Back to Special Relativity

Contents

1. Relativistic Mechanics
2. Relativistic Quantum Mechanics
3. Kinetic Energy in Classical and Relativistic Mechanics
   1. Derivation of Kinetic Energy from Momentum
   2. Kinetic Energy in Classical Mechanics
   3. Kinetic Energy in Relativistic Mechanics
   4. E = mc2
   5. E = mc2 and Approximate Kinetic Energy
   6. E = mc2 in Action: Particle Annihilation

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.)

Relativistic Mechanics

• Momentum
  • p = mv in Classical Mechanics is replaced by p = γmv in Relativistic Mechanics, where γ is the Lorentz Factor 1/√(1 – v²/c²).
• Equation of Motion
  • F = ma is replaced by F = dp/dt, where p is momentum
• Kinetic Energy
  • K = ½ mv² is replaced by K = γmc², where γ is the Lorentz Factor 1/√(1 – v²/c²).

Relativistic Quantum Mechanics

• 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.”
    • britannica.com/biography/Paul-Dirac

Kinetic Energy in Classical and Relativistic Mechanics

Derivation of Kinetic Energy from Momentum

1. KE = ∫ F dx
   • KE = the kinetic energy of a particle = the quantity of work required to increase the velocity of the particle from zero to v
   • Work = force x distance (F dx)
   • ∫ F dx = the integral of F dx, that is, (very) roughly, the sum of the force F times each tiny piece of distance.
2. KE = ∫ dp/dt dx
   • p = momentum
   • From 1: force = the time rate of change of momentum: F = dp/dt
3. KE = ∫ dp/dt v dt
   • From 2: distance = velocity times time: dx = v dt
4. KE = ∫ dp/dv dv/dt v dt
   • From 3: differential chain rule: dp/dt = dp/dv dv/dt
5. KE = ∫ dp/dv a v dt
   • From 4: acceleration = time rate of change of velocity: dv/dt = a
6. KE = ∫ dp/dv v dv
   • From 5: velocity = acceleration times time: a dt = dv
7. See below

Kinetic Energy in Classical Mechanics

• 7. KE = ∫ d(mv)/dv v dv
  • From 6: momentum = mass times velocity: p = mv
• 8. KE = 1/2 mv²
  • Mathematica
    • Integrate[D[m v, v] v, v] = mv²/2

Kinetic Energy in Relativistic Mechanics

• 7. KE = ∫ d(mvγ)/dv v dv
  • From 6: momentum = mass times velocity times the Lorentz Factor : p = mvγ
    • where the Lorentz Factor γ = 1/√(1 – v²/c²)
• 8. KE = γmc² = mc²/√(1 – v²/c²)
  • Mathematica
    • FullSimplify[Integrate[D[γ m v, v] v, v]]
      • = mc²/Sqrt[1 – v²/c²]

E = mc²

• E = mc² says that the energy of a particle at rest equals its mass times the speed of light squared.
• The equation follows directly from the relativistic kinetic energy of particle, γmc².
  • Mathematica
    • KE = γmc² = mc²/√(1 – v²/c²)
    • v = 0
    • Therefore, γmc² = mc²/√(1 – 0/c²) = mc²
• That is, mc² is the “kinetic” energy of a particle at rest.

E = mc² and Approximate Kinetic Energy

• A useful approximation of 1/√(1 – x²), where x is small, is 1 + x²/2.
  • For example, where x = 1/1000:
    • 1/√(1 – x²) = 1.000000500000375000312500
    • 1 + x²/2 = 1.000000500000000000000000
• KE = mc²/√(1 – v²/c²)
• Therefore, KE ≅ mc² (1 + (v/c)²/2) = mc² + 1/2 mv²
  • Mathematica
    • Expand[m c^2 (1 + (v/c)^2/2)]
      • = c^2 m + (m v^2)/2
• Thus, the energy of a particle is its rest energy plus its approximate kinetic energy in Classical Mechanics.

E = mc² in Action: Particle Annihilation