PHYS20040:
From Classical to Modern Physics
1 The geometry of relativity
1.1 Einstein’s postulates
Principle of relativity
The laws of Physics apply in all inertial frames of reference.
Universal speed of light
The speed of light in vacuum is the same for all inertial observers, regardless of the motion of the light source.
1.2 Thought experiments
Let’s do some thought experiments:
- Simultaneity
- Time dilation
- Lorentz contraction
1.3 Simultaneity
- Imagine a car travelling at constant speed.
- Turn on the light in the middle of the wagon.
- Observer in the train sees light reach two ends at the same time.
1.4 Simultaneity
- Observer on the ground sees light reach point (a) before point (b)
1.5 Time dilation
- Consider the light ray that reaches the floor of the wagon.
- How long does it take?
- On the train: \[\Delta \bar t = \frac{h}{c} \tag{1}\]
1.6 Time dilation
- From the ground: \[\Delta t = \frac{ \sqrt{h^2 + (v\Delta t^2)} }{c} \tag{2}\]
- Solve Equation 2 for \(\Delta t\): \[\Delta t = \frac{h}{c} \frac{1}{\sqrt{1-v^2/c^2} } \tag{3}\]
1.7 Time dilation
Therefore:
\[\Delta \bar t = \sqrt{1-v^2/c^2} \cdot \Delta t \tag{4}\]
Remember: \(\Delta \bar t\) is the time measured from inside the train.
1.8 Definitions
\[ \beta = \frac{v}{c} \tag{5}\]
\[ \begin{aligned} \gamma &= \frac{1}{\sqrt{1-v^2/c^2}} \\ &= \frac{1}{\sqrt{1-\beta^2}} \end{aligned} \tag{6}\]
1.9 \(\beta\)-factor
\[ \beta = \frac{v}{c},\;0 < \beta < 1 \]
1.10 \(\gamma\)-factor or Lorentz factor
\[ \gamma = \frac{1}{\sqrt{1-v^2/c^2}},\;\; \gamma>1\]
1.11 Exercise
| Velocity | m/s | \(\beta\) | \(\gamma\) | |
|---|---|---|---|---|
| Usaine Bolt | 9.58s / 100m | |||
| Bicycle | 30 km/h | |||
| Car | 70 mph | |||
| Bullet train | 320 km/h | |||
| Plane | 800 km/h | |||
| Sound | 1,235 km/h | |||
| Concorde | 2,179 km/h | |||
| Falcon-9 | 9.31 km/s |
1.12 Time dilation
\[ \Delta \bar t = \sqrt{1-v^2/c^2} \cdot \Delta t = \Delta t/ \gamma \]
\[ \gamma > 1 \Rightarrow \Delta \bar t > \Delta t \]
Remember: \(\Delta \bar t\) is the time measured from inside the train.
1.13 Time dilation
Moving clocks run slow
1.14 Time dilation
Does this contradict Postulate 1?
1.15 Measure time dilation
A train is passing through Bristol Temple Meads. At which speed does the time, \(\Delta \bar t\), as measured on the train, differs from the time , \(\Delta t\), as measured from Platform 1 by 0.1%?
1.16 Measure time dilation
A train is passing through Bristol Temple Meads. At which speed does the time, \(\Delta \bar t\), as measured on the train, differs from the time , \(\Delta t\), as measured from Platform 1 by 0.1%?
\[ \Delta \bar t = \frac{\Delta t}{\gamma} = 1.001 \Delta t \]
1.17 Measure time dilation
A train is passing through Bristol Temple Meads. At which speed does the time, \(\Delta \bar t\), as measured on the train, differs from the time , \(\Delta t\), as measured from Platform 1 by 0.1%?
\[ \Delta \bar t = \frac{\Delta t}{\gamma} = 1.001 \Delta t \]
Solve for v: \[ v = \sqrt{1-(\frac{1}{1.001})^2}\;c = 0.045 c = 1.3 \times 10^7\; m/s\]
1.18 Twin paradox
A twin leaves the Earth on a rocket ship travelling close to the speed of light.
He comes back to reunite with his sister after 10 years.
Do they have a different age?
Why is this a terrible example for a paradox?
1.19 Muons
- Muons are relativistic particles (\(\gamma=25.33\)).
- Their lifetime is measured to be \(2.10\mu s\).
- How far can they travel in the Earth’s atmosphere?
- Why do we detect them on the ground?
1.20 Muons
1.21 Lorentz contraction
- Let’s set a lamp on one side of the wagon, a mirror on the other side.
- Observers on the train measure \[ \Delta \bar t = 2 \frac{\Delta \bar x}{c} \] for the light to bounce back.
\(\bar x\) is the length of the wagon
1.22 Lorentz contraction
- Observers on the ground measure \[ \Delta t_1 = \frac{\Delta x+v\Delta t_1}{c} \] \[ \Delta t_2 = \frac{\Delta x-v\Delta t_2}{c} \]
1.23 Lorentz contraction
Solve for \(\Delta t_1\), \(\Delta t_2\) :
\[ \Delta t_1 = \frac{\Delta x}{c-v},\;\; \Delta t_2 = \frac{\Delta x}{c+v} \]
So, round trip takes: \[ \Delta t = \Delta t_1 + \Delta t_2 = 2 \frac{\Delta x}{c}\frac{1}{1-v^2/c^2} \]
1.24 Lorentz contraction
From time dilation:
\[ \Delta \bar t = \sqrt{1-v^2/c^2} \Delta t\]
Therefore:
\[ \Delta \bar x = \frac{1}{\sqrt{1-v^2/c^2}} \Delta x = \gamma \Delta x\]
1.25 Lorentz contraction
Moving objects are shorter
1.26 Muons again
- Assume they are created at 10km above the ground.
- How do the muons ‘experience’ their travel through the atmosphere?
- Why do they reach the ground according to their frame of reference?