Special relativity

Hyperbolic motion in special relativity is rectilinear motion with constant proper acceleration. A representative positive-$x$ world line is $x=c{t}_0\cosh (\tau /t_0)$, $t=t_0\sinh (\tau /t_0)$, with proper acceleration $c/t_0$. This describes a hyperbola in spacetime, not a hyperbolic Kepler orbit.

Proper length is spatial length measured using the induced metric on a specified spacelike slice. For a moving extended object, specify the simultaneity convention; local rest lengths need not coincide with a coordinate length.

Spacetime combines spatial position and time into one geometric setting for physical events. In special relativity it is modeled by Minkowski spacetime; in general relativity its geometry is dynamical.

Proper time is the time measured by a clock following a timelike worldline. With metric signature $(-,+,+,+)$ and units $c=1$,along that worldline.

A nonzero spacetime vector $k$ is null when its Minkowski norm vanishes, $k^a{k}_a=0$. Null tangent vectors describe the local directions of light rays.

Special relativity describes inertial frames using Lorentz transformations and Minkowski spacetime.

In Minkowski spacetime with metric signature $(-,+,+,+)$, write two nonzero future-pointing null vectors as $A=a(1,\mathbf{n})$ and $B=b(1,\mathbf{m})$, where $a,b>0$ and $\mathbf{n},\mathbf{m}$ are unit vectors. Their Lorentzian inner product givesThus their sum is a future-pointing causal vector. It is a null vector exactly when the spatial directions coincide, and a timelike vector otherwise. The positivity of the time components is essential: adding past- and future-pointing null vectors can instead produce a spacelike vector.

The null condition for a nonzero vector $k$ in a Lorentzian metric is $g(k,k)=0$. Applied to the tangent of a curve it makes the curve a null curve. Nullity alone does not imply the geodesic equation; an additional zero or tangent-parallel covariant acceleration is required.

For speed $v<c$, the Lorentz factor is

A rod of rest length $L_0$ moving parallel to its length has simultaneous laboratory length $L_0/\gamma$, where $\gamma$ is the Lorentz factor. Transverse lengths are unchanged by the corresponding Lorentz transformation.

If a clock measures proper-time interval $\Delta \tau$ while moving at constant speed $v$ in an inertial frame, that frame measures $\Delta t=\gamma \Delta \tau$.

The twin paradox compares proper time along the different world lines of a travelling observer and a stay-at-home observer who reunite. Reciprocal inertial time dilation produces an apparent contradiction only if the traveller's change of inertial frame is ignored.

An ultrarelativistic particle has energy much larger than its rest energy. Its dispersion relation is then approximated by $E=pc$.

For inertial frames with relative velocity $u$ along the $x$-axis, the Galilean transformation is

The laws of Newtonian mechanics have the same form in every inertial frame related by a Galilean transformation.

A Lorentz transformation preserves the Minkowski metric. A boost of speed $u$ along the $x$-axis uses $\beta =u/c$ and

Spatially separated simultaneous events in one inertial frame are generally not simultaneous in a relatively moving frame. The Lorentz transformation explicitly mixes position into time. This is essential when comparing distances between moving light pulses: one must compare their positions at the same time in the measuring frame.

For two photons emitted simultaneously a distance $d$ apart in one frame and moving in the same positive direction, the world-line intercept $x-ct$ transforms to $\gamma (1+v/c)(x-ct)$. Their equal-time separation in the boosted frame is therefore the displayed constant $d^{\prime }$. It is different from length contraction of the emitters, because the emission events are not simultaneous after the boost.

A physical quantity is Lorentz invariant when its value is unchanged under Lorentz transformations. Tensor equations can be Lorentz covariant while their components transform. Fixed-vector gauge choices need not make covariance manifest, although physical gauge-invariant predictions retain the symmetry.

