What is the coordinate-free expression for doppler effect?

Question

I'm interested in the general formula in terms of a position vector velocity and inner products, without reference to any coordinates.

Answer 1

Nonrelativistic Doppler effect in a medium

_{(Plagiarized from another answer of mine)}

Work in the rest frame of the medium. Suppose the source emits a pulse at time $0$ and position $\mathbf 0$, and another at time $δt_s$ and position $\mathbf v_s δt_s$, and the observer detects them at times $t$ and $t+δt_o$ and positions $\mathbf x$ and $\mathbf x + \mathbf v_o δt_o$. Then you have

$$\begin{eqnarray} \lVert \mathbf x \rVert &=& ct \\ (\mathbf x + \mathbf v_o δt_o - \mathbf v_s δt_s)^2 &=& c^2 (t+δt_o-δt_s)^2 \end{eqnarray}$$

where $c$ is the speed of sound. Substitute $t \to \lVert \mathbf x \rVert / c$ in the second equation, expand it, and discard terms that are second order in $δt_s$ and $δt_o$, to get

$$\mathbf x \cdot \mathbf v_o\,δt_o - \mathbf x \cdot \mathbf v_s\,δt_s = \lVert \mathbf x \rVert c (δt_o-δt_s)$$

Solve for $δt_s/δt_o = f_o/f_s$ to get

$$\boxed { \frac{f_o}{f_s} = \frac{c - \mathbf v_o\cdot \mathbf{\hat x}}{c - \mathbf v_s\cdot \mathbf{\hat x}} }$$

where $\mathbf{\hat x} = \mathbf x / \lVert \mathbf x \rVert$ is a unit vector pointing from the location of the source at the time of emission toward the location of the observer at the time of reception.

Special-relativistic Doppler effect for light

Suppose the source emits a pulse at spacetime position $\mathbf 0$ and another at $\mathbf v_s δτ_s$ (where $\mathbf v$ is a four-velocity and $τ$ is proper time), and the observer detects them at $\mathbf x$ and $\mathbf x + \mathbf v_o δτ_o$. Then you have

$$\begin{eqnarray} \mathbf x^2 &=& 0 \\ (\mathbf x + \mathbf v_o δτ_o - \mathbf v_s δτ_s)^2 &=& 0 \end{eqnarray}$$

Expand the second equation, substitute the first into it, and discard terms that are second order in $δτ_s$ and $δτ_o$, to get

$$\mathbf x \cdot \mathbf v_o\,δτ_o - \mathbf x \cdot \mathbf v_s\,δτ_s = 0$$

Solve for $δτ_s/δτ_o = f_o/f_s$ to get

$$\boxed { \displaystyle \frac{f_o}{f_s} = \frac{\mathbf v_o\cdot \mathbf x}{\mathbf v_s\cdot \mathbf x} }$$