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Friday, March 24, 2023

# Relativity on Rotated Graph Paper (a graphical motivation)

Within the penultimate draft, I had a non-algebraic motivating argument (which additionally motivates time dilation and size contraction)
that needed to be disregarded as a result of the article was already too lengthy.

This argument now seems in Introducing relativity on rotated graph paper (Ch 7), my contribution to a not too long ago revealed ebook: Instructing Einsteinian Physics in Faculties
Kersting and Blair, Routledge 2021, https://doi.org/10.4324/9781003161721
(has supplementary materials on the backside: https://www.routledge.com/Instructing-Einsteinian-Physics-in-Faculties-An-Important-Information-for-Academics/Kersting-Blair/p/ebook/9781003161721 )

## The storylineâ€¦ ranging from Einsteinâ€™s ideas

The storyline goes like this:

• The Velocity of Gentle Precept and Bobâ€™s Velocity offers the form of transferring observer Bobâ€™s light-clock diamond (with edges parallel to the sunshine cone, to the rotated graph paper).
• development:
Draw Bobâ€™s worldline.
Bobâ€™s diamonds could have a diagonal alongside Bobâ€™s worldline and edges parallel to the rotated grid.
The dimensions of the diamonds corresponds to the spacing of the mirror worldlines, equidistant from and parallel to Bobâ€™s worldline.
However what determines the dimensions of the clock diamonds? (What occasion F on Bobâ€™s worldline marks the signal-reflections?)
• The Relativity Precept determines the dimension (the scaling) of Bobâ€™s light-clock diamond.
(That is what I name the â€śCalibration Drawbackâ€ť.)
• SIGNAL-EXCHANGE EXPERIMENT: Two inertial observers meet at an occasion O.
2 seconds after they meet, they ship a sign to the opposite.
We anticipate they’ve the identical outcomes, in accord with the relativity precept.
(I may have chosen â€ś1 secâ€ťâ€¦ however the diagram is extra cluttered.)

## An instance: theÂ  ##v=(3/5)c## case

• Take ##v=(3/5)c## for simplicity.
The makes an attempt:

[* possible analysis: Since the round-trips are equal in both of the previous cases, we might
expect the geometric mean ##sqrt{(3.2)(5)}=4## to be the expected result of their signal-exchange experiment.]

Do that for ##v=(4/5)c##.

Play with these utilizing
https://www.geogebra.org/m/HYD7hB9v#materials/UBXdQaz4 (ensure BOBâ€™s diamonds are proven)
https://www.geogebra.org/m/kvfsq664 (up to date)â€¦ (ensure BOBâ€™s diamonds are proven)
[You can manually adjust Bobâ€™s velocity and â€ślengths [in the lab frame]â€ť of the sunshine clocks.]

To acquire the textbook components relating the time-dilation elements and relative velocity, we might to proceed alongside the traces of my AJP article.

However the level is