Kinematics, Dynamics, and the Relativistic Particle

Manual: General · Subject: Physics

Explore time dilation, length contraction, four-vectors, and the relativistic action principle for particles.

Relativistic kinematics

Time dilation and proper time

The proper time along a worldline is $d\tau = \frac{1}{c}\sqrt{c^2dt^2-d\mathbf{x}^2}.$ For a clock moving at speed $v$ , the relation is $d\tau = dt/\gamma$ , so moving clocks accumulate less proper time.

Length contraction and velocity addition

A rod at rest in its own frame has proper length $L_0$ . Observers moving relative to the rod measure $L = L_0/\gamma$ along the direction of motion. Velocities combine by $u' = \frac{u-v}{1-uv/c^2}$ , ensuring no material object exceeds $c$ .

Four-velocity and four-momentum

The four-velocity is $u^\mu = dx^\mu/d\tau$ , satisfying $u^\mu u_\mu = c^2$ . The four-momentum is $p^\mu = mu^\mu = (E/c,\mathbf{p})$ , leading to the dispersion relation $E^2 = p^2c^2 + m^2c^4.$

💡

Mass and energy

Rest energy $E_0 = mc^2$ is not a separate substance; it is the energy of a particle at zero three-momentum in spacetime.

A particle with rest mass $m$ and momentum $\mathbf{p}$ satisfies which relation?

E = pc

for all massive particles

E^2 = p^2c^2 + m^2c^4

E = m c^2 - p^2/2m

exactly

p = mv

exactly at relativistic speeds

What is proper time, and why is it important?

Common relativistic formulas

Quantity	Expression
Lorentz factor	$\gamma = 1/\sqrt{1-v^2/c^2}$
Time dilation	$\Delta t = \gamma\Delta\tau$
Length contraction	$L = L_0/\gamma$
Energy-momentum	$E^2 = p^2c^2 + m^2c^4$

Why is $E=mc^2$ incomplete as a statement of relativistic mechanics?

Neuroanatomy Explorer

Memory Deck

Match the Pairs

Concept Constellation

Kinematics, Dynamics, and the Relativistic Particle

Relativistic kinematics

Time dilation and proper time

Length contraction and velocity addition

Four-velocity and four-momentum