Mass-Energy

Summary

• The special theory of relativity is a founded on two postulates (assumed truths).
  1. The laws of physics are invariant in all inertial reference frames.
     • An invariant property is one that does not change under any circumstances.
     • An inertial reference frame is one where the observer is not accelerating.
  2. The speed of light in a vacuum is the same for all sources and observers
     • The speed of light is not affected by the motion of the source or the observer.
• This results in two new equations for the energy and momentum of a moving object.
  • Relativistic energy

    E =	mc2
    	√(1 − v2/c2)

    or using the Lorentz gamma notation…

    E = γmc²

    For objects at rest (v = 0, γ = 1) this reduces to the mass-energy equivalence.

    E = mc²

  • Relativistic momentum

    p =	mv
    	√(1 − v2/c2)

    or using the Lorentz gamma notation…

    p = γmv

    At low speeds (v → 0, γ → 1) this reduces to the classical equation for momentum.

    p = mv

• These two equations can be combined mathematically to yield a relativistic energy-momentum relation.
  • In general, for massive particles in motion, the equation is written as…

    E² = p²c² + m²c⁴

  • For massive particles at rest, the equation reduces to the mass-energy equivalence…

    E = mc²

  • For massless particles (e.g. photons) moving at the speed of light, the equation reduces to…

    E = pc