λ 1 ν F μ ν = Λ 0 1 Λ 1 0 F 10 + Λ 0 0 Λ 1 1 F 01 = ( − γ β ) ( − γ β ) ( − E x ) + γ γ E x = − γ 2 β 2 ( E x ) + γ 2 E x = E x ( 1 − β 2 ) γ 2 = E x , E y ′ = F 0 ′ 2 ′ = Λ 0 μ Λ 2 ν F μ ν = Λ 0 μ Λ 2 2 F μ 2 = Λ 0 0 Λ 2 2 F 02 + Λ 0 1 Λ 2 2 F 12 = γ × 1 × E y + ( − β γ ) × 1 × B z = γ E y − β γ B z = γ ( E + β × B ) y E z ′ = F 0 ′ 3 ′ = Λ 0 μ Λ 3 ν F μ ν = Λ 0 μ Λ 3 3 F μ 3 = Λ 0 0 Λ 3 3 F 03 + Λ 0 1 Λ 3 3 F 13 = γ × 1 × E z − β γ × 1 × ( − B y ) = γ E z + β γ B y = γ ( E + β × B ) z . {\displaystyle {\begin{aligned}E_{x'}&=F^{0'1'}={\Lambda ^{0}}_{\mu }{\Lambda ^{1}}_{\nu }F^{\mu \nu }={\Lambda ^{0}}_{1}{\Lambda ^{1}}_{0}F^{10}+{\Lambda ^{0}}_{0}{\Lambda ^{1}}_{1}F^{01}\\&=(-\gamma \beta )(-\gamma \beta )(-E_{x})+\gamma \gamma E_{x}=-\gamma ^{2}\beta ^{2}(E_{x})+\gamma ^{2}E_{x}=E_{x}(1-\beta ^{2})\gamma ^{2}\\&=E_{x},\\E_{y'}&=F^{0'2'}={\Lambda ^{0}}_{\mu }{\Lambda ^{2}}_{\nu }F^{\mu \nu }={\Lambda ^{0}}_{\mu }{\Lambda ^{2}}_{2}F^{\mu 2}={\Lambda ^{0}}_{0}{\Lambda ^{2}}_{2}F^{02}+{\Lambda ^{0}}_{1}{\Lambda ^{2}}_{2}F^{12}\\&=\gamma \times 1\times E_{y}+(-\beta \gamma )\times 1\times B_{z}=\gamma E_{y}-\beta \gamma B_{z}\\&=\gamma \left(\mathbf {E} +{\boldsymbol {\beta }}\times \mathbf {B} \right)_{y}\\E_{z'}&=F^{0'3'}={\Lambda ^{0}}_{\mu }{\Lambda ^{3}}_{\nu }F^{\mu \nu }={\Lambda ^{0}}_{\mu }{\Lambda ^{3}}_{3}F^{\mu 3}={\Lambda ^{0}}_{0}{\Lambda ^{3}}_{3}F^{03}+{\Lambda ^{0}}_{1}{\Lambda ^{3}}_{3}F^{13}\\&=\gamma \times 1\times E_{z}-\beta \gamma \times 1\times (-B_{y})=\gamma E_{z}+\beta \gamma B_{y}\\&=\gamma \left(\mathbf {E} +{\boldsymbol {\beta }}\times \mathbf {B} \right)_{z}.\end{aligned}}}
Here, β = (β, 0, 0) is used. These results can be summarized by
E ∥ ′ = E ∥ B ∥ ′ = B ∥ E ⊥ ′ = γ ( E ⊥ + β × B ⊥ ) = γ ( E + β × B ) ⊥ , B ⊥ ′ = γ ( B ⊥ − β × E ⊥ ) = γ ( B − β × E ) ⊥ , {\displaystyle {\begin{aligned}\mathbf {E} _{\parallel '}&=\mathbf {E} _{\parallel }\\\mathbf {B} _{\parallel '}&=\mathbf {B} _{\parallel }\\\mathbf {E} _{\bot '}&=\gamma \left(\mathbf {E} _{\bot }+{\boldsymbol {\beta }}\times \mathbf {B} _{\bot }\right)=\gamma \left(\mathbf {E} +{\boldsymbol {\beta }}\times \mathbf {B} \right)_{\bot },\\\mathbf {B} _{\bot '}&=\gamma \left(\mathbf {B} _{\bot }-{\boldsymbol {\beta }}\times \mathbf {E} _{\bot }\right)=\gamma \left(\mathbf {B} -{\boldsymbol {\beta }}\times \mathbf {E} \right)_{\bot },\end{aligned}}}
and are independent of the metric signature. For SI units, substitute E → E⁄c. Misner, Thorne & Wheeler (1973) refer to this last form as the 3 + 1 view as opposed to the geometric view represented by the tensor expression
F μ ′ ν ′ = Λ μ ′ μ Λ ν ′ ν F μ ν , {\displaystyle F^{\mu '\nu '}={\Lambda ^{\mu '}}_{\mu }{\Lambda ^{\nu '}}_{\nu }F^{\mu \nu },}
and make a strong point of the ease with which results that are difficult to achieve using the 3 + 1 view can be obtained and understood. Only objects that have well defined Lorentz transformation properties (in fact under any smooth coordinate transformation) are geometric objects. In the geometric view, the electromagnetic field is a six-dimensional geometric object in spacetime as opposed to two interdependent, but separate, 3-vector fields in space and time. The fields E (alone) and B (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 the same event in spacetime. Thus the complete equation with spacetime dependence is
F μ ′ ν ′ ( x ′ ) = Λ μ ′ μ Λ ν ′ ν F μ ν ( Λ − 1 x ′ ) = Λ μ ′ μ Λ ν ′ ν F μ ν ( x ) . {\displaystyle F^{\mu '\nu '}\left(x'\right)={\Lambda ^{\mu '}}_{\mu }{\Lambda ^{\nu '}}_{\nu }F^{\mu \nu }\left(\Lambda ^{-1}x'\right)={\Lambda ^{\mu '}}_{\mu }{\Lambda ^{\nu '}}_{\nu }F^{\mu \nu }(x).}
Length contraction has an effect on charge density ρ and current density J, 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,
j ′ = j − γ ρ v n + ( γ − 1 ) ( j ⋅ n ) n ρ ′ = γ ( ρ − j ⋅ v n c 2 ) , {\displaystyle {\begin{aligned}\mathbf {j} '&=\mathbf {j} -\gamma \rho v\mathbf {n} +\left(\gamma -1\right)(\mathbf {j} \cdot \mathbf {n} )\mathbf {n} \\\rho '&=\gamma \left(\rho -\mathbf {j} \cdot {\frac {v\mathbf {n} }{c^{2}}}\right),\end{aligned}}}
or, in the simpler geometric view,
j μ ′ = Λ μ ′ μ j μ . {\displaystyle j^{\mu ^{\prime }}={\Lambda ^{\mu '}}_{\mu }j^{\mu }.}
One says that charge density transforms as the time component of a four-vector. It is a rotational scalar. The current density is a 3-vector.
The Maxwell equations are invariant under Lorentz transformations.
Spinors[edit]
Equation (T1) hold unmodified for any representation of the Lorentz group, including the bispinor representation. In (T2) one simply replaces all occurrences of Λ by the bispinor representation Π(Λ),
