page_title: Bondi k-calculus
section_title: The twins paradox
hierarchical_section_title: Bondi k-calculus / The twins paradox
context_page_description: Bondi k-calculus is a method of teaching special relativity popularised by Professor Sir Hermann Bondi, and now common in university and college-level physics classes.
  The usefulness of the k-calculus is its simplicity. It has been successfully used to teach special relativity to young children and also in 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, 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. From this he explains the twins paradox, and the relativity of simultaneity, time dilation, and length contraction, all in terms of. It is not until later in the exposition that he provides a link between velocity and the fundamental ratio.
context_section_description: Consider, now, a fourth inertial observer Carol who travels from Dave to Alice at exactly  the same speed as Bob travels from Alice to Dave. Carol's journey is timed such that she leaves Dave at exactly the same time as Bob arrives. Denote times recorded by Alice's, Bob's and Carol's clocks by .
  When Bob passes Alice, they both synchronise their clocks to . When Carol passes Bob, she synchronises her clock to Bob's, . Finally, as Carol passes Alice, they compare their clocks against each other. In Newtonian physics, the expectation would be that, at the final comparison, Alice's and Carol's clock would agree, . It will be shown below that in relativity this is not true. This is a version of the well-known "twins paradox" in which identical twins separate and reunite, only to find that one is now older than the other.
  If Alice sends a flash of light at time  towards Bob, then, by the definition of the k-factor, it will be received by Bob at time . The flash is timed so that it arrives at Bob just at the moment that Bob meets Carol, so Carol synchronises her clock to read .
  Also, when Bob and Carol meet, they both simultaneously send flashes to Alice, which are received simultaneously by Alice. Considering, first, Bob's flash, sent at time , it must be received by Alice at time , using the fact that the k-factor from Alice to Bob is the same as the k-factor from Bob to Alice.
  As Bob's outward journey had a duration of , 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 , by her clock, and so when Carol meets Alice, Carol's clock reads . The k-factor for this leg of the journey must be the reciprocal  (as discussed earlier), so, considering Carol's flash towards Alice, a transmission interval of  corresponds to a reception interval of . This means that the final time on Alice's clock, when Carol and Alice meet, is . This is larger than Carol's clock time  since
  provided  and .
   
caption: Spacetime diagram for the twins paradox    Alice   Bob   Carol   Dave   Flash of light