In physics, the Lorentz transformation (or transformations) is named after the Dutch physicist Hendrik Lorentz. It was 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 transformations describe how measurements related to events in space and time by two observers, in inertial frames moving at constant velocity with respect to each other, are related. They reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events. 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.

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 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, so 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 Joseph Larmor, and Hendrik Lorentz himself had been discussing the physics implied by these equations since 1887.[1] 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 ether. 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 ether-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.[2] Their explanation was widely known before 1905.[3]

Lorentz (1892–1904) and Larmor (1897–1900), who believed the luminiferous ether hypothesis, were also seeking the transformation under which Maxwell's equations are invariant when transformed from the ether 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) as the consequence of clock synchronization, under the assumption that the speed of light is constant in moving frames.[4] Larmor is credited to have been the first to understand the crucial time dilation property inherent in his equations.[5]

In 1905, Poincaré was the first to recognize that the transformation has the properties of a mathematical group, and named it after Lorentz.[6] 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 mechanical aether.[7]

Derivation

From Einstein's second postulate of relativity follows immediately

c^2(t_2 - t_1)^2 - (x_2 - x_1)^2 - (y_2 - y_1)^2 - (z_2 - z_1)^2 = 0

in all reference frames for events connected by light signals. An event is something which happens at a certain place and certain time, and in any inertial frame can be defined by a time coordinate t and some set of position coordinates, here Cartesian coordinates x, y, z are used. The quantity on the left is called the spacetime interval. The interval between any two reference frames is in fact invariant, as is shown here (where one can also find several more explicit derivations than presently given). 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.

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. Notice immediately in general that

t_2' - t_1' \neq t_2 - t_1 \,,

and the possibility of time progressing at different rates, depending on which frame one measures from, is allowed. This is in stark contrast to Newtonian mechanics and Galilean relativity in which time is absolute and progresses at the same rate for all observers. Now one observes that a linear solution to the simpler problem

c^2t^2 - x^2 - y^2 - z^2 = c^2t'^2 - x'^2 - y'^2 - z'^2

will solve the general problem too. This is just a matter of look-up in the theory of classical groups that preserve bilinear forms of various signature. The Lorentz transformation is thus an element of the group O(3, 1) or, for those that prefer the other metric signature, O(1, 3).

The relations between the primed and unprimed spacetime coordinates are the Lorentz transformations, each coordinate in one frame is a 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, other parameters will enter the transformation equations.

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" usually just refers to transformations between inertial frames in the context of special relativity. Relative motion with constant velocity is called a "boost", and the relative velocity between the frames is the parameter of the transformation. It is also possible to consider rotations through an angle about an axis, these are also inertial frames since there is no relative motion, the frames are simply tilted, and in this case quantities defining the rotation are the parameters of the transformation (e.g. axis–angle representation, or Euler angles, etc.).

The Lorentz transformations are linear transformations, and can be expressed in matrix form using a transformation matrix.

Boosts

Below, the Lorentz transformations are called "boosts" in the stated directions. A "boost" means relative motion with constant (uniform) velocity, and should not be conflated with mere displacements in spacetime (in this case, the coordinate systems are simply shifted and there is no relative motion).

Boost in the Cartesian directions

Velocity parametrization

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.[8]

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.

What is the conversion between these coordinate systems? If an observer in F records an event t, x, y, z, then an observer in F records the same event with coordinates[9]

Lorentz boost (x direction)
\begin{align} t' &= \gamma \left( t - \frac{v x}{c^2} \right) \\ x' &= \gamma \left( x - v t \right)\\ y' &= y \\ z' &= z \end{align}

where v is the relative velocity between frames in the x-direction, c is the speed of light, and

\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 in general can take a continuous range of values. In the setup used here, positive relative velocity v > 0 is motion along the standard setup (positive directions of the xx axes), zero relative velocity v = 0 is no relative motion, while negative relative velocity v < 0 is relative motion in the opposite direction to the standard configuration, i.e. along the negative directions of the xx axes. The magnitude of relative velocity v cannot equal or exceed c, so that c < v < c. The corresponding range of γ is 1 ≤ γ < ∞. The transformations are well-defined if these ranges hold.

The transformations are not defined if v is outside these limits. If v = c, the transformations are undefined because γ is infinite. For v > c, the Lorentz factor is a complex number, and the transformations make no sense because they are complex-valued. The space and time coordinates are measurable quantities and numerically must be real numbers, not complex.

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.

Another important property is for relative speeds much less than the speed of light, the Lorentz transformations reduce to the Galilean transformation in accordance with the correspondence principle. Mathematically, as v → 0, c → ∞. In words, as relative velocity approaches 0, the speed of light (seems to) approach infinity. Hence, it is sometimes said that nonrelativistic physics is a physics of "instantaneous action at a distance".[10]

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, but it's very tedious. A much 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.

For spatial differences and time intervals, 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;

\Delta t' = \gamma \left( \Delta t - \frac{v \Delta x}{c^2} \right) \,,
\Delta x' = \gamma \left( \Delta x - v \Delta t \right) \,,

with inverse relations

\Delta t = \gamma \left( \Delta t' + \frac{v \Delta x'}{c^2} \right) \,,
\Delta x = \gamma \left( \Delta x' + v \Delta t' \right) \,.

where Δ (capital Delta) indicates a difference of quantities, e.g. Δx = x2x1 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 iterated 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 = xx0, Δx′ = x′ − x0, etc.

The Lorentz transformations have two implications which, although counterintuitive, are correct within the scope of special relativity.

  • Time dilation. In a frame F boosted relative to another frame F, time intervals are longer in F than those in F. If a time interval is measured at the same point in F, so that Δx = 0, then Δt′ = γΔt.
  • Length contraction. In a frame F boosted relative to another frame F, spatial lengths intervals are shorter in F than those in F. If a spatial length is measured at an instant of time in F, so that Δt′ = 0, then Δx = γΔx.

Sometimes it is more convenient to use β = v/c (lowercase beta) instead of v, so that

\begin{align} ct' &= \gamma \left( ct - \beta x \right) \,, \\ x' &= \gamma \left( x - \beta ct \right) \,, \\ \end{align}

which shows clearer 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.

Lorentz boosts in the Cartesian directions.

The above collection of equations apply only for a boost in the x-direction. The standard configuration works equally well in the y or z directions, so the results are similar. The coordinates perpendicular to the motion do not change, while those in the direction of relative motion do change with the time coordinate. Thus for a boost along the yy axes, using β = v/c for compactness,

\begin{align} ct' & = \gamma ( ct - \beta y) \\ x' &= x \\ y' &= \gamma ( y - \beta c t ) \\ z' & = z \end{align}

and for a boost along the zz axes

\begin{align} c t' &= \gamma ( c t - \beta z ) \\ x' &= x \\ y' &= y \\ z' &= \gamma ( z - \beta c t ) \end{align}

The inverse transformations are always found by exchanging primed and unprimed quantities and negating β.

