Intro to Special Relativity : Seminar Saturday

Spacetime Overview

• Example of surveyors: invariant distance
  

  $(\mathrm{\Delta }{x}_1{)}^2=(\mathrm{\Delta }{x}_2{)}^2$

• Example of spacetime: invariant interval
  

  $(c\mathrm{\Delta }{t}_1{)}^2-(\mathrm{\Delta }{x}_1{)}^2=(c\mathrm{\Delta }{t}_2{)}^2-(\mathrm{\Delta }{x}_2{)}^2=(\text{interval}{)}^2$

• 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

$$\begin{array}{rl}10\text{ft}& =3.048\text{m}\\ ⟹\frac{10\text{ft}}{3.048\text{m}}& =\frac{3.048\text{m}}{10\text{ft}}=1[\text{unitless}]\end{array}$$

If we want to convert 3.14 feet to meters:

$$\begin{array}{rl}3.14\text{m}& =3.14\text{m}\ast 1[\text{unitless}]\\ & =3.14\text{m}\ast \frac{10\text{ft}}{3.048\text{m}}\\ & =3.14\ast \frac{10\text{ft}}{3.048}\\ & =\frac{3.14\ast 10}{3.048}\text{ft}\\ & =0.957072\text{ft}\end{array}$$

Meters and seconds example

$$\begin{array}{rl}1\text{s}& =c\text{ m}\\ ⟹\frac{1\text{s}}{c\text{ m}}& =\frac{c\text{ m}}{1\text{s}}=1[\text{unitless}]\end{array}$$

If we want to convert 3.14 s to meters:

$$\begin{array}{rl}3.14\text{s}& =3.14\text{s}\ast 1[\text{unitless}]\\ & =3.14\text{s}\ast \frac{3\times {10}^8\text{m}}{1\text{s}}\\ & =3.14\ast \frac{3\times {10}^8\text{m}}{1}\\ & =\frac{3.14\ast 3\times {10}^8}{1}\text{m}\\ & =9.42\times {10}^8\text{m}\end{array}$$

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 =

$\frac{2m}{(3c/4)m{s}^{-1}}$ =

$\frac{8}{3c}s$ of time =

$8.89\times {10}^{-9}s$ =

$\frac{8}{3c}s\times \frac{cm}{1s}$ =

$\frac{8}{3}m$ of time

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

Time in proton frame

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

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)