In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz. The most common form of the transformation, parametrized by the real constant v , {\displaystyle v,} representing a velocity confined to the x-direction, is expressed as
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 (t, x, y, z) and (t′, x′, y′, z′) are the coordinates of an event in two frames with the spatial origins coinciding at t = t′ = 0, where the primed frame is seen from the unprimed frame as moving with speed v along the x-axis, where c is the speed of light, and
γ = 1 1 − v 2 / c 2 {\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}
is the Lorentz factor. When speed v is much smaller than c, the Lorentz factor is negligibly different from 1, but as v approaches c, γ {\displaystyle \gamma } grows without bound. The value of v must be smaller than c for the transformation to make sense. Expressing the speed as a fraction of the speed of light, β = v / c , {\textstyle \beta =v/c,} an equivalent form of the transformation is
c t ′ = γ ( c t − β x ) x ′ = γ ( x − β c t ) y ′ = y z ′ = z . {\displaystyle {\begin{aligned}ct'&=\gamma \left(ct-\beta x\right)\\x'&=\gamma \left(x-\beta ct\right)\\y'&=y\\z'&=z.\end{aligned}}}
Frames of reference can be divided into two groups: inertial (relative motion with constant velocity) and non-inertial (accelerating, moving 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 (usually Cartesian coordinates in this context) to measure lengths, and a clock to measure time intervals. 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. 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 less 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 transformations later became a cornerstone for 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. 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.
History
In the 1890s, Hendrik Lorentz began to develop theories of electrodynamics based on a pervasive luminiferous aether uncoupled from matter. In a series of paper from 1892 to 1904 he used transformations which would come to bear his name as mathematical aids in analyzing these theories. Woldemar Voigt had developed similar transformations when studying Doppler shift physics in 1887. In that year, the aether-wind experiment of Michelson and Morley was repeated with better accuracy, leading George FitzGerald to conjecture that bodies in motion through aether are being contracted. In 1892, Lorentz independently presented the same idea in a more detailed manner, which was subsequently called FitzGerald–Lorentz contraction hypothesis. This hypothesis was widely known before 1905. Many other physicists were also involved in the development and understanding of these transformation. Lorentz indirectly discovered time dilation in 1892, but Joseph Larmor was the first to explicit mention that the 'scale of time is enlarged'. The time coordinate in these transformation came to be called ("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. In 1905, Poincaré was the first to recognize that the transformation has the properties of a mathematical group, and he named it after Lorentz. All of the work up to 1905 assumed that the constant speed of light was a consequence of an aether and hypothesized the transformation. In that 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.
Derivation of the group of Lorentz transformations
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 (invariance of c) it follows that:
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:
where (t, x, y, z) are the spacetime coordinates used to define events in one frame, and (t′, x′, y′, z′) are the coordinates in another frame. First one observes that (D2) is satisfied if an 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 solves the general problem too:
(a solution satisfying the first formula automatically satisfies the second one as well; 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. First equation in (D3) can be written more compactly as:
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 O(1, 3), the Lorentz group or, for those that prefer the other metric signature, O(3, 1) (also called the Lorentz group). One has:
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 Poincaré group.
Generalities 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 Lorentz boosts or simply boosts, and the relative velocity between the frames is the parameter of the transformation. The other basic type of Lorentz transformation is only a rotation in the spatial coordinates. Unlike boosts, these are inertial transformations since there is no relative motion, the frames are simply tilted (and not continuously rotating), and in this case quantities defining the rotation are the parameters of the transformation (e.g., axis–angle representation, or Euler angles, etc.). A combination of a rotation and a 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
Coordinate transformation
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
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
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. In the case of three spatial dimensions [ct, x, y, z], where the boost β {\displaystyle \beta } is in the x direction, the eigenstates of the transformation are [1, 1, 0, 0] with eigenvalue ( 1 − β ) / ( 1 + β ) {\displaystyle {\sqrt {(1-\beta )/(1+\beta )}}} , [1, −1, 0, 0] with eigenvalue ( 1 + β ) / ( 1 − β ) {\displaystyle {\sqrt {(1+\beta )/(1-\beta )}}} , and [0, 0, 1, 0] and [0, 0, 0, 1], the latter two with eigenvalue 1. When the boost velocity v {\displaystyle {\boldsymbol {v}}} is in an arbitrary vector direction with the boost vector β = v / c {\displaystyle {\boldsymbol {\beta }}={\boldsymbol {v}}/c} , then the transformation from an unprimed spacetime coordinate system to a primed coordinate system is given by
[ c t ′ − γ β x x ′ 1 + γ 2 1 + γ β x 2 y ′ γ 2 1 + γ β x β y z ′ γ 2 1 + γ β y β z ] = [ γ − γ β x − γ β y − γ β z − γ β x 1 + γ 2 1 + γ β x 2 γ 2 1 + γ β x β y γ 2 1 + γ β x β z − γ β y γ 2 1 + γ β x β y 1 + γ 2 1 + γ β y 2 γ 2 1 + γ β y β z − γ β z γ 2 1 + γ β x β z γ 2 1 + γ β y β z 1 + γ 2 1 + γ β z 2 ] [ c t − γ β x x 1 + γ 2 1 + γ β x 2 y γ 2 1 + γ β x β y z γ 2 1 + γ β y β z ] , {\displaystyle {\begin{bmatrix}ct'{\vphantom {-\gamma \beta _{\text{x}}}}\\x'{\vphantom {1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}^{2}}}\\y'{\vphantom {{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}\beta _{\text{y}}}}\\z'{\vphantom {{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{y}}\beta _{\text{z}}}}\end{bmatrix}}={\begin{bmatrix}\gamma &-\gamma \beta _{\text{x}}&-\gamma \beta _{\text{y}}&-\gamma \beta _{\text{z}}\\-\gamma \beta _{\text{x}}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}^{2}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}\beta _{\text{y}}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}\beta _{\text{z}}\\-\gamma \beta _{\text{y}}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}\beta _{\text{y}}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{y}}^{2}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{y}}\beta _{\text{z}}\\-\gamma \beta _{\text{z}}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}\beta _{\text{z}}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{y}}\beta _{\text{z}}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{z}}^{2}\\\end{bmatrix}}{\begin{bmatrix}ct{\vphantom {-\gamma \beta _{\text{x}}}}\\x{\vphantom {1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}^{2}}}\\y{\vphantom {{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}\beta _{\text{y}}}}\\z{\vphantom {{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{y}}\beta _{\text{z}}}}\end{bmatrix}},}
where the Lorentz factor is γ = 1 / 1 − β 2 {\displaystyle \gamma =1/{\sqrt {1-{\boldsymbol {\beta }}^{2}}}} . The determinant of the transformation matrix is +1 and its trace is 2 ( 1 + γ ) {\displaystyle 2(1+\gamma )} . The inverse of the transformation is given by reversing the sign of β {\displaystyle {\boldsymbol {\beta }}} . The quantity c 2 t 2 − x 2 − y 2 − z 2 {\displaystyle c^{2}t^{2}-x^{2}-y^{2}-z^{2}} is invariant under the transformation: namely ( c 2 t ′ 2 − x ′ 2 − y ′ 2 − z ′ 2 ) = ( c 2 t 2 − x 2 − y 2 − z 2 ) {\displaystyle (c^{2}t'^{2}-x'^{2}-y'^{2}-z'^{2})=(c^{2}t^{2}-x^{2}-y^{2}-z^{2})} . The Lorentz transformations can also be derived in a way that resembles circular rotations in 3-dimensional space using the hyperbolic functions. For the boost in the x direct