A Lorentz boost relates inertial frames moving at a constant relative velocity without an additional spatial rotation. In units with light speed one, the displayed transformation preserves $t^2-x^2$, with Lorentz factor $\gamma =(1-v^2{)}^{-1/2}$. Applying it to a static classical field-theory soliton produces a uniformly moving solution in a Lorentz-invariant theory.

Two noncollinear Lorentz boosts produce a Lorentz transformation with both a boost and a rotation. In the SL(2,C) lift, the product of two positive Hermitian boost matrices generally ceases to be Hermitian. Its polar decomposition of an invertible complex matrix separates a positive Hermitian boost factor and a unitary rotation factor. The rotation depends on both boosts and their order, and is called the Wigner rotation.

For boost matrices $A=cI+s\mathbf{n}\cdot \mathbf{\sigma }$ and $A^{\prime }=c^{\prime }I+s^{\prime }{\mathbf{n}}^{\prime }\cdot \mathbf{\sigma }$, the Pauli matrix multiplication law gives the displayed anti-Hermitian part. If both rapidities are nonzero and the axes are noncollinear, it cannot vanish, while either lift of a pure boost is Hermitian. Thus an additional Wigner rotation is necessary. A zero boost is a degenerate exception.

For one spatial dimension and $c=1$, a massive particle has $v=\tanh \theta$, $E=M\cosh \theta$ and $p=M\sinh \theta$. A Lorentz boost adds a constant to all rapidities, so a rapidity difference is invariant. Rapidity is useful for factorized scattering, where two-body S-matrices depend only on that difference.

The relativistic velocity-addition formula becomes addition of rapidities for collinear Lorentz boosts. Starting from rest, applying the same positive rest-frame velocity increment $c\tanh \alpha$ repeatedly gives $u_n=c\tanh (n\alpha )$. For increment $c/2$, $\alpha =\frac{1}{2}\log 3$ and $u_n=c(3^n-1)/(3^n+1)$. Every finite sequence remains below the speed of light.

An active transformation keeps the coordinate argument fixed and changes a contravariant vector field by $A^{\prime \mu }(x)={\Lambda }^{\mu }{}_{\nu }{A}^{\nu }({\Lambda }^{-1}x)$. Infinitesimally, $\delta A^{\mu }=\alpha {\omega }^{\mu }{}_{\nu }{A}^{\nu }-\alpha {\omega }^{\rho }{}_{\sigma }{x}^{\sigma }{\partial }_{\rho }{A}^{\mu }$. For a Lorentz scalar density, the spin-index term is absent, giving $\delta \mathcal{L}=-\alpha {\omega }^{\rho }{}_{\sigma }{x}^{\sigma }{\partial }_{\rho }\mathcal{L}$.

A Lorentz scalar is unchanged under a Lorentz transformation. Contracting every Lorentz index in a tensor expression produces a scalar when the constituent fields transform covariantly.

A Lorentz tensor transforms by one Lorentz matrix for each contravariant index and one inverse matrix for each covariant index. Complete contraction of all indices produces a Lorentz scalar.

The Lorentz group consists of real matrices preserving the Minkowski metric: ${\Lambda }^T\eta \Lambda =\eta$.

For $M=\mathrm{diag}(-1,1,1,1)$,Every such matrix has the block formso the Lie algebra and the Lorentz group have dimension six. At $A\in O(1,3)$, the tangent space is $A\mathfrak{o}(1,3)$.

The orthochronous Lorentz group consists of Lorentz transformations preserving the future sheet of the unit timelike hyperboloid. Equivalently, $A_{00}>0$. This is an open subgroup of $O(1,3)$ and hence a full-dimensional submanifold.

The proper orthochronous Lorentz group is the identity component of the Lorentz group. Its elements have determinant one and preserve the future light cone.

