Bondi k-calculus

Bondi k-calculus is a method of teaching special relativity popularised by Sir Hermann Bondi, that has been used in university-level physics classes (e.g. at the University of Oxford[1]), and in some relativity textbooks.

Many introductions to relativity begin with the concept of velocity and a derivation of the Lorentz transformation.

Other concepts such as time dilation, length contraction, the relativity of simultaneity, the resolution of the twins paradox and the relativistic Doppler effect are then derived from the Lorentz transformation, all as functions of velocity.

Bondi, in his book Relativity and Common Sense,[4] first published in 1964 and based on articles published in The Illustrated London News in 1962, reverses the order of presentation.

He begins with what he calls "a fundamental ratio" denoted by the letter

[3]: 40 From this he explains the twins paradox, and the relativity of simultaneity, time dilation, and length contraction, all in terms of

[4]: 109 Consider two inertial observers, Alice and Bob, moving directly away from each other at constant relative velocity.

Alice sends a flash of blue light towards Bob once every

This means that the time interval between Bob receiving the flashes, as measured by his clock, is greater than

(If Alice and Bob were, instead, moving directly towards each other, a similar argument would apply, but in that case

So Bondi's k-factor is another name for the Doppler factor (when source Alice and observer Bob are moving directly away from or towards each other).

This means that Dave receives Alice's blue flashes at a rate of once every

Einstein's second postulate, that the speed of light is independent of the motion of its source, implies that Alice's blue flash and Bob's red flash both travel at the same speed, neither overtaking the other, and therefore arrive at Dave at the same time.

[4]: 80 This establishes that the k-factor for observers moving directly apart (red shift) is the reciprocal of the k-factor for observers moving directly towards each other at the same speed (blue shift).

Denote times recorded by Alice's, Bob's and Carol's clocks by

In Newtonian physics, the expectation would be that, at the final comparison, Alice's and Carol's clock would agree,

, by his clock, it follows by symmetry that Carol's return journey over the same distance at the same speed must also have a duration of

(as discussed earlier), so, considering Carol's flash towards Alice, a transmission interval of

In the k-calculus methodology, distances are measured using radar.

An observer sends a radar pulse towards a target and receives an echo from it.

are times recorded by the observer's clock at transmission and reception of the radar pulse.

In the particular case where the radar observer is Alice and the target is Bob (momentarily co-located with Dave) as described previously, by k-calculus we have

Consider three inertial observers Alice, Bob and Ed, arranged in that order and moving at different speeds along the same straight line.

will be used to denote the k-factor from Alice to Bob (and similarly between other pairs of observers).

As before, Alice sends a blue flash towards Bob and Ed every

Using the radar method described previously, inertial observer Alice assigns coordinates

Now, applying the k-calculus method to the signal that travels from Alice to Bob

Similarly, applying the k-calculus method to the signal that travels from Bob to Alice

in the previous section can be solved as simultaneous equations to obtain:[4]: 118 [2]: 67