# 4: Dynamics

- Page ID
- 3444

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

- 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.

- 4.7: 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.