# What is the coordinate-free expression for doppler effect?

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

## 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

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