Since the Lorentz transformations are a set of coupled linear equations, in other words a linear transformation, they can be written in a single matrix equation (see matrix product for the multiplication of these matrices). The separate algebraic equations are often used in practical calculations, but for theoretical purposes it is useful to collect all the separate equations into one matrix equation. For for a boost along the xx axes with velocity v, again using β = v/c for compactness, the transformation and its inverse are

\begin{bmatrix} c t' \\ x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} \gamma & -\beta \gamma & 0 & 0\\ -\beta \gamma & \gamma & 0 & 0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}

likewise for a boost along the yy axes

\begin{bmatrix} c t' \\ x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} \gamma&0&-\beta \gamma&0\\ 0&1&0&0\\ -\beta \gamma&0&\gamma&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}

and a boost along the zz axes

\begin{bmatrix} c t' \\ x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} \gamma&0&0&-\beta \gamma\\ 0&1&0&0\\ 0&0&1&0\\ -\beta \gamma&0&0&\gamma\\ \end{bmatrix} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}

where the coordinates are separated into column vectors, and the quantities defining the relative motion contained in the transformation matrix.

In the inverse transformations the transformation matrix will be the matrix inverse of the original transformation, for the boost along the xx axes

\begin{bmatrix} c t \\ x \\ y \\ z \end{bmatrix} = \begin{bmatrix} \gamma & \beta \gamma & 0 & 0\\ \beta \gamma & \gamma & 0 & 0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} c t' \\ x' \\ y' \\ z' \end{bmatrix}

and so on for the other directions. So instead of solving two or more equations algebraically, the original matrix equation can be inverted to find the inverse transformation.

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. A "composition" of two or more boosts can be made as follows. Suppose there are three frames instead of two, all three in standard configuration. If a frame F is boosted with velocity v1 relative to frame F along the xx axes, and another frame F′′ is boosted with velocity v2 relative to F along the xx′′ axes, then (suppressing the irrelevant y, z, y′, z′, y′′, z′′ coordinates)

\begin{bmatrix} c t'' \\ x'' \end{bmatrix} = \begin{bmatrix} \gamma_2 & -\gamma_2\beta_2 \\ -\gamma_2\beta_2 & \gamma_2 \end{bmatrix} \begin{bmatrix} c t' \\ x' \end{bmatrix} \,,\quad \begin{bmatrix} c t' \\ x' \end{bmatrix} = \begin{bmatrix} \gamma_1 & -\gamma_1\beta_1 \\ -\gamma_1\beta_1 & \gamma_1 \end{bmatrix} \begin{bmatrix} c t \\ x \end{bmatrix} \,,

where

\beta_1 = \frac{v_1}{c}\,,\quad \gamma_1 = \frac{1}{\sqrt{1-\beta_1^2}}\,,\quad \beta_2 = \frac{v_2}{c}\,,\quad \gamma_2 = \frac{1}{\sqrt{1-\beta_2^2}} \,.

Now, the relation between the frames F′′ and F must also a Lorentz transformation since these frames are simply boosted relative to each other along the same direction, just with different relative velocity v, so that

\begin{bmatrix}ct''\\ x''\end{bmatrix} =\begin{bmatrix}\gamma & -\gamma\beta\\ -\gamma\beta & \gamma \end{bmatrix}\begin{bmatrix}ct\\ x \end{bmatrix} =\begin{bmatrix}\gamma_1\gamma_2(1+\beta_2\beta_1) & -\gamma_1\gamma_2(\beta_1+\beta_2)\\ -\gamma_1\gamma_2(\beta_1+\beta_2) & \gamma_1\gamma_2(1+\beta_2\beta_1) \end{bmatrix}\begin{bmatrix}ct\\ x \end{bmatrix} \,,

from which

\gamma=\gamma_1\gamma_2(1+\beta_2\beta_1) \,,
\gamma\beta=\gamma_1\gamma_2(\beta_1+\beta_2)\,,

and after dividing the second equation by the first, it is clear the relative velocity of F′′ to F is nonlinearly determined from the separate relative velocities according to

\beta=\frac{\beta_1+\beta_2}{1+\beta_2\beta_1} \,.

This holds if the boosts are collinear as they are here (not just along the common x directions of each frame, but any direction). The relative velocities can be in the same or opposite directions, but must be collinear.

For two or more consecutive boosts in different directions the situation is more complicated. Suppose again there are three frames instead of two, all three in standard configuration, but this time if a frame F is boosted with velocity v1 relative to frame F along the yy axes, and another frame F′′ is boosted with velocity v2 relative to F along the zz′′ axes, then

\begin{bmatrix} c t'' \\ x'' \\ y'' \\ z'' \end{bmatrix} = \begin{bmatrix} \gamma_2&0&0&-\beta_2 \gamma_2\\ 0&1&0&0\\ 0&0&1&0\\ -\beta_2 \gamma_2&0&0&\gamma_2\\ \end{bmatrix} \begin{bmatrix} c t' \\ x' \\ y' \\ z' \end{bmatrix} \,, \quad \begin{bmatrix} c t' \\ x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} \gamma_1&0&-\beta_1 \gamma_1&0\\ 0&1&0&0\\ -\beta_1 \gamma_1&0&\gamma_1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} c t \\ x \\ y \\ z \end{bmatrix}

and the relation between the frames F′′ and F is

\begin{bmatrix} c t'' \\ x'' \\ y'' \\ z'' \end{bmatrix} = \begin{bmatrix}\gamma_1\gamma_2 & 0 & -\beta_1\gamma_1\gamma_2 & -\beta_2\gamma_2\\ 0 & 1 & 0 & 0\\ -\beta_1\gamma_1 & 0 & \gamma_1 & 0\\ -\beta_2\gamma_1\gamma_2 & 0 & \beta_1\beta_2\gamma_1\gamma_2 & \gamma_2 \end{bmatrix} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}

which cannot be rewritten in the form of a single boost alone. Unlike a single boost, this transformation matrix has no symmetry. It turns out this is not just a single boost, but a boost followed or preceded by a rotation. See velocity-addition formula and Thomas precession.

