Relativistic Doppler Effect

Supposed that a light source

• emits $N$ waves during the proper time interval $T_0^{\prime }$, which is the time observed from the moving light source
• For $T_0^{\prime }$ in the light source’s frame, the stationary observer experiences coordinate time $T$ ($T=\gamma T_0^{\prime }$ from Lorentz Transformation)

$$N={\nu }_0{T}_0^{\prime }=\frac{{\nu }_{0}T}{\gamma }\text{where }{\nu }_0\text{ is the source frequency}$$

In the stationary observer’s frame

$$\text{Wave Propogation Distance(T)}=cT+\text{v}T$$ $$\lambda =\frac{cT+\text{v}T}{N}⟹\nu =\frac{cN}{cT+\text{v}T}=\cdots ={\nu }_0\sqrt{\frac{1-\frac{v}{c}}{1+\frac{v}{c}}}$$ $$\nu =\frac{cN}{cT+\text{v}T}={\nu }_0\sqrt{\frac{1-\frac{v}{c}}{1+\frac{v}{c}}}={\nu }_0\frac{\sqrt{1-\frac{{\text{v}}^2}{c^2}}}{1+\cos (\theta )\frac{v}{c}}\approx \left({\nu }_0\frac{1-\frac{1}{2}\frac{{\text{v}}^2}{c^2}}{1+\cos (\theta )\frac{v}{c}}\theta \not\approx 90^{\circ },\text{v}\not\approx c\right)$$

• Use $\beta =\frac{v}{c}$ as > 0 when the source and receiver are receding from each other (red shift to lower frequency [this is the current signs for $\beta$])
• Use $\beta =\frac{v}{c}$ as < 0 when the source and receiver are approaching each other (blue shift to higher frequency)