Coordinate transformation A "stationary" observer in frame defines events with coordinates , , , . Another frame moves with velocity relative to , and an observer in this "moving" frame defines events using the coordinates , , , . The coordinate axes in each frame are parallel (the and axes are parallel, the and axes are parallel, and the and axes are parallel), remain mutually perpendicular, and relative motion is along the coincident axes. At , the origins of both coordinate systems are the same, . In other words, the times and positions are coincident at this event. If all these hold, then the coordinate systems are said to be in
standard configuration, or
synchronized. If an observer in records an event , , , , then an observer in records the
same event with coordinates {{Equation box 1 \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 is the relative velocity between frames in the -direction, is the
speed of light, and \gamma = \frac{1}{ \sqrt{1 - \frac{v^2}{c^2}}} (lowercase
gamma) is the
Lorentz factor. Here, is the
parameter of the transformation, for a given boost it is a constant number, but can take a continuous range of values. In the setup used here, positive relative velocity is motion along the positive directions of the axes, zero relative velocity is no relative motion, while negative relative velocity is relative motion along the negative directions of the axes. The magnitude of relative velocity cannot equal or exceed , so only subluminal speeds are allowed. The corresponding range of is . The transformations are not defined if is outside these limits. At the speed of light () is infinite, and
faster than light () 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. As an
active transformation, an observer in notices the coordinates of the event to be "boosted" in the negative directions of the axes, because of the in the transformations. This has the equivalent effect of the
coordinate system boosted in the positive directions of the axes, while the event does not change and is simply represented in another coordinate system, a
passive transformation. The inverse relations (, , , in terms of , , , ) can be found by algebraically solving the original set of equations. A more efficient way is to use physical principles. Here is the "stationary" frame while is the "moving" frame. According to the principle of relativity, there is no privileged frame of reference, so the transformations from to must take exactly the same form as the transformations from to . The only difference is moves with velocity relative to (i.e., the relative velocity has the same magnitude but is oppositely directed). Thus if an observer in notes an event , , , , then an observer in notes the
same event with coordinates {{Equation box 1 \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 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. Sometimes it is more convenient to use (lowercase
beta) instead of , so that \begin{align} ct' &= \gamma \left( ct - \beta x \right) \,, \\ x' &= \gamma \left( x - \beta ct \right) \,, \\ \end{align} which shows much more clearly the symmetry in the transformation. From the allowed ranges of and the definition of , it follows . The use of and is standard throughout the literature. In the case of three spatial dimensions , where the boost \beta is in the
x direction, the
eigenstates of the transformation are with eigenvalue \sqrt{(1-\beta)/(1+\beta)}, with eigenvalue \sqrt{(1+\beta)/(1-\beta)}, and and , the latter two with eigenvalue 1. When the boost velocity \boldsymbol{v} is in an arbitrary vector direction with the boost vector \boldsymbol{\beta}=\boldsymbol{v}/c, then the transformation from an unprimed spacetime coordinate system to a primed coordinate system is given by \begin{bmatrix} ct' \vphantom{-\gamma\beta _\text{x}} \\ x' \vphantom{1+\frac{\gamma^2}{1+\gamma}\beta _\text{x}^2} \\ y' \vphantom{ \frac{\gamma^2}{1+\gamma}\beta _\text{x} \beta _\text{y} } \\ z' \vphantom{\frac{\gamma^2}{1+\gamma}\beta _\text{y}\beta _\text{z}} \end{bmatrix} = \begin{bmatrix} \gamma & -\gamma\beta _\text{x} & -\gamma\beta _\text{y} & -\gamma\beta _\text{z} \\ -\gamma\beta _\text{x} & 1+\frac{\gamma^2}{1+\gamma}\beta _\text{x}^2 & \frac{\gamma^2}{1+\gamma}\beta _\text{x} \beta _\text{y}& \frac{\gamma^2}{1+\gamma}\beta _\text{x}\beta _\text{z} \\ -\gamma\beta _\text{y} & \frac{\gamma^2}{1+\gamma}\beta _\text{x} \beta _\text{y} & 1+\frac{\gamma^2}{1+\gamma}\beta _\text{y}^2 & \frac{\gamma^2}{1+\gamma}\beta _\text{y}\beta _\text{z} \\ -\gamma\beta _\text{z} & \frac{\gamma^2}{1+\gamma}\beta _\text{x}\beta _\text{z} & \frac{\gamma^2}{1+\gamma}\beta _\text{y}\beta _\text{z} & 1+\frac{\gamma^2}{1+\gamma}\beta _\text{z}^2 \\ \end{bmatrix} \begin{bmatrix} ct \vphantom{-\gamma\beta _\text{x}} \\ x \vphantom{1+\frac{\gamma^2}{1+\gamma}\beta _\text{x}^2} \\ y \vphantom{ \frac{\gamma^2}{1+\gamma}\beta _\text{x} \beta _\text{y} } \\ z \vphantom{\frac{\gamma^2}{1+\gamma}\beta _\text{y}\beta _\text{z}} \end{bmatrix}, where the
Lorentz factor is \gamma =1/\sqrt{1 - \boldsymbol{\beta}^2} . The
determinant of the transformation matrix is +1 and its
trace is 2(1+\gamma). The inverse of the transformation is given by reversing the sign of \boldsymbol{\beta}. The quantity c^2t^2-x^2-y^2-z^2 is invariant under the transformation: namely (ct'^2-x'^2-y'^2-z'^2)=(ct^2-x^2-y^2-z^2). The Lorentz transformations can also be derived in a way that resembles circular rotations in 3-dimensional space using the
hyperbolic functions. For the boost in the direction, the results are {{Equation box 1 \begin{align} ct' &= ct \cosh\zeta - x \sinh\zeta \\ x' &= x \cosh\zeta - ct \sinh\zeta \\ y' &= y \\ z' &= z \end{align} where (lowercase
zeta) is a parameter called
rapidity (many other symbols are used, including , , , , , ). Given the strong resemblance to rotations of spatial coordinates in 3-dimensional space in the Cartesian , , and planes, a Lorentz boost can be thought of as a
hyperbolic rotation of spacetime coordinates in the xt, yt, and zt Cartesian-time planes of 4-dimensional
Minkowski space. The parameter is 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 spacetime interval, rather than a sum. The geometric significance of the hyperbolic functions can be visualized by taking or in the transformations. Squaring and subtracting the results, one can derive hyperbolic curves of constant coordinate values but varying , which parametrizes the curves according to the identity \cosh^2\zeta - \sinh^2\zeta = 1 \,. Conversely the and axes can be constructed for varying coordinates but constant . The definition \tanh\zeta = \frac{\sinh\zeta}{\cosh\zeta} \,, provides the link between a constant value of rapidity, and the
slope of the axis in spacetime. A consequence these two hyperbolic formulae 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 , , and are \begin{align} \beta &= \tanh\zeta \,, \\ \gamma &= \cosh\zeta \,, \\ \beta \gamma &= \sinh\zeta \,. \end{align} Taking the inverse hyperbolic tangent gives the rapidity \zeta = \tanh^{-1}\beta \,. Since , it follows . From the relation between and , positive rapidity is motion along the positive directions of the axes, zero rapidity is no relative motion, while negative rapidity is relative motion along the negative directions of the axes. The inverse transformations are obtained by exchanging primed and unprimed quantities to switch the coordinate frames, and negating rapidity since this is equivalent to negating the relative velocity. Therefore, {{Equation box 1 \begin{align} ct & = ct' \cosh\zeta + x' \sinh\zeta \\ x &= x' \cosh\zeta + ct' \sinh\zeta \\ y &= y' \\ z &= z' \end{align} The inverse transformations can be similarly visualized by considering the cases when and . So far the Lorentz transformations have been applied 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: \begin{align} \Delta t' &= \gamma \left( \Delta t - \frac{v \, \Delta x}{c^2} \right) \,, \\ \Delta x' &= \gamma \left( \Delta x - v \, \Delta t \right) \,, \end{align} with inverse relations \begin{align} \Delta t &= \gamma \left( \Delta t' + \frac{v \, \Delta x'}{c^2} \right) \,, \\ \Delta x &= \gamma \left( \Delta x' + v \, \Delta t' \right) \,. \end{align} where (uppercase
delta) indicates a difference of quantities; e.g., for two values of 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 in and in , 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., , , etc.