Suppose this setup is repeated, but the boosts are exchanged so that F is boosted with velocity v2 relative to frame F along the zz axes, and another frame F′′ is boosted with velocity v1 relative to F along the yy′′ axes, then

\begin{bmatrix} c t'' \\ x'' \\ y'' \\ z'' \end{bmatrix} = \begin{bmatrix} \gamma_1&0&-\beta_1 \gamma_1&0\\ 0&1&0&0\\ -\beta_1 \gamma_1&0&\gamma_1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} c t' \\ x' \\ y' \\ z' \end{bmatrix} \,, \quad \begin{bmatrix} c t' \\ x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} \gamma_2&0&0&-\beta_2 \gamma_2\\ 0&1&0&0\\ 0&0&1&0\\ -\beta_2 \gamma_2&0&0&\gamma_2\\ \end{bmatrix} \begin{bmatrix} c t \\ x \\ y \\ z \end{bmatrix}

and the relation between the frames F′′ and F is

\begin{bmatrix}ct''\\ x''\\ y''\\ z'' \end{bmatrix}=\begin{bmatrix}\gamma_1\gamma_2 & 0 & -\beta_1\gamma_1 & -\beta_2\gamma_1\gamma_2\\ 0 & 1 & 0 & 0\\ -\beta_1\gamma_1\gamma_2 & 0 & \gamma_1 & \beta_1\beta_2\gamma_1\gamma_2\\ -\beta_2\gamma_2 & 0 & 0 & \gamma_2 \end{bmatrix}\begin{bmatrix}ct\\ x\\ y\\ z \end{bmatrix}

which again is not a single boost, but a boost followed or preceded by a rotation. This transformation is not the same as the previous composition, which shows Lorentz boosts along different directions do not commute, changing their order changes the resultant transformation. It is, however, simply the transpose of the other matrix which follows from the transpose properties of the matrix product.[nb 1]

Rapidity parametrization

The momentarily co-moving inertial frames along the world line of a rapidly accelerating observer (center). The vertical direction indicates time, while the horizontal indicates distance, the dashed line is the spacetime trajectory ("world line") of the observer. The small dots are specific events in spacetime. If one imagines these events to be the flashing of a light, then the events that pass the two diagonal lines in the bottom half of the image (the past light cone of the observer in the origin) are the events visible to the observer. The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the momentarily co-moving inertial frame changes when the observer accelerates.

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)
\begin{align} ct' &= ct \cosh\zeta - x \sinh\zeta \\ x' &= x \cosh\zeta - ct \sinh\zeta \\ y' &= y \\ z' &= z \end{align}

where ζ (lowercase zeta) is a parameter called rapidity (many other symbols are used, including ϕ, φ, η, ψ, ξ). Given the strong resemblance to rotations of spatial coordinates in 3d space (in the Cartesian planes, or about the Cartesian axes), the Lorentz transformation can be thought of as a hyperbolic rotation of spacetime coordinates in 4d Minkowski space (in the Cartesian-time planes, here the xt plane). The parameter ζ represents 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 equation for a light pulse, according to the identity

\cosh^2\zeta - \sinh^2\zeta = 1 \,,

Using the definition

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

a consequence these two hyperbolic formaule is an identity which matches the Lorentz factor

\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

\beta = \tanh\zeta \,,
\gamma = \cosh\zeta \,,
\beta \gamma = \sinh\zeta \,.

Taking the inverse hyperbolic tangent gives the rapidity

\zeta = \tanh^{-1}\beta \,.

Since −1 < β < 1, it follows −∞ < ζ < ∞. Positive rapidity ζ > 0 is motion according to the standard setup (along the positive directions of the xx axes), zero rapidity ζ = 0 is no relative motion, while negative rapidity ζ < 0 is relative motion in the opposite direction (along the negative directions of the xx axes).

This diagram actually shows the inverse configuration of F "stationary" while F is boosted away along the negative x direction, although it correctly gives the original transformation since the coordinates ct, x of F are projected onto the coordinates ct′, x of F. The event (ct, x) = (8, 6) in F corresponds to approximately (ct′, x′) ≈ (5.55, 1.67) in F, with rapidity ζ ≈ −0.66. Notice the difference in length and time scales, such that the speed of light is invariant.

The geometric significance of the hyperbolic functions can be visualized as follows (the irrelevant y, z, y′, z coordinates will be suppressed). For x = 0 (events at the origin in F),

ct' = ct \cosh\zeta
x' = - ct \sinh\zeta

then squaring each and taking the difference gives the equation of a family of hyperbolae

(ct')^2 - {x'}^2 = (ct)^2

which is not just one curve but infinitely many. There is one hyperbola for each constant value of ct, while rapidity varies since it parametrizes each hyperbola. The ct axis is the set of ct values where x = 0, and this axis deviates from the ct axis by a constant value of rapidity, the value corresponding to the boost according to

\frac{x'}{ct'} = -\tanh\zeta = -\beta \,.

Likewise for ct = 0,

ct' = - x \sinh\zeta
x' = x \cosh\zeta

the hyperbolae corresponding to each x coordinate are

{x'}^2 - (ct')^2 = x^2

and the x axis is the set of x coordinates such that ct = 0, again deviated by the same value of rapidity.

The inverse transformation in the rapidity parametrization is straightforwards; as well as exchanging primed and unprimed quantities, negating rapidity ζ → −ζ is equivalent to negating the relative velocity, which follows from the relation between ζ and β. Therefore,

Inverse Lorentz boost (x direction with rapidity ζ)
\begin{align} ct & = ct' \cosh\zeta + x' \sinh\zeta \\ x &= x' \cosh\zeta + ct' \sinh\zeta \\ y &= y' \\ z &= z' \end{align}
This diagram shows the original configuration of F "stationary" while F is boosted away along the positive x direction, although it correctly gives the inverse transformation since the coordinates ct′, x of F are projected onto the coordinates ct, x of F. The event (ct′, x′) = (8, 6) in F corresponds to approximately (ct, x) ≈ (14.3, 13.28) in F, with rapidity ζ ≈ +0.66. Again, the difference in length and time scales is such that the speed of light is invariant.

The inverse transformations can be similarly visualized by considering the case when x′ = 0 (events at the origin in F) so that

ct = ct' \cosh\zeta
x = ct' \sinh\zeta

so the family of hyperbolae are

(ct)^2 - x^2 = (ct')^2

one for each constant value of ct but varying rapidity. The ct axis represents variable ct where x′ = 0, and constant rapidity given by

\frac{x}{ct} = \tanh\zeta = \beta \,.

Similarly, for ct′ = 0,

ct = x' \sinh\zeta
x = x' \cosh\zeta

the hyperbolae corresponding to each x coordinate are

x^2 - (ct)^2 = {x'}^2

and the x axis has constant rapidity and variable x for which ct′ = 0.

