# Lorentz transformation
The **Lorentz transformation** is the set of formulas of special relativity that defines how to convert the space and time coordinates of an event between two inertial reference frames in relative motion: given the coordinates one observer assigns to an event, it yields the coordinates a second observer moving at constant velocity would assign to the same event.[1](https://www.einstein-online.info/en/explandict/lorentz-transformation-2/) It replaces the Galilean transformation of Newtonian mechanics and is the mathematical core of special relativity.
| Key fact | Detail |
|---|---|
| Standard form (boost along x) | \( t' = \gamma(t - vx/c^{2}) \), \( x' = \gamma(x - vt) \), \( y' = y \), \( z' = z \), with \( \gamma = 1/\sqrt{1 - v^{2}/c^{2}} \)[2](https://openstax.org/books/university-physics-volume-3/pages/5-5-the-lorentz-transformation) |
| Inverse | Obtained by exchanging primed and unprimed variables and replacing \( v \) with \( -v \)[2](https://openstax.org/books/university-physics-volume-3/pages/5-5-the-lorentz-transformation) |
| Invariant quantity | The spacetime interval \( \Delta s^{2} = \Delta x^{2} + \Delta y^{2} + \Delta z^{2} - c^{2}\Delta t^{2} \) is unchanged by the transformation; proper time is likewise invariant[2](https://openstax.org/books/university-physics-volume-3/pages/5-5-the-lorentz-transformation) |
| Velocity addition | \( u = (u' + v)/(1 + u'v/c^{2}) \); if \( u' = c \) then \( u = c \), so light has the same speed in every frame[3](https://www.physics.ucla.edu/~pheuer/content/docs/PhysNotes.pdf) |
| Group structure | The transformations form the Lorentz group; boosts in different directions do not commute, and two non-collinear boosts compose as a boost plus a Wigner rotation[4](https://www.pks.mpg.de/~haque/maynooth_MP352/lecturenotes/MP352notes_LorentzGroup_c_01.pdf) |
| Rapidity | Writing \( \beta = v/c \) and \( \eta = \tanh^{-1}\beta \), a boost acts like a rotation by a hyperbolic angle; collinear boosts simply add rapidities[5](https://ocw.mit.edu/courses/8-033-relativity-fall-2006/909e205adee074e5401e0a7c7cce06c0_lecture6_kinem3.pdf) |
| Electromagnetism | Maxwell's equations are invariant under the transformation, and electric and magnetic fields mix under it[6](https://www.feynmanlectures.caltech.edu/II_26.html) |
## The transformation equations
Consider two inertial frames S and S′, with S′ moving at velocity \( v \) along the x-axis of S and their origins coinciding at \( t = t' = 0 \). The [Galilean transformation](https://www.edgepedia.org/galilean-transformation), \( x = x' + vt \) with \( t = t' \), cannot preserve a spherical light pulse, which must have radius \( r = ct \) in one frame and \( r' = ct' \) in the other; the condition \( x^{2} - c^{2}t^{2} = x'^{2} - c^{2}t'^{2} \) rules it out for any nonzero relative velocity.[2](https://openstax.org/books/university-physics-volume-3/pages/5-5-the-lorentz-transformation) The transformation that satisfies it is
\[ t' = \frac{t - vx/c^{2}}{\sqrt{1 - v^{2}/c^{2}}}, \qquad x' = \frac{x - vt}{\sqrt{1 - v^{2}/c^{2}}}, \qquad y' = y, \qquad z' = z, \]
where \( \gamma = 1/\sqrt{1 - v^{2}/c^{2}} \geq 1 \). The inverse transformation follows by interchanging primed and unprimed variables and replacing \( v \) with \( -v \).[2](https://openstax.org/books/university-physics-volume-3/pages/5-5-the-lorentz-transformation)
In matrix form the boost mixes \( x \) and \( ct \) exactly as a rotation mixes \( x \) and \( y \), with \( \gamma \) and \( \beta\gamma \) playing the roles of cosine and sine:[2](https://openstax.org/books/university-physics-volume-3/pages/5-5-the-lorentz-transformation)
\[ x' = \gamma x - \beta\gamma\, ct, \qquad ct' = -\beta\gamma\, x + \gamma\, ct. \]
The analogy is not exact: because the metric of spacetime has a minus sign on the time term, a Lorentz transformation involving the time axis does not keep the axes perpendicular or preserve the length scale along each axis, as an ordinary spatial rotation would.[2](https://openstax.org/books/university-physics-volume-3/pages/5-5-the-lorentz-transformation)
