PX155 - H2 - length contraction

• consider a rod at rest in frame $S^{\prime }$ along the $x^{\prime }$ axis. two ends at $x_A^{\prime }$ and $x_B^{\prime }$
• length of rod is $x_B^{\prime }-x_A^{\prime }=L^{\prime }$
• in frame a $S$, the rod is moving at $+u$, so, it must measure $x_A$ and $x_B$ at the same time, otherwise not the same length
  $x_A^{\prime }=\gamma (x_A-ut)$
  $x_B^{\prime }=\gamma (x_B-ut)$
  $L^{\prime }=x_B^{\prime }-x_A^{\prime }=\gamma (x_B-ut)-\gamma (x_A-ut)=\gamma L$
  • for a rod at rest in $S$:

• what if the rod is at rest in $S$?
  $x_A=\gamma (x_A^{\prime }+u{t}^{\prime })$
  $x_B=\gamma (x_B^{\prime }+u{t}^{\prime })$
  • NB: must be measured at the same $t^{\prime }$
    $L=x_B-x_A=\gamma (x_A^{\prime }+u{t}^{\prime })-\gamma (x_B^{\prime }+u{t}^{\prime })=\gamma L^{\prime }$
  • for a rod at rest in $S$:

• define $L_0$ as "proper length" as length in the rest frame of the rod. $L_u$ as length in the frame where the rod moves at $u$
• length contraction:

- the direction of $u$ doesn't matter as $L_u$ depends on $u^2$

• eg: our galaxy is $100,000$ light years across its own rest frame. to an UHECR proton with $\gamma =5\times {10}^{10}$, the galaxy is contracting
  • $L_u=\frac{L_0}{\gamma }=\frac{9.5\times {10}^{20}}{5\times {10}^{10}}=1.9\times {10}^{10}m\approx 13\mathrm{\%}$ of the earth-sun distance