# Lorentz transformation

In physics, the Lorentz transformation (or transformations) are coordinate transformations between two inertial frames that move at constant velocity relative to each other.

In each reference frame, an observer can use a local coordinate system (most exclusively Cartesian coordinates in this context) to measure lengths, and a clock to measure time intervals. An observer is a real or imaginary entity that can take measurements, say humans, or any other living organism—or even robots and computers. 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 smaller 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 Lorentz transformation is in accordance with special relativity, but was derived before special relativity. The transformations are named after the Dutch physicist Hendrik Lorentz.

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 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, also looked for 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

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 t 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 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 inertial frames for events connected by light signals. The quantity on the left is called the spacetime interval between events (t1, x1, y1, z1) and (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 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. Now one observes that a linear solution to the simpler problem

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

solves the general problem too. 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 Lorentz transformation is thus an element of the group O(3, 1) or, for those that prefer the other metric signature, O(1, 3).[nb 2]

## 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.

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" only refers to transformations between inertial frames, usually in the context of special relativity.

Transformations describing relative motion with constant velocity and without rotation of the space coordinate axes are called a boosts, and the relative velocity between the frames is the parameter of the transformation. The other basic type of Lorentz transformations is rotations in the spatial coordinates only, these are also inertial frames 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.).

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).

### Coordinate transformation

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, in other words only subluminal speeds c < v < c are allowed. 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. At luminal speed (v = c) γ is infinite, and for superluminal speeds (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, 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 → ∞, γ → 1, so that

$t'\approx t$
$x'\approx x - vt$

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 Lorentz transformation can be viewed as an active and passive 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 (an active transformation). 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. Here the Lorentz transformations are employed as passive transformations.

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

 Inverse 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}

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.

The above Lorentz transformations apply to one event. If there are two events, there is a spatial separation and time interval between them. 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 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 = xx0, Δx′ = x′ − x0, etc.

The Lorentz transformations have two implications that, though counter-intuitive, 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.

The above collection of equations apply only for a boost in the x-direction. The boosts in the y or z directions are almost identical, 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.

### Vector transformations

Further information: Euclidean vector and vector projection
Boost in an arbitrary direction, the position vector as measured in each frame is split into components parallel and perpendicular to the relative velocity vector v. Left: Standard configuration. Right: Inverse configuration.

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. For standard configuration, at t = t′ = 0, the frames coincide at the origin, r = r′ = 0.

It is convenient to decompose the spatial position vector r as measured in F, and r as measured in F′, each into components perpendicular and parallel to v,

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

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

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

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 into the direction of relative motion

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

and the perpendicular component by vector rejection from the direction of relative motion

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

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 for a single boost, allows either v or β to be reinstated when convenient, and makes alternative parametrizations easier. It is not convenient for multiple boosts. The relative velocity is v = vn with magnitude v and direction n. Combining the results gives

 Lorentz boost (in direction n with magnitude v) $t' = \gamma \left(t - \frac{v\mathbf{n}\cdot \mathbf{r}}{c^2} \right) \,,$ $\mathbf{r}' = \mathbf{r} + (\gamma-1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n} - \gamma t v\mathbf{n} \,.$

Three numbers define the Lorentz boost in any direction: one for the magnitude v and two[nb 3] for the direction n, or in Cartesian components the three components of the relative velocity vector v = (vx, vy, vz).

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) $t = \gamma \left(t' + \frac{\mathbf{r}' \cdot v\mathbf{n}}{c^{2}} \right) \,,$ $\mathbf{r} = \mathbf{r}' + (\gamma-1)(\mathbf{r}'\cdot\mathbf{n})\mathbf{n} + \gamma t' v\mathbf{n} \,,$

### Matrix transformations

Since the Lorentz transformations are 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.

Returning to the general case, introduce 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), then 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

$\begin{bmatrix} c\,t' \\ x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} \gamma&-\gamma\beta n_x&-\gamma\beta n_y&-\gamma\beta n_z\\ -\gamma\beta n_x&1+(\gamma-1)n_x^2&(\gamma-1)n_x n_y&(\gamma-1)n_x n_z\\ -\gamma\beta n_y&(\gamma-1)n_y n_x&1+(\gamma-1)n_y^2&(\gamma-1)n_y n_z\\ -\gamma\beta n_z&(\gamma-1)n_z n_x&(\gamma-1)n_z n_y&1+(\gamma-1)n_z^2\\ \end{bmatrix} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}\,.$

The boost matrix equals its transpose, it is a symmetric matrix. 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 is the matrix inverse of the original transformation. As always, the inverse transformation matrix is simply obtained by negating the relative velocity v.

