- The **Bondi k-calculus** (an algebra-based method) develops the basic ideas of special-relativity in $tx$-plane (without using the transverse direction). This approach is presented in his book “Relativity and Common Sense” (1962, 1964). In addition, Bondi presented “E=mc2: Thinking Relativity Through”, a series of ten lectures on BBC TV running from Oct 5 to Dec 7, 1963. (Read more at my contributed article to a blog at https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/ ) I have used it to develop the "longitudinal light clock" without appealing to the standard textbook "transverse light clock". In particular, I draw the light signals in the longitudinal light clock to form the "**light-clock diamonds**" (the "**causal diamond**" between consecutive tick events). All light-clock diamonds for all inertial observers have equal area since the Lorentz boost transformation has determinant one. This is developed using the Bondi k-calculus in my article:<BR> *Relativity on rotated graph paper*,<BR> AmJPhy 84, 344 (2016); https://doi.org/10.1119/1.4943251 .<BR> An early draft is at https://arxiv.org/abs/1111.7254 . Rather than use "time-dilation" from the transverse clock,<BR> I use the Principle of Relativity in the $tx$-plane. The key figure (using $v=(3/5)c$) is<BR> [![robphy-RRGP-diamonds][1]][1] Once the light-clock diamond size is established (using the Bondi k-calculus in the above), the spacelike-diagonal of the light-clock diamond determines the reflection events on the longitudinal light-clock. So, draw through those events parallels to the timelike diagonal. This shows length-contraction as viewed in the lab frame. (The construction can be shown to be symmetric between the observers.) ---------- - I have a different approach (that doesn't use the Bondi $k$-calculus directly) which appears in my recent contributed chapter<P> *Introducing relativity on rotated graph paper*<BR> Ch 7 in **Teaching Einsteinian Physics in Schools**<BR> Kersting and Blair, Routledge 2021, https://doi.org/10.4324/9781003161721 I will describe it below.<BR> You can play with the ideas in this visualization: https://www.geogebra.org/m/HYD7hB9v#material/UBXdQaz4 <BR> The key idea is that the diamond **size** is determined by the Principle of Relativity. (The diamond **shape** is determined by the Speed of Light Principle and the velocity of the observer.) The two observers perform the same experiment and should expect the same results:<BR> **2 seconds after they meet, send a light signal to the other.** Assuming absolute time and absolute space fails to satisfy the principle, but the third configuration works. Assuming absolute time (so the elapsed times are equal),<BR> Alice (red) receives Bob's signal at 3.2, whereas Bob (blue) receives it at 5.<BR> Bob's ticks need to be scaled up from this size. This hints that there is time-dilation... but by how much?<BR> [![robphy-RRGP-absoluteTime][2]][2] Assuming absolute space (so the length of the light-clock is the same),<BR> Alice (red) receives Bob's signal at 5, whereas Bob (blue) receives it at 3.<BR> Bob's ticks need to be scaled down from this size. This hints that there is length-contraction... but by how much?<BR> [![robphy-RRGP-absoluteSpace][3]][3] By playing around (taking a hint from the geometric mean?),<BR> we get agreement with the Principle of Relativity by each receiving the other signal at 4 ticks.<BR> <BR> [![robphy-RRGP-relativity][4]][4] <BR> The ratio $(4\mbox{ ticks})/(2\mbox{ ticks})$ is the Doppler factor $k=2$ for $v=(3/5)c$, where we have used the Principle of Relativity and the Speed of Light Principle... <BR> ...and, as a consequence, we now know the factor for time-dilation and length-contraction. <BR> Try it for $v=(4/5)c$. <BR> https://www.geogebra.org/m/kvfsq664 (updated)... make sure BOB's diamonds are shown ---------- For more information, consult my article and chapter above.<BR> See also <BR> https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/ <BR> https://www.physicsforums.com/insights/relativity-rotated-graph-paper/ [1]: https://i.sstatic.net/AOhaVm.png [2]: https://i.sstatic.net/WIQsdm.png [3]: https://i.sstatic.net/9tzrym.png [4]: https://i.sstatic.net/IVKczm.png