Fundamentals of Time and Relativity

Doppler Effect

In the situation described on the previous page, we consider a second observer $O^{'}$ , called a receiver, moving with four-velocity $v = γ (v) (1, v)$ with respect to the source $S$ . This receiver will see light rays $k = (κ^{'}, k^{'})$ arriving at an angle $θ_{S}$ with respect to his velocity $v$ , with different frequency $κ^{'} = k \cdot v$ . The ratio of the received and emitted frequencies is given by:

6.1

D^{- 1} = \frac{κ^{'}}{κ} = γ (v) (1 - \frac{k}{κ} \cdot v) = γ (v) (1 - v \cos θ_{S})

which is the (inverse) relativistic Doppler factor as derived by Einstein in 1905. The difference with the classical Doppler-shift is the Lorentz factor which accounts for time dilatation. The sign convention is that $v : = \pm | v |$ is negative if receiver and source are moving towards each other.

Equation (6.1) relates the observed and emitted frequencies of electromagnetic radiation. The observed frequency is higher when the velocity $v$ is negative. This is because the receiver moves into the wave train, but also because his clocks are running slow. We can equally say that waves from an approaching light-source have higher frequency than waves from an identical stationary source. In fact, time dilation and classical Doppler effect combine in precisely the right way to make this true.

Einsteins equation (6.1) may suggest that the rest frame of the source is a preferred frame of reference. However, we may generalize by allowing the source to have an arbitrary four-velocity $u = γ (u) (1, u)$ . Then the Doppler shift equation takes the Lorentz-invariant form

6.2

\frac{κ^{'}}{κ} = \frac{k \cdot v}{k \cdot u} = \frac{γ (v)}{γ (u)} \frac{1 - (k / κ) \cdot v}{1 - (k / κ) \cdot u}

valid for arbitrary motions of source and receiver. There is no preferred frame in accordance with the Einstein’s principle of relativity.

The Lorentz invariance of the Doppler factor (6.2) implies that the relativistic Doppler effect depends only on the relative velocity between source and receiver.
If a source is moving exactly orthogonal to the direction from which its radiation is observed, equation (6.1) predicts a transversal Doppler shift as a pure consequence of time dilatation.