Everything from an exchange of pulses

Special relativity is usually introduced through a transformation between coordinate systems, which requires deciding in advance what it means for two distant events to happen at the same time. That decision is a convention, it is defensible, and it is a strange thing to have to make before any physics has been done.

Bondi’s alternative, from a set of lectures published in 1962, starts from something an experimenter can do without any conventions at all: send a signal and time when the answer comes back.

The one number #

Let $A$ flash a light every second by $A$’s own clock. Let $B$ recede at a constant speed and record how often the flashes arrive, by $B$’s own clock. The answer is a constant interval $k$ seconds, and $k$ is what the whole construction is made of.

Three things are needed to get started and all three are modest.

The interval is constant. Because the relative speed is constant, each successive flash has a slightly longer journey than the last by the same amount, so successive arrivals are equally spaced. This uses nothing but the constancy of the speed of light.

The same $k$ works in the other direction. If $B$ flashes every second, $A$ receives them every $k$ seconds — the same $k$. That is the relativity principle: neither observer is distinguished, so there is no way to make the two ratios different without saying which of them is the one that is really moving.

And a round trip multiplies by $k$ twice. A flash from $A$ reflected by $B$ and returning arrives with its interval multiplied by $k$ on the way out and by $k$ again on the way back, giving $k^2$. That is exactly the double shift a radar measures, and it is the only compound quantity the method needs.

Time dilation, in four lines #

Now do a radar measurement. $A$ sends a flash at time $t_1$ by $A$’s clock, it reaches $B$, is reflected, and returns to $A$ at time $t_2$.

$B$’s clock, when the flash arrives there, reads $k t_1$ — that is the definition of $k$, applied to a signal sent at $t_1$.

The return journey multiplies again, so $t_2 = k^2 t_1$.

$A$ now assigns a time and a distance to the reflection event by the radar convention: the event happened halfway between sending and receiving, at $\tfrac12(t_1 + t_2)$, and at a distance $\tfrac12 c(t_2 - t_1)$.

Substituting $t_2 = k^2 t_1$ gives $A$’s assigned time $\tfrac12 t_1(1 + k^2)$ and $B$’s own reading $kt_1$. Their ratio is

$$\gamma = \frac{t_1(1+k^2)/2}{kt_1} = \frac{k + 1/k}{2},$$

and the speed follows from the assigned distance over the assigned time:

$$\beta = \frac{k^2 - 1}{k^2 + 1}.$$

Invert the second and $k = \sqrt{(1+\beta)/(1-\beta)}$, which is the relativistic Doppler factor. Substituting into the first gives $\gamma = 1/\sqrt{1-\beta^2}$, which is the Lorentz factor.

The order of presentation has been inverted, and that is the point. The Lorentz factor, which is usually the primitive object, has turned out to be the arithmetic mean of two things an observer can measure directly; the Doppler factors, usually derived at the end of a chapter, are the primitives.

What each observer sees, before any correction #

The k factor is worth separating from the Lorentz factor with some care, because conflating them produces the most durable confusion in the subject.

$k$ is what an observer sees: the ratio of received interval to sent interval, with light travel time still in it. It is $2$ at $\beta = 0.6$ receding and $\tfrac12$ approaching — one is twice the other’s reciprocal, and neither is the Lorentz factor of $1.25$.

$\gamma$ is what an observer infers after subtracting the travel time, and it is the same on approach and on recession, because it is the average of $k$ and $1/k$ and the average does not care which is which.

The famous puzzle — each observer finds the other’s clock slow, which sounds like a contradiction — dissolves once the two are kept apart. What each sees is asymmetric in an obvious way: a receding clock appears slowed by $k$ and an approaching one appears fast by $1/k$, and any pair of observers agrees about which is happening. What each infers is symmetric, and there is no contradiction in two observers each inferring that the other’s clock runs slow, because the inference involves a convention about distant simultaneity and the two are using different ones.

Bondi’s method makes this hard to get wrong, because it starts with the thing that is seen and derives the thing that is inferred, in that order. A presentation that starts with the transformation has already made the convention and cannot easily point to it.

Velocities compose because factors multiply #

Three observers in a line, each receding from the one behind. Let $A$’s flashes reach $B$ stretched by $k_{AB}$, and let $B$’s flashes reach $C$ stretched by $k_{BC}$.

Now let $A$ flash directly at $C$. The signal passes $B$ on the way and nothing happens to it there, but the stretching is a property of the relative motion and composes: the interval as received by $C$ is $A$’s interval multiplied by $k_{AB}$ and then by $k_{BC}$.

$$k_{AC} = k_{AB}\,k_{BC}.$$

Velocities do not add and $k$ factors multiply, which is one line rather than a page. Converting back with $\beta = (k^2-1)/(k^2+1)$ turns the multiplication into

$$\beta_{AC} = \frac{\beta_{AB} + \beta_{BC}}{1 + \beta_{AB}\beta_{BC}},$$

which is the composition law, obtained without writing down a transformation.