u ⊗ v → Π ( Λ ) u ⊗ Π ( Λ ) v = Π ( Λ ) α β u β ⊗ Π ( Λ ) ρ σ v σ = Π ( Λ ) α β Π ( Λ ) ρ σ u β ⊗ v σ ≡ Π ( Λ ) α β Π ( Λ ) ρ σ w α β . {\displaystyle u\otimes v\rightarrow \Pi (\Lambda )u\otimes \Pi (\Lambda )v={\Pi (\Lambda )^{\alpha }}_{\beta }u^{\beta }\otimes {\Pi (\Lambda )^{\rho }}_{\sigma }v^{\sigma }={\Pi (\Lambda )^{\alpha }}_{\beta }{\Pi (\Lambda )^{\rho }}_{\sigma }u^{\beta }\otimes v^{\sigma }\equiv {\Pi (\Lambda )^{\alpha }}_{\beta }{\Pi (\Lambda )^{\rho }}_{\sigma }w^{\alpha \beta }.} (T4)
The above equation could, for instance, be the transformation of a state in Fock space describing two free electrons.
Transformation of general fields[edit]
A general noninteracting multi-particle state (Fock space state) in quantum field theory transforms according to the rule[24]
U ( Λ , a ) Ψ p 1 σ 1 n 1 ; p 2 σ 2 n 2 ; ⋯ = e − i a μ [ ( Λ p 1 ) μ + ( Λ p 2 ) μ + ⋯ ] ( Λ p 1 ) 0 ( Λ p 2 ) 0 ⋯ p 1 0 p 2 0 ⋯ ( ∑ σ 1 ′ σ 2 ′ ⋯ D σ 1 ′ σ 1 ( j 1 ) [ W ( Λ , p 1 ) ] D σ 2 ′ σ 2 ( j 2 ) [ W ( Λ , p 2 ) ] ⋯ ) Ψ Λ p 1 σ 1 ′ n 1 ; Λ p 2 σ 2 ′ n 2 ; ⋯ , {\displaystyle {\begin{aligned}&U(\Lambda ,a)\Psi _{p_{1}\sigma _{1}n_{1};p_{2}\sigma _{2}n_{2};\cdots }\\={}&e^{-ia_{\mu }\left[(\Lambda p_{1})^{\mu }+(\Lambda p_{2})^{\mu }+\cdots \right]}{\sqrt {\frac {(\Lambda p_{1})^{0}(\Lambda p_{2})^{0}\cdots }{p_{1}^{0}p_{2}^{0}\cdots }}}\left(\sum _{\sigma _{1}'\sigma _{2}'\cdots }D_{\sigma _{1}'\sigma _{1}}^{(j_{1})}\left[W(\Lambda ,p_{1})\right]D_{\sigma _{2}'\sigma _{2}}^{(j_{2})}\left[W(\Lambda ,p_{2})\right]\cdots \right)\Psi _{\Lambda p_{1}\sigma _{1}'n_{1};\Lambda p_{2}\sigma _{2}'n_{2};\cdots },\end{aligned}}}
( 1 )
where W(Λ, p) is the Wigner rotation and D(j) is the (2j + 1)-dimensional representation of SO(3).
See also[edit]
Ricci calculus
Electromagnetic field
Galilean transformation
Hyperbolic rotation
Invariance mechanics
Lorentz group
Representation theory of the Lorentz group
Principle of relativity
Velocity-addition formula
Algebra of physical space
Relativistic aberration
Prandtl–Glauert transformation
Split-complex number
Gyrovector space
Footnotes[edit]
Jump up ^ One can imagine that in each inertial frame there are observers positioned throughout space, each endowed with a synchronized clock and at rest in the particular inertial frame. These observers then report to a central office, where a report is collected. When one speaks of a particular observer, one refers to someone having, at least in principle, a copy of this report. See, e.g., Sard (1970).
Jump up ^ It should be noted that 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 the Poincare group or the inhomogeneous Lorentz group. The first equation (or the second restricted to lightlike separation) leads to a yet larger group, the conformal group of spacetime.
Jump up ^ The groups O(3, 1) and O(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 to O(3, 1) and O(1, 3) respectively, e.g., the Clifford algebras corresponding to the different signatures of the bilinear form associated to the two groups, are non-isomorphic.
Jump up ^ For two square matrices A and B, det(AB) = det(A)det(B)
Jump up ^ Explicitly,
ζ ⋅ K = ζ x K x + ζ y K y + ζ z K z {\displaystyle {\boldsymbol {\zeta }}\cdot \mathbf {K} =\zeta _{x}K_{x}+\zeta _{y}K_{y}+\zeta _{z}K_{z}}
θ ⋅ J = θ x J x + θ y J y + θ z J z {\displaystyle {\boldsymbol {\theta }}\cdot \mathbf {J} =\theta _{x}J_{x}+\theta _{y}J_{y}+\theta _{z}J_{z}}
Jump up ^ In quantum mechanics, relativistic quantum mechanics, and quantum field theory, a different convention is used for these matrices; the right hand sides are all multiplied by a factor of the imaginary unit i = √−1.
Jump up ^ Until now the term "vector" has exclusively referred to "Euclidean vector", examples are position r, velocity v, etc. The term "vector" applies much more broadly than Euclidean vectors, row or column vectors, etc., see linear algebra and vector space for details. The generators of a Lie group also form a vector space over a field of numbers (e.g. real numbers, complex numbers), since a linear combination of the generators is also a generator. They just live in a different space to the position vectors in ordinary 3d space.
Jump up ^ In ordinary 3d position space, the position vector r = xex + yey + zez is expressed as a linear combination of the Cartesian unit vectors ex, ey, ez which form a basis, and the Cartesian coordinates x, y, z are coordinates with respect to this basis.
Notes[edit]
Jump up ^ Lorentz, Hendrik Antoon (1904), "Electromagnetic phenomena in a system moving with any velocity smaller than that of light", Proceedings of the Royal Netherlands Academy of Arts and Sciences, 6: 809–831
