Lorentz transformation
Coordinate transformation

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.

If an observer in F records an event t, x, y, z, then an observer in F ′ records the same event with coordinates

t ′ = γ (t − v x c 2) x ′ = γ (x − v t) y ′ = y z ′ = z {\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\\y'&=y\\z'&=z\end{aligned}}}

where v is the relative velocity between frames in the x -direction, c is the speed of light, and γ = 1 1 − v 2 c 2 {\displaystyle \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 can take a continuous range of values. In the setup used here, positive relative velocity v > 0 is motion along the positive directions of the xx ′ axes, zero relative velocity v = 0 is no relative motion, while negative relative velocity v < 0 is relative motion along the negative directions of the xx ′ axes. The magnitude of relative velocity v cannot equal or exceed c, so only subluminal speeds − c < v < c are allowed. The corresponding range of γ is 1 ≤ γ < ∞.

The transformations are not defined if v is outside these limits. At the speed of light (v = c) γ is infinite, and faster than light (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.

As an active 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. 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, a passive transformation.

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. A 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

t = γ (t ′ + v x ′ c 2) x = γ (x ′ + v t ′) y = y ′ z = z ′, {\displaystyle {\begin{aligned}t&=\gamma \left(t'+{\frac {vx'}{c^{2}}}\right)\\x&=\gamma \left(x'+vt'\right)\\y&=y'\\z&=z',\end{aligned}}}

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.

Sometimes it is more convenient to use β = v / c (lowercase beta) instead of v, so that c t ′ = γ (c t − β x), x ′ = γ (x − β c t), {\displaystyle {\begin{aligned}ct'&=\gamma \left(ct-\beta x\right)\,,\\x'&=\gamma \left(x-\beta ct\right)\,,\\\end{aligned}}} which shows much more clearly 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.

When the boost velocity v {\displaystyle {\boldsymbol {v}}} is in an arbitrary vector direction with the boost vector β = v / c {\displaystyle {\boldsymbol {\beta }}={\boldsymbol {v}}/c}, then the transformation from an unprimed spacetime coordinate system to a primed coordinate system is given by

=, {\displaystyle {\begin{bmatrix}ct'\\x'\\y'\\z'\\\end{bmatrix}}={\begin{bmatrix}\gamma &-\gamma \beta _{x}&-\gamma \beta _{y}&-\gamma \beta _{z}\\-\gamma \beta _{x}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}^{2}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{y}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{z}\\-\gamma \beta _{y}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{y}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{y}^{2}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{y}\beta _{z}\\-\gamma \beta _{z}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{z}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{y}\beta _{z}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{z}^{2}\\\end{bmatrix}}{\begin{bmatrix}ct\\x\\y\\z\end{bmatrix}},}

where the Lorentz factor is γ = 1 / 1 − β 2 {\displaystyle \gamma =1/{\sqrt {1-{\boldsymbol {\beta }}^{2}}}}. The determinant of the transformation matrix is +1 and its trace is 2 (1 + γ) {\displaystyle 2(1+\gamma)}. The inverse of the transformation is given by reversing the sign of β {\displaystyle {\boldsymbol {\beta }}}.

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

c t ′ = c t cosh ζ − x sinh ζ x ′ = x cosh ζ − c t sinh ζ y ′ = y z ′ = z {\displaystyle {\begin{aligned}ct'&=ct\cosh \zeta -x\sinh \zeta \\x'&=x\cosh \zeta -ct\sinh \zeta \\y'&=y\\z'&=z\end{aligned}}}

where ζ (lowercase zeta) is a parameter called rapidity (many other symbols are used, including θ, ϕ, φ, η, ψ, ξ). Given the strong resemblance to rotations of spatial coordinates in 3d space in the Cartesian xy, yz, and zx 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 4d Minkowski space. The parameter ζ 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 x = 0 or ct = 0 in the transformations. Squaring and subtracting the results, one can derive hyperbolic curves of constant coordinate values but varying ζ, which parametrizes the curves according to the identity cosh 2 ζ − sinh 2 ζ = 1. {\displaystyle \cosh ^{2}\zeta -\sinh ^{2}\zeta =1\,.}

Conversely the ct and x axes can be constructed for varying coordinates but constant ζ. The definition tanh ζ = sinh ζ cosh ζ, {\displaystyle \tanh \zeta ={\frac {\sinh \zeta }{\cosh \zeta }}\,,} provides the link between a constant value of rapidity, and the slope of the ct axis in spacetime. A consequence these two hyperbolic formulae is an identity that matches the Lorentz factor cosh ζ = 1 1 − tanh 2 ζ. {\displaystyle \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 β = tanh ζ, γ = cosh ζ, β γ = sinh ζ. {\displaystyle {\begin{aligned}\beta &=\tanh \zeta \,,\\\gamma &=\cosh \zeta \,,\\\beta \gamma &=\sinh \zeta \,.\end{aligned}}}

Taking the inverse hyperbolic tangent gives the rapidity ζ = tanh − 1 β. {\displaystyle \zeta =\tanh ^{-1}\beta \,.}

Since −1 < β < 1, it follows −∞ < ζ < ∞. From the relation between ζ and β, positive rapidity ζ > 0 is motion along the positive directions of the xx ′ axes, zero rapidity ζ = 0 is no relative motion, while negative rapidity ζ < 0 is relative motion along the negative directions of the xx ′ axes.

The inverse transformations are obtained by exchanging primed and unprimed quantities to switch the coordinate frames, and negating rapidity ζ → − ζ since this is equivalent to negating the relative velocity. Therefore,

c t = c t ′ cosh ζ + x ′ sinh ζ x = x ′ cosh ζ + c t ′ sinh ζ y = y ′ z = z ′ {\displaystyle {\begin{aligned}ct&=ct'\cosh \zeta +x'\sinh \zeta \\x&=x'\cosh \zeta +ct'\sinh \zeta \\y&=y'\\z&=z'\end{aligned}}}

The inverse transformations can be similarly visualized by considering the cases when x ′ = 0 and ct ′ = 0.

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;

Δ t ′ = γ (Δ t − v Δ x c 2), Δ x ′ = γ (Δ x − v Δ t), {\displaystyle {\begin{aligned}\Delta t'&=\gamma \left(\Delta t-{\frac {v\,\Delta x}{c^{2}}}\right)\,,\\\Delta x'&=\gamma \left(\Delta x-v\,\Delta t\right)\,,\end{aligned}}} with inverse relations Δ t = γ (Δ t ′ + v Δ x ′ c 2), Δ x = γ (Δ x ′ + v Δ t ′). {\displaystyle {\begin{aligned}\Delta t&=\gamma \left(\Delta t'+{\frac {v\,\Delta x'}{c^{2}}}\right)\,,\\\Delta x&=\gamma \left(\Delta x'+v\,\Delta t'\right)\,.\end{aligned}}}

where Δ (uppercase delta) indicates a difference of quantities; e.g., Δ x = x − x 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:
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