For for a boost along the xx axes with velocity v, simply take nx = 1 and ny = nz = 0 to obtain

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

with similar matrices for the y and z directions. The inverse boost along the xx axes is

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

and so on for the other directions.

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 that are not collinear but in different directions, the result is still a Lorentz transformation, but not a single boost. Moreover, Lorentz boosts along different directions do not commute, changing their order changes the resultant transformation.

The non-commutativity of Lorentz boosts is another unintuitive feature of special relativity that is unlike Galilean relativity. In Newtonian mechanics any pair of Galilean boosts can be performed in either order, and the result is the same Galilean transformation. The general problem of two boosts each in any direction is outlined later.

## Rapidity parametrization

The momentarily co-moving inertial frames along the trajectory ("world line") of a rapidly accelerating observer (center). The vertical direction indicates time, while the horizontal indicates distance, the dashed line is the spacetime of the observer. The small dots are specific events in spacetime. 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 that 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 by taking x = 0 or ct = 0 in the transformations, see diagram (the irrelevant y, z, y′, z coordinates are suppressed). From them, one can derive hyperbolic curves of constant coordinate values but varying ζ, which parametrizes the curves. Conversely the ct and x axes can be constructed for varying coordinates but constant ζ.

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 cases when x′ = 0 and 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.

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.

## Transformation of velocities

Further information: differential of a function
The ordering of operands in the definition is chosen to reflect the ordering of the addition of velocities, first v (the velocity of F′ relative to F) then u (the velocity of X relative to F′) to obtain u = vu (the velocity of X relative to F).
If the velocities are switched in the velocity addition, a different composite velocity is obtained.

So far only the Lorentz transformation of position and time coordinates have been given. Since the concept of a Lorentz transformation depends on specifying inertial frames, and the constant relative velocity between any two, it is instructive to look at the transformation of velocities between inertial frames.

For the boost in the x direction, the differentials in the coordinates and time are

$dx' = \gamma ( dx - v dt ) \,, \quad dy' = dy \,, \quad dz' = dz$
$dt' = \gamma \left( dt - \frac{v dx}{c^2} \right)$

and defining the Cartesian components of the coordinate velocity as measured in F by

$u_x = \frac{dx}{dt} \,, \quad u_y = \frac{dy}{dt} \,, \quad u_z = \frac{dz}{dt}$

the corresponding definitions in F′ are found by dividing the coordinate differentials by the time differentials

$u_x' = \frac{dx'}{dt'} = \frac{u_x - v}{1 - \dfrac{u_x v}{c^2}}$
$u_y' = \frac{dy'}{dt'} = \frac{u_y}{\gamma \left( 1 - \dfrac{u_x v}{c^2} \right)}$
$u_z' = \frac{dz'}{dt'} = \frac{u_z}{\gamma \left( 1 - \dfrac{u_x v}{c^2} \right)}$

This is the Lorentz transformation of the velocity components for an x-boost only. The Lorentz factor applies to the relative velocity v between the frames, not the coordinate velocities. The inverse transformations follow by negating v and exchange the primed and unprimed quantities, just as for the coordinate transformations.

Taking this to the full extent; u is the coordinate velocity measured in F and u the corresponding velocity in F', and F' is boosted with velocity v relative to F. Defining

$\mathbf{u} = \frac{d\mathbf{r}}{dt} \,,\quad \mathbf{u}' = \frac{d\mathbf{r}'}{dt'} \,,\quad \gamma_\mathbf{v} = \frac{1}{\sqrt{1-\dfrac{v^2}{c^2}}} = \frac{1}{\sqrt{1-\dfrac{\mathbf{v}\cdot\mathbf{v}}{c^2}}}$

and repeating the process of taking differentials and dividing equations leads to