The rapidity relations can be substituted into the boost matrices along the Cartesian directions in the previous section. An additional detail is that rapidities can be added to obtain the overall rapidity, unlike relative velocities. If a frame F is boosted with rapidity ζ1 relative to frame F along the xx axes, and another frame F′′ is boosted with rapidity ζ2 relative to F along the xx′′ axes, so that (suppressing the irrelevant y, z, y′, z′, y′′, z′′ coordinates)

\begin{bmatrix} c t'' \\ x'' \end{bmatrix} = \begin{bmatrix} \cosh\zeta_2 &-\sinh\zeta_2 \\ -\sinh\zeta_2 & \cosh\zeta_2 \end{bmatrix} \begin{bmatrix} c t' \\ x' \end{bmatrix} \,,\quad \begin{bmatrix} c t' \\ x' \end{bmatrix} = \begin{bmatrix} \cosh\zeta_1 &-\sinh\zeta_1 \\ -\sinh\zeta_1 & \cosh\zeta_1 \end{bmatrix} \begin{bmatrix} c t \\ x \end{bmatrix} \,,

then ζ1 + ζ2 is the rapidity of the overall boost of F′′ relative to F,

\begin{bmatrix} c t'' \\ x'' \end{bmatrix} = \begin{bmatrix} \cosh(\zeta_1+\zeta_2) &-\sinh(\zeta_1+\zeta_2) \\ -\sinh(\zeta_1+\zeta_2) & \cosh(\zeta_1+\zeta_2) \end{bmatrix} \begin{bmatrix} c t \\ x \end{bmatrix} \,,

and the relative velocities are related to the rapidities by

\beta = \tanh(\zeta_1+\zeta_2) \,,\quad \beta_1 = \tanh\zeta_1 \,,\quad \beta_2 = \tanh\zeta_2 \,.

This holds if the boosts are along the same direction as they are here. Moreover, the hyperbolic identity

\tanh(\zeta_1+\zeta_2) = \frac{\tanh\zeta_1 + \tanh\zeta_2}{1+\tanh\zeta_1 \tanh\zeta_2}

coincides with the resultant relative velocity of the two relative velocities along the same direction.

Boost in any direction

Lorentz boost in a general direction defined by a unit vector n.
Boost in an arbitrary direction, with vector projection and rejection of the position vector as measured in each frame.

A boost in an arbitrary direction now depends on the full relative velocity vector v which has magnitude |v| = v. An observer in frame F observes F to move with relative velocity v, while an observer in F observes F to move with relative velocity v. The coordinate axes of each frame are still parallel and orthogonal. The magnitude of relative velocity |v| = v cannot equal or exceed c, so that 0 ≤ v < c. The vector analogue of β is simply β = v/c, and correspondingly its magnitude |β| = β cannot equal or exceed 1, so that 0 ≤ β < 1.

Again in the following, standard configuration is assumed, so at t = t′ = 0, the frames coincide at the origin, r = r′ = 0.

For the transformations in the x, y, and z directions, the coordinates perpendicular to the relative motion remain unchanged, while those parallel to the relative motion do change along with the time coordinate. For this reason, it is convenient to decompose the spatial position vector r = (x, y, z) as measured in F, and r′ = (x′, y′, z′) as measured in F′, each into components perpendicular and parallel to v = (vx, vy, vz),

\mathbf{r}=\mathbf{r}_\perp+\mathbf{r}_\|\,,\quad \mathbf{r}' = \mathbf{r}_\perp' + \mathbf{r}_\|' \,,

where ‖ means "parallel" to v and ⊥ means "perpendicular" to v.

The transition from the boost in any of the Cartesian directions, say the x direction, to a boost in any direction can be made from the identifications[nb 2]

\mathbf{v} = v\mathbf{e}_x \,,\quad \mathbf{r}_\parallel = x\mathbf{e}_x \,,\quad \mathbf{r}_\perp = y\mathbf{e}_y + z\mathbf{e}_z \,,

where ex, ey, ez are the Cartesian basis vectors, a set of mutually perpendicular unit vectors along their indicated directions. Then the Lorentz transformations take the form

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

where • indicates 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 v and not the direction.

These transformations are vector equations and therefore true in any direction. The transformations between the entire position vectors r and r can be constructed from these also. The parallel component can be found by vector projection

\mathbf{r}_\parallel = (\mathbf{r}\cdot\mathbf{n})\mathbf{n}

and the perpendicular component by vector rejection

\mathbf{r}_\perp = \mathbf{r} - (\mathbf{r}\cdot\mathbf{n})\mathbf{n}

where n = v/v = β/β is a unit vector in the direction of v. The procedure for r is identical. The unit vector has the advantage of simplifying equations and makes alternative parametrizations easier. The relative velocity is v = vn with magnitude v and direction n. Combining the results gives[nb 3]

Lorentz boost (in direction n with magnitude v)
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 t v\mathbf{n} \,.

There are three numbers which define the Lorentz boost in any direction, one for the magnitude v and two[nb 4] for the direction n, or in Cartesian components the three components of the relative velocity vector v = (vx, vy, vz).

Introducing the row and column vectors

\mathbf{r}' = \begin{bmatrix} x' \\ y' \\ z' \\ \end{bmatrix} \,, \quad \mathbf{n} = \begin{bmatrix} n_x \\ n_y \\ n_z \\ \end{bmatrix} \,, \quad \mathbf{n}^\mathrm{T} = \begin{bmatrix}n_x & n_y & n_z\end{bmatrix} \,, \quad \mathbf{r} = \begin{bmatrix} x \\ y \\ z \\ \end{bmatrix}

where the T indicates the matrix transpose (switch rows for columns and vice versa), the matrix form of the dot product is

\mathbf{n}\cdot\mathbf{r}\leftrightarrow \mathbf{n}^\mathrm{T}\mathbf{r}

and the Lorentz transformation can be written in block matrix form thus

\begin{bmatrix} c t' \\ \mathbf{r}' \end{bmatrix} = \begin{bmatrix} \gamma & - \gamma \beta\mathbf{n}^\mathrm{T} \\ -\gamma\beta\mathbf{n} & \mathbf{I} + (\gamma-1) \mathbf{n}\mathbf{n}^\mathrm{T} \\ \end{bmatrix} \begin{bmatrix} c t \\ \mathbf{r} \end{bmatrix}\,

where I is the 3×3 identity matrix. The column and row vectors n and nT and their product nnT have an origin in boost generators, as shown later. This block matrix version is useful for displaying the general form compactly, and illustrates the dependence on direction and the magnitude of the boost. For reference, the full form is explicitly

\begin{bmatrix} c\,t' \\ x' \\ y' \\ z' \end{bmatrix} = \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} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}\,.

This transformation is only a "boost", i.e., a transformation between two frames whose x, y, and z axis are parallel and whose spacetime origins coincide. The most general proper Lorentz transformation also contains a rotation of the three axes, because the composition of two boosts is not a pure boost but is a boost followed by a rotation. The rotation gives rise to Thomas precession. The boost is given by a symmetric matrix, but the general Lorentz transformation matrix need not be symmetric.

The inverse transformations are easy to obtain, as always exchange primed for unprimed indices and negate the relative velocity (which is relative motion in the opposite direction), v → −v, which also amounts to simply negating the unit vector n → −n since the magnitude v is always positive,

Inverse Lorentz boost (in direction n with magnitude v)
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} \,,

which in matrix form is

\begin{bmatrix} c t \\ \mathbf{r} \end{bmatrix} = \begin{bmatrix} \gamma & \gamma \beta\mathbf{n}^\mathrm{T} \\ \gamma\beta\mathbf{n} & \mathbf{I} + (\gamma-1) \mathbf{n}\mathbf{n}^\mathrm{T} \\ \end{bmatrix} \begin{bmatrix} c t' \\ \mathbf{r}' \end{bmatrix}\,.