Physical implications 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 the equation for a pulse of light along the direction is , then in the Lorentz transformations give , and vice versa, for any . For relative speeds much less than the speed of light, the Lorentz transformations reduce to the
Galilean transformation: \begin{align} t' &\approx t \\ x' &\approx x - vt \end{align} in accordance with the
correspondence principle. It is sometimes said that nonrelativistic physics is a physics of "instantaneous action at a distance". Three counterintuitive, but correct, predictions of the transformations are: ;
Relativity of simultaneity : Suppose two events occur along the x axis simultaneously () in , but separated by a nonzero displacement . Then in , we find that \Delta t' = -\gamma {v\,\Delta x}/{c^2} , so the events are no longer simultaneous according to a moving observer. ;
Time dilation : Suppose there is a clock at rest in . If a time interval is measured at the same point in that frame, so that , then the transformations give this interval in by . Conversely, suppose there is a clock at rest in . If an interval is measured at the same point in that frame, so that , then the transformations give this interval in by . Either way, each observer measures the time interval between ticks of a moving clock to be longer by a factor than the time interval between ticks of his own clock. ;
Length contraction : Suppose there is a rod at rest in aligned along the axis, with length . In , the rod moves with velocity , so its length must be measured by taking two simultaneous () measurements at opposite ends. Under these conditions, the inverse Lorentz transform shows that . In the two measurements are no longer simultaneous, but this does not matter because the rod is at rest in . So each observer measures the distance between the end points of a moving rod to be shorter by a factor than the end points of an identical rod at rest in his own frame. Length contraction affects any geometric quantity related to lengths, so from the perspective of a moving observer, areas and volumes will also appear to shrink along the direction of motion.
Vector transformations The use of vectors allows positions and velocities to be expressed in arbitrary directions compactly. A single boost in any direction depends on the full relative
velocity vector with a magnitude that cannot equal or exceed , so that . Only time and the coordinates parallel to the direction of relative motion change, while those coordinates perpendicular do not. With this in mind, split the spatial
position vector as measured in , and as measured in , each into components perpendicular () and parallel () to , \mathbf{r}=\mathbf{r}_\perp+\mathbf{r}_\|\,,\quad \mathbf{r}' = \mathbf{r}_\perp' + \mathbf{r}_\|' \,, then the transformations are \begin{align} 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 \end{align} where is the
dot product. The Lorentz factor retains its definition for a boost in any direction, since it depends only on the magnitude of the relative velocity. The definition with magnitude is also used by some authors. Introducing a
unit vector in the direction of relative motion, the relative velocity is with magnitude and direction , and
vector projection and rejection give respectively \mathbf{r}_\parallel = (\mathbf{r}\cdot\mathbf{n})\mathbf{n}\,,\quad \mathbf{r}_\perp = \mathbf{r} - (\mathbf{r}\cdot\mathbf{n})\mathbf{n} Accumulating the results gives the full transformations, {{Equation box 1 \begin{align} 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} \,. \end{align} The projection and rejection also applies to . For the inverse transformations, exchange and to switch observed coordinates, and negate the relative velocity (or simply the unit vector since the magnitude is always positive) to obtain {{Equation box 1 \begin{align} 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} \,, \end{align} The unit vector has the advantage of simplifying equations for a single boost, allows either or to be reinstated when convenient, and the rapidity parametrization is immediately obtained by replacing and . It is not convenient for multiple boosts. The vectorial relation between relative velocity and rapidity is \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 is the absolute value of the rapidity scalar confined to , which agrees with the range .
