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# 4: Dynamics

• 4.1: Ultrarelativistic particles
Ultrarelativistic objects are objects moving at nearly c . A good way of thinking about an ultrarelativistic particle is that it’s a particle with a very small mass. For example, the subatomic particle called the neutrino has a very small mass, thousands of times smaller than that of the electron.
• 4.2: E=mc²
We now know the relativistic expression for kinetic energy in the limiting case of an ultrarelativistic particle: its energy is proportional to the “stretch factor” D of the Lorentz transformation. What about intermediate cases?
• 4.3: Relativistic Momentum
Since mass and energy are equivalent, we must stop talking about a material object’s kinetic energy and consider instead its total energy E, which includes a contribution from its mass. Massless particles always move at v=c , while massive ones always move at v<c .
• 4.4: Systems with internal structure
E=mc2 and the four-vector nature of p are both valid for systems with ﬁnite spatial extent, provided that the systems are isolated.
• 4.5: Force
Force is a concept that is seldom needed in relativity, and that’s why this section is optional.
• 4.6: Two Applications
• 4.7: Tachyons and Faster-than-Light (FTL)
A tachyons are hypothetical particle that always moves faster than light. Most physicists believe that faster-than-light (FTL) particles cannot exist because superluminal transmission of information would violate causality, since it would allow a causal relationship between events that were spacelike in relation to one another, and the timeordering of such events is diﬀerent according to diﬀerent observers.
• 4.E: Dynamics (Exercises)