Special relativity

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

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

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

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

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.

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.

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

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 $x=\pm ct$.

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 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 a particle of rest mass $m$,

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 $m_1\ge m_2+m_3$.

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 $c=1$.

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

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

Proper acceleration is the acceleration measured in the particle’s instantaneous rest frame.

A uniformly accelerated observer has a null boundary beyond which light signals can never reach the observer.