$\mathbf{u}'=\frac{1}{ 1-\frac{\mathbf{v}\cdot\mathbf{u}}{c^2} }\left[\frac{\mathbf{u}}{\gamma_\mathbf{v}}-\mathbf{v}+\frac{1}{c^2}\frac{\gamma_\mathbf{v}}{\gamma_\mathbf{v}+1}\left(\mathbf{u}\cdot\mathbf{v}\right)\mathbf{v}\right]$

Negating v and exchanging the primed and unprimed vectors gives the inverse transformation. In the nonrelativistic limit v0, γ → 1, and the Galilean transformations of velocity are recovered

$\mathbf{u}'\approx\mathbf{u}-\mathbf{v}\,,\quad\mathbf{u}\approx\mathbf{u}'+\mathbf{v}$

Only in the non-relativistic limit can we use ordinary vector addition of two velocities to obtain the resultant velocity. The general rule for combining velocities in relativistic mechanics can be inferred from the transformation, and it turns out to be substantially more complicated for two arbitrary velocities. Nevertheless, it is useful for the general case of two Lorentz boosts in the next section.

The velocities u and u are the velocity of some massive object, or a third inertial frame (say F′′) provided for this case they are constant. Denote either entity by X. Then X moves with velocity u relative to F, or equivalently with velocity u relative to F′, in turn F′ moves with velocity v relative to F. This can motivate the operation and notation for the relativistic velocity addition formula of two 3-velocites using the inverse transformation[nb 4]

$\mathbf{v}\oplus\mathbf{u}'=\mathbf{u}=\frac{1}{1+\frac{\mathbf{v}\cdot\mathbf{u}'}{c^2}}\left[\frac{\mathbf{u}'}{\gamma_\mathbf{v}}+\mathbf{v}+\frac{1}{c^2}\frac{\gamma_\mathbf{v}}{\gamma_\mathbf{v}+1}(\mathbf{u}'\cdot\mathbf{v})\mathbf{v}\right]$

The inverse transformation is used since this gives the desired velocity u (that of X relative to F), and correctly reduces to the Galilean sum of velocities for speeds much less than c. If the original transformation was used, a difference in velocities would be obtained and we would have u instead. Although the relativistic velocity addition has the unfortunate properties of being nonlinear, non-commutative, and non-associative, it does correctly obtain a velocity with a magnitude less than c. If ordinary vector addition was used, then it would be possible to obtain a velocity with magnitude larger than c, which is unphysical for Lorentz transformations.

If X is a third frame F′′ and all three are in standard configuration, then the trajectory of F′′ must pass through the origins of F and F′ at the initial time as measured in all of these frames (which synchronizes them), i.e., r′′ = r′ = r = 0 at t′′ = t′ = t = 0.

The Lorentz transformations of other kinematic or dynamic quantities, such as momentum and acceleration, can be obtained from the above results.

## Two general boosts

So far, all the Lorentz transformations are only boosts, 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.

### Standard configuration

Suppose again there are three frames all in standard configuration. If F′′ is boosted with velocity β2 relative to F', and F' boosted with velocity β1 relative to F, then the separate boosts (rewritten to resemble velocity addition) are

$\begin{bmatrix}c\,t''\\ \mathbf{r}'' \end{bmatrix}=\begin{bmatrix}\gamma_2 & -\gamma_2\boldsymbol{\beta}_2^\mathrm{T}\\ -\gamma_2\boldsymbol{\beta}_2 & \mathbf{I}+\frac{\gamma_2^2}{\gamma_2+1}\boldsymbol{\beta}_2\boldsymbol{\beta}_2^\mathrm{T} \end{bmatrix}\begin{bmatrix}c\,t'\\ \mathbf{r}' \end{bmatrix}\,\quad\begin{bmatrix}c\,t'\\ \mathbf{r}' \end{bmatrix}=\begin{bmatrix}\gamma_1 & -\gamma_1\boldsymbol{\beta}_1^\mathrm{T}\\ -\gamma_1\boldsymbol{\beta}_1 & \mathbf{I}+\frac{\gamma_1^2}{\gamma_1+1}\boldsymbol{\beta}_1\boldsymbol{\beta}_1^\mathrm{T} \end{bmatrix}\begin{bmatrix}c\,t\\ \mathbf{r} \end{bmatrix}$

where

$\boldsymbol{\beta}_1 = \frac{\mathbf{v}_1}{c} \,, \quad \boldsymbol{\beta}_2 = \frac{\mathbf{v}_2}{c} \,, \quad \gamma_1 = \frac{1}{\sqrt{1-\beta_1^2}} \,, \quad \gamma_2 = \frac{1}{\sqrt{1-\beta_2^2}}$

