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