Identify Minkowski vectors with Hermitian matrices $X=x^{0}I+x^a{\sigma }_a$. The action $X\mapsto SX{S}^{†}$ preserves $\det X=(x^0{)}^2-|\mathbf{x}{|}^2$. It gives a surjective homomorphism from $SL(2,\mathbb{C})$ to the Proper orthochronous Lorentz group with kernel $\{I,-I\}$. Polar decomposition retracts the covering group onto SU(2), making it simply connected. This cover permits half-integer spin group representations that do not descend to the Lorentz group itself.

A real four-vector corresponds to a two-by-two Hermitian matrix with determinant equal to its Minkowski squared norm. The action $X\mapsto NX{N}^{†}$ for $N\in SL(2,\mathbb{C})$ preserves that determinant and the future cone. Expanding with $\frac{1}{2}\mathrm{Tr}({\bar{\sigma }}^{\mu }{\sigma }_{\nu })={\delta }^{\mu }{}_{\nu }$ yields ${\Lambda }^{\mu }{}_{\nu }(N)=\frac{1}{2}\mathrm{Tr}({\bar{\sigma }}^{\mu }N{\sigma }_{\nu }{N}^{†})$. The map has kernel $\{\pm I\}$ and image $S{O}^{+}(1,3)$.

The Poincare group is the semidirect product of spacetime translations by the Lorentz group. It is the isometry group of Minkowski spacetime.

The Poincare algebra is generated by translations $P^{\mu }$ and Lorentz transformations $M^{\mu \nu }$. Its mixed commutator is $[M^{\mu \nu },P^{\rho }]=i({\eta }^{\nu \rho }{P}^{\mu }-{\eta }^{\mu \rho }{P}^{\nu })$.

The Pauli-Lubanski pseudovector is $W_{\mu }={ϵ}_{\mu \nu \rho \sigma }{M}^{\nu \rho }{P}^{\sigma }/2$, up to an overall sign convention. Its square is a Casimir of the Poincare algebra and measures spin in a massive irreducible representation.

With signature $(+---)$ and ${ϵ}_{0123}=+1$, a momentum $(k,0,0,k)$ and helicity $j_3$ give lower components $W_0=k{j}_3$ and $W_3=-k{j}_3$. They obey $W_{\mu }{P}^{\mu }=k(W_0+W_3)=0$. These longitudinal identities require only the momentum and rotation-projection eigenvalues, not a statement about transverse little-group translations.

With signature $(+---)$, ${ϵ}_{0123}=+1$, $J_3=M^{12}$ and the lower-index definition of $W_{\mu }$, a rest momentum $(m,0,0,0)$ gives $W_0=0$ and $W_3=-m{J}_3$ on that momentum eigenspace. A Hermitian spin projection $j_3$ consequently gives eigenvalues $0$ and $-m{j}_3$. Raising the third index changes its sign.

The little group of a momentum $p$ is its stabilizer in the Lorentz group. Its unitary irreducible representations label the internal degrees of freedom of one-particle states on the orbit of $p$.

A particle polarization specifies a physical spin state at fixed four-momentum. Massive spin states transform under the rotation little group $SO(D-1)$; finite-spin massless states transform under its transverse rotation part $SO(D-2)$. A massive two-form field has $\left(\genfrac{}{}{0ex}{}{D-1}{2}\right)$ such states, while a two-form gauge field has $\left(\genfrac{}{}{0ex}{}{D-2}{2}\right)$. Polarization components must be counted after field equations and any gauge equivalence are imposed.

The stabilizer of a nonzero null four-momentum has connected group $ISO(D-2)$. For ordinary finite-polarization particles its translation part acts trivially, leaving a representation of $SO(D-2)$.

An ordinary massless spin-one particle has transverse vector polarizations under $SO(D-2)$. A covariant polarization satisfies $p\cdot ϵ=0$ and is identified under $ϵ\mapsto ϵ+\lambda p$. The quotient has $D-2$ positive polarizations.

The stabilizer of a timelike four-momentum in $D$-dimensional Minkowski spacetime has connected group $SO(D-1)$. Massive particle polarizations form representations of this group.