The Lorentz transformation from frame F to F′′ is[12]

$\begin{bmatrix}c\,t''\\ \mathbf{r}'' \end{bmatrix}=\begin{bmatrix}\gamma & -\mathbf{a}^\mathrm{T}\\ -\mathbf{b} & \mathbf{M} \end{bmatrix}\begin{bmatrix}c\,t\\ \mathbf{r} \end{bmatrix}$

where the result of the matrix product is not written for its enormous size, instead the following abbreviations are used,

$\gamma=\gamma_2\gamma_1(1+\boldsymbol{\beta}_2^\mathrm{T}\boldsymbol{\beta}_1)$
$\mathbf{a}^\mathrm{T}=\gamma_2\left(\gamma_1\boldsymbol{\beta}_1^\mathrm{T}+\boldsymbol{\beta}_2^\mathrm{T}+\frac{\gamma_1^2}{\gamma_1+1}(\boldsymbol{\beta}_2^\mathrm{T}\boldsymbol{\beta}_1)\boldsymbol{\beta}_1^\mathrm{T}\right)=\gamma(\boldsymbol{\beta}_1\oplus\boldsymbol{\beta}_2)^\mathrm{T}$
$\mathbf{b}=\gamma_1\left(\gamma_2\boldsymbol{\beta}_2+\boldsymbol{\beta}_1+\frac{\gamma_2^2}{\gamma_2+1}\boldsymbol{\beta}_2(\boldsymbol{\beta}_2^\mathrm{T}\boldsymbol{\beta}_1)\right)=\gamma(\boldsymbol{\beta}_2\oplus\boldsymbol{\beta}_1)$
$\mathbf{M} =\mathbf{I}+\frac{\gamma_1^2}{\gamma_1+1}\boldsymbol{\beta}_1\boldsymbol{\beta}_1^\mathrm{T}+\frac{\gamma_2^2}{\gamma_2+1}\boldsymbol{\beta}_2\boldsymbol{\beta}_2^\mathrm{T}+\gamma_2\gamma_1\left[1+(\boldsymbol{\beta}_2^\mathrm{T}\boldsymbol{\beta}_1)\frac{\gamma_2}{\gamma_2+1}\frac{\gamma_1}{\gamma_1+1}\right]\boldsymbol{\beta}_2\boldsymbol{\beta}_1^\mathrm{T}$

Like the other boldface vectors in the matrix equations, the vectors a and b are column vectors. Also, M a 3×3 square matrix. The first equation is the Lorentz factor of the composite transformation, the second two give the two composite velocities found from the relativistic velocity addition ⊕. The magnitudes of a and b are equal

$|\mathbf{a}| = |\mathbf{b}| = \sqrt{\gamma^2 - 1}$

but clearly the vectors are different, so the directions are different, suggesting one is a rotated copy of the other.

The overall Lorentz transformation cannot be a single boost alone as the matrix is not symmetric, and the next simplest assumption is that it equals a boost followed or preceded by a rotation. While a boost mixes all the space and time coordinates together, a rotation mixes together the spatial coordinates only. The 4d matrix to perform the rotation is simply

$R = \begin{bmatrix} 1 & 0 \\ 0 & \mathbf{R} \end{bmatrix}$

where R is a 3d rotation matrix, and for the purposes of this article the formulae and conventions from the axis-angle representation will be used here. The inverse of R corresponds to rotations using the same axis and angle, but in the opposite sense. The rotation matrix is orthogonal, so the transpose equals the inverse.

