# Intro to Special Relativity : Seminar Saturday

## Spacetime Overview
- Example of surveyors: invariant distance
$(\Delta x_1)^2 = (\Delta x_2)^2$

- Example of spacetime: invariant interval
$(c\Delta t_1)^2 - (\Delta x_1)^2 = (c\Delta t_2)^2 - (\Delta x_2)^2 = (\text{interval})^2$

- Why do we need invariants?

## Events and Intervals
- Wristwatch time, proper time, local time
- Space-like vs time-like intervals

## Units of space and time
- Space and time have the **same** units in our framework.
- $c$ is a conversion factor, like factor between feet and metres, accident of history

### Feet and Metre example
$$\begin{align}
10 \text{ft} &= 3.048 \text{m} \\
\implies \frac{10 \text{ft}}{3.048 \text{m}} &= \frac{3.048 \text{m}}{10 \text{ft}} = 1 [\text{unitless}]
\end{align}$$

If we want to convert 3.14 feet to meters:
$$\begin{align}
3.14 \text{m} &= 3.14 \text{m} * 1[\text{unitless}] \\
&= 3.14 \text{m} * \frac{10 \text{ft}}{3.048 \text{m}} \\
&= 3.14 * \frac{10 \text{ft}}{3.048} \\
&= \frac{3.14 * 10}{3.048} \text{ft} \\
&= 0.957072 \text{ft}\\
\end{align}
$$

### Meters and seconds example
$$\begin{align}
1 \text{s} &= c \text{ m} \\
\implies \frac{1 \text{s}}{c\text{ m}} &= \frac{c\text{ m}}{1 \text{s}} = 1 [\text{unitless}]
\end{align}$$

If we want to convert 3.14 s to meters:
$$\begin{align}
3.14 \text{s} &= 3.14 \text{s} * 1[\text{unitless}] \\
&= 3.14 \text{s} * \frac{3 \times 10^8 \text{m}}{1 \text{s}} \\
&= 3.14 * \frac{3 \times 10^8 \text{m}}{1} \\
&= \frac{3.14 * 3 \times 10^8}{1} \text{m} \\
&= 9.42 \times 10^8 \text{m}\\
\end{align}
$$

### Example with same units of time and space
A proton moving at 3/4 light speed passes through 2 detectors in a laboratory placed 2 metres apart. The events 1 and 2 are the transits through the detectors.

#### Interval
Lab time between events = $\frac{2m}{(3c/4) ms^{-1}}$ = $\frac{8}{3c}s$ **of time** = $8.89 \times 10^{-9} s$ = $\frac{8}{3c}s \times \frac{c m}{1s}$ = $\frac{8}{3} m$ **of time**

$$\begin{align}
(\text{proton interval})^2 &= (\text{lab interval})^2 \\
(\text{lab interval})^2 &= (\frac{8}{3}m)^2 - (2m)^2 \\
&= (2.66667m)^2 - (2m)^2 \\
&= (7.1111 - 4) (\text{metres})^2 \\
&= 3.1111 (\text{metres})^2 \\
\end{align}
$$

#### Time in proton frame
$$\begin{align}
(\text{proton interval})^2 &= (\text{lab interval})^2 \\
(\text{proton time})^2 - (\text{proton distance})^2 &= 3.1111 m^2 \\
(\text{proton time})^2 - (0 m)^2 &= 3.1111 m^2 \\
\text{proton time} &= 1.764 m \\
\text{proton time} &= 1.764 m \times \frac{1s}{c \text{ m}} \\
\text{proton time} &= 5.88 \times 10^{-9} s \\
\end{align}
$$

### Comments
- Time dilation follows from invariance of interval (which is a consequence of invariance of light speed)
- Space and time have the _same units_ but are not the same in quality, as seen by the sign of the metric

## Measuring space time
- Framework of metre sticks and clocks
- Coordinating all the clocks using a light pulse
- Events at the "same place" versus at the "same time"

## Free float frames
- Free float, inertial frame
- Any frame in which the Law of Inertia applies
- Free float frames are local in nature
- Local in time and local in space

## Special Relativity
All inertial frames are equivalent.
- An observer can verify physics by making observations from his own frame.
- All inertial observers will see the same laws of physics.

## Lorentz Transform (some math)