Jump up ^ John & O'Connor 1996
Jump up ^ Brown 2003
Jump up ^ Rothman 2006, pp. 112f.
Jump up ^ Darrigol 2005, pp. 1–22
Jump up ^ Macrossan 1986, pp. 232–34
Jump up ^ The reference is within the following paper:Poincaré 1905, pp. 1504–1508
Jump up ^ Einstein 1905, pp. 891–921
Jump up ^ Young & Freedman 2008
Jump up ^ Forshaw & Smith 2009
Jump up ^ Einstein 1916
Jump up ^ Barut 1964, p. 18–19
Jump up ^ Chaichian & Hagedorn 1997, p. 239
Jump up ^ Weinberg 1972
Jump up ^ Weinberg 2005, pp. 55–58
Jump up ^ Ohlsson 2011, p. 3–9
Jump up ^ Dennery, Philippe; Krzywicki, André (2012). Mathematics for Physicists. Courier Corporation. p. 138. ISBN 978-0-486-15712-2. Extract of page 138
Jump up ^ Hall 2003, Chapter 4
Jump up ^ Carroll 2004, p. 22
Jump up ^ Grant & Phillips 2008
Jump up ^ Griffiths 2007
Jump up ^ Jackson 1999
Jump up ^ Misner, Thorne & Wheeler 1973
Jump up ^ Weinberg 2002, Chapter 3
References[edit]
Websites[edit]
O'Connor, John J.; Robertson, Edmund F. (1996), A History of Special Relativity
Brown, Harvey R. (2003), Michelson, FitzGerald and Lorentz: the Origins of Relativity Revisited
Papers[edit]
Cushing, J. T. (1967). "Vector Lorentz transformations". American Journal of Physics. 35: 858–862. Bibcode:1967AmJPh..35..858C. doi:10.1119/1.1974267.
Macfarlane, A. J. (1962). "On the Restricted Lorentz Group and Groups Homomorphically Related to It". Journal of Mathematical Physics. 3 (6): 1116–1129. Bibcode:1962JMP.....3.1116M. doi:10.1063/1.1703854.
Rothman, Tony (2006), "Lost in Einstein's Shadow" (PDF), American Scientist, 94 (2): 112f.
Darrigol, Olivier (2005), "The Genesis of the theory of relativity" (PDF), Séminaire Poincaré, 1: 1–22, doi:10.1007/3-7643-7436-5_1
Macrossan, Michael N. (1986), "A Note on Relativity Before Einstein", Brit. Journal Philos. Science, 37: 232–34, CiteSeerX 10.1.1.679.5898  , doi:10.1093/bjps/37.2.232
Poincaré, Henri (1905), "On the Dynamics of the Electron", Comptes rendus hebdomadaires des séances de l'Académie des sciences, 140: 1504–1508
Einstein, Albert (1905), "Zur Elektrodynamik bewegter Körper" (PDF), Annalen der Physik, 322 (10): 891–921, Bibcode:1905AnP...322..891E, doi:10.1002/andp.19053221004 . See also: English translation.
Einstein, A. (1916). "Relativity: The Special and General Theory" (PDF). Retrieved 2012-01-23.
Ungar, A. A. (1988). "Thomas rotation and the parameterization of the Lorentz transformation group". Foundations of Physics Letters. Kluwer Academic Publishers-Plenum Publishers. 1 (1): 55–89. Bibcode:1988FoPhL...1...57U. doi:10.1007/BF00661317. ISSN 0894-9875. (Subscription required (help)). eqn (55).
Ungar, A. A. (1989). "The relativistic velocity composition paradox and the Thomas rotation". Foundations of Physics. 19: 1385–1396. Bibcode:1989FoPh...19.1385U. doi:10.1007/BF00732759.
Ungar, A. A. (2000). "The relativistic composite-velocity reciprocity principle". Foundations of Physics. Springer. 30 (2): 331–342. CiteSeerX 10.1.1.35.1131  .
Mocanu, C. I. (1986). "Some difficulties within the framework of relativistic electrodynamics". Archiv für Elektrotechnik. Springer. 69: 97–110. doi:10.1007/bf01574845.
Mocanu, C. I. (1992). "On the relativistic velocity composition paradox and the Thomas rotation". Foundations of Physics. Plenum. 5: 443–456. Bibcode:1992FoPhL...5..443M. doi:10.1007/bf00690425.
Weinberg, S. (2002). The Quantum Theory of Fields, vol I. Cambridge University Press. ISBN 0-521-55001-7.
Books[edit]
Young, H. D.; Freedman, R. A. (2008). University Physics – With Modern Physics (12th ed.). ISBN 0-321-50130-6.
Halpern, A. (1988). 3000 Solved Problems in Physics. Schaum Series. Mc Graw Hill. p. 688. ISBN 978-0-07-025734-4.
Forshaw, J. R.; Smith, A. G. (2009). Dynamics and Relativity. Manchester Physics Series. John Wiley & Sons Ltd. pp. 124–126. ISBN 978-0-470-01460-8.
Wheeler, J. A.; Taylor, E. F (1971). Spacetime Physics. Freeman. ISBN 0-7167-0336-X.
Wheeler, J. A.; Thorne, K. S.; Misner, C. W. (1973). Gravitation. Freeman. ISBN 0-7167-0344-0.
Carroll, S. M. (2004). Spacetime and Geometry: An Introduction to General Relativity (illustrated ed.). Addison Wesley. p. 22. ISBN 0-8053-8732-3.
Grant, I. S.; Phillips, W. R. (2008). "14". Electromagnetism. Manchester Physics (2nd ed.). John Wiley & Sons. ISBN 0-471-92712-0.
Griffiths, D. J. (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley,. ISBN 81-7758-293-3.
Hall, Brian C. (2003). Lie Groups, Lie Algebras, and Representations An Elementary Introduction. Springer Publishing. ISBN 0-387-40122-9.
Weinberg, S. (2008), Cosmology, Wiley, ISBN 978-0-19-852682-7
