ローレンツ変換

Starting point: the postulates of special relativity

1. The laws of physics take the same form in every inertial frame (relativity).
2. The speed of light in vacuum, $c$, is the same in every inertial frame.

1D boost

For a frame $S'$ moving with velocity $v$ along the $x$-axis:

$$\begin{aligned} t' &= \gamma\!\left(t - \frac{v x}{c^2}\right), \\ x' &= \gamma\,(x - v t), \\ y' &= y,\quad z' = z, \end{aligned}$$

with the Lorentz factor

$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$$

The invariant: spacetime interval

$$ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$$

is preserved under any Lorentz transformation. With the Minkowski metric

$$\eta_{\mu\nu} = \mathrm{diag}(-1,+1,+1,+1),$$

it is $ds^2 = \eta_{\mu\nu}\, dx^\mu dx^\nu$.

In terms of rapidity

Set $\tanh\eta = v/c$. The boost becomes

$$\begin{pmatrix} ct' \\ x' \end{pmatrix} = \begin{pmatrix} \cosh\eta & -\sinh\eta \\ -\sinh\eta & \cosh\eta \end{pmatrix} \begin{pmatrix} ct \\ x \end{pmatrix}$$

— a hyperbolic rotation in spacetime. Rapidity is additive, simplifying the velocity-addition rule.

Velocity addition

If a particle moves with velocity $u'$ in $S'$ and $S'$ moves at $v$ in $S$, then in $S$:

$$u = \frac{u' + v}{1 + u' v / c^2}.$$

If $u', v < c$, then $u < c$ — nothing exceeds the speed of light.

Time dilation and length contraction

• Time dilation: for proper time $\tau$, $\Delta t = \gamma\,\Delta\tau$.
• Length contraction: for proper length $L_0$, $L = L_0/\gamma$.

Two faces of one Lorentz transformation.

Energy-momentum 4-vector

With $p^\mu = (E/c, \mathbf p)$,

$$p^\mu p_\mu = -m^2 c^2, \quad E^2 = (pc)^2 + (mc^2)^2.$$

At rest, $E = mc^2$. This connects directly to Maxwell’s equations and the field-theory framework.

A physical example

GM v₀ リセット

Mercury’s perihelion precession deviates from Newtonian prediction. Including special-relativistic time dilation and the spacetime curvature of general relativity yields the observed $43''/$ century — historic confirmation of the theory.

Related

• Maxwell’s Equations — the original Lorentz-invariant field theory.
• Lagrangian Mechanics — the relativistic action $S = -mc^2\int d\tau$.