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 -direction, is expressed as Equation 6-3.24, page 210
where and are the coordinates of an event in two frames with the spatial origins coinciding at , where the primed frame is seen from the unprimed frame as moving with speed along the -axis, where is the speed of light, and
is the Lorentz factor. When speed is much smaller than , the Lorentz factor is negligibly different from 1, but as approaches , grows without bound. The value of must be smaller than for the transformation to make sense.
Expressing the speed as a fraction of the speed of light, an equivalent form of the transformation is
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.One can imagine that in each inertial frame there are observers positioned throughout space, each with a synchronized clock and at rest in the particular inertial frame. These observers then report to a central office, where all reports are collected. When one speaks of a particular observer, one refers to someone having, at least in principle, a copy of this report. See, e.g., .
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 , 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.The reference is within the following paper: 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 and a set of Cartesian coordinates , , to specify position in space in that frame. Subscripts label individual events.
From Einstein's second postulate of relativity (invariance of ) 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 and . 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 are the spacetime coordinates used to define events in one frame, and are the coordinates in another frame. First one observes that () is satisfied if an arbitrary -tuple of numbers are added to events and . 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.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 Poincaré 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. First equation in () can be written more compactly as:
where refers to the bilinear form of signature on exposed by the right hand side formula in (). The alternative notation defined on the right is referred to as the relativistic dot product. Spacetime mathematically viewed as endowed with this bilinear form is known as Minkowski space . The Lorentz transformation is thus an element of the group , the Lorentz group or, for those that prefer the other metric signature, (also called the Lorentz group).The groups and are isomorphic. It is widely believed that the choice between the two metric signatures has no physical relevance, even though some objects related to and respectively, e.g., the Clifford algebras corresponding to the different signatures of the bilinear form associated to the two groups, are non-isomorphic. One has:
which is precisely preservation of the bilinear form () which implies (by linearity of and bilinearity of the form) that () 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 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 defines events with coordinates , , , . Another frame moves with velocity relative to , and an observer in this "moving" frame defines events using the coordinates , , , .
The coordinate axes in each frame are parallel (the and axes are parallel, the and axes are parallel, and the and axes are parallel), remain mutually perpendicular, and relative motion is along the coincident axes. At , the origins of both coordinate systems are the same, . 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 records an event , , , , then an observer in records the same event with coordinates