Weinberg, S. (2005), The quantum theory of fields (3 vol.), 1, Cambridge University Press, ISBN 978-0-521-67053-1
Ohlsson, T. (2011), Relativistic Quantum Physics, Cambridge University Press, ISBN 978-0-521-76726-2
Goldstein, H. (1980) [1950]. Classical Mechanics (2nd ed.). Reading MA: Addison-Wesley. ISBN 0-201-02918-9.
Jackson, J. D. (1975) [1962]. "Chapter 11". Classical Electrodynamics (2nd ed.). John Wiley & Sons. pp. 542–545. ISBN 0-471-43132-X.
Landau, L. D.; Lifshitz, E. M. (2002) [1939]. The Classical Theory of Fields. Course of Theoretical Physics. 2 (4th ed.). Butterworth–Heinemann. pp. 9–12. ISBN 0 7506 2768 9.
Feynman, R. P.; Leighton, R. B.; Sands, M. (1977) [1963]. "15". The Feynman Lectures on Physics. 1. Addison Wesley. ISBN 0-201-02117-X.
Feynman, R. P.; Leighton, R. B.; Sands, M. (1977) [1964]. "13". The Feynman Lectures on Physics. 2. Addison Wesley. ISBN 0-201-02117-X.
Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 978-0-7167-0344-0.
Rindler, W. (2006) [2001]. "Chapter 9". Relativity Special, General and Cosmological (2nd ed.). Dallas: Oxford University Press. ISBN 978-0-19-856732-5.
Ryder, L. H. (1996) [1985]. Quantum Field Theory (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-0521478144.
Sard, R. D. (1970). Relativistic Mechanics - Special Relativity and Classical Particle Dynamics. New York: W. A. Benjamin. ISBN 978-0805384918.
R. U. Sexl, H. K. Urbantke (2001) [1992]. Relativity, Groups Particles. Special Relativity and Relativistic Symmetry in Field and Particle Physics. Springer. ISBN 978-3211834435.
Gourgoulhon, Eric (2013). Special Relativity in General Frames: From Particles to Astrophysics. Springer. p. 213. ISBN 978-3-642-37276-6.
Chaichian, Masud; Hagedorn, Rolf (1997). Symmetry in quantum mechanics:From angular momentum to supersymmetry. IoP. p. 239. ISBN 0-7503-0408-1.
Landau, L.D.; Lifshitz, E.M. (2002) [1939]. The Classical Theory of Fields. Course of Theoretical Physics. 2 (4th ed.). Butterworth–Heinemann. ISBN 0 7506 2768 9.
Further reading[edit]
Einstein, Albert (1961), Relativity: The Special and the General Theory, New York: Three Rivers Press (published 1995), ISBN 0-517-88441-0
Ernst, A.; Hsu, J.-P. (2001), "First proposal of the universal speed of light by Voigt 1887" (PDF), Chinese Journal of Physics, 39 (3): 211–230, Bibcode:2001ChJPh..39..211E, archived from the original (PDF) on 2011-07-16
Thornton, Stephen T.; Marion, Jerry B. (2004), Classical dynamics of particles and systems (5th ed.), Belmont, [CA.]: Brooks/Cole, pp. 546–579, ISBN 0-534-40896-6
Voigt, Woldemar (1887), "Über das Doppler'sche princip", Nachrichten von der Königlicher Gesellschaft den Wissenschaft zu Göttingen, 2: 41–51
External links[edit]
Wikisource has original works on the topic: Relativity
Wikibooks has a book on the topic of: special relativity
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.
Relativity – a chapter from an online textbook
Warp Special Relativity Simulator. A computer program demonstrating the Lorentz transformations on everyday objects.
Animation clip on YouTube visualizing the Lorentz transformation.
Lorentz Frames Animated from John de Pillis. Online Flash animations of Galilean and Lorentz frames, various paradoxes, EM wave phenomena, etc.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Lorentz_transformation&oldid=804386957"
Categories:
Special relativity
Theoretical physics
Mathematical physics
Spacetime
Coordinate systems
Hendrik Lorentz
Hidden categories:
Pages containing links to subscription-only content
Navigation menu
Personal tools
Not logged in
Talk
Contributions
Create account
Log in
Namespaces
Article
Talk
Variants
Views
Read
Edit
View history
More
Search
Navigation
Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Wikipedia store
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact page
Tools
What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Wikidata item
Cite this page
Print/export
Create a book
Download as PDF
Printable version
In other projects
Wikimedia Commons
Languages
Alemannisch
العربية
Беларуская
Беларуская (тарашкевіца)
Български
Català
Čeština
Dansk
Deutsch
Eesti
Ελληνικά
Español
Esperanto
فارسی
Français
Galego
한국어
Հայերեն
हिन्दी
Italiano
עברית
Қазақша
Magyar
Македонски
Nederlands
日本語
Norsk nynorsk
ਪੰਜਾਬੀ
Piemontèis
Polski
Português
Română
Русский
Simple English
Slovenčina
Slovenščina
Српски / srpski
Suomi
Svenska
Tagalog
Türkçe
Українська
Tiếng Việt
中文
Edit links
This page was last edited on 8 October 2017, at 17:37.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Developers
Cookie statement
Mobile view
Enable previews