## Consequences
**Time dilation and length contraction.** For a clock moving at velocity \( v \), the transformation gives \( T = \gamma T_{0} \): more time passes in the rest frame than on the moving clock, the statement that moving clocks run slow. For a rod of proper length \( L_{0} \) moving lengthwise, the rest-frame measurement gives \( L = L_{0}/\gamma \). The contraction occurs only along the direction of motion; the y and z dimensions are not distorted.[3](https://www.physics.ucla.edu/~pheuer/content/docs/PhysNotes.pdf)
**Relativity of simultaneity.** Events simultaneous in one frame are generally not simultaneous in another, because the time coordinate depends on position through the \( vx/c^{2} \) term. Whether two events are time-like separated (some frame exists in which they occur at the same place) or space-like separated (some frame exists in which they are simultaneous) is agreed on by all observers, since \( \Delta s^{2} \) is invariant.[2](https://openstax.org/books/university-physics-volume-3/pages/5-5-the-lorentz-transformation)
**Velocity addition.** Dividing the transformed displacement by the transformed time gives the Einstein velocity addition rule,
\[ u = \frac{u' + v}{1 + u'v/c^{2}}. \]
For \( u', v \ll c \) it reduces to the classical \( u = u' + v \), and if \( u' = c \) the result is \( u = c \): light moves at the same velocity in every reference frame.[3](https://www.physics.ucla.edu/~pheuer/content/docs/PhysNotes.pdf)
## Rapidity and the Lorentz group
Because \( \gamma^{2} - \beta^{2}\gamma^{2} = 1 \) matches the identity \( \cosh^{2}\eta - \sinh^{2}\eta = 1 \), a boost can be written with hyperbolic functions, where \( \eta = \tanh^{-1}\beta \) is called the rapidity:[5](https://ocw.mit.edu/courses/8-033-relativity-fall-2006/909e205adee074e5401e0a7c7cce06c0_lecture6_kinem3.pdf)
\[ \Lambda = \begin{pmatrix} \cosh\eta & 0 & 0 & \sinh\eta \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \sinh\eta & 0 & 0 & \cosh\eta \end{pmatrix}. \]
While \( \beta \) is bounded between −1 and 1, the rapidity runs from −∞ to ∞, and composing collinear boosts amounts to adding rapidities, which is simpler than multiplying the velocity-addition formula.[5](https://ocw.mit.edu/courses/8-033-relativity-fall-2006/909e205adee074e5401e0a7c7cce06c0_lecture6_kinem3.pdf)
The set of all Lorentz transformations satisfying \( \Lambda^{T} g \Lambda = g \), with \( g \) the Minkowski metric, forms the [Lorentz group](https://www.edgepedia.org/lorentz-group). The restricted (proper, orthochronous) group \( SO^{+}(3,1) \) contains boosts, rotations, and their combinations, and any of its elements can be decomposed into a pure boost and a pure rotation, in either order.[4](https://www.pks.mpg.de/~haque/maynooth_MP352/lecturenotes/MP352notes_LorentzGroup_c_01.pdf) Boosts alone do not form a group: two successive boosts give a pure boost only if they are collinear. Collinear boosts in one direction do form an abelian subgroup, \( O(1,1) \) or \( SO^{+}(1,1) \), whereas \( SO^{+}(1,3) \) is non-abelian because boosts in different directions, and boosts with rotations, do not commute.[4](https://www.pks.mpg.de/~haque/maynooth_MP352/lecturenotes/MP352notes_LorentzGroup_c_01.pdf)
**Wigner rotation and Thomas precession.** Composing two boosts in different directions yields not a pure boost but a boost combined with a rotation, called the [Wigner rotation](https://www.edgepedia.org/wigner-rotation) (or Thomas rotation), named after [Eugene Wigner](https://www.edgepedia.org/eugene-wigner) and [Llewellyn Thomas](https://www.edgepedia.org/llewellyn-thomas). An object in rapid circular motion, such as an electron orbiting a nucleus, undergoes a sequence of non-collinear boosts that returns it to its original velocity each period; the residual rotations accumulate into a net rotation per orbit, so the spin axis precesses in space. This is Thomas precession, a purely relativistic effect with no Newtonian counterpart for a gyroscope pushed through its center.[4](https://www.pks.mpg.de/~haque/maynooth_MP352/lecturenotes/MP352notes_LorentzGroup_c_01.pdf) • [7](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_%28Crowell%29/08%3A_Rotation/8.03%3A_Boosts_and_Rotations)
## Electromagnetic fields
[Maxwell's equations](https://www.edgepedia.org/maxwells-equations) are not invariant under the Galilean transformation but are Lorentz covariant, meaning their form is unaltered by Lorentz transformations; this invariance was the physical motivation for the transformation in the first place.[8](https://eagle.phys.utk.edu/guidry/phys541/book/EM.pdf) The electric and magnetic fields themselves are not individually invariant; they mix. For a boost at speed \( v \) along x (with \( c = 1 \)), the Feynman Lectures give[6](https://www.feynmanlectures.caltech.edu/II_26.html)
