The energy-momentum relation

Connecting energy and momentum

Relativistic energy and momentum are

$E=gamma mc^2$

and

ec{p}=gamma mec{v}

These quantities are not independent. They satisfy the invariant energy-momentum relation:

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

where p=|ec{p}|.

This equation is one of the most important formulas in relativity.

Rest frame check

In the particle's rest frame, $p=0$. The relation becomes

$E^2=m^2c^4$

so

$E=mc^2$

This recovers rest energy.

High-momentum limit

When a particle's momentum is very large compared with $mc$, the term $p^2c^2$ dominates, and

$Eapprox pc$

This is exactly true for massless particles, such as photons.

Deriving the relation

Starting with

$E=gamma mc^2$

and

$p=gamma mv$

compute

$E^2-p^2c^2=gamma^2m^2c^4-gamma^2m^2v^2c^2$

Factor:

ight)$$ Since $$gamma^2=rac{1}{1-v^2/c^2}$$ we get $$E^2-p^2c^2=m^2c^4$$ Rearranging gives $$E^2=p^2c^2+m^2c^4$$ ## Invariant mass The quantity $$E^2-p^2c^2$$ has the same value in all inertial frames and equals $m^2c^4$. This is why rest mass is invariant. Energy and momentum separately depend on frame, but their combination does not. ## Kinetic energy from momentum The relation can be used to find kinetic energy if momentum is known: $$K=E-mc^2$$ with $$E=sqrt{p^2c^2+m^2c^4}$$ At low momentum, this reduces to the classical approximation $$Kapproxrac{p^2}{2m}$$ At high momentum, the relationship becomes very different. ## Particle physics use Particle collisions are often analyzed using energy and momentum conservation together with the energy-momentum relation. New particles can be created if enough energy is available. The invariant mass of a system can be found from total energy and total momentum: $$M^2c^4=E*{total}^2-p*{total}^2c^2$$ This is central in accelerator experiments. ## The big idea The energy-momentum relation unifies rest energy, kinetic energy, and momentum. The invariant equation $E^2=p^2c^2+m^2c^4$ applies to all particles and reduces to $E=mc^2$ at rest and $E=pc$ for massless particles.