### Inverse configuration

For the inverse transformation, F is boosted with velocity −β1 relative to F′, and F′ boosted with velocity −β2 relative to F′′, so that

$\begin{bmatrix}c\,t\\\mathbf{r}\end{bmatrix}=\begin{bmatrix}\gamma_1 & \gamma_1\boldsymbol{\beta}_1^\mathrm{T}\\ \gamma_1\boldsymbol{\beta}_1 & \mathbf{I}+\frac{\gamma_1^2}{\gamma_1+1}\boldsymbol{\beta}_1\boldsymbol{\beta}_1^\mathrm{T} \end{bmatrix} \begin{bmatrix}c\,t'\\ \mathbf{r}'\end{bmatrix} \,,\quad \begin{bmatrix}c\,t'\\\mathbf{r}'\end{bmatrix} =\begin{bmatrix}\gamma_2 & \gamma_2\boldsymbol{\beta}_2^\mathrm{T}\\ \gamma_2\boldsymbol{\beta}_2 & \mathbf{I}+\frac{\gamma_2^2}{\gamma_2+1}\boldsymbol{\beta}_2\boldsymbol{\beta}_2^\mathrm{T} \end{bmatrix}\begin{bmatrix}c\,t''\\ \mathbf{r}''\end{bmatrix}$

Combining these and carrying out the block matrix multiplication, using the previous abbreviations the result is

$\begin{bmatrix}c\,t\\\mathbf{r}\end{bmatrix}=\begin{bmatrix}\gamma & \mathbf{b}^\mathrm{T}\\ \mathbf{a} & \mathbf{M}^\mathrm{T} \end{bmatrix}\begin{bmatrix}c\,t''\\ \mathbf{r}'' \end{bmatrix}$

In the separate boosts, the double primed and unprimed quantities can be switched, and velocities negated. This cannot be done for the overall Lorentz transformation, given the inverse Lorentz transformation matrix.

### Boost followed by a rotation

Setup of the original configuration, F′′ is boosted with velocity v2 relative to F′, and F′ is boosted with velocity v1 relative to F. An observer in F′ notices F and F′′ to be boosted with coordinate axes parallel. However, an observer in frame F will notice the frame F′′ to be rotated clockwise and boosted.
The inverse configuration of the original configuration. F′ is boosted with velocity v2 relative to F′′, and F is boosted with velocity v1 relative to F′. An observer in F′ still notices F and F′′ to be boosted with coordinate axes parallel. However, an observer in frame F′′ will notice the frame F to be rotated anticlockwise (about the same axis), and boosted with a rotated composite velocity.

Decompose the Lorentz transformation into a boost followed by a rotation ,

$\begin{bmatrix}\gamma & -\mathbf{a}^\mathrm{T}\\ -\mathbf{b} & \mathbf{M} \end{bmatrix} = \begin{bmatrix}1 & 0\\ 0 & \mathbf{R} \end{bmatrix}\begin{bmatrix}\gamma & -\gamma\boldsymbol{\beta}^\mathrm{T}\\ -\gamma\boldsymbol{\beta} & \mathbf{I}+\frac{\gamma^2}{\gamma+1}\boldsymbol{\beta}\boldsymbol{\beta}^\mathrm{T} \end{bmatrix}$

and it follows the composite velocity β, the velocity of F′′ relative to F, is found from a,

$\mathbf{a}^\mathrm{T}=\gamma\boldsymbol{\beta}^\mathrm{T} \quad\Rightarrow\quad \boldsymbol{\beta} = \boldsymbol{\beta}_1\oplus\boldsymbol{\beta}_2 \,.$

while the other possible composite velocity is a rotation of β,

$\mathbf{b}=\gamma\mathbf{R}\boldsymbol{\beta} \quad\Rightarrow\quad \mathbf{b}=\mathbf{R}\mathbf{a}$

Is the velocity of F relative to F′′ simply the negative of β? The answer is no.

To see this, look at the inverse configuration; the frame F is rotated anticlockwise through the same angle and axis. The inverse decomposition is thus a boost (the exact nature to be determined, owing to the complication of relativistic velocity addition) followed by an inverse rotation

$\begin{bmatrix}\gamma & \mathbf{b}^\mathrm{T}\\ \mathbf{a} & \mathbf{M}^\mathrm{T} \end{bmatrix}=\begin{bmatrix}1 & 0\\ 0 & \mathbf{R}^\mathrm{T} \end{bmatrix}\begin{bmatrix}\gamma & -\gamma\boldsymbol{\beta}^\mathrm{T}\\ -\gamma\boldsymbol{\beta} & \mathbf{I}+\frac{\gamma^2}{\gamma+1}\boldsymbol{\beta}\boldsymbol{\beta}^\mathrm{T} \end{bmatrix}$

so this time the composite velocity β, the velocity of F relative to F′′, is this time found from b,

