Relativistic Dynamics and Conservation Laws

Manual: General · Subject: Physics

Derives relativistic momentum, energy, mass-energy equivalence, and conservation-law structure.

Energy, Momentum, and Mass

Relativistic Momentum

The relativistic three-momentum is $\mathbf{p} = \gamma m\mathbf{v}.$ This reduces to Newtonian momentum $m\mathbf{v}$ when $v \ll c$ but differs strongly at high speed because $\gamma$ grows rapidly.

Relativistic Energy

The total energy is $E = \gamma mc^2,$ and the kinetic energy is $K = (\gamma - 1)mc^2.$ The energy-momentum relation is $E^2 = p^2c^2 + m^2c^4.$

Mass-Energy Equivalence

Einstein’s relation $E_0 = mc^2$ identifies rest energy as an intrinsic property of matter. It implies that rest mass is a measure of internal energy content, not simply a mechanical bookkeeping parameter.

ℹ️

Important Distinction

Modern treatments usually reserve $m$ for invariant rest mass and avoid the obsolete notion of velocity-dependent relativistic mass.

Which relation is correct for a free massive particle?

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

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

E = pc

for all particles

p = mv

exactly at all speeds

What is the kinetic energy of a particle in special relativity?

Dynamics Formulas

Quantity	Expression	Low-Speed Limit
Momentum	$\mathbf{p} = \gamma m\mathbf{v}$	$m\mathbf{v}$
Total energy	$E = \gamma mc^2$	$mc^2 + \frac{1}{2}mv^2$
Kinetic energy	$K = (\gamma-1)mc^2$	$\frac{1}{2}mv^2$
Invariant relation	$E^2 = p^2c^2 + m^2c^4$	$E \approx mc^2$ for small $p$

Why does a massive particle’s speed never reach $c$ under finite energy input?

Because

\gamma

approaches infinity as

v \to c

Because momentum becomes negative Because rest mass vanishes Because spacetime ceases to exist

Conservation-Law Procedure

1

Step 1: Write energy and momentum in four-vector form.
2

Step 2: Use symmetry or interaction assumptions to define conserved quantities.
3

Step 3: Apply the invariant mass-shell relation.
4

Step 4: Take the nonrelativistic limit to verify consistency with classical mechanics.

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