Chapter 2: Special Theory of Relativity

PHYS2710

Galilean Transformations

x^{\prime}=x-vt, y^{\prime}=y , z^{\prime}\equiv z

Inverse Galilean Transformations

x=x^{\prime}+vt , y=y^{\prime} , z=z^{\prime}

Lorentz Tranformations

x^{\prime}=\gamma\left(x-\beta ct\right) , y^{\prime}=y , z^{\prime}\equiv z , t^{\prime}=\gamma\left(t-\frac{\beta x}{c}\right)

Inverse Lorentz Transformations

x=\gamma\left(x^{\prime}+\beta ct^{\prime}\right) , y=y^{\prime} , z=z^{\prime} , t=\gamma\left(t^{\prime}-\frac{\beta x^{\prime}}{c}\right)

Time Dilation

T^{\prime}=\gamma T_0

Length Contractions

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

Lorentz Velocity Transformations

u_{x}^{\prime}=\frac{u_{x}-v}{1-\left(\frac{\beta}{c}\right)u_{x}} , u_{y}^{\prime}=\frac{u_{y}}{\gamma\left\lbrack1-\left(\frac{\beta}{c}\right)u_{x}\right\rbrack} , u_{z}^{\prime}=\frac{u_{z}}{\gamma\left\lbrack1-\left(\frac{\beta}{c}\right)u_{x}\right\rbrack}

Inverse Lorentz Velocity Transformations

u_{x}=\frac{u_{x}^{\prime}+v}{1+\left(\frac{\beta}{c}\right)u_{x}^{\prime}} , u_{y}^{\prime}=\frac{u_{y}^{\prime}}{\gamma\left\lbrack1+\left(\frac{\beta}{c}\right)u_{x}^{\prime}\right\rbrack} , u_{z}^{\prime}=\frac{u_{z}^{\prime}}{\gamma\left\lbrack1+\left(\frac{\beta}{c}\right)u_{x}^{\prime}\right\rbrack}

Relativistic Doppler Effect for a Source and Receiver Approaching

f_{obs}=\frac{\sqrt{1+\beta}}{\sqrt{1-\beta}}f_0

Relativistic Doppler Effect for a Source and Receiver Receding

f_{obs}=\frac{\sqrt{1+\beta}}{\sqrt{1-\beta}}f_0

Relativistic Momentum

\overrightarrow{p}=\gamma m\overrightarrow{u}

Relativistic Kinetic Energy

K=\gamma mc^2-mc^2=mc^2\left(\gamma-1\right)

Rest Energy

E_0=mc^2

Total Energy

E=\gamma mc^2=K+E_0

Momentum-Energy Relation

E^2=p^2c^2+E_0^2=p^2c^2+m^4c^2

Binding Energy

E_{B}=\sum m_{i}c^2-M_{bound}c^2