key: html

type: local

value: div

attrs: id class

values: mw-page-base noprint

values: mw-head-base noprint

value: a

attrs: id class lang

values: firstHeading firstHeading en

values: siteSub noprint

values: jump-to-nav mw-jump

value: td

value: tr

attrs: href title

value: th

value: img

attrs: alt src width height srcset data-file-width data-file-height

attrs: href class

attrs: class id href

values: NavToggle NavToggle1 #

attrs: class style

values: font-size: 105%; padding: 0.2em 0px 0.4em; text-align: center; display: none;

values: Introduction to the mathematics of general relativity

values: Derivations of the Lorentz transformations

values: float:right;clear:right;width:22.0em;margin:0 0 1.0em 1.0em;background:#f9f9f9;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%

values:
/wiki/Coordinate_transformation
mw-redirect
Coordinate transformation

values:
/wiki/Coordinate_frame
mw-redirect
Coordinate frame

values:
/wiki/Cartesian_coordinates
mw-redirect
Cartesian coordinates

values:
/wiki/Newtonian_physics
mw-redirect
Newtonian physics

values:
/wiki/Galilean_relativity
mw-redirect
Galilean relativity

values:
/wiki/Linear_transformation
mw-redirect
Linear transformation

values:
/wiki/Spacetime_interval
mw-redirect
Spacetime interval

values:
/wiki/Hyperbolic_rotation
mw-redirect
Hyperbolic rotation

values:
/wiki/George_FitzGerald
mw-redirect
George FitzGerald

values:
/wiki/Spherical_symmetry
mw-redirect
Spherical symmetry

values:
/wiki/Inertial_reference_frame
mw-redirect
Inertial reference frame

values: Edit section: Derivation of the group of Lorentz transformations

values:
/wiki/Cartesian_coordinate
mw-redirect
Cartesian coordinate

values: vertical-align: -0.838ex; width:89.289ex; height:3.176ex;

values: border-left:none; border-top:0px none #e5e5e5; border-right:none; border-bottom:none; padding:0.08em;

values: border-collapse:collapse; background:none; margin:0; border:none; width:99%;

values: vertical-align:middle; width:99%; border:none; padding:0.08em;

values:
math_D1
reference nourlexpansion
font-weight:bold; font-style:italic;

values: border-collapse:collapse; background:none; margin:0; border:none;

values: vertical-align: -1.005ex; width:123.536ex; height:3.343ex;

values:
math_D2
reference nourlexpansion
font-weight:bold; font-style:italic;

values: vertical-align: -1.005ex; width:110.614ex; height:3.343ex;

values:
math_D3
reference nourlexpansion
font-weight:bold; font-style:italic;

values: vertical-align: -0.838ex; width:35.434ex; height:3.009ex;

values: {\displaystyle (a,a)=(a',a')\quad {\text{or}}\quad a\cdot a=a'\cdot a',}

values:
math_D4
reference nourlexpansion
font-weight:bold; font-style:italic;

values:
/wiki/Signature_(quadratic_form)
mw-redirect
Signature (quadratic form)

values: vertical-align: -0.838ex; width:55.34ex; height:3.009ex;

values:
math_D5
reference nourlexpansion
font-weight:bold; font-style:italic;

values:
/wiki/Poincare_group
mw-redirect
Poincare group

values:
/wiki/Euler_angle
mw-redirect
Euler angle

values: Edit section: Physical formulation of Lorentz boosts

attrs: alt src width height class srcset data-file-width data-file-height

attrs: columnalign rowspacing columnspacing displaystyle

values: right left right left right left right left right left right left

values: 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em

values: vertical-align: -7.171ex; width:18.01ex; height:15.509ex;

values: border:1px solid black;background: white; text-align: center;

values: vertical-align: -4.671ex; width:14.054ex; height:8.009ex;

values:
/wiki/Faster_than_light
mw-redirect
Faster than light

values:
/wiki/Active_transformation
mw-redirect
Active transformation

values:
/wiki/Passive_transformation
mw-redirect
Passive transformation

values: vertical-align: -6.906ex; margin-bottom: -0.265ex; width:18.702ex; height:15.509ex;

values: vertical-align: -2.505ex; width:18.742ex; height:6.176ex;

values: vertical-align: -5.505ex; width:24.873ex; height:12.176ex;

values: vertical-align: -0.671ex; width:21.921ex; height:3.009ex;

values: vertical-align: -2.505ex; width:17.114ex; height:6.176ex;

values: \tanh \zeta ={\frac {\sinh \zeta }{\cosh \zeta }}\,,

values: vertical-align: -4.671ex; width:24.534ex; height:8.009ex;

values: vertical-align: -4.005ex; width:13.707ex; height:9.176ex;

values: vertical-align: -0.671ex; width:14.03ex; height:3.009ex;

values: vertical-align: -5.505ex; width:25.565ex; height:12.176ex;

values: vertical-align: -4.005ex; width:25.28ex; height:9.176ex;

values: vertical-align: -4.005ex; width:25.972ex; height:9.176ex;

values: vertical-align: -2.338ex; width:12.072ex; height:5.843ex;

values: {\displaystyle {\begin{aligned}t'&\approx t\\x'&\approx x-vt\end{aligned}}}

values: vertical-align: -2.171ex; width:14.949ex; height:5.676ex;

