The Doppler effect and special relativity

Sound is a wave in air. Its crests travel at a fixed speed $u$ relative to the air, whoever made the sound. Let $\Delta t_S$ be the time between crests at the source (one period) and $\Delta t_O$ the time between crests arriving at the observer. The question is how they are related when something is moving.

Case 1: the source moves away

The observer and the air are at rest; the source moves away at speed $v$. Between emitting one crest and the next, the source moves a distance $\Delta\ell = v\,\Delta t_S$ farther away. The second crest therefore has an extra distance $\Delta\ell$ to travel, which takes an extra time $\Delta\ell/u$:

Astronomers describe the stretching by the redshift $z$, defined so that $z = 0$ means no shift at all:

So for a moving source of sound, $z = v/u$.

Case 2: the observer moves away

Now the source and the air are at rest and the observer moves away at $v$. Between receiving one crest and the next, the observer moves $\Delta\ell = v\,\Delta t_O$, and the second crest must chase it. That gives an equation with $\Delta t_O$ on both sides:

For slow motion both answers are close to $z \approx v/u$. They differ by $(v/u)^2/(1 - v/u)$: for $v/u = 1/1000$ the difference is about one part in a million. They differ at all because the air is a real, physical frame. Moving through the air and having the air move past you are genuinely different situations.

Light is different. In the 1860s Maxwell completed the equations of electromagnetism, and they predict a definite speed of light $c$ — but relative to what? For decades physicists assumed an invisible medium, the ether, with light moving at $c$ relative to it. Einstein's insight in 1905 was to drop the ether and instead change how measurements in different frames are related, so that light moves at $c$ for every inertial observer. The rules for that change are the Lorentz transformations of special relativity.

Without a medium, "the source moves" and "the observer moves" are the same physical situation seen from two frames. So the two calculations must agree. Checking that they do is a good test of special relativity.

The key ingredient is that a moving clock runs slow. If the moving clock itself records a time $\Delta t_\text{clock}$ between two of its ticks, the frame it moves through measures a longer time between those ticks:

Guth stresses what "appears" means here. It is not what your eye sees at one instant, which is affected by light travel times. It is what is measured by a whole frame of rulers with synchronised clocks at every point. With that care, a clock moving at speed $v$ ticks once every $\gamma$ of the frame's seconds.

Source moving. Repeat case 1 with $u = c$, but the source's clock now runs slow: in the observer's frame, the time between emissions is $\gamma\,\Delta t_S$, and the source moves $\Delta\ell = \gamma v\,\Delta t_S$ in that time. Then $\Delta t_O = \gamma\,\Delta t_S\,(1 + \beta)$.

Observer moving. Repeat case 2 in the source's frame, where the time between receptions is $\Delta t' = \Delta t_S/(1 - \beta)$. The observer's clock runs slow, so it records only $\Delta t_O = \Delta t'/\gamma$.

Using $1 - \beta^2 = (1+\beta)(1-\beta)$, both give exactly the same answer:

For small $\beta$ this is $z \approx \beta + \beta^2/2$, between the two sound results. As $\beta \to 1$ the redshift grows without limit.

Special relativity is sometimes said to fail for accelerated motion. Guth corrects this: acceleration is handled by special relativity plus the relevant forces; general relativity is a theory of gravity. What special relativity cannot say is how a real clock responds to being shaken — a wristwatch survives, a clock smashed into a wall stops. Physicists therefore define an ideal clock: one whose rate at each moment equals that of an unaccelerated clock moving alongside it, so only its speed matters.

Everything special relativity says about measuring times and distances comes down to three effects:

1. Time dilation: a moving clock runs slow by a factor $\gamma$.
2. Lorentz–FitzGerald contraction: a moving rod is shorter by a factor $\gamma$ along its motion; lengths across the motion are unchanged.
3. Relativity of simultaneity: two clocks at the ends of a moving rod of rest length $\ell_0$, synchronised in the rod's own frame, are not synchronised in yours — the trailing clock reads later.

The third effect resolves an apparent paradox: two observers moving past each other each find the other's clocks running slow. There is no contradiction, because they disagree about which distant events are simultaneous.

• Redshift is defined by $1 + z = \lambda_O/\lambda_S = \Delta t_O/\Delta t_S$.
• For sound, source motion gives $z = v/u$ and observer motion $z = (v/u)/(1 - v/u)$: the medium makes them different.
• For light there is no medium; with time dilation both cases give $1 + z = \sqrt{(1+\beta)/(1-\beta)}$.
• Special relativity's kinematics is three effects: time dilation, length contraction and the relativity of simultaneity.