A massive spin-one particle transforms as a vector under its massive particle little group $SO(D-1)$. Its $D-1$ polarizations are orthogonal to its timelike four-momentum. Under transverse $SO(D-2)$ rotations they split into a vector and a scalar.

The symmetry used to classify an open string must preserve its endpoint boundary background as well as its momentum. A Neumann-Dirichlet open-string boundary condition breaks target Lorentz transformations mixing ND and common directions with Neumann boundary conditions. For a D1–D25 pair with 24 ND directions, the common worldvolume has Lorentz group $SO(1,1)$, whose connected massive little group is trivial. The transverse $SO(24)$ remains as an internal rotation group. Thus a massive 24-component oscillator multiplet is not an $SO(25)$ vector of the full 26-dimensional massive little group; a vector of that full group would have 25 components.

A massive vector in $D$ spacetime dimensions has $D-1$ polarizations perpendicular to its timelike momentum. A massless vector has transverse polarizations modulo its null longitudinal polarization, leaving $D-2$ physical degrees of freedom.

Helicity is the projection of angular momentum onto the momentum direction. For a finite-helicity massless particle, the Pauli-Lubanski vector satisfies $W^{\mu }=\lambda P^{\mu }$.

For $\mathbf{k}=k\hat{z}$, these complex vectors form an orthonormal basis of the transverse plane using the Hermitian inner product. They have opposite phases under rotations about $\hat{z}$, so they describe the two helicity components. Since $\mathbf{e}\cdot m^{(\pm )}=\sin \theta e^{\pm i\phi }/\sqrt{2}$, transverse-vector sources carry only the angular sectors $m=\pm 1$ of spherical harmonics. Streaming by $ik\cos \theta$ preserves $m$ and couples neighboring angular degrees.

A continuous-spin representation is a massless unitary irreducible representation of the Poincare group in which the translation subgroup of the $ISO(2)$ little group acts nontrivially. It contains infinitely many helicities.

Wigner's classification obtains irreducible positive-energy representations of the Poincare group by inducing from unitary irreducible representations of the little group of a standard momentum.

The determinant-one Lorentz group in one space and one time dimension has two connected components. Its identity component consists of boostsand the other component is its negative.

A boost of rapidity $\phi$ acts on null coordinates by $u\mapsto e^{\phi }u$ and $v\mapsto e^{-\phi }v$, preserving $uv=x^2-y^2$.

In the $(t,z)$ plane, $u=t-z$ is retarded time and $v=t+z$ is advanced time. The radial part of the Minkowski metric is $-d{t}^2+d{z}^2=-dudv$.

For $u=t-z$ and $v=t+z$, the four-dimensional Minkowski wave operator isEvery profile depending only on one null coordinate solves the wave equation when it is uniform in the transverse coordinates.

The nonrelativistic limit takes characteristic speeds much smaller than $c$. Lorentz transformations then reduce to Galilean transformations at leading order.

For a boost of speed $V$ along $x$,

A four-vector transforms as $U^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{U}^{\nu }$ under every Lorentz transformation.

In metric signature $(+---)$, a future-pointing timelike four-vector has positive time component larger than its spatial magnitude. Two such vectors have strictly positive Minkowski metric scalar product, since $P\cdot Q=P^0{Q}^0-\mathbf{P}\cdot \mathbf{Q}\ge P^0{Q}^0-|\mathbf{P}||\mathbf{Q}|>0$. In the opposite signature the scalar-product sign reverses, while the causal condition is the same. The future timelike cone is closed under addition, which rules out a null photon momentum equalling a sum of two massive future four-momenta.

The four-divergence of a four-vector field is the Lorentz scalar contraction ${\partial }_{\mu }{V}^{\mu }$. With $x^{\mu }=(ct,\mathbf{x})$, it equals $c^{-1}{\partial }_t{V}^0+\nabla \cdot \mathbf{V}$. The four-current has zero four-divergence, expressing local charge conservation.

