Physics 616

 
 
 
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Outline

Units

Special relativity

Indices

Metric

Distance "interval" between two points

Space-time diagrams

  • Proper time between two events is time ticked off by a clock which passes through both events
  • Proper time, \( \Delta \tau^2 = - \Delta s^2 \)
Schutz
Strickland (2014)

Space-time diagrams II

Schutz

Length Contraction

  • World path of a rod at rest in \( \bar{\cal O} \)
    Length in \( \bar{\cal O} \) is \( \Delta s_{AC}^2 \) and in \( {\cal O} \) is \( \Delta s_{AB}^2 \)
    In \( \bar{\cal O} \), the coordinates of C are $$ \bar{t}_C=0, \quad \bar{x}_C=\ell $$ while in \( {\cal O} \), the coordinates of C are $$ x_C = \ell/\sqrt{1-v^2} \quad t_C= \ell v / \sqrt{1-v^2} $$ but the interval is the same in both systems so $$ x_C^2 - t_C^2 = \ell^2 $$ and $$ t_C = v x_C $$ Also, $$ \frac{x_C-x_B}{t_C-t_B}=v $$
Schutz
  • We want \( x_B \) when \( t_B =0 \) $$ x_B = x_C - v t_C = \ell \sqrt{1-v^2} $$
  • This is length contraction

SR results

Invariant Hyperbolae

  • Curves with fixed intervals
  • What might these curves be useful for?
Schutz

Group Work