Doppler Effect for Light — Question 1

Question 1

A wavelength ratio gives an exact recession speed

A spectral line has source-rest wavelength $\lambda_0=400\,\mathrm{nm}$ and is received at $\lambda=600\,\mathrm{nm}$. Assume purely longitudinal recession in flat spacetime, with no other shift.

1. Invert $\lambda/\lambda_0=\sqrt{(1+\beta)/(1-\beta)}$ for the signed recession speed.

2. Find the corresponding frequency ratio and compare with the low-speed estimate $\beta\simeq z$, where $z=(\lambda-\lambda_0)/\lambda_0$.

Question 1 - Solution

A wavelength ratio gives an exact recession speed

(a) Method and calculation

$$R=\lambda/\lambda_0=3/2,\quad R^2(1-\beta)=1+\beta.$$$$\boxed{\beta=\frac{R^2-1}{R^2+1}=5/13\simeq0.384615}.$$

(b) Interpretation and further calculation

$$\boxed{f/f_0=1/R=2/3}.$$$$z=0.500,\qquad\beta\ne z\quad\text{at this shift}.$$

Physical check

The longer wavelength indicates recession; a cosmological or gravitational shift would require a different physical model.