Special Relativity

Postulates of special relativity

Laws of physics are constant in all inertial reference frames
Speed of light $c$ is the same to all observers (proved by Michelson-Morley)

$\therefore$ , $t$ must dilate as speed changes.

Time and space - four dimensional relationship

Inertial reference frame - not accelerating

Proper time / length - measured by observer in same frame as events

Lorentz factor

$\gamma = {1 \over \sqrt{1-{v^2 \over c^2}}}$

Proper (time|length) denoted by ( $t|l_0$ ) is time observed from moving frame.

Applied:

$t = t_0 \gamma$ (time - longer in moving frame)
$l={l_0 \over \gamma}$ (length - only contracts in direction of motion - shorter in moving frame)
$m=m_0 \gamma$ (mass dilation)

$v=c \sqrt{1-{1 \over \gamma ^2}}$

( $c=3 \times 10^8$ m / s)

Mass-energy equivalence

$E_0=mc^2$ (rest energy)

$E=\gamma mc^2$ (total energy)

mass is a dense form of energy

Relativistic momentum

$\rho = {mv \over {\sqrt{1-{v^2 \over c^2}}}} = \gamma mv=\gamma \rho_0$

Low speeds - little difference between relativistic & non-relativistic momentum
$\rho$ approaches $\infty$ as speed approaches $c$
Impossible to reach speed of light (speed $c$ requires $\infty$ energy)
Photon has no mass - zero space or time

Total energy of an object

$E_{total}=E_k+E_{rest}=\rho mc^2$

Kinetic energy

$E_k=(\gamma - 1)mc^2$

(takes relativity into account - use for fast objects)

Rest energy

$E_{rest}=mc^2$

(where $v<0.1c$ ) - this is the innate mass energy (proper energy)

Work

$W = \Delta E = \Delta mc^2$

Velocity

$v={\rho \over {m \sqrt{1+{p^2 \over {m^2c^2}}}}}$

Fusion / Fission equations

eV to J: multiply by $1.6\times 10^{-19}$ (charge of an electron)

High altitude muons

Time dilation - more muons reach Earth than expected
Normal half-life is $2.2 \mu s$ in stationary frame of reference
At close to $c$ , muons observed from Earth have lifetime $> 2.2 \mu s$
Slower time - more time to travel, so more muons reach surface