The impossibility of exceeding the speed of light appears here in an unusually clean form. A $k$ factor is a ratio of two positive durations and is therefore positive. Multiplying positive numbers gives a positive number, and $\beta = (k^2-1)/(k^2+1)$ is less than one for every positive $k$. There is no arrangement of finite $k$ factors that produces $\beta \ge 1$, and no limit-taking is required to see it.

The quantity that does add is $\ln k$, and it has a name: the rapidity. That $k$ factors multiply and rapidities add are the same statement, and it is the quantity that keeps adding when speeds refuse to; it is worth noticing that the natural composition law is the multiplicative one — the additive quantity is a logarithm of it, which is the reverse of how the subject is usually taught.

The twins, by counting #

The twin problem becomes a counting exercise with no paradox left in it.

$B$ travels away at $\beta$ for a while, turns round, and comes back. Both twins flash a light once a second by their own clocks throughout, and both count the flashes they receive.

Suppose $B$’s outward and return legs each take $N$ seconds by $B$’s clock, so $B$ ages $2N$ and sends $2N$ flashes. During the outward leg $B$ receives $A$’s flashes at intervals of $k$, so $B$ receives $N/k$ of them; during the return leg the recession becomes approach and the factor becomes $1/k$, so $B$ receives $Nk$. In total $B$ receives $N(k + 1/k)$ flashes, and since $A$ sent one per second of $A$’s time,

$$\frac{T_A}{T_B} = \frac{N(k+1/k)}{2N} = \frac{k+1/k}{2} = \gamma.$$

$A$ ages more, by exactly the Lorentz factor, and the calculation used the turnaround only to say that it happened.

The asymmetry is now visible without any appeal to acceleration. $B$ switches from receiving at $k$ to receiving at $1/k$ at the midpoint of the journey — the moment $B$ turns. $A$ switches from receiving at $k$ to receiving at $1/k$ much later, because the last signal $B$ sent before turning is still in flight and takes a long time to arrive. The two twins do not have symmetric experiences of the exchange, and one can say precisely how they differ by counting: $A$ spends most of the trip receiving slowly, $B$ spends exactly half.

That is the whole resolution, and the thing that makes it convincing is that no observer needs to know anything about the other’s clock — each is counting flashes that arrive at their own eye.

A worked number #

Take $\beta = 0.6$, so $k = \sqrt{1.6/0.4} = 2$ exactly. That is a convenient speed and the reason it appears in every textbook.

$A$ flashes every second. $B$, receding, receives them every two seconds. $B$ flashes every second and $A$ receives them every two seconds. A radar pulse sent at $t = 1$ s returns at $t = 4$ s, so $A$ places the reflection at $2.5$ s and at $1.5$ light-seconds — a speed of $0.6c$, consistently.

$B$’s clock at the reflection read $2$ s. So $A$ assigns $2.5$ s to an event at which $B$’s clock read $2$: a factor of $1.25$, which is $\gamma$ at $0.6c$, and which is $(2 + \tfrac12)/2$.

Now the twin trip. $B$ goes out for $N = 10$ s of $B$’s time and returns in another $10$, ageing $20$. Outbound, $B$ receives $A$’s flashes every $2$ s, so $5$ arrive; inbound, every half second, so $20$ arrive. Total $25$, which is how many seconds $A$ aged — and $25/20 = 1.25$.

Every number in that paragraph is an integer or a simple fraction, and none of it required a coordinate.

What has been assumed #

The method is unusually explicit about its inputs, which is its best feature.

It assumes the speed of light is the same for both observers, used when the ratio of arrivals was said to be constant. It assumes the relativity principle, used when the same $k$ was claimed in both directions. And it assumes the radar convention: that the reflection event happened halfway, in time, between sending and receiving.

That last one is a convention, and it is worth being clear that it has not been eliminated. What has changed is where it appears. In the standard development it is buried in the definition of a coordinate system before anything is measured; here it appears once, explicitly, in the one step that assigns a time to a distant event — and everything that does not depend on that step is convention-free.

What has been assumed is worth separating out, because it is easy to smuggle. The k factor is measured: it is a ratio of two intervals, both read on one clock, and it needs no convention at all. Assigning a time to a distant event is a different act, and different conventions for doing it are consistent with exactly the same measurements. Keeping the two apart is the whole methodological point of running the argument this way round — the ratio is a fact, the simultaneity is a choice, and the standard derivation mixes them in its first line.

The quantities that are convention-free are exactly the ones that involve a round trip at one place: the total elapsed proper time of a twin, the count of flashes received, the round-trip radar time. The one-way speed of light is not among them, which is why no experiment has ever measured it and why the convention survives.

Where it stops #

Everything above is one-dimensional. All three observers were on a line and every signal went along it. Extending the method to motion at an angle requires a Doppler factor that depends on direction, and the composition of $k$ factors along different directions is no longer a plain multiplication — the loss of commutativity there is what produces the Thomas rotation, which the one-dimensional method cannot see.

And every observer is inertial. The derivation of $k$ used a constant relative speed, and the twin calculation treated the turnaround as instantaneous. Making it gradual requires the assumption that a clock’s rate depends on its instantaneous speed and not on its acceleration, which is an assumption rather than a theorem, and the k-calculus does not supply it.