To get the rapidity parametrization in any direction, the expressions γ = coshζ and γβ = sinhζ can be inserted into all the velocity-parametrized formulae above. Two additional details are that, using the same unit vector n = v/v = β/β, the vectorial relation between relative velocity and rapidity is[11]

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

and the "rapidity vector" can be defined as

\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 |ζ| = |ζ| confined to 0 ≤ ζ < ∞, which agrees with the range 0 ≤ β < 1. The direction of ζ is always parallel to n, and reversed relative velocity still corresponds to reversing the direction of n and hence ζ.

Rotations of frames (static)

For the simpler case

x^2 +y^2 + z^2 = {x'}^2 + {z'}^2 + {z'}^2\,,
t'= t \,,

the transformation between the coordinates are related by purely spatial rotations, so one frame is simply tilted relative to the other, and there is no relative motion. These also count as Lorentz transformations since they leave the line element invariant.

Throughout the axis–angle representation, one of many rotation formalisms in three dimensions, is used on the spatial position coordinates.

Rotations about the Cartesian axes

Rotation anticlockwise through angle θ of a Cartesian coordinate system about the three possible pairs of common Cartesian axes; when x = x, y = y, and z = z. The blue circles indicate the plane of rotation.

If a frame F with coordinates x′, y′, z′, t has its z axis coincident with the z axis of another frame F with coordinates x, y, z, t, each aligned in the same direction, and is rotated about the common zz axis (or equivalently in the coincident xy and xy planes) through an angle θ anticlockwise, the transformations of coordinates are

\begin{bmatrix}ct' \\ x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta & 0 \\ 0 & \sin \theta & \cos \theta & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} \begin{bmatrix}ct \\ x \\ y \\ z \end{bmatrix}

(The factor of the speed of light c is included for consistency with the Lorentz boosts, it is not essential to include here). The transformation matrix is parametrized by the angle θ, for a given rotation it is constant, but its values can be chosen from a continuous range (in radians) 0 ≤ θ ≤ 2π.

For a similar setup but with rotations about the common xx axes, we have

\begin{bmatrix}ct' \\ x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos \theta & -\sin \theta \\ 0 & 0 & \sin \theta & \cos \theta \\ \end{bmatrix} \begin{bmatrix}ct \\ x \\ y \\ z \end{bmatrix} \,,

as for the common yy axes

\begin{bmatrix}ct' \\ x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos \theta & 0 & \sin \theta \\ 0 & 0 & 1 & 0 \\ 0 & -\sin \theta & 0 & \cos \theta \\ \end{bmatrix} \begin{bmatrix}ct \\ x \\ y \\ z \end{bmatrix} \,.

All of these matrix equations can be partitioned into block matrix form

\begin{bmatrix} ct' \\ \mathbf{r}' \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & \mathbf{R} \end{bmatrix} \begin{bmatrix} ct \\ \mathbf{r}\end{bmatrix}\,,

making apparent which parts of the transformation matrix act on the time and spatial coordinates. The submatrix R is any one of the 3d rotation matrices about the Cartesian axes.

Also for each case, since frame F′ will measure the rotation to be through the same angle but in the opposite sense, simply negating the angle θ → −θ while leaving the direction of the rotation axis unchanged, and exchanging unprimed and primed coordinates, gives the inverse transformation.

Rotations about an axis in any direction

Rotation anticlockwise through angle θ of a Cartesian coordinate system about an axis defined from a unit vector a.
Decomposition of r into parallel and perpendicular components, and rotation of r anticlockwise through angle θ about a unit vector a defining an axis of rotation. The angle of rotation and axis vector can be combined into an "axis–angle vector" θ = θa.

If F is rotated about a unit vector a which defines the rotation axes, through an angle θ anticlockwise, the position vector r in this frame will rotate with the frame through the same angle about the same axis. The full transformation of position can be found by splitting the position vectors r and its rotated counterpart r into components parallel and perpendicular to a thus

\mathbf{r}=\mathbf{r}_\perp+\mathbf{r}_\|\,,\quad \mathbf{r}' = \mathbf{r}_\perp' + \mathbf{r}_\|' \,,

where ‖ means "parallel" to a and ⊥ means "perpendicular" to a. The parallel component will not change magnitude or direction,

\mathbf{r}_\parallel ' = \mathbf{r}_\parallel \,,

only the component perpendicular to the axis will retain its magnitude, but change direction, according to the rotation (see diagram)

\mathbf{r}_\perp ' = \mathbf{r}_\perp \cos\theta + \mathbf{a}\times\mathbf{r}_\perp \sin\theta \,.

The parallel component can be found from vector projection

\mathbf{r}_\parallel = (\mathbf{a}\cdot\mathbf{r})\mathbf{a} \,,

while the perpendicular component can be found from vector rejection

\mathbf{r}_\perp = \mathbf{r} - (\mathbf{a}\cdot\mathbf{r})\mathbf{a} = - \mathbf{a} \times (\mathbf{a}\times\mathbf{r}) \,.

Combining the results gives

Rotation (about axis a through angle θ)
t' = t
\mathbf{r}' = \mathbf{r} + \mathbf{a}\times(\mathbf{a}\times\mathbf{r})(1-\cos\theta) + \mathbf{a}\times\mathbf{r}\sin\theta \,.

To put this into matrix frorm, the components of the vector equations can be written out in full then organized into a matrix equation, but a more efficient way, which also connects with the Lie algebraic generators of rotation, is to introduce the cross product matrix

\mathbf{A} = \begin{bmatrix} 0 & -a_z & a_y \\ a_z & 0 & -a_x \\ -a_y & a_x & 0 \end{bmatrix}

so that

\mathbf{a}\times\mathbf{r} \leftrightarrow \mathbf{A}\mathbf{r} \,,
\mathbf{a}\times(\mathbf{a}\times\mathbf{r}) \leftrightarrow \mathbf{A}^2\mathbf{r} \,,

where again r is a column vector. In addition, the 3×3 identity matrix I allows the term containing r only to be written r = Ir. Then in block matrix form

\begin{bmatrix} ct' \\ \mathbf{r}' \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & \mathbf{I} + \mathbf{A}^2(1-\cos\theta) + \mathbf{A}\sin\theta \end{bmatrix} \begin{bmatrix} ct \\ \mathbf{r}\end{bmatrix}\,.