values: {\displaystyle \Delta t'=\gamma {\frac {-v\Delta x}{c^{2}}}}

values:
/wiki/Velocity_vector
mw-redirect
Velocity vector

values:
/wiki/Position_vector
mw-redirect
Position vector

values: vertical-align: -1.505ex; width:29.245ex; height:3.676ex;

values: vertical-align: -6.005ex; width:21.415ex; height:13.176ex;

values: vertical-align: -1.171ex; width:33.988ex; height:3.176ex;

values: vertical-align: -4.005ex; width:33.031ex; height:9.176ex;

values: vertical-align: -4.005ex; width:34.415ex; height:9.176ex;

values: vertical-align: -0.671ex; width:19.714ex; height:2.509ex;

values: {\boldsymbol {\beta }}=\beta \mathbf {n} =\mathbf {n} \tanh \zeta \,,

values: vertical-align: -0.671ex; width:21.739ex; height:3.009ex;

values: {\boldsymbol {\zeta }}=\zeta \mathbf {n} =\mathbf {n} \tanh ^{-1}\beta \,,

values: Edit section: Transformation of velocities

values:
/wiki/Velocity_addition_formula
mw-redirect
Velocity addition formula

values: vertical-align: -6.005ex; width:42.575ex; height:9.676ex;

values: vertical-align: -3.338ex; width:45.654ex; height:7.009ex;

values:
/wiki/Stellar_aberration
mw-redirect
Stellar aberration

values: Edit section: Transformation of other quantities

values: vertical-align: -0.338ex; width:26.474ex; height:3.009ex;

values: vertical-align: -5.338ex; width:36.475ex; height:11.843ex;

values:
/wiki/Four_vector
mw-redirect
Four vector

values:
/wiki/Four_momentum
mw-redirect
Four momentum

values:
/wiki/Four_current
mw-redirect
Four current

values:
/wiki/Electromagnetic_four_potential
mw-redirect
Electromagnetic four potential

values:
/wiki/Mass_density
mw-redirect
Mass density

values:
/wiki/Rest_energy
mw-redirect
Rest energy

values:
/wiki/Matrix_product
mw-redirect
Matrix product

values:
/wiki/Minkowski_metric
mw-redirect
Minkowski metric

values: vertical-align: -5.671ex; width:53.003ex; height:12.509ex;

values: vertical-align: -0.838ex; width:26.497ex; height:3.509ex;

values: X\cdot X=X^{\mathrm {T} }\eta X={X'}^{\mathrm {T} }\eta {X'}

values: vertical-align: -0.338ex; width:9.422ex; height:2.509ex;

values: vertical-align: -0.338ex; width:1.614ex; height:2.176ex;

values:
/wiki/Matrix_Lie_group
mw-redirect
Matrix Lie group

values: vertical-align: -0.838ex; width:10.142ex; height:3.176ex;

values: \eta =\Lambda ^{\mathrm {T} }\eta \Lambda

values: vertical-align: -0.838ex; width:34.457ex; height:3.176ex;

values: [\det(\Lambda )]^{2}=1\quad \Rightarrow \quad \det(\Lambda )=\pm 1

values: vertical-align: -2.505ex; width:36.953ex; height:6.176ex;

values: vertical-align: -0.505ex; width:14.075ex; height:2.843ex;

values: {\displaystyle \Gamma ^{2}=1+\mathbf {b} ^{\mathrm {T} }\mathbf {b} }

values: vertical-align: -0.671ex; width:32.139ex; height:3.009ex;

values: \Gamma ^{2}\geq 1\quad \Rightarrow \quad \Gamma \leq -1\,,\quad \Gamma \geq 1

values: vertical-align: -0.838ex; width:20.04ex; height:3.343ex;

values: {\displaystyle {\mathcal {L}}^{\downarrow }=\{\Lambda \,:\,\Gamma \leq -1\}}

values: vertical-align: -0.838ex; width:18.221ex; height:3.343ex;

values: {\displaystyle {\mathcal {L}}^{\uparrow }=\{\Lambda \,:\,\Gamma \geq 1\}}

values: vertical-align: -0.838ex; width:25.747ex; height:2.843ex;

values: {\displaystyle {\mathcal {L}}_{+}=\{\Lambda \,:\,\det(\Lambda )=+1\}}

values: vertical-align: -1.005ex; width:14.643ex; height:3.509ex;

values: {\mathcal {L}}_{+}^{\uparrow }={\mathcal {L}}_{+}\cap {\mathcal {L}}^{\uparrow }

values: {\displaystyle {\mathcal {L}}_{-}=\{\Lambda \,:\,\det(\Lambda )=-1\}}

values: vertical-align: -0.213ex; margin-bottom: -0.625ex; width:14.643ex; height:3.343ex;

values: {\mathcal {L}}_{-}^{\uparrow }={\mathcal {L}}_{-}\cap {\mathcal {L}}^{\uparrow }

values:
/wiki/Disjoint_set
mw-redirect
Disjoint set

values: vertical-align: -1.005ex; width:25.032ex; height:3.509ex;

values: vertical-align: -1.005ex; width:3.132ex; height:3.509ex;

values: {\displaystyle {\mathcal {L}}_{+}^{\uparrow }}

values: vertical-align: -0.671ex; width:3.132ex; height:2.509ex;

values: vertical-align: -0.338ex; width:2.676ex; height:2.843ex;

values: {\displaystyle {\mathcal {L}}^{\uparrow }}

values: {\displaystyle {\mathcal {L}}_{+}^{\downarrow }}

values: vertical-align: -0.213ex; margin-bottom: -0.625ex; width:3.132ex; height:3.343ex;

values: {\displaystyle {\mathcal {L}}_{-}^{\downarrow }}

values: {\displaystyle {\mathcal {L}}_{-}^{\uparrow }}

values: vertical-align: -0.838ex; width:12.825ex; height:3.009ex;

values: vertical-align: -6.338ex; width:68.009ex; height:13.843ex;

values: vertical-align: -0.838ex; width:30.647ex; height:3.009ex;

