AcademyRelativity
Academy
Relativistic Doppler Effect
Level 1 - Physics topic page in Relativity.
Principle
Relative motion changes measured light frequency because time intervals and wavefront spacing transform together.
Notation
\(f_s\)
frequency in the source rest frame
\(\mathrm{Hz}\)
\(f_o\)
frequency measured by the observer
\(\mathrm{Hz}\)
\(\lambda_s\)
wavelength in the source rest frame
\(\mathrm{m}\)
\(\lambda_o\)
wavelength measured by the observer
\(\mathrm{m}\)
\(\beta\)
relative speed divided by \(c\)
\(z\)
redshift
Method
Derivation 1: Use the longitudinal shift
For source and observer moving directly toward each other, the received frequency is increased. For recession, it is decreased.
Approaching
\[f_o=f_s\sqrt{\frac{1+\beta}{1-\beta}}\]
Receding
\[f_o=f_s\sqrt{\frac{1-\beta}{1+\beta}}\]
Derivation 2: Convert frequency to wavelength
Light always satisfies \(c=f\\lambda\) in vacuum, so a lower observed frequency means a longer observed wavelength.
Vacuum wave relation
\[c=f_o\lambda_o=f_s\lambda_s\]
Redshift
\[1+z=\frac{\lambda_o}{\lambda_s}=\frac{f_s}{f_o}\]
Derivation 3: Keep sign convention explicit
Use the approaching formula only when separation is decreasing. Use the receding formula when separation is increasing.
Rules
Approaching frequency
\[f_o=f_s\sqrt{\frac{1+\beta}{1-\beta}}\]
Receding frequency
\[f_o=f_s\sqrt{\frac{1-\beta}{1+\beta}}\]
Redshift
\[z=\frac{\lambda_o-\lambda_s}{\lambda_s}\]
Wavelength ratio
\[\frac{\lambda_o}{\lambda_s}=\frac{f_s}{f_o}\]
Examples
Question
A source emitting
\[5.00\times10^{14}\,\mathrm{Hz}\]
approaches at \[0.200c\]
Find the observed frequency.Answer
\[f_o=f_s\sqrt{\frac{1+0.200}{1-0.200}}=(5.00\times10^{14})(1.225)=6.12\times10^{14}\,\mathrm{Hz}\]
Checks
- Approaching sources are blueshifted: frequency up, wavelength down.
- Receding sources are redshifted: frequency down, wavelength up.
- The relativistic shift is symmetric between source and observer.
- The sound Doppler formula is not valid for light in vacuum.