This matrix is neither symmetric nor antisymmetric. The inverse transformations are as always found by negating the angle and switching unprimed and primed quantities,

Inverse rotation (about axis a through angle θ)
t' = t
\mathbf{r} = \mathbf{r}' + \mathbf{a}\times(\mathbf{a}\times\mathbf{r}')(1-\cos\theta) - \mathbf{a}\times\mathbf{r}'\sin\theta \,,

in block matrix form

\begin{bmatrix} ct \\ \mathbf{r} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & \mathbf{I} + \mathbf{A}^2(1-\cos\theta) - \mathbf{A}\sin\theta \end{bmatrix} \begin{bmatrix} ct' \\ \mathbf{r}' \end{bmatrix}\,.

General Lorentz transformations: combined boosts and rotations

Another way to obtain the boost in an arbitrary direction is to rotate the coordinates into a boost along a direction for which the Lorentz transformation is simple and known, then perform that Lorentz transformation, then rotate back; summarized by[12]

B' = R_1 B R_2

where

R_1 = \begin{bmatrix} 1 & 0 \\ 0 & \mathbf{R}_1 \end{bmatrix}\,,\quad R_2 = \begin{bmatrix} 1 & 0 \\ 0 & \mathbf{R}_2 \end{bmatrix} \,,

in which R1 and R2 are 3d rotation matrices on the spatial coordinates only, leaving the time components unchanged. For example, B could be the Lorentz boost along any of the x, y, or z directions.

Boost and rotation matrices

In the matrix form of the Lorentz transformations, the components of the four positions are arranged into column vectors, and the Lorentz transformation denoted Λ (Greek capital Lambda) can always be compactly written as a single matrix equation of the form

X' = \Lambda X \,.

All "pure" Lorentz transformations take this form, those which do not include additional displacement in spacetime (this more general case is detailed later).

The general rotation matrix is (in block form for compactness)

R(\mathbf{a},\theta) = \begin{bmatrix} 1 & 0 \\ 0 & \mathbf{I} + \mathbf{A}^2(1-\cos\theta) + \mathbf{A}\sin\theta \end{bmatrix} \,,

and general boost matrix is

B(\mathbf{n},\zeta) = \begin{bmatrix} \cosh\zeta & - \mathbf{n}^\mathrm{T} \sinh\zeta \\ -\mathbf{n}\sinh\zeta & \mathbf{I} + (\cosh\zeta-1)\mathbf{n}\mathbf{n}^\mathrm{T} \\ \end{bmatrix} \,.

The general inverse rotation matrix is

R (\mathbf{a} ,\theta)^{-1} = R (\mathbf{a} ,-\theta) = R (-\mathbf{a} , \theta) \,.

similarly the general inverse boost matrix is

B (\mathbf{n} ,\zeta)^{-1} = B (\mathbf{n} ,-\zeta) = R (-\mathbf{n} , \zeta) \,.

Successive transformations are applied on the left. Two rotations are another rotation

R_2(\mathbf{a}_2,\theta_2) R_1(\mathbf{a}_1,\theta_1) = R_3(\mathbf{a}_3,\theta_3) \,,

but they are not commutative unless the rotations share the same axis,

R_2(\mathbf{a},\theta_2) R_1(\mathbf{a},\theta_1) = R_1(\mathbf{a},\theta_1) R_2(\mathbf{a},\theta_2) = R(\mathbf{a},\theta_1+\theta_2) \,.

Two boosts along different directions (not collinear with each other) form a boost followed or preceded by a rotation, instead of an another single boost,

B_2(\mathbf{n}_2,\zeta_2) B_1(\mathbf{n}_1,\zeta_1) = B(\mathbf{n},\zeta)R(\mathbf{a},\theta) \,,

however two boosts along the same direction form a boost along that direction without rotation, and are commutative,

B_2(\mathbf{n},\zeta_2) B_1(\mathbf{n},\zeta_1) = B_1(\mathbf{n},\zeta_1) B_2(\mathbf{n},\zeta_2) = B(\mathbf{n},\zeta_1+\zeta_2) \,,

The most general Lorentz transformation Λ is a boost and rotation, either can be performed before the other, but the results are different since the boost and rotation matrices do not commute. To show this explicitly, the boost followed by a rotation is

\Lambda_{RB} = R(\mathbf{a},\theta)B(\mathbf{n},\zeta) = \begin{bmatrix}\cosh\zeta & -\mathbf{n}^{\mathrm{T}}\sinh\zeta\\ -\mathbf{R}\mathbf{n}\sinh\zeta & \mathbf{R}(\mathbf{I}+(\cosh\zeta-1)\mathbf{n}\mathbf{n}^{\mathrm{T}}) \end{bmatrix}

while the rotation followed by a boost is

\Lambda_{BR} = B(\mathbf{n},\zeta)R(\mathbf{a},\theta) =\begin{bmatrix} \cosh\zeta & -\mathbf{n}^\mathrm{T}\mathbf{R}\sinh\zeta\\ -\mathbf{n}\sinh\zeta & (\mathbf{I}+(\cosh\zeta-1)\mathbf{n}\mathbf{n}^\mathrm{T})\mathbf{R} \end{bmatrix}

where again R = I + A2(1 − cosθ) + Asinθ is the general rotation matrix. It follows in general

\Lambda_{RB} \neq \Lambda_{BR} \,.

One important aspect of the boost and rotation matrices is they form a group \mathcal{L}, since

  • the operation of composition can be defined (here matrix multiplication),
  • boosts and rotations are Lorentz transformations, the product of any two (two rotations, two boosts, one rotation and boost) is also a Lorentz transformation, so the set of these matrices is closed under this operation of composition,
R(\mathbf{a},\theta) \in \mathcal{L} \,, \quad B(\mathbf{n},\zeta) \in \mathcal{L} \,,
R(\mathbf{a},\theta)B(\mathbf{n},\zeta) \in \mathcal{L} \,, \quad B(\mathbf{n},\zeta)R(\mathbf{a},\theta) \in \mathcal{L} \,,
R(\mathbf{a},0) = B(\mathbf{n},0) = \mathbf{I} \in \mathcal{L} \,,
  • there are inverse elements (those which correspond to the inverse Lorentz transformations, i.e. a rotation which "undoes" another rotation, likewise for boosts),
R(\mathbf{a},-\theta)R(\mathbf{a},\theta) = \mathbf{I} \,, \quad R(\mathbf{a},-\theta) \in \mathcal{L} \,,
B(\mathbf{n},-\zeta)B(\mathbf{n},\zeta) = \mathbf{I} \,, \quad B(\mathbf{n},-\zeta) \in \mathcal{L} \,,
  • given three transformations, each of which may be a rotation and/or boost, the composition is associative, for example
R_2(B R_1) = (R_2 B)R_1 = R_2 B R_1 \,.

The R and B matrices are the elements of the Lorentz group, an example of a Lie group. The parameters are continuous variables. The number of parameters in the group is six, since three are for the boost and three for the rotation, therefore the Lorentz group is six-dimensional.