values: {\displaystyle X''=B(\mathbf {v} )X'\,,\quad X'=B(\mathbf {u} )X}

values: vertical-align: -0.838ex; width:19.43ex; height:3.009ex;

values: {\displaystyle X''=B(\mathbf {v} )B(\mathbf {u} )X\,.}

values:
/wiki/Axis-angle_representation
mw-redirect
Axis-angle representation

values: vertical-align: -2.505ex; width:23.513ex; height:6.176ex;

values: vertical-align: -0.338ex; width:6.762ex; height:2.176ex;

values: {\displaystyle {\boldsymbol {\theta }}=\theta \mathbf {e} }

values:
/wiki/Matrix_inverse
mw-redirect
Matrix inverse

values:
/wiki/Identity_transformation
mw-redirect
Identity transformation

values: vertical-align: -0.838ex; width:21.219ex; height:2.843ex;

values:
/wiki/Taylor_expansion
mw-redirect
Taylor expansion

values: vertical-align: -2.838ex; width:25.692ex; height:6.509ex;

values: vertical-align: -2.838ex; width:18.041ex; height:6.509ex;

values: vertical-align: -2.505ex; width:35.304ex; height:6.676ex;

values: Characterizations of the exponential function

values: vertical-align: -0.838ex; width:29.899ex; height:3.176ex;

values: vertical-align: -5.671ex; width:72.628ex; height:12.509ex;

values: vertical-align: -5.671ex; width:76.036ex; height:12.509ex;

values: vertical-align: -0.838ex; width:47.595ex; height:3.176ex;

values: vertical-align: -0.838ex; width:44.173ex; height:3.176ex;

values: vertical-align: -4.005ex; width:37.793ex; height:9.176ex;

values: vertical-align: -0.838ex; width:19.897ex; height:3.176ex;

values: vertical-align: -0.838ex; width:21.995ex; height:3.176ex;

values: vertical-align: -0.838ex; width:19.516ex; height:2.843ex;

values:
/wiki/Commutation_relation
mw-redirect
Commutation relation

values: vertical-align: -1.005ex; width:50.949ex; height:3.009ex;

values: vertical-align: -0.838ex; margin-left: -0.03ex; width:7.438ex; height:2.843ex;

values:
/wiki/Lie_bracket
mw-redirect
Lie bracket

values: vertical-align: -0.838ex; width:26.338ex; height:2.843ex;

values: \mathrm {exp} \,:\,{\mathfrak {so}}(3,1)\rightarrow \mathrm {SO} (3,1),

values:
/wiki/Parity_inversion
mw-redirect
Parity inversion

values: vertical-align: -2.505ex; width:14.43ex; height:6.176ex;

values: {\displaystyle P={\begin{bmatrix}1&0\\0&-\mathbf {I} \end{bmatrix}}}

values: vertical-align: -2.505ex; width:14.47ex; height:6.176ex;

values: {\displaystyle T={\begin{bmatrix}-1&0\\0&\mathbf {I} \end{bmatrix}}}

values: Edit section: Inhomogeneous Lorentz group

values: vertical-align: -0.505ex; width:14.05ex; height:2.676ex;

values:
/wiki/Poincar%C3%A9_transformation
mw-redirect
Poincaré transformation

values: Representation theory of the Lorentz group

values: vertical-align: -6.505ex; width:42.178ex; height:14.176ex;

values:
/wiki/Tensor_index_notation
mw-redirect
Tensor index notation

values: vertical-align: -1.005ex; width:13.445ex; height:3.343ex;

values: {\displaystyle {x^{\prime }}^{\nu }={\Lambda ^{\nu }}_{\mu }x^{\mu },}

values: Covariance and contravariance of vectors

values:
/wiki/Summation_convention
mw-redirect
Summation convention

values: vertical-align: -1.005ex; width:14.659ex; height:3.343ex;

values: {\displaystyle {A^{\prime }}^{\nu }={\Lambda ^{\nu }}_{\mu }A^{\mu }\,.}

values: vertical-align: -1.005ex; width:15.078ex; height:3.509ex;

values: {\displaystyle A^{\nu '}={\Lambda ^{\nu '}}_{\mu }A^{\mu }\,.}

values: vertical-align: -1.338ex; width:19.013ex; height:3.676ex;

values: {\displaystyle {X'}^{\alpha }={\Pi (\Lambda )^{\alpha }}_{\beta }X^{\beta }\,,}

values: vertical-align: -1.005ex; width:12.064ex; height:3.009ex;

values: vertical-align: -0.838ex; width:12.082ex; height:2.843ex;

values: vertical-align: -1.005ex; width:20.697ex; height:3.176ex;

values: vertical-align: -1.171ex; width:22.619ex; height:3.676ex;

values: vertical-align: -1.171ex; width:16.195ex; height:3.676ex;

values: vertical-align: -1.005ex; width:14.272ex; height:3.176ex;

values: {\displaystyle {A'}_{\nu }={\Lambda _{\nu }}^{\mu }A_{\mu }.}

values: vertical-align: -1.171ex; width:28.172ex; height:3.676ex;

values: vertical-align: -1.171ex; width:15.513ex; height:4.009ex;

values: {\displaystyle A'=\left(\Lambda ^{-1}\right)^{\mathrm {T} }A.}

values: vertical-align: -0.838ex; width:62.163ex; height:2.843ex;

values:
math_.28T1.29
reference nourlexpansion
font-weight:bold; font-style:italic;

values: border:2px solid #0073CF;background: #ECFCF4; text-align: center;

values: vertical-align: -0.671ex; width:70.061ex; height:2.676ex;

values:
math_.28T2.29
reference nourlexpansion
font-weight:bold; font-style:italic;

values: vertical-align: -1.338ex; width:51.976ex; height:4.176ex;

values:
math_.28T3.29
reference nourlexpansion
font-weight:bold; font-style:italic;

values: Edit section: Transformation of the electromagnetic field

values: Classical electromagnetism and special relativity

values:
/wiki/Electromagnetic_force
mw-redirect
Electromagnetic force

values: vertical-align: -7.005ex; width:75.129ex; height:15.176ex;

