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

{\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 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

{\textstyle \gamma =\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{-1}}

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

,

\gamma

grows without bound. The value of

v

must be smaller than

c

for the transformation to make sense.

Expressing the speed as ${\textstyle \beta ={\frac {v}{c}},}$ an equivalent form of the transformation is^[3]

{\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.^{[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.

History

Many physicists—including Woldemar Voigt, George FitzGerald, Joseph Larmor, and Hendrik Lorentz^[4] himself—had been discussing the physics implied by these equations since 1887.^[5] 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 luminiferous 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.^[6] Their explanation was widely known before 1905.^[7]

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.^[8] Larmor is credited to have been the first to understand the crucial time dilation property inherent in his equations.^[9]

In 1905, Poincaré was the first to recognize that the transformation has the properties of a mathematical group, and he named it after Lorentz.^[10] 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.^[11]

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:

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 $a 1 = (t 1, x 1, y 1, z 1)$ and $a 2 = (t 2, x 2, y 2, z 2)$ . 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:

{\begin{aligned}&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)}}.\end{aligned}}

[1]

[2]

[3]

[nb 1]

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