$\mathbf{b}^\mathrm{T}=-\gamma\boldsymbol{\beta}^\mathrm{T}\quad\Rightarrow\quad\boldsymbol{\beta} = - \boldsymbol{\beta}_2\oplus\boldsymbol{\beta}_1$

and the other composite velocity is the inverse rotation,

$\mathbf{a}=-\gamma\mathbf{R}^\mathrm{T}\boldsymbol{\beta} \quad \Rightarrow \quad \mathbf{a}=\mathbf{R}^\mathrm{T}\mathbf{b}$

The vectors a and b are indeed related by a rotation. The rotation matrix R is the same for the rotation between a and b, as well as the coordinates r and r′′.

This peculiar result of two relative velocities seems to contradict the symmetry of relative motion between two frames, as found for a single boost between two inertial frames, but there is no paradox. The two velocities are geometrically different owing to their magnitude and directions. Algebraically, they arise from the non-commutativity of velocity addition and Lorentz boosts. Physically from the perspectives of the observers, the relative rotation between F and F′′ means the composite velocity should appear to be different. An observer in F notices F′′ to move with velocity β1β2 relative to F, conversely an observer in F′′ notices F to move with velocity β2β1 relative to F′′.

So why does the frame F′ not appear to rotate from the perspective of either F or F′′, if the axes of this frame are parallel to F and F′′ ? The frames F and F′′ are boosted relative to each other with composite velocities that have different directions. The frame F′ is only boosted with one velocity relative to the other frames, not two, so there is no rotation.

### Rotation followed by a boost

If the velocities are exchanged, an observer in F will notice F′′ to be rotated through the same angle and axis, but boosted with a different composite velocity, that is a rotation of the other composite velocity.
The inverse of the original configuration with exchanged velocities.

It is also possible to decompose the Lorentz transformation into a rotation followed by a boost,

$\begin{bmatrix}\gamma & -\mathbf{a}^\mathrm{T}\\ -\mathbf{b} & \mathbf{M}\end{bmatrix} = \begin{bmatrix}\gamma & -\gamma\boldsymbol{\beta}^\mathrm{T}\\ -\gamma\boldsymbol{\beta} & \mathbf{I}+\frac{\gamma^2}{\gamma+1}\boldsymbol{\beta}\boldsymbol{\beta}^\mathrm{T} \end{bmatrix}\begin{bmatrix}1 & 0\\0 & \mathbf{R}\end{bmatrix}$

so the composite velocity β, the velocity of F′′ relative to F, is this time found from b,

$\boldsymbol{\beta} = \boldsymbol{\beta}_2\oplus\boldsymbol{\beta}_1$

The inverse transformation follows a similar decomposition procedure,

$\begin{bmatrix}\gamma & \mathbf{b}^\mathrm{T}\\ \mathbf{a} & \mathbf{M}^\mathrm{T} \end{bmatrix}=\begin{bmatrix}\gamma & -\gamma\boldsymbol{\beta}^\mathrm{T}\\ -\gamma\boldsymbol{\beta} & \mathbf{I}+\frac{\gamma^2}{\gamma+1}\boldsymbol{\beta}\boldsymbol{\beta}^\mathrm{T} \end{bmatrix}\begin{bmatrix}1 & 0\\ 0 & \mathbf{R}^\mathrm{T} \end{bmatrix}$

and the composite velocity β, the velocity of F′′ relative to F, is found from a,

$\boldsymbol{\beta}= -\boldsymbol{\beta}_1\oplus\boldsymbol{\beta}_2$

The vectors a and b are still related by the same rotation. The key effect in exchanging the rotation and boost is the separate boost velocities are exchanged in the final composite velocity.

### Thomas rotation

The above formulae constitute the relativistic velocity addition and the Thomas rotation explicitly in the general Lorentz transformations. Throughout, the important formula

$\mathbf{M}=\mathbf{R}+\frac1{\gamma+1}\mathbf{b}\mathbf{a}^\mathrm{T}$

holds, allowing the rotation matrix (and hence the axis and angle) to be defined completely in terms of the relative velocities β1 and β2.