The light clock is the other standard route to the same factor, and it is worth having both because they assume different things. That argument needs a particular mechanism — a pulse bouncing between mirrors — and gets the answer from the geometry of the resulting triangle. This one assumes only that the clock ticks, whatever it is made of. When two arguments with different premises reach the same number, the number is more secure than either.

The k factor is a ratio of proper times and nothing else. It is defined by two clocks each reading their own time and by signals travelling between them, so it carries no information about lengths, and the method reaches length contraction only by a further step — measuring the ends of a moving rod by radar and combining, which reintroduces the simultaneity convention because the two ends are at different places. That is why the construction gives time dilation in four lines and length contraction in rather more, and it is a fair reflection of which of the two is the more primitive.

The method is a derivation and not an experiment. Everything in it is checkable and almost none of it has been checked in the form given — the measured evidence for time dilation comes from decaying particles, atomic clocks in aeroplanes and the Mössbauer effect, none of which involves anybody counting flashes. The virtue of the construction is pedagogical and structural: it shows what the theory rests on, in the smallest number of pieces.

The geometrical statement of the same content is the invariant hyperbola, which is what calibrates the tilted axes of a spacetime diagram: it marks the events at unit interval from the origin, and it is the curve every observer agrees on. The k factor is what stretches one null coordinate while shrinking the other, and the hyperbola is the locus that stretching preserves — which is why $k$ and $1/k$ multiply to one and why the interval is the thing that survives.

Why anybody bothered #

Bondi wrote Relativity and Common Sense in 1962 for readers with school algebra, and the method has a reputation as a teaching device. That undersells it, and the reason is in what it does not need.

The standard derivation requires setting up two coordinate systems, defining what it means for clocks at different places to be synchronised, and only then deriving the transformation between them. Every step is defensible and the whole apparatus has to be in place before the first physical conclusion is reached. A reader who is uneasy about the synchronisation convention — and the unease is warranted — has to hold the discomfort through the entire derivation.

The k-calculus reaches time dilation, velocity composition and the twin result before any of that is required. It uses one measurable ratio and the relativity principle, and the convention enters at exactly one identifiable step, which can then be examined on its own. That separation is not merely pedagogical: it is what makes it clear which results of special relativity are conventional and which are not, a distinction that occupied Reichenbach and Grünbaum for decades and is much harder to see from inside the transformation.

There is also a practical descendant. Radar ranging in the solar system, spacecraft navigation and pulsar timing all work by exchanging signals and timing round trips, and none of them ever assigns a time to a distant event by anything other than the convention above. The bookkeeping those systems use is Bondi’s, because it is the bookkeeping the measurement actually permits.

Where the k factor sits in the geometry #

There is a way of seeing why $k$ is the natural object, and it is worth a paragraph because it explains why the arithmetic is so simple.

Use light-cone coordinates: $u = t - x/c$ and $v = t + x/c$, which label the two families of light rays. A Lorentz boost does not mix them. It multiplies one by $k$ and the other by $1/k$, and that is the entire transformation — a stretch along one null direction and an equal squeeze along the other.

Everything above follows from that single sentence. The product of the two factors is one, which is why the interval $uv = t^2 - x^2/c^2$ is invariant. The mean of the two factors is $\gamma$. Composing boosts multiplies the stretches, which is why $k$ factors multiply. And the areas in a spacetime diagram are preserved, because a stretch and an equal squeeze leave an area alone.

Bondi’s method is that geometry expressed without any geometry — the stretch factor measured with a stopwatch instead of drawn on a diagram.

What the method makes obvious that the transformation hides #

Two facts about relativity are awkward to see in the usual formulation and fall out here.

The first is that the two Doppler factors are reciprocal, $k \cdot (1/k) = 1$, at every speed. In the transformation language that is a small algebraic identity; here it is the statement that going out and coming back returns a signal to its original rate, which is obviously true and is the reason a round trip is the natural measurement. Every quantity in the construction that is convention-free is a round-trip quantity, and the identity is why.

The second is that the composition of boosts is a multiplication rather than an addition. Written as matrices the Lorentz transformations compose by matrix multiplication and the fact that the result is another boost takes a calculation; written as $k$ factors it is the multiplication of two positive numbers, which cannot leave the range and cannot produce a speed at or above $c$. The group structure is visible in one line, and the impossibility of exceeding the speed of light is a property of the arithmetic rather than a limit to be taken.

The ladder from here #

Later rungs on this anchor: the k-calculus in two dimensions, and where it stops being simple; the relativistic aberration formula obtained the same way, by asking about the direction rather than the rate of arrival; the Doppler factor as the single unknown that beaming, variability and apparent superluminal motion all constrain in an astrophysical jet; and the connection to the conformal group, of which the null-coordinate stretch is a one-parameter subgroup.

The neighbouring ladders are the clock that has to slow, which is the same factor derived from a mechanism; the twin who comes back younger, which is the counting argument’s subject; and the shift a mirror gives twice, where the round-trip factor is an instrument’s constant rather than a foundation.