Generators of the Lorentz group

All the group elements can be derived from the Lie-algebraic generators and parameters of the group. In this context, the generators of the Lorentz group are operators which correspond to important symmetries in spacetime: the rotation generators are physically angular momentum,

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}

and the boost generators correspond to the motion of the system in spacetime,

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}

In quantum mechanics, relativistic quantum mechanics, and quantum field theory, a different convention is used for the generators; they are all multiplied by a factor of the imaginary unit i = −1.

The general boost matrix can be derived from the generators as follows

B(\mathbf{n},\zeta) = e^{-\zeta\mathbf{n}\cdot\mathbf{K}} = I -\sinh\zeta(\mathbf{n}\cdot\mathbf{K})+(\cosh\zeta-1)(\mathbf{n}\cdot\mathbf{K})^2

and negating the rapidity in the exponential gives the inverse

B(\mathbf{n},-\zeta) = e^{\zeta\mathbf{n}\cdot\mathbf{K}} = I +\sinh\zeta(\mathbf{n}\cdot\mathbf{K})+(\cosh\zeta-1)(\mathbf{n}\cdot\mathbf{K})^2 \,.

The general rotation matrix is equivalent to Rodrigues' rotation formula

R(\mathbf{a},\theta) = e^{\theta\mathbf{a}\cdot\mathbf{J}}= I +\sin\theta(\mathbf{a}\cdot\mathbf{J})+(1-\cos\theta)(\mathbf{a}\cdot\mathbf{J})^2

and negating the angle gives the inverse

R(\mathbf{a},-\theta) = e^{-\theta\mathbf{a}\cdot\mathbf{J}} = I -\sin\theta(\mathbf{a}\cdot\mathbf{J})+(1-\cos\theta)(\mathbf{a}\cdot\mathbf{J})^2 \,.

Transformation of other physical quantities

Writing the general matrix transformation

\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}

in tensor index notation allows the transformation of other physical quantities which cannot be expressed as four-vectors, e.g. tensors or spinors in 4d spacetime, to be defined,

{X'}^\alpha = \Lambda^\alpha {}_\beta X^\beta \,,

where upper and lower indices label covariant and contravariant components respectively, and the summation convention is applied. It is a standard convention to use Greek indices which 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.

For rotations, the rotation matrix entries Λαβ = Rαβ can be found by comparing the full matrix equation with the Lorentz rotation about any axis to obtain[13]

R^0{}_0 = 1
R^0{}_i = R^i{}_0 = 0
R^i{}_j = a_i a_j + (\delta_{ij} - a_i a_j) \cos\theta - \varepsilon_{ijk} a_k \sin\theta

where δij is the Kronecker delta, εijk is the three-dimensional Levi-Civita symbol, and the correspondence between Cartesian components and spatial indices is made as follows, a1 = ax, a2 = ay, a3 = az, and similarly for other 3d vectors.

Similarly the boost matrix entries Λαβ = Bαβ can be found by comparing the full matrix equation with the Lorentz boost in any direction to obtain[14]

B^0{}_0 = \gamma = \cosh\zeta \,,
B^0{}_i = B^i{}_0 = - \gamma \beta n_i = - n_i \sinh\zeta \,,
B^i{}_j = B^j{}_i = \delta_{ij} + ( \gamma - 1 ) n_i n_j = \delta_{ij} + ( \cosh\zeta - 1 ) n_i n_j\,,

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^{\alpha'} = \Lambda^{\alpha'}{}_\alpha A^\alpha \,.

in which the primed indices denote the indices of A in the primed frame.

More generally, the transformation of any tensor quantity T is given by:[15]

T^{\alpha' \beta' \cdots \zeta'}_{\theta' \iota' \cdots \kappa'} = \Lambda^{\alpha'}{}_{\mu} \Lambda^{\beta'}{}_{\nu} \cdots \Lambda^{\zeta'}{}_{\rho} \Lambda_{\theta'}{}^{\sigma} \Lambda_{\iota'}{}^{\upsilon} \cdots \Lambda_{\kappa'}{}^{\zeta} T^{\mu \nu \cdots \rho}_{\sigma \upsilon \cdots \zeta}

where Λχ′ψ is the inverse matrix of Λχ′ψ.

Transformation of the electromagnetic field

Lorentz transformations can also be used to illustrate that magnetic and electric fields are simply different aspects of the same force — the electromagnetic force, as a consequence of relative motion between electric charges and observers.[16] The fact that the electromagnetic field shows relativistic effects becomes clear by carrying out a simple thought experiment.[17]

  • Consider an observer measuring a charge at rest in a reference 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 will not observe any magnetic field.
  • Consider another observer in frame F′ moving at relative velocity v (relative to F and the charge). This observer will see a different electric field because the charge is moving at velocity −v in their rest frame. Further, in frame F′ the moving charge constitutes an electric current, and thus the observer in frame F′ will also see a magnetic field.

This shows that the Lorentz transformation also applies to electromagnetic field quantities when changing the frame of reference, given below in vector form.

Spacetime interval

In a given coordinate system xμ, if two events 1 and 2 are separated by

(\Delta t, \Delta x, \Delta y, \Delta z) = (t_2-t_1, x_2-x_1, y_2-y_1, z_2-z_1)\ ,

the spacetime interval between them is given by

s^2 = - c^2(\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2\ .

This can be written in another form using the Minkowski metric. In this coordinate system,

\eta_{\mu\nu} = \begin{bmatrix} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix}\ .

Then, we can write

s^2 = \begin{bmatrix}c \Delta t & \Delta x & \Delta y & \Delta z \end{bmatrix} \begin{bmatrix} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix} \begin{bmatrix} c \Delta t \\ \Delta x \\ \Delta y \\ \Delta z \end{bmatrix}

or, using the Einstein summation convention,

s^2= \eta_{\mu\nu} x^\mu x^\nu\ .

Now suppose that we make a coordinate transformation xμxμ. Then, the interval in this coordinate system is given by

s'^2 = \begin{bmatrix}c \Delta t' & \Delta x' & \Delta y' & \Delta z' \end{bmatrix} \begin{bmatrix} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix} \begin{bmatrix} c \Delta t' \\ \Delta x' \\ \Delta y' \\ \Delta z' \end{bmatrix}

or

s'^2= \eta_{\mu\nu} x'^\mu x'^\nu\ .

It is a result of special relativity that the interval is an invariant. That is, s2 = s2, see invariance of interval. For this to hold, it can be shown[18] that it is necessary and sufficient for the coordinate transformation to be of the form

x'^\mu = x^\nu \Lambda^\mu_\nu + C^\mu\ ,

where Cμ is a constant vector and Λμν a constant matrix, where we require that

\eta_{\mu\nu}\Lambda^\mu_\alpha \Lambda^\nu_\beta = \eta_{\alpha\beta}\ .

Such a transformation is called a Poincaré transformation or an inhomogeneous Lorentz transformation.[19][20] The Ca represents a spacetime translation. When Ca = 0, the transformation is called an homogeneous Lorentz transformation, or simply a Lorentz transformation.

