special relativity

Galilean transformations

$$x^{\prime}=x-vt$$

$$t^{\prime}=t$$

$$u^{\prime}=u$$

$$a^{\prime}=a$$

Time difference from beam splitter

$$\Delta t=\frac{L}{c}\frac{v^2}{c^2}$$

Path length difference from beam splitter

$L\frac{v^2}{c^2}$ =path length difference

Lorentz transformations (on formula sheet)

$$x^{\prime}=\gamma(x-vt)$$

$$y^{\prime}=y$$

$$z^{\prime}=z$$

$$t^{\prime}=\gamma(t-\frac{v}{c^2}x)$$

Lorentz factor (on formula sheet)

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

Inverse Lorentz transformations $x^{\prime}=\gamma(x-vt)\Rightarrow x=\gamma(x^{\prime}+vt^{\prime})$

Just reverse + sign and primed factors

$$x^{\prime}=\gamma(x-vt)\Rightarrow x=\gamma(x^{\prime}+vt^{\prime})$$

time difference between 2 events (failure of simultaneity)

$$\Delta t^{\prime}=\gamma\frac{v}{c^2}(x_1-x_2)$$

Length contraction (on formula sheet)

$$L^{\prime}=\frac{L_0}{\gamma}$$

the body at rest will perceive shortening of the moving body

Time dilation (on formula sheet)

$$T^{\prime}=\gamma T_0$$

In the rest inertial frame time appears longer

Mass energy equivalence (on formula sheet)

$$E=mc^2$$

where m is relativistic mass

relationship between rest mass and relativistic mass (on formula sheet)

$$m=\gamma m_0$$

m is relativistic mass

$m_0$ is rest mass

Kinetic energy (on formula sheet)

$$KE=c^2(m-m_0)$$

Relationship between energy and momentum (on formula sheet)

$$E_0^2=E^2-\rho^2c^2$$