\[ E_{\parallel}' = E_{\parallel}, \qquad E_{\perp}' = \frac{(\mathbf{E} + \mathbf{v} \times \mathbf{B})_{\perp}}{\sqrt{1 - v^{2}/c^{2}}}, \qquad B_{\parallel}' = B_{\parallel}, \qquad B_{\perp}' = \frac{\left(\mathbf{B} - \dfrac{\mathbf{v} \times \mathbf{E}}{c^{2}}\right)_{\perp}}{\sqrt{1 - v^{2}/c^{2}}}, \]
where the subscripts denote components parallel and perpendicular to the relative velocity. A purely electric field in one frame is therefore seen as a combination of electric and magnetic fields in another; transforming the static Coulomb field of a charge at rest produces the magnetic field of a moving charge.[6](https://www.feynmanlectures.caltech.edu/II_26.html) • [9](https://hepweb.ucsd.edu/ph110b/110b_notes/node69.html) Compact covariant formulations collect the six field components into the electromagnetic field tensor \( F_{\mu\nu} \), whose components transform by the Lorentz transformation of the indices.[6](https://www.feynmanlectures.caltech.edu/II_26.html)
## History
The equations are named for H.A. Lorentz (1853–1928), who first proposed them, though he justified them on a hypothesis later found to be fallacious; the correct theoretical basis is Einstein's special theory of relativity.[2](https://openstax.org/books/university-physics-volume-3/pages/5-5-the-lorentz-transformation) According to a historical study of the 1905 paper, [Henri Poincaré](https://www.edgepedia.org/henri-poincare) gave the equations the name "Lorentz transformations", although [Woldemar Voigt](https://www.edgepedia.org/woldemar-voigt) had published equivalent equations in 1887.[10](https://www.martinezwritings.com/m/Relativity_files/1905.pdf) In his 1905 paper *On the Electrodynamics of Moving Bodies* (Zur Elektrodynamik bewegter Körper, Annalen der Physik 17:891), Einstein derived the transformation from the two principles of relativity and the constancy of the velocity of light, and showed that a spherical light wave \( x^{2} + y^{2} + z^{2} = c^{2}t^{2} \) transforms into the corresponding spherical form \( \xi^{2} + \eta^{2} + \zeta^{2} = c^{2}\tau^{2} \).[11](https://www.fourmilab.ch/etexts/einstein/specrel/www/)
## References
1. [Lorentz transformation, Einstein Online (Max Planck Institute for Gravitational Physics)](https://www.einstein-online.info/en/explandict/lorentz-transformation-2/)
2. [5.5 The Lorentz Transformation, University Physics Volume 3, OpenStax](https://openstax.org/books/university-physics-volume-3/pages/5-5-the-lorentz-transformation)
3. [Notes on Physics, P. Heuer, UCLA](https://www.physics.ucla.edu/~pheuer/content/docs/PhysNotes.pdf)
4. [Rotation group, Lorentz group, Poincaré group, lecture notes, Maynooth MP352 / Max Planck Institute for the Physics of Complex Systems](https://www.pks.mpg.de/~haque/maynooth_MP352/lecturenotes/MP352notes_LorentzGroup_c_01.pdf)
5. [Lecture 6: Kinematics, 8.033 Relativity, MIT OpenCourseWare](https://ocw.mit.edu/courses/8-033-relativity-fall-2006/909e205adee074e5401e0a7c7cce06c0_lecture6_kinem3.pdf)
6. [The Feynman Lectures on Physics Vol. II Ch. 26: Lorentz Transformations of the Fields](https://www.feynmanlectures.caltech.edu/II_26.html)
7. [8.3: Boosts and Rotations, Special Relativity (Crowell), Physics LibreTexts](https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_%28Crowell%29/08%3A_Rotation/8.03%3A_Boosts_and_Rotations)
8. [Classical Electromagnetic Theory, M. Guidry, University of Tennessee](https://eagle.phys.utk.edu/guidry/phys541/book/EM.pdf)
9. [Lorentz Transformation of the Fields, UCSD Physics 110B notes](https://hepweb.ucsd.edu/ph110b/110b_notes/node69.html)
10. [AJP Journal: commentary on Einstein's 1905 derivation of the transformation equations](https://www.martinezwritings.com/m/Relativity_files/1905.pdf)
11. [On the Electrodynamics of Moving Bodies, A. Einstein (1905), English translation](https://www.fourmilab.ch/etexts/einstein/specrel/www/)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › Special relativity › Relativistic kinematics › Lorentz transformations and interval geometry*
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