In the axis–angle representation, the general 3d rotation matrix is[nb 5]

$\mathbf{R} = \mathbf{I} + \mathbf{E}\sin\epsilon + \mathbf{E}^2(1-\cos\epsilon) \,,$

where the components of a unit vector e parallel to the axis are arranged into the antisymmetric matrix

$\mathbf{E} = \begin{bmatrix} 0 & -e_z & e_y \\ e_z & 0 & -e_x \\ -e_y & e_x & 0 \end{bmatrix}$

(which should not be confused for the Cartesian unit vectors ex, ey, ez). Here the right-handed convention for the spatial coordinates is used (see orientation (vector space)), so that rotations are positive in the anticlockwise sense according to the right-hand rule, and negative in the clockwise sense. This matrix rotates any 3d vector about the axis e through angle ε anticlockwise (an active transformation), which has the equivalent effect of rotating the coordinate frame clockwise about the same axis through the same angle (a passive transformation).

Starting from a, the matrix R rotates this into b anticlockwise, it follows their cross product (in the right-hand convention)

$\mathbf{a}\times\mathbf{b}= \frac{\gamma_1\gamma_2(\gamma^2 -1)(\gamma+\gamma_2+\gamma_1+1)}{(\gamma_2+1)(\gamma_1+1)(\gamma+1)} \boldsymbol{\beta}_1\times\boldsymbol{\beta}_2$

defines the axis correctly, therefore the axis is also parallel to β1×β2, geometrically this is perpendicular to the plane of the boost velocities. Since the magnitude of β1×β2 is neither interesting nor important, only the direction is, it is customary to normalize the vector into the unit vector above thus

$\mathbf{e} = \frac{\boldsymbol{\beta}_1\times\boldsymbol{\beta}_2}{|\boldsymbol{\beta}_1\times\boldsymbol{\beta}_2|}$

which still completely defines the direction of the axis without loss of information.

The angle of a rotation matrix in the axis–angle representation can be found from the trace of the rotation matrix, the general result for any axis is

$\cos\epsilon = \frac{\mathrm{tr}(\mathbf{R}) - 1}{2}$

and using the relation between M, R, aT, b, the angle ε that F′′ is rotated clockwise about e is determined completely by the velocities,[13][14][15]

$\cos\epsilon = \frac{(1+\gamma+\gamma_1+\gamma_2)^2}{(1+\gamma)(1+\gamma_1)(1+\gamma_2)} - 1$

The angle ε between a and b is not the same as the angle α between β1 and β2. The two cross products are

$\mathbf{a}\times \mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin\epsilon \mathbf{e}$
$\boldsymbol{\beta}_1 \times \boldsymbol{\beta}_2 = |\boldsymbol{\beta}_1||\boldsymbol{\beta}_2|\sin\alpha \mathbf{e}$

hence the angles are related by[16]

$\sin\epsilon = |\boldsymbol{\beta}_1||\boldsymbol{\beta}_2|\sin\alpha \frac{\gamma_1\gamma_2 (\gamma+\gamma_2+\gamma_1+1)}{(\gamma_2+1)(\gamma_1+1)(\gamma+1)}$

so they cannot be equal in general.

The rotation is simply a "static" rotation and there is no relative rotational motion between the frames, there is relative translational motion in the boost. However, if the frames accelerate, then the rotated frame rotates with an angular velocity. This effect is known as the Thomas precession, and arises purely from the kinematics of successive Lorentz boosts.

## Tensor formulation

For the notation used, see Ricci calculus.

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 that 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 that 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.

### 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[17] 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.[18][19] 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 is a proper Lorentz transformation.

From the above definition of Λ it can be shown that (Λ00)2 ≥ 1, so either

• Λ00 ≥ 1, called orthochronous transformations or
• Λ00 ≤ −1, called non-orthochronous transformations.

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.

### Transformation of other physical quantities

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:[20]

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

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

### Transformation of the electromagnetic field

For the transformation rules, see classical electromagnetism and special relativity.