Taking the determinant of

\eta_{\mu\nu}{\Lambda^\mu}_\alpha{\Lambda^\nu}_\beta = \eta_{\alpha\beta}

gives us

\det (\Lambda^a_b) = \pm 1\ .

The cases are:

  • Proper Lorentz transformations have det(Λμν) = +1, and form a subgroup called the special orthogonal group SO(1,3).
  • Improper Lorentz transformations are det(Λμν) = −1, which do not form a subgroup, as the product of any two improper Lorentz transformations will be a proper Lorentz transformation.

From the above definition of Λ it can be shown that (Λ00)2 ≥ 1, so either Λ00 ≥ 1 or Λ00 ≤ −1, called orthochronous and non-orthochronous respectively. An important subgroup of the proper Lorentz transformations are the proper orthochronous Lorentz transformations which consist purely of boosts and rotations. Any Lorentz transform can be written as a proper orthochronous, together with one or both of the two discrete transformations; space inversion P and time reversal T, whose non-zero elements are:

P^0_0=1, P^1_1=P^2_2=P^3_3=-1
T^0_0=-1, T^1_1=T^2_2=T^3_3=1

The set of Poincaré transformations satisfies the properties of a group and is called the Poincaré group. Under the Erlangen program, Minkowski space can be viewed as the geometry defined by the Poincaré group, which combines Lorentz transformations with translations. In a similar way, the set of all Lorentz transformations forms a group, called the Lorentz group.

A quantity invariant under Lorentz transformations is known as a Lorentz scalar.

Alternative formalisms

The usual way to formulate the Lorentz transformation is tensor analysis and group theory as shown above. Other formalisms are shown below.

Gyrovector formalism

The composition of two Lorentz boosts B(u) and B(v) of velocities u and v is given by[21][22]

B(\mathbf{u})B(\mathbf{v})=B\left ( \mathbf{u}\oplus\mathbf{v} \right )\mathrm{Gyr}\left [ \mathbf{u},\mathbf{v}\right ]=\mathrm{Gyr}\left [\mathbf{u},\mathbf{v} \right ]B \left ( \mathbf{v}\oplus\mathbf{u} \right ),

where

  • B(v) is the 4 × 4 matrix that uses the components of v, i.e. v1, v2, v3 in the entries of the matrix, or rather the components of v/c in the representation that is used above,
  • \mathbf{u}\oplus\mathbf{v} is the velocity-addition,
  • Gyr[u,v] (capital G) is the rotation arising from the composition. If the 3 × 3 matrix form of the rotation applied to spatial coordinates is given by gyr[u,v], then the 4 × 4 matrix rotation applied to 4-coordinates is given by[23]
\mathrm{Gyr}[\mathbf{u},\mathbf{v}]= \begin{pmatrix} 1 & 0 \\ 0 & \mathrm{gyr}[\mathbf{u},\mathbf{v}] \end{pmatrix}\,
  • gyr (lower case g) is the gyrovector space abstraction of the gyroscopic Thomas precession, defined as an operator on a velocity w in terms of velocity addition:
\text{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u} \oplus \mathbf{v}) \oplus (\mathbf{u} \oplus (\mathbf{v} \oplus \mathbf{w}))
for all w.

The composition of two Lorentz transformations L(u, U) and L(v, V) which include rotations U and V is given by:[24]

L(\mathbf{u},U)L(\mathbf{v},V)=L(\mathbf{u}\oplus U\mathbf{v}, \mathrm{gyr}[\mathbf{u},U\mathbf{v}]UV)

See also

Footnotes

  1. ^ If A and B are n×n square matrices, the transpose of their product is the product of the transposes in the reverse order, (AB)T = BTAT. If these are symmetric matrices so that by definition, AT = A and BT = B, it follows (AB)T = BA.
  2. ^ Starting from the boost in the y direction
    \mathbf{v} = v\mathbf{e}_y \,,\quad \mathbf{r}_\parallel = y\mathbf{e}_y \,,\quad \mathbf{r}_\perp = x\mathbf{e}_x + z\mathbf{e}_z \,,
    or z direction
    \mathbf{v} = v\mathbf{e}_z \,,\quad \mathbf{r}_\parallel = z\mathbf{e}_z \,,\quad \mathbf{r}_\perp = x\mathbf{e}_x + y\mathbf{e}_y \,,
    obtains the same vectorial result.
  3. ^ In the position transformation, r can be neatly factorized using the dot and dyadic products of vectors
    \mathbf{r}' = - \gamma t v\mathbf{n} + ( \mathbf{I} + (\gamma-1)\mathbf{n}\mathbf{n})\cdot\mathbf{r} \,,
    where I is the unit dyadic and nn is the dyadic product of n with itself. The identities I · r = r and nn · r = n(n · r) follow from the definitions of the dot and dyadic products. However, dyadic tensors are an archaic formalism almost never used in this context.
  4. ^ Since n is a unit vector, only two components of the vector are independent, the absolute value of the third is given by its unit magnitude
    |\mathbf{n}|^2 = n_x^2 + n_y^2 + n_z^2 = 1
    (the sign must be chosen appropriately after taking the root to point in the correct direction). To see this another way, express the unit vector in the spherical polar angles in the Cartesian basis
    \mathbf{n} = \sin\theta_v(\cos\phi_v\mathbf{e}_x + \sin\phi_v\mathbf{e}_y) + \cos\theta_v\mathbf{e}_z
    where the subscript "v" refers to the relative velocity. This clearly shows two numbers uniquely define the direction.

Notes

  1. ^ John & O'Connor 1996
  2. ^ Brown 2003
  3. ^ Rothman 2006, pp. 112f.
  4. ^ Darrigol 2005, pp. 1–22
  5. ^ Macrossan 1986, pp. 232–34
  6. ^ The reference is within the following paper:Poincaré 1905, pp. 1504–1508
  7. ^ Einstein 1905, pp. 891–921
  8. ^ Young & Freedman 2008
  9. ^ Forshaw & Smith 2009
  10. ^ Einstein 1916
  11. ^ Barut 1964, p. 18–19
  12. ^ Barut 1964, p. 18–19
  13. ^
  14. ^ Misner, Thorne & Wheeler 1973 These authors give the results in terms of βi, not the unit vector used here. The substitution ni = βi obtains their result.
  15. ^ Carroll 2004, p. 22
  16. ^ Grant & Phillips 2008
  17. ^ Griffiths 2007
  18. ^ Weinberg 1972
  19. ^ Weinberg 2005, pp. 55–58
  20. ^ Ohlsson 2011, p. 3–9
  21. ^ Ungar 1989, pp. 1385–1396
  22. ^ Ungar 2000, pp. 331–342
  23. ^ Ungar 1989, pp. 1385–1396
  24. ^ Ungar 1988

References

Websites

Papers

  • . See also: English translation.