Theory of Special Relativity

Abstract

In Mechanics, one first investigates the motion of particles. In classical Newtonian mechanics, a path is a map from ${\mathbb R}$ to ${\mathbb R}^3$ which gives the position $\bar{r}(t)$ of the particle at time t. A path is thus a parameter representation of a curve in ${\mathbb R}^3$. This is depicted in Fig. 5.1, but one dimension is suppressed. In order to be able to compare this Newtonian viewpoint with the relativistic one, we rephrase the above. We understand the world as a Cartesian product, where the first component of its elements is time, and the second is position, and instead of the path $\bar{r}(t)$, we use the map $\hat{r}:{\mathbb R}\rightarrow {\mathbb R}\times {\mathbb R}^3$ given by $\hat{r}(t) =(t,\bar{r}(t))$. This creates a curve in ${\mathbb R}\times {\mathbb R}^3$ (Fig. 5.2). The velocity vector then has the form $\hat{r}'(t)=(1, \bar{r}'(t))$. This again lies in ${\mathbb R}\times {\mathbb R}^3$, and so every tangent space can also be identified with ${\mathbb R}\times {\mathbb R}^3$.