Modifications to Mechanics

Back to Special Relativity

Contents

Classical Mechanics and Quantum Mechanics presuppose that distances and intervals are the same across inertial reference frames. The theories had to be modified to make them compatible with Special Relativity. (Maxwell’s Theory of Electromagnetism required no modification.)

Classical Mechanics, Modified (Relativistic Mechanics)
  • p = γmv replaces p = mv, the formula for momentum.
  • F = dp/dt replaces F = ma, the classic equation of motion
  • KE = γmc2 – mc2 replaces KE = ½ mv2, the equation for kinetic energy.
Quantum Mechanics, Modified
  • In 1928 Paul A.M. Dirac set forth a wave equation for the electron that combined Special Relativity with Quantum Mechanics
  • The theory explained the spin of the electron that had been earlier theorized.
  • It also predicted antimatter.
    • “From the beginning, Dirac was aware that his theory had a problem: it had an extra set of solutions that made no physical sense, as they corresponded to negative values of energy. After failing to make sense of this negative energy, Dirac admitted in 1931 that his theory, if true, implied the existence of “a new kind of particle, unknown to experimental physics, having the same mass and opposite charge to an electron.” One year later, to the astonishment of physicists, this particle—the antielectron, or positron—was accidentally discovered in cosmic rays by Carl Anderson of the United States.”
Kinetic Energy in Classical Mechanics
  • The kinetic energy of a moving object is the quantity of work required to increase its velocity from zero to v
  • The classical formula for kinetic energy is KE = ½ mv2
    • where m is the mass of the object
    • and v its velocity
  • The formula is derived from
    • W = FD (work = force X distance)
    • F = MA (force = mass X acceleration)
  • For example, KE = 500 joules
    • if m = 10 kg (22 lbs)
    • and v = 10 m/s (22.4 mph)
Kinetic Energy in Relativistic Mechanics
  • The formula for kinetic energy in Relativistic Mechanics is:
    • KE = γmc2 – mc2
      • where γ is the Lorentz Factor (γ ≥ 1)
  • The formula is derived from
    • W = FD (work = force X distance)
    • F = d(γmv)/dt (force = the rate of change of relativistic momentum)
  • For example, KE = 512 joules
    • where m = 10 kg (22 lbs)
    • and v = 10 m/s (22.4 mph)
    • γmc2 = 8.98755178736818176 * 1017
    • mc2 = 8.98755178736817664 * 1017
    • KE = 8.98755178736818176 * 1017 – 8.98755178736817664 * 1017  = 512 

Physicists no longer use the terms relativistic mass and rest mass.  The single word mass is used to refer to the mass of a particle at rest.

E = mc2
  • Kinetic energy in Relativistic Mechanics is:
    • KE = γmc2 – mc2
      • where γ is the Lorentz Factor (γ ≥ 1)
  • For a particle at rest, γ = 1 and KE = mc2 – mc2 = 0.
  • For a particle in motion, the kinetic energy is greater than zero because γ is positive. 
  • Unlike ½ mv2, the relativistic formula subtracts one quantity of energy from another.  It’s natural to regard γmc2 as the total energy of the particle and mc2 as its energy at rest.
  • Thus E = mc2 means that the energy E of a particle at rest equals its mass m times the speed of light squared, c2.
E = mc2 in Action
Particle Annihilation
  • If a particle and its antiparticle collide, they annihilate each other, releasing energy.
  • Positrons (i.e. positive electrons) don’t live long because they are attracted to negatively-charged electrons.  When they collide (at V1 in the diagram), the masses of the particles are converted into energy in the form of a high-energy photon γ per E=mc2.  The photon γ then splits (at V2) into a muon (𝜇) and an antimuon (𝜇+).
  • Charge, energy, and momentum are conserved in the interaction.
  • (Arrows for antimatter particles are reversed in Feynman Diagrams.)