physics / relativity.mdon commit [spec] minor clarifications to calculus section of sac notes (065aae3)
   1# Special Relativity
   2
   3## Postulates of special relativity
   4
   51. Laws of physics are constant in all inertial reference frames
   6
   72. Speed of light $c$ is the same to all observers (proved by Michelson-Morley)
   8
   9$\therefore$, $t$ must dilate as speed changes.
  10
  11Time and space - four dimensional relationship
  12
  13**Inertial reference frame** - not accelerating
  14
  15**Proper time / length** - measured by observer in same frame as events
  16
  17## Lorentz factor
  18
  19$$\gamma = {1 \over \sqrt{1-{v^2 \over c^2}}}$$
  20
  21Proper (time|length) denoted by ($t|l_0$) is time observed from moving frame.
  22
  23Applied:
  24
  25$t = t_0 \gamma$ (time - longer in moving frame)
  26$l={l_0 \over \gamma}$ (length - only contracts in direction of motion - shorter in moving frame)
  27$m=m_0 \gamma$ (mass dilation)
  28
  29$$v=c \sqrt{1-{1 \over \gamma ^2}}$$
  30
  31($c=3 \times 10^8$ m / s)
  32
  33## Mass-energy equivalence
  34
  35$E_0=mc^2$ (rest energy)
  36
  37$E=\gamma mc^2$ (total energy)
  38
  39- mass is a dense form of energy
  40
  41### Relativistic momentum
  42$$\rho = {mv \over {\sqrt{1-{v^2 \over c^2}}}} = \gamma mv=\gamma \rho_0$$
  43
  44Low speeds - little difference between relativistic & non-relativistic momentum
  45$\rho$ approaches $\infty$ as speed approaches $c$
  46Impossible to reach speed of light (speed $c$ requires $\infty$ energy)
  47Photon has no mass - zero space or time
  48
  49### Total energy of an object
  50
  51$$E_{total}=E_k+E_{rest}=\rho mc^2$$
  52
  53### Kinetic energy
  54
  55$$E_k=(\gamma - 1)mc^2$$
  56
  57(takes relativity into account - use for fast objects)
  58
  59### Rest energy
  60$$E_{rest}=mc^2$$
  61(where $v<0.1c$) - this is the innate mass energy (proper energy)
  62
  63### Work
  64
  65$$W = \Delta E = \Delta mc^2$$
  66
  67### Velocity
  68
  69$$v={\rho \over {m \sqrt{1+{p^2 \over {m^2c^2}}}}}$$
  70
  71### Fusion / Fission equations
  72
  73eV to J: multiply by $1.6\times 10^{-19}$ (charge of an electron)
  74
  75## High altitude muons
  76- Time dilation - more muons reach Earth than expected
  77- Normal half-life is $2.2 \mu s$ in stationary frame of reference
  78- At close to $c$, muons observed from Earth have lifetime $> 2.2 \mu s$
  79- Slower time - more time to travel, so more muons reach surface