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Lorentz transformation

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 representing a velocity confined to the x-direction, is expressed as[1][2]


Expressing the speed as an equivalent form of the transformation is[3]


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.[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 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. 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.

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 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 rotation in the spatial coordinates only, these like boosts 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 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.

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.

: 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)

inverses

for no relative motion/rotation: B(0) = R(0) = I

identity transformation

unit : det(B) = det(R) = +1. This property makes them proper transformations.

determinant

: B is symmetric (equals transpose), while R is nonsymmetric but orthogonal (transpose equals inverse, RT = R−1).

matrix symmetry

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 , and thus the observer in frame F′ also sees a magnetic field.

electric current

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

Ernst, A.; Hsu, J.-P. (2001), (PDF), Chinese Journal of Physics, 39 (3): 211–230, Bibcode:2001ChJPh..39..211E, archived from the original (PDF) on 2011-07-16

"First proposal of the universal speed of light by Voigt 1887"

Thornton, Stephen T.; Marion, Jerry B. (2004), Classical dynamics of particles and systems (5th ed.), Belmont, [CA.]: Brooks/Cole, pp. 546–579,  978-0-534-40896-1

ISBN

(1887), "Über das Doppler'sche princip", Nachrichten von der Königlicher Gesellschaft den Wissenschaft zu Göttingen, 2: 41–51

Voigt, Woldemar

. This web page contains a more detailed derivation of the Lorentz transformation with special emphasis on group properties.

Derivation of the Lorentz transformations

. This webpage poses a problem, the solution of which is the Lorentz transformation, which is presented graphically in its next page.

The Paradox of Special Relativity

Archived 2011-08-29 at the Wayback Machine – a chapter from an online textbook

Relativity

. A computer program demonstrating the Lorentz transformations on everyday objects.

Warp Special Relativity Simulator

on YouTube visualizing the Lorentz transformation.

Animation clip

on YouTube explaining and visualizing the Lorentz transformation with a mechanical Minkowski diagram

MinutePhysics video

on Desmos (graphing) showing Lorentz transformations with a virtual Minkowski diagram

Interactive graph

on Desmos showing Lorentz transformations with points and hyperbolas

Interactive graph

from John de Pillis. Online Flash animations of Galilean and Lorentz frames, various paradoxes, EM wave phenomena, etc.

Lorentz Frames Animated