Relativistic Dynamics

Kinematics fixes how coordinates transform; dynamics fixes how energy, momentum, and force behave. Building four-vectors from the proper time and demanding Lorentz covariance (Lorentz transformations) yields four-momentum, $E=m{c}^2$, the dispersion relation, and the conservation laws governing collisions. Units: $c=1$ unless restored; $\eta =\mathrm{diag}(+,-,-,-)$.

Four-velocity and four-momentum

Differentiating the worldline by proper time gives the four-velocity $u^{\mu }=d{x}^{\mu }/d\tau =\gamma (c,\mathbf{v})$, normalized as $u^{\mu }{u}_{\mu }=c^2$. Multiplying by rest mass $m$ gives the four-momentum

$$p^{\mu }=m{u}^{\mu }=(E/c,\mathbf{p}),E=\gamma m{c}^2,\mathbf{p}=\gamma m\mathbf{v}.$$

Both transform as four-vectors, so conservation of $p^{\mu }$ holds in every inertial frame.

Energy, rest energy, and the limits

At rest ($\gamma =1$) the energy is the rest energy

$$E=m{c}^2,$$

while for $v\ll c$ expansion recovers $E\approx m{c}^2+\frac{1}{2}m{v}^2$ — rest energy plus Newtonian kinetic energy. The kinetic energy is $T=(\gamma -1)m{c}^2$.

The dispersion relation

The invariant norm of $p^{\mu }$ is the rest mass:

$$p_{\mu }{p}^{\mu }={\left(\frac{E}{c}\right)}^2-|\mathbf{p}{|}^2=(mc{)}^2\Rightarrow E^2=(pc{)}^2+(m{c}^2{)}^2.$$

Special cases: massless particles ($m=0$) obey $E=pc$; non-relativistically $E\approx m{c}^2+p^2/2m$. This is the on-shell condition $p^2=m^2$ that reappears in QFT as the free-propagator pole; promoting $E\to i{\partial }_t$, $\mathbf{p}\to -i\nabla$ gives the Klein–Gordon equation.

Collisions and conservation

Total $p^{\mu }$ is conserved; rest mass is not additive. Useful invariants: the Mandelstam $s=(p_1+p_2{)}^2$ fixes thresholds; threshold energy for producing mass $M$ has minimum $s=M^2$. Compton scattering gives ${\lambda }^{\prime }-\lambda =(h/m_{e}c)(1-\cos \theta )$. Massless quanta still carry momentum $p=E/c$.

Force and acceleration

The four-force $f^{\mu }=d{p}^{\mu }/d\tau$ generalizes Newton's law; because $\mathbf{p}=\gamma m\mathbf{v}$, force and acceleration are generally non-parallel, splitting into longitudinal (${\gamma }^{3}m$) and transverse ($\gamma m$) responses. Covariant electromagnetism realizes this via the Lorentz force $f^{\mu }=q{F}^{\mu \nu }{u}_{\nu }$; see fields/electromagnetism.md.

Summary

Next

The covariant field bridge is in electromagnetism; the route to quantum theory in bridge to QFT.