Glossary

7.3 Conservation laws in special relativity

3 min read•august 7, 2024

Special relativity reshapes our understanding of conservation laws. Energy and momentum are now intertwined, with at the core. This means the 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 and become crucial concepts.

Conservation Laws

Energy and Momentum Conservation

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  • 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
  • states that the of an isolated system remains constant over time
  • is defined as p=γmvp = \gamma mv, where γ=1/1−v2/c2\gamma = 1/\sqrt{1-v^2/c^2} is the , mm is the , and vv is the velocity

Mass-Energy Equivalence

  • 's famous equation E=mc2E = 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 , 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 , 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 c2c^2

Invariant Mass

  • The invariant mass of a system is a quantity that is independent of the reference frame
  • It is defined as minv=E2/c4−p2/c2m_{inv} = \sqrt{E^2/c^4 - p^2/c^2}, where EE is the total energy and pp 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
  • Examples:
    • The invariant mass of a proton is its rest mass, approximately 938 MeV/c2c^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)

Key Terms to Review (15)

Center of mass frame: The center of mass frame is a reference frame in which the center of mass of a system is at rest. This concept is crucial for analyzing interactions and collisions in special relativity, as it simplifies the calculations of momentum and energy by eliminating the effects of motion from the perspective of the center of mass.
Conservation of energy: Conservation of energy is a fundamental principle stating that the total energy of an isolated system remains constant over time, meaning energy cannot be created or destroyed, only transformed from one form to another. This principle is essential in understanding how energy behaves in relativistic scenarios, particularly when examining how momentum and energy are interrelated and conserved in special relativity.
Conservation of momentum: Conservation of momentum is a fundamental principle stating that the total momentum of a closed system remains constant if no external forces act on it. This principle is crucial in understanding interactions between objects in relativistic physics, where momentum behaves differently than in classical mechanics. It ties closely to the concepts of relativistic momentum and energy, as well as the conservation laws that govern physical processes at high speeds.
E=mc²: The equation e=mc², formulated by Albert Einstein, expresses the equivalence of mass (m) and energy (e), stating that energy is equal to mass multiplied by the speed of light squared (c²). This fundamental concept shows that mass can be converted into energy and vice versa, which is crucial for understanding various physical phenomena, including conservation laws, the behavior of particles at high speeds, and the implications of special relativity.
Einstein: Albert Einstein was a theoretical physicist known for developing the theory of relativity, which fundamentally changed our understanding of space, time, and gravity. His work laid the groundwork for modern physics, particularly in understanding how conservation laws are viewed in the context of special relativity, where mass and energy are interconnected through the famous equation $$E=mc^2$$.
Elastic collision: An elastic collision is a type of collision in which both momentum and kinetic energy are conserved. During such a collision, the objects involved rebound off each other without any permanent deformation or generation of heat. This concept is crucial in understanding interactions at the fundamental level, especially in the context of conservation laws in special relativity, where the behavior of particles and energy transformations are analyzed.
Inelastic collision: An inelastic collision is a type of collision in which the kinetic energy of the system is not conserved, although momentum is conserved. During this process, some of the kinetic energy is transformed into other forms of energy, such as heat or sound, and the colliding bodies may stick together or deform. This characteristic is crucial for understanding conservation laws in special relativity as it highlights the interplay between energy and momentum in high-speed interactions.
Invariant mass: Invariant mass is a measure of the mass of a system that remains constant regardless of the system's velocity or the reference frame from which it is observed. It is crucial for understanding the behavior of particles and their interactions in relativistic contexts, linking energy and momentum to a consistent description of mass across different frames of reference.
Lorentz Factor: The Lorentz factor is a mathematical term that arises in the theory of relativity, represented as $$ rac{1}{ ext{sqrt}(1 - rac{v^2}{c^2})}$$, where $$v$$ is the velocity of an object and $$c$$ is the speed of light. It quantifies how much time, length, and relativistic mass increase as an object approaches the speed of light. Understanding the Lorentz factor is crucial for grasping the effects of time dilation, length contraction, and relativistic measurements that occur in high-speed scenarios.
Mass-energy equivalence: Mass-energy equivalence is the principle that mass and energy are interchangeable; they are different forms of the same thing. This concept is famously encapsulated in the equation $$E = mc^2$$, which shows that energy (E) is equal to mass (m) multiplied by the speed of light (c) squared. This relationship implies that a small amount of mass can be converted into a large amount of energy, connecting the fundamental concepts of energy, mass, and their roles in physical processes.
Particle Physics: Particle physics is the branch of physics that studies the fundamental constituents of matter and radiation, along with their interactions. It explores the smallest building blocks of the universe, such as quarks, leptons, and bosons, and helps us understand the fundamental forces that govern their behavior. The insights gained from particle physics connect closely with concepts like relativistic velocity, conservation laws, and the constancy of light speed.
Relativistic momentum: Relativistic momentum is the momentum of an object when it is moving at a significant fraction of the speed of light, taking into account the effects of special relativity. It differs from classical momentum because it incorporates the Lorentz factor, which adjusts for time dilation and length contraction experienced at high velocities. This concept is essential for understanding how objects behave as they approach the speed of light, especially in terms of energy and conservation laws.
Rest Mass: Rest mass is the intrinsic mass of an object when it is at rest, which is invariant regardless of its state of motion. It serves as a fundamental property that plays a critical role in understanding the relationship between mass and energy, as well as how particles interact in different frames of reference. The concept helps bridge ideas about how objects behave when they are not influenced by external forces or relative velocities.
Total energy: Total energy is the sum of all forms of energy present in a system, including kinetic, potential, and rest mass energy, and is a crucial concept in understanding the behavior of objects in motion, especially at relativistic speeds. In the realm of physics, total energy is conserved, meaning that it remains constant in an isolated system, reflecting the interplay between different types of energy as they transform from one form to another. This principle is key to comprehending how energy behaves under special relativity and informs the design and operation of high-energy particle accelerators.
Total momentum: Total momentum refers to the vector sum of the momenta of all objects in a system, which remains constant in an isolated system due to the law of conservation of momentum. This principle is crucial in understanding interactions between particles and how they exchange energy and momentum during collisions and other processes, particularly in the context of special relativity where the relativistic effects come into play as speeds approach the speed of light.
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