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