Causality requires an effect at an event to depend only on sources in its past light cone. Relativistic field equations implement this choice through retarded Green functions.

Minkowski spacetime is the flat spacetime of special relativity.

For $\rho >0$, these coordinates cover a wedge of Minkowski spacetime and give $d{s}^2=-{\kappa }^2{\rho }^{2}d{t}^2+d{\rho }^2$ in two dimensions. A constant-$\rho$ observer has proper acceleration $1/\rho$ and the Unruh effect temperature $1/(2\pi \rho )$ in natural units. The apparent degeneration at $\rho =0$ is a coordinate horizon.

With the mostly-plus convention, the Minkowski metric is ${\eta }_{\mu \nu }=\mathrm{diag}(-1,1,1,1)$.

For mostly-plus signature, $x^{+}=(x^{D-1}+x^0)/\sqrt{2}$ and $x^{-}=(x^{D-1}-x^0)/\sqrt{2}$ give $d{s}^2=2d{x}^{+}d{x}^{-}+d{x}^{I}d{x}^I$. In this convention raising a plus or minus derivative exchanges the two labels.

The Minkowski norm is the Lorentz-invariant quadratic form $U\cdot U={\eta }_{\mu \nu }{U}^{\mu }{U}^{\nu }$.

The light cone through an event consists of null displacement vectors. In one spatial dimension its two directions satisfy $x=\pm ct$.

The past light cone of an event consists of earlier events connected to it by past-directed null curves. A retarded field at the event depends on sources where their worldlines intersect this cone.

The speed of light $c$ is the invariant speed of every inertial frame and the limiting speed of causal propagation in special relativity.

Null directions are tangent directions of the light cone; their Minkowski squared norm is zero.

A null curve has tangent $u$ satisfying $g(u,u)=0$. It represents the possible spacetime path of a light signal.

A null geodesic is both a null curve and a geodesic. In geometric optics, light rays follow null geodesics.

A radial null geodesic in a spherically symmetric spacetime has zero angular momentum and fixed angular coordinates. Its projection is a null curve of the two-dimensional radial Lorentzian metric.

A world line is the path of an object through spacetime. A massive observer follows a timelike curve.

A timelike curve has an everywhere timelike tangent vector and can represent the world line of a massive observer.

A subluminal speed has magnitude strictly below the speed of light: $|v|<c$. Massive particles follow timelike worldlines and therefore move subluminally in every inertial frame.

The sum of two future-pointing timelike four-momenta is timelike. It therefore cannot equal a photon's null four-momentum, which forbids spontaneous photon decay into a massive particle pair in vacuum.

A massive particle has four-momentumFor a massless particle, $P^{\mu }=(E/c,\mathbf{p})$ with $E=c|\mathbf{p}|$ and $P^2=0$.

For future-directed particles with total four-momentum $P$ and metric signature $(-+++)$, the positive square root of $-P_{\mu }{P}^{\mu }$ is their energy in the frame where total spatial momentum vanishes. Its invariant definition remains valid at a collision in spacetime.

The mass shell is the locus of four-momenta satisfying the relativistic dispersion relation for a fixed mass $m$. With mostly-minus Minkowski metric, $p^2=m^2$; with mostly-plus Minkowski metric, $p^2=-m^2$. The positive-energy sheet represents particle four-momenta. For $m=0$ this is the light cone. The condition of being on shell refers more generally to satisfying the field equations.

In an isolated relativistic interaction, the sum of incoming four-momenta equals the sum of outgoing four-momenta.

When an electron of Lorentz factor $\gamma >1$ annihilates with a stationary equal-mass positron, transverse four-momentum conservation makes the photons' transverse components opposite. Eliminating their positive frequencies gives the displayed relation for noncollinear photon angles to the incoming electron. The exactly collinear forward/backward configuration is physically allowed but makes the quotient $0/0$; its undivided cross-multiplied identity remains valid. The angle relation thus requires nondegeneracy or an appropriate limit, rather than an assertion that every allowed angle pair defines the quotient.