Transformation of velocities , the ordering of vectors is chosen to reflect the ordering of the addition of velocities; first (the velocity of relative to ) then (the velocity of relative to ) to obtain (the velocity of relative to ). Defining the coordinate velocities and Lorentz factor by \mathbf{u} = \frac{d\mathbf{r}}{dt} \,,\quad \mathbf{u}' = \frac{d\mathbf{r}'}{dt'} \,,\quad \gamma_\mathbf{v} = \frac{1}{\sqrt{1-\dfrac{\mathbf{v}\cdot\mathbf{v}}{c^2}}} taking the differentials in the coordinates and time of the vector transformations, then 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] The velocities and are the velocity of some massive object. They can also be for a third inertial frame (say ), in which case they must be
constant. Denote either entity by . Then moves with velocity relative to , or equivalently with velocity relative to , in turn moves with velocity relative to . The inverse transformations can be obtained in a similar way, or as with position coordinates exchange and , and change to . The transformation of velocity is useful in
stellar aberration, the
Fizeau experiment, and the
relativistic Doppler effect. The
Lorentz transformations of acceleration can be similarly obtained by taking differentials in the velocity vectors, and dividing these by the time differential.
Transformation of other quantities In general, given four quantities and and their Lorentz-boosted counterparts and , a relation of the form A^2 - \mathbf{Z}\cdot\mathbf{Z} = {A'}^2 - \mathbf{Z}'\cdot\mathbf{Z}' implies the quantities transform under Lorentz transformations similar to the transformation of spacetime coordinates; \begin{align} A' &= \gamma \left(A - \frac{v\mathbf{n}\cdot \mathbf{Z}}{c} \right) \,, \\ \mathbf{Z}' &= \mathbf{Z} + (\gamma-1)(\mathbf{Z}\cdot\mathbf{n})\mathbf{n} - \frac{\gamma A v\mathbf{n}}{c} \,. \end{align} The decomposition of (and ) into components perpendicular and parallel to is exactly the same as for the position vector, as is the process of obtaining the inverse transformations (exchange and to switch observed quantities, and reverse the direction of relative motion by the substitution ). The quantities collectively make up a
four-vector, where is the "timelike component", and the "spacelike component". Examples of and are the following: For a given object (e.g., particle, fluid, field, material), if or correspond to properties specific to the object like its
charge density,
mass density,
spin, etc., its properties can be fixed in the rest frame of that object. Then the Lorentz transformations give the corresponding properties in a frame moving relative to the object with constant velocity. This breaks some notions taken for granted in non-relativistic physics. For example, the energy of an object is a scalar in non-relativistic mechanics, but not in relativistic mechanics because energy changes under Lorentz transformations; its value is different for various inertial frames. In the rest frame of an object, it has a
rest energy and zero momentum. In a boosted frame its energy is different and it appears to have a momentum. Similarly, in non-relativistic quantum mechanics the spin of a particle is a constant vector, but in
relativistic quantum mechanics spin depends on relative motion. In the rest frame of the particle, the spin pseudovector can be fixed to be its ordinary non-relativistic spin with a zero timelike quantity , however a boosted observer will perceive a nonzero timelike component and an altered spin. Not all quantities are invariant in the form as shown above, for example orbital
angular momentum does not have a timelike quantity, and neither does the
electric field nor the
magnetic field . The definition of angular momentum is , and in a boosted frame the altered angular momentum is . Applying this definition using the transformations of coordinates and momentum leads to the transformation of angular momentum. It turns out transforms with another vector quantity related to boosts, see
Relativistic angular momentum for details. For the case of the and fields, the transformations cannot be obtained as directly using vector algebra. The
Lorentz force is the definition of these fields, and in it is while in it is . A method of deriving the EM field transformations in an efficient way which also illustrates the unit of the electromagnetic field uses tensor algebra,
given below. == Mathematical formulation ==