Special relativity reshapes our understanding of conservation laws. Energy and momentum are now intertwined, with mass-energy equivalence at the core. This means the total energy of a system includes both rest energy and kinetic energy.

Collisions in special relativity follow similar principles to classical mechanics, but with important differences. Elastic collisions conserve both energy and momentum, while inelastic collisions only conserve momentum. The center of mass frame and invariant mass become crucial concepts.

Conservation Laws

Energy and Momentum Conservation

• Conservation of energy states that the total energy of an isolated system remains constant over time
• In special relativity, the total energy includes both rest energy and kinetic energy
• Conservation of momentum states that the total momentum of an isolated system remains constant over time
• Relativistic momentum is defined as $p = \gamma mv$, where $\gamma = 1/\sqrt{1-v^2/c^2}$ is the Lorentz factor, $m$ is the rest mass, and $v$ is the velocity

Mass-Energy Equivalence

• Einstein's famous equation $E = mc^2$ relates mass and energy
• Mass and energy are interchangeable and can be converted into each other
• In nuclear reactions and particle collisions, mass can be converted into energy and vice versa
• The total mass-energy of an isolated system is conserved
• Examples:
  • Nuclear fission in power plants converts a small amount of mass into a large amount of energy
  • In particle accelerators, high-energy collisions can create new particles with mass from the kinetic energy of the colliding particles

Collision Types

Elastic Collisions

• In an elastic collision, both the total kinetic energy and total momentum of the system are conserved
• The particles involved in the collision do not undergo any internal changes or deformations
• Kinetic energy is transferred between the particles, but the total kinetic energy remains the same before and after the collision
• Examples:
  • Collisions between billiard balls on a pool table
  • Collisions between atoms in an ideal gas

Inelastic Collisions

• In an inelastic collision, the total momentum is conserved, but the total kinetic energy is not
• Some of the kinetic energy is converted into other forms of energy, such as heat or deformation of the particles
• The particles may stick together or break apart after the collision
• Examples:
  • Collisions between cars in a traffic accident, where the cars deform and kinetic energy is lost to heat and sound
  • Collisions between atoms in a solid, where the atoms can vibrate and transfer energy to the lattice

Reference Frames and Mass

Center of Mass Frame

• The center of mass frame is a reference frame in which the total momentum of the system is zero
• In this frame, the center of mass of the system appears stationary
• Calculations involving collisions and interactions between particles are often simplified in the center of mass frame
• The total energy of the system in the center of mass frame is equal to the invariant mass of the system multiplied by $c^2$

Invariant Mass

• The invariant mass of a system is a quantity that is independent of the reference frame
• It is defined as $m_{inv} = \sqrt{E^2/c^4 - p^2/c^2}$, where $E$ is the total energy and $p$ is the magnitude of the total momentum
• For a single particle, the invariant mass is equal to its rest mass
• For a system of particles, the invariant mass is the rest mass of an equivalent single particle that has the same total energy and momentum as the system
• The invariant mass is conserved in collisions and decays, making it a useful quantity in particle physics
• Examples:
  • The invariant mass of a proton is its rest mass, approximately 938 MeV/$c^2$
  • The invariant mass of a system of two photons can be calculated from their total energy and momentum, and is often used to identify particle decays (such as the decay of a Higgs boson into two photons)