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.[21] The fact that the electromagnetic field shows relativistic effects becomes clear by carrying out a simple thought experiment.[22]

• 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 does not observe any magnetic field.
• Consider another observer in frame F′ moving at relative 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. Further, in frame F′ the moving charge constitutes an electric current, and thus the observer in frame F′ also sees 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.

## Footnotes

1. ^ One can imagine that in each inertial frame there are observers positioned throughout space, each endowed with a synchronized clock and at rest in the particular inertial frame. These observers then report to a central office, where a report is 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., Sard (1970).
2. ^ The groups O(3, 1) and O(1, 3) are isomorphic. It is widely believed that the choice between the two metric signatures has no physical relevance, even though some objects related to O(3, 1) and O(1, 3) respectively, e.g., the Clifford algebras corresponding to the different signatures of the bilinear form associated to the two groups, are non-isomorphic.
3. ^ 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.
4. ^ Notations and conventions for the velocity addition vary from author to author. Different symbols may be used for the operation, and the operands may be switched for the same expression. Since the velocity addition is non-commutative, we cannot naïvely switch the operands without changing the result. For the same expression
$\frac{1}{1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^2}}\left[\mathbf{v}+\mathbf{u}+\frac{1}{c^2}\frac{\gamma_\mathbf{u}}{\gamma_\mathbf{u}+1}\mathbf{u}\times(\mathbf{u}\times\mathbf{v})\right]$
the different symbolic conventions are
Left-to-right ordering of operands
• Ungar (1988, 1989), u*v
Right-to-left ordering of operands
• Sexl & Urbantke (1992) $\mathbf{v}\circ\mathbf{u}$
5. ^ In the literature, the 3d rotation matrix may be denoted by other symbols like D, others use a name and the relative velocity vectors u, v involved, e.g., tom[u, v] for "Thomas rotation" or gyr[u, v] for "gyration" (see gyrovector space). These are nothing more than alternative names for the same R in this article. Correspondingly the 4d rotation matrices may be denoted
$\mathrm{Tom} = \begin{bmatrix}1 & 0 \\ 0 & \mathrm{tom}[\mathbf{u}, \mathbf{v}] \end{bmatrix} \quad \text{or} \quad \mathrm{Gyr} = \begin{bmatrix}1 & 0 \\ 0 & \mathrm{gyr}[\mathbf{u}, \mathbf{v}] \end{bmatrix}$
which are nothing more than alternative names for the same R (non-bold italic) in this article.

## 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. ^ Sexl & Urbantke 1992, pp. 40 The notation and form of the general Lorentz transformation matrix in this article follows these authors, but not the convention for velocity addition, this article uses a more common convention and notation.
13. ^ Macfarlane 1962
14. ^ Sexl & Urbantke 1992, pp. 4, 11, 41
15. ^ Gourgoulhon 2013, pp. 213
16. ^ Ungar 1989, p. 170
17. ^ Weinberg 1972
18. ^ Weinberg 2005, pp. 55–58
19. ^ Ohlsson 2011, p. 3–9
20. ^ Carroll 2004, p. 22
21. ^ Grant & Phillips 2008
22. ^ Griffiths 2007

## References

### Papers

• Mocanu, C. I. (1986). "Some difficulties within the framework of relativistic electrodynamics". Archiv für elektrotechnik (Springer) 69: 97–110.
• Mocanu, C. I. (1992). "On the relativistic velocity composition paradox and the Thomas rotation". Foundations of Physics (Plenum) 5: 443–456.

### Books

• Young, H. D.; Freedman, R. A. (2008). University Physics – With Modern Physics (12th ed.). ISBN 0-321-50130-6.
• Halpern, A. (1988). 3000 Solved Problems in Physics. Schaum Series. Mc Graw Hill. p. 688. ISBN 978-0-07-025734-4.
• Forshaw, J. R.; Smith, A. G. (2009). Dynamics and Relativity. Manchester Physics Series. John Wiley & Sons Ltd. p. 124–126. ISBN 978-0-470-01460-8.
• Grant, I. S.; Phillips, W. R. (2008). "14". Electromagnetism. Manchester Physics (2nd ed.). John Wiley & Sons. ISBN 0-471-92712-0.
• Sard, R. D. (1970). Relativistic Mechanics - Special Relativity and Classical Particle Dynamics. New York: W. A. Benjamin. ISBN 978-0805384918.