special-relativity.bigb

= Special relativity
{wiki}

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

= Lorentz factor
{title2=$\gamma$}
{parent=Special relativity}
{c}
{wiki}

For speed $v<c$, the Lorentz factor is
$$
\gamma=\frac1{\sqrt{1-v^2/c^2}}.
$$

= Ultrarelativistic particle
{parent=Special relativity}

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

= Galilean transformation
{parent=Special relativity}
{wiki}

For inertial frames with relative velocity $u$ along the $x$-axis, the Galilean transformation is
$$
t'=t,\qquad x'=x-ut.
$$

= Lorentz transformation
{title2=$\Lambda$}
{parent=Special relativity}
{c}
{wiki}

A Lorentz transformation preserves the <Minkowski metric>. A boost of speed $u$ along the $x$-axis uses $\beta=u/c$ and
$$
ct'=\gamma(ct-\beta x),\qquad
x'=\gamma(x-\beta ct).
$$

= Nonrelativistic limit
{parent=Lorentz transformation}
{wiki=Classical_limit}

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

= Relativistic velocity-addition formula
{parent=Lorentz transformation}
{wiki=Velocity-addition_formula}

For a boost of speed $V$ along $x$,
$$
u'_x=\frac{u_x-V}{1-Vu_x/c^2},
\qquad
u'_y=\frac{u_y}{\gamma_V(1-Vu_x/c^2)}.
$$

= Four-vector
{parent=Special relativity}
{wiki=Four-vector}

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

= Causality
{parent=Special relativity}
{wiki=Causality_(physics)}

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
{c}
{parent=Special relativity}
{wiki}

Minkowski spacetime is the flat spacetime of special relativity.

= Minkowski metric
{c}
{parent=Minkowski spacetime}
{wiki}

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

= Minkowski norm
{title2=$U\cdot U$}
{c}
{parent=Minkowski metric}
{wiki=Minkowski_space}

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

= Light cone
{parent=Minkowski metric}
{wiki}

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

= Null directions
{parent=Light cone}
{wiki=Lightlike}

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

= Null curve
{parent=Minkowski metric}
{wiki=Lightlike}

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

= Null geodesic
{parent=Null curve}
{wiki=Geodesic#General_relativity}

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

= Subluminal speed
{parent=Special relativity}
{wiki=Speed_of_light#Upper_limit_on_speeds}

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.

= Impossibility of photon decay into two massive particles
{parent=Special relativity}

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.

= Four-momentum
{parent=Special relativity}
{wiki}

A massive particle has four-momentum
$$
P^\mu=(E/c,\mathbf p)=m\gamma(c,\mathbf v),
\qquad P^2=m^2c^2.
$$
For a massless particle, $P^\mu=(E/c,\mathbf p)$ with $E=c|\mathbf p|$ and $P^2=0$.

= Relativistic energy-momentum relation
{parent=Four-momentum}
{wiki=Energy%E2%80%93momentum_relation}

For a particle of rest mass $m$,
$$
\mathcal E^2=m^2c^4+c^2|\mathbf p|^2.
$$

= Conservation of relativistic energy
{parent=Relativistic energy-momentum relation}
{wiki=Energy_conservation}

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

= Relativistic motion in a constant electric field with transverse momentum
{parent=Relativistic energy-momentum relation}

For $\mathbf E=(0,0,E)$, $\mathbf B=0$, and initial momentum $(p_0,0,0)$,
$$
p_x=p_0,\qquad p_z=qEt,\qquad
\mathcal E(t)=\sqrt{\mathcal E_0^2+c^2q^2E^2t^2}.
$$
With the initial <position> at the origin,
$$
z(t)=\frac{\mathcal E(t)-\mathcal E_0}{qE},
\qquad
x(t)=\frac{cp_0}{qE}
\operatorname{arsinh}\frac{cqEt}{\mathcal E_0}.
$$

= Relativistic force
{title2=$F=dp/dt$}
{parent=Four-momentum}
{wiki=Four-force}

For three-momentum $p=\gamma mv$, force and acceleration satisfy
$$
F=m\gamma\left(a+\frac{\gamma^2}{c^2}(v\mathbin{\cdot}a)v\right),
\qquad
a=\frac1{m\gamma}\left(F-\frac{F\mathbin{\cdot}v}{c^2}v\right).
$$

= Relativistic motion under a constant force from rest
{parent=Relativistic force}

Under a constant force $F$ with $p(0)=0$, momentum is $p=Ft$ and
$$
v(t)=\frac{Ft/m}{\sqrt{1+|F|^2t^2/(m^2c^2)}}.
$$
The speed approaches $c$ without reaching it at finite time.

= Relativistic two-body decay
{parent=Four-momentum}
{wiki}

For a rest-frame decay $1\to2+3$,
$$
E_2=\frac{m_1^2+m_2^2-m_3^2}{2m_1}c^2,
\qquad
E_3=\frac{m_1^2+m_3^2-m_2^2}{2m_1}c^2.
$$
The decay is allowed exactly when $m_1\geq m_2+m_3$.

= Mandelstam variables
{parent=Four-momentum}
{c}
{wiki}

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

= Four-velocity
{parent=Special relativity}
{wiki}

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

= Four-acceleration
{parent=Special relativity}
{wiki}

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

= Proper acceleration
{parent=Four-acceleration}
{wiki}

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

= Rindler horizon
{parent=Proper acceleration}
{c}
{wiki}

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