When two massive particles coalesce without additional outgoing radiation, four-momentum conservation makes the final four-momentum their sum. For collinear incident velocities, its rest mass is the positive square root of the displayed invariant, and its final velocity equals the center-of-momentum frame velocity $c^2{p}_{\mathrm{total}}/E_{\mathrm{total}}$. Relative kinetic energy can become internal rest energy, so the final rest mass is generally larger than the sum of the incident rest masses.

The rest energy of a particle of invariant mass $m$ is $m{c}^2$, the energy measured in its rest frame.

The center-of-momentum frame is an inertial frame in which the total spatial momentum of a system vanishes. For a two-particle initial state, the momenta are equal and opposite and the total invariant energy is $\sqrt{s}$.

The centre-of-momentum energy is the total energy measured in the centre-of-momentum frame. For a scattering process with total four-momentum $P$, it is the Lorentz-invariant quantity $\sqrt{P^2}=\sqrt{s}$ in the mostly-minus convention.

For a particle of rest mass $m$,

Rest mass is the invariant mass $m$ in the relativistic energy-momentum relation. It equals the energy measured in the particle's rest frame divided by $c^2$.

In an isolated relativistic collision, total four-momentum is conserved; its time component states conservation of total relativistic energy.

For $\mathbf{E}=(0,0,E)$, $\mathbf{B}=0$, and initial momentum $(p_0,0,0)$,With the initial position at the origin,

For three-momentum $p=\gamma mv$, force and acceleration satisfy

Under a constant force $F$ with $p(0)=0$, momentum is $p=Ft$ andThe speed approaches $c$ without reaching it at finite time.

For a rest-frame decay $1\to 2+3$,The decay is allowed exactly when $m_1\ge m_2+m_3$.

A stationary particle of rest mass $m$ absorbing a photon of energy $E$ acquires total invariant massThis follows from four-momentum conservation, since the incoming photon has momentum magnitude $E/c$. If the system splits into two equal particles, the largest kinematically permitted daughter rest mass is $M/2$, attained when both are at rest in the centre-of-momentum frame. Their common laboratory speed at threshold is $cE/(m{c}^2+E)$. Above this daughter-mass threshold there is insufficient centre-of-momentum energy; below it, relative kinetic energy is available.

The Källén function controls two-body relativistic kinematics. In the rest frame of total invariant mass $\sqrt{s}$, the magnitude of each final momentum is $\sqrt{\kappa (s,m_1^2,m_2^2)}/(2\sqrt{s})$. It also determines the invariant flux factor in a two-particle collision.

For two-to-two scattering $A+B\to C+D$,They satisfy $s+t+u=m_A^2+m_B^2+m_C^2+m_D^2$ in units $c=1$.

With a specified ordering of external legs, this crossed channel carries the difference between an incoming and an outgoing energy-momentum vector. A one-particle exchange pole requires that difference to be on the exchanged particle's mass shell. Ordered amplitudes in one spatial dimension can label the crossed invariant using the difference of the two rapidity-parameterized external momenta; the external-leg convention fixes which crossed invariant is named $t$.

The direct channel combines the two incoming four-momentum vectors. A one-particle pole in this channel has the intermediate particle's squared mass equal to $s$. Imaginary rapidity differences can satisfy this mass relation for a bound state even though there is no pole at real physical scattering rapidity.

Four-velocity is the proper-time derivative of position and has invariant squared norm c^2.

With signature $(-,+,+,+)$ and units $c=1$, parametrizing a timelike worldline by proper time makes its four-velocity satisfy $U^a{U}_a=-1$.

Four-acceleration is the proper-time derivative of four-velocity and is orthogonal to it.