values: vertical-align: -5.932ex; margin-bottom: -0.239ex; width:112.394ex; height:13.509ex;

values: vertical-align: -1.005ex; width:21.752ex; height:3.509ex;

values: vertical-align: -12.908ex; margin-bottom: -0.263ex; width:69.935ex; height:27.509ex;

values: vertical-align: -14.593ex; margin-bottom: -0.245ex; width:73.85ex; height:30.843ex;

values: vertical-align: -6.338ex; width:44.592ex; height:13.676ex;

values: vertical-align: -1.171ex; width:53.78ex; height:3.676ex;

values: vertical-align: -3.849ex; margin-bottom: -0.323ex; width:32.264ex; height:9.509ex;

values: vertical-align: -1.005ex; margin-left: -0.016ex; width:13.341ex; height:3.509ex;

values: {\displaystyle j^{\mu ^{\prime }}={\Lambda ^{\mu '}}_{\mu }j^{\mu }.}

values:
/wiki/Maxwell_equations
mw-redirect
Maxwell equations

values: vertical-align: -1.338ex; width:99.448ex; height:3.676ex;

values:
math_.28T4.29
reference nourlexpansion
font-weight:bold; font-style:italic;

values: Edit section: Transformation of general fields

values: vertical-align: -5.273ex; margin-bottom: -0.231ex; width:109.61ex; height:12.176ex;

values:
math_1
reference nourlexpansion
font-weight:bold; font-style:italic;

values: div-col columns column-count column-count-2

values: -moz-column-count: 2; -webkit-column-count: 2; column-count: 2;

values: vertical-align: -1.005ex; width:29.355ex; height:3.009ex;

values: vertical-align: -1.005ex; width:26.904ex; height:2.843ex;

values:
/wiki/Position_space
mw-redirect
Position space

values:
reflist mw-references-wrap
list-style-type: decimal;

values: -moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;

values:
nofollow
external text
//doi.org/10.1119%2F1.1974267

values:
nofollow
external text
//doi.org/10.1063%2F1.1703854

values:
nofollow
external text
http://www.bourbaphy.fr/darrigol2.pdf

attrs: alt src title width height srcset data-file-width data-file-height

values:
nofollow
external text
//doi.org/10.1093%2Fbjps%2F37.2.232

values: s:Translation:On the Dynamics of the Electron (June)

values:
nofollow
external text
//doi.org/10.1002%2Fandp.19053221004

values:
nofollow
external text
//doi.org/10.1007%2FBF00661317

values:
nofollow
external text
//www.worldcat.org/issn/0894-9875

values: The site requires a paid subscription to access this page.

values:
nofollow
external text
//doi.org/10.1007%2FBF00732759

values:
nofollow
external text
//doi.org/10.1007%2Fbf01574845

values:
nofollow
external text
//doi.org/10.1007%2Fbf00690425

values: refbegin columns references-column-count references-column-count-3

values: -moz-column-count: 3; -webkit-column-count: 3; column-count: 3;

values:
/wiki/David_Griffiths_(physicist)
mw-redirect
David Griffiths (physicist)

values:
/wiki/Addison-Wesley_Publishing_Company
mw-redirect
Addison-Wesley Publishing Company

values:
/wiki/W._H._Freeman
mw-redirect
W. H. Freeman

values:
nofollow
http://www.adamauton.com/warp/
external text

values: Category:Pages containing links to subscription-only content

values: Discussion about edits from this IP address [ctrl-alt-n]

values: A list of edits made from this IP address [ctrl-alt-y]

values: You're encouraged to log in; however, it's not mandatory. [ctrl-alt-o]

values:
/wiki/Lorentz_transformation
View the content page [ctrl-alt-c]
c

values: Discussion about the content page [ctrl-alt-t]

values: p-namespaces navigation vectorTabs p-namespaces-label

values:
p-variants
navigation
vectorMenu emptyPortlet
p-variants-label

values: Past revisions of this page [ctrl-alt-h]

values: p-views navigation vectorTabs p-views-label

values:
p-cactions
navigation
vectorMenu emptyPortlet
p-cactions-label

values:
mw-wiki-logo
/wiki/Main_Page
Visit the main page

values:
/wiki/Main_Page
Visit the main page [ctrl-alt-z]
z

values: Featured content – the best of Wikipedia

values: Find background information on current events

values: portal navigation p-wikibase-otherprojects p-wikibase-otherprojects-label

values:
uls-settings-trigger
Language settings
0
button
true

values: تحويلات لورينتز – Arabic

values: Пераўтварэнні Лорэнца – Belarusian

values: Пераўтварэньні Лёрэнца – Belarusian (Taraškievica orthography)

values: Лоренцови трансформации – Bulgarian

values: Μετασχηματισμοί Λόρεντζ – Greek

values: تبدیل لورنتس – Persian

values: 로런츠 변환 – Korean

values: Լորենցի ձևափոխություններ – Armenian

values: लोरेन्ट्स रूपांतरण – Hindi

values: טרנספורמציות לורנץ – Hebrew

values: Лоренц түрлендірулері – Kazakh

values: Лоренцови трансформации – Macedonian

values: ローレンツ変換 – Japanese

values: Lorentztransformasjon – Norwegian Nynorsk

values: ਲੌਰੰਟਜ਼ ਪਰਿਵਰਤਨ – Punjabi

values: Преобразования Лоренца – Russian

values: Лоренцове трансформације – Serbian

values: Перетворення Лоренца – Ukrainian

values: /static/images/wikimedia-button-1.5x.png 1.5x, /static/images/wikimedia-button-2x.png 2x

values: /static/images/poweredby_mediawiki_132x47.png 1.5x, /static/images/poweredby_mediawiki_176x62.png 2x

values:
suggestions
display: none; font-size: 13px;

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Lorentz_transformation rootpage-Lorentz_transformation vector-nav-directionality skin-vector action-view