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S1.6 Relativity: Einstein`s view

This section is included for general interest. It is not part of the examinable programme of the course.

First steps

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First steps

[A] A worrying experiment

  • A particle (a π0-meson) at rest in the laboratory decays, emitting two photons (γ-rays) which travel off in opposite directions, at speed c:
    Graphic - No title - decayatrest
  • Experiment is repeated using particles accelerated to high speeds:
    • what we might expect:
      Graphic - No title - decayinmotiona
    • what we find:
      Graphic - No title - decayinmotionb
  • Implications

    • light (a photon) always has speed c irrespective of frame of measurer
    • common sense (Galilean) velocity addition formula is wrong at high speeds.

[B] Where does common sense argument go wrong?

  • Recall the argument:
    • We took

      $$ \vec{r}_{PA} = \vec{r}_{PB} + \vec{r}_{BA} $$

    • We differentiated w.r.t ‘time’ to get

      $$ \vec{v}_{PA} = \vec{v}_{PB} + \vec{v}_{BA}$$

  • The assumption: ‘time’ means the same for both frames
  • The mistake: it doesn’t !

[C] Putting it right …Einstein’s theory

  • Key assumptions of Einstein’s theory

    • the universality of c: measured speed of light is always c

      A small lie

      A small lie

      We have missed out a bit of small print: the claim is actually that the speed oflight is always the same in a vacuum; the speed of light through a medium depends on the properties of that medium.
       
    • the Principle of Relativity: the Laws of Physics are the same in all reference frames

      strictly, inertial reference frames

  • Implications for a special clock

    Consider an idealised clock utilising light beams:

    • light is reflected back and forth between mirrors
    • clock ‘ticks’ every time light hits lower mirror
    Graphic - No title - lbclock

    Analysis of one cycle of clock viewed at rest:

    • distance covered by light: 2L0
    • speed of light: c
    • hence time between ticks:
      τ0=2L0/c

    Consider a light-beam clock moving at speed v:

    Graphic - No title - movinglbclock

    Analysis of one cycle of clock viewed in motion:

    A little more....

    A little more....

    Can you stand a painful Truth?

    The analysis that follows supposes that the distance between the mirrors (the length of the clock axis) remains L0 when the clock is viewed in motion. This is common sense, of course; but that does not mean it is right! In this case however common sense is correct: it is possible to use the Principle of Relativity to prove it. But the argument rests on the fact that the clock is supposed to be moving at right angles to its axis. If, instead, the clock were moving parallel to its axis, the axis length would not remain the same: it would be shorter!
     
    • distance covered by light: $2\left[L_0 ^2 + \left( \frac{v \tau}{2}\right)^2\right ] ^{1/2}$
    • speed of light: c
    • hence time between ticks: $\tau = 2\left[L_0 ^2 + \left( \frac{v \tau}{2}\right)^2\right ] ^{1/2} /c$
    • cross-multiply, square, and substitute:
      \[ \left(\frac{c\tau}{2}\right)^2 = L_0^2 + \left(\frac{v\tau}{2}\right)^2 =\left(\frac{c\tau_0}{2}\right)^2 +\left(\frac{v\tau}{2}\right)^2 \]
    • Reorganise:
      \[ \tau^2 \left[ c^2 -v ^2 \right ] = \tau_0^2 c^2 \hspace{0.5cm}\mbox{\rm or}\hspace{0.5cm} \tau = \tau_0/\sqrt{1-v^2/c^2} \]
  • Implications for time

    • We showed
      $$ \tau = \tau_0/\sqrt{1-v^2/c^2}$$
      is true for a light beam clock
    • Principle of relativity implies it must be true for all clocks.
    • Since τ is necessarily bigger that τ0 we may say:

      A moving clock takes longer to complete its tick cycle

      or, more plainly,

      A moving clock runs slow

  • The moral: Time is not simple!
  • The burning question: Is it REALLY like that?
 

Learning Resources

Textbook: HRW Chapter 37 takes you through the first steps in Special Relativity.