PX155 - H3 - time dilation

• consider a clock at rest in frame $S^{\prime }$
• 2 subsequent ticks take place as events $A$ and $B$ with coordinates $x_A^{\prime },t_A^{\prime }$, and $x_B^{\prime },t_B^{\prime }$
• time coordinates in $S$?
  $t_A=\gamma (t_A^{\prime }+\frac{u{x}_A^{\prime }}{c^2})$
  $t_B=\gamma (t_A^{\prime }+\frac{u{x}_B^{\prime }}{c^2})$
  • at rest in $S^{\prime }$, so $x_A^{\prime }=x_B^{\prime }$
  • so, $t_B-t_A=\gamma (t_B^{\prime }-t_A^{\prime })$
    • for a clock at rest in $S^{\prime }$:

• time between ticks is dilated in frame where clock moves
• similarly, $\mathrm{\Delta }{t}^{\prime }=\gamma \mathrm{\Delta }t$ for a clock at rest in $S$
• define $\mathrm{\Delta }{t}_0$ as "proper time" interval, ie: the interval between ticks in rest frame of the clock

atmospheric muons and time dialation

• rossi and hall (1941)
• cosmic rays hit the upper atmosphere and generate muons heading towards the ground at $\approx c$
• muons:

• number of muons: $N=N_0{e}^{-\frac{t}{\tau }}$ with $\tau =2.2\mu s$
• measure downward flux at the top of a $600m$ mountain and again at the bottom
• expect

• instead measure $0.7$
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