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:
- Experiment is repeated using particles accelerated to high speeds:
- what we might expect:
- what we find:
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
We differentiated w.r.t ‘time’ to get
- 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
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
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:
Analysis of one cycle of clock viewed in motion:
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:
- speed of light: c
- hence time between ticks:
- cross-multiply, square, and substitute:
- Reorganise:
Implications for time
- We showed
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
- We showed
- The moral: Time is not simple!
- The burning question: Is it REALLY like that?
Learning Resources
![]() | HRW Chapter 37 takes you through the first steps in Special Relativity. |