Relativistic collisions using four-momentum
PHYS 401 · Four-Vectors and Minkowski Spacetime
Four-momentum conservation provides a powerful method for analyzing relativistic collisions and particle reactions. This lesson explains invariant mass, threshold energy, and center-of-momentum reasoning.
Key equations
p^mu=left( rac{E}{c}, ec{p}
ight)sum p^mu_{initial}=sum p^mu_{final}P^mu_{total}=sum_i p_i^muP^mu P_mu=-M^2c^2M^2c^4=E_{total}^2-p_{total}^2c^2 ec{p}*{total}=0E*{CM}=Mc^2P_i^mu=(Mc, ec{0})P_i^mu=p_1^mu+p_2^muE_{CM}=sum_f m_fc^2Learning objectives
- State conservation of four-momentum.
- Define invariant mass of a system.
- Use the center-of-momentum frame conceptually.
- Explain how kinetic energy can become rest energy.
- Describe threshold energy for particle creation.
Conservation in spacetime form
In relativity, energy and momentum are conserved together as four-momentum. For each particle,
ight)$$ For an isolated collision or decay, $$sum p^mu_{initial}=sum p^mu_{final}$$ This single four-vector equation includes energy conservation and three components of momentum conservation. ## Why four-momentum helps Energy and momentum separately depend on frame, but the total four-momentum has an invariant magnitude. If $$P^mu_{total}=sum_i p_i^mu$$ then the invariant mass $M$ of the system satisfies $$P^mu P_mu=-M^2c^2$$ or $$M^2c^4=E_{total}^2-p_{total}^2c^2$$ This invariant can be computed in any frame. ## Center-of-momentum frame The center-of-momentum frame is the frame where total spatial momentum is zero: $$ ec{p}*{total}=0$$ In that frame, $$E*{CM}=Mc^2$$ This frame is often the simplest for collision analysis because energy is not tied up in overall motion of the system. ## Decay example Suppose a particle of mass $M$ at rest decays into two particles. Initial four-momentum is $$P_i^mu=(Mc, ec{0})$$ Four-momentum conservation gives $$P_i^mu=p_1^mu+p_2^mu$$ The daughter particles must have equal and opposite momenta in the parent rest frame. Their energies are determined by conservation and the energy-momentum relation. ## Inelastic collision In relativity, kinetic energy can become rest energy. If two particles collide and stick together, the final composite object's rest mass can exceed the sum of the original rest masses. The extra rest mass comes from internal energy. Classically, lost kinetic energy becomes heat or deformation. Relativistically, that internal energy contributes to invariant mass. ## Particle creation threshold New particles can be created when enough invariant energy is available. It is not enough for one frame to assign a large kinetic energy; the invariant mass-energy of the system must meet the final rest-energy requirement. At threshold in the center-of-momentum frame, final particles have no kinetic energy relative to one another, so available energy just equals total rest energy: $$E_{CM}=sum_f m_fc^2$$ ## Lab versus collider efficiency In a fixed-target experiment, much of the incoming particle energy becomes motion of the center of mass. In a collider with equal and opposite beams, total momentum can be zero, so more energy is available for creating new particles. This is why high-energy colliders are powerful tools for particle physics. ## The big idea Relativistic collisions are best analyzed by conserving total four-momentum. The invariant mass of a system connects total energy and total momentum in a frame-independent way. Four-momentum methods reveal how kinetic energy, rest energy, internal energy, and particle creation fit into one conservation law.Ask your AI physics guide
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