Special Relativity

Coordinatizations of spacetime

Intuitively, an “event” is “something which happens at a definite place at a definite time”.

The set $\mathbf E$ of all events is called spacetime.

coordinatization of $\mathbf E$

E onto the four-dimensional real vector space $R^4$

$$e(t_e,x_e,y_e,z_e)$$ $$t_e:=t_0-\frac{1}{c}(x_e^2+y_e^2+z_e^2)^{1/2}$$

Lorentz coordinatizations

Properties

(1)For any stationary standard clock with coordinates $(t(\tau),x_0,y_0,z_0),t(\tau)=\tau$ for all $\tau$.

(2)Light always moves in straight lines with unit velocity.The function $t\longmapsto\overrightarrow{r}(t)$ is of the form $\overrightarrow{r}(t)=\overrightarrow{v}t+\overrightarrow{r_0}$, where $\overrightarrow{v},\overrightarrow{r_0}\in R^3$ are constant vectors, and $\overrightarrow{v}$ is a unit vector.

Minkowski space

$$u=(u^0,u^1,u^2,u^3)$$ $$v=(v^0,v^1,v^2,v^3)$$

Minkowski metric

$$<u,v>:=u^0v^0-u^1v^1-u^2v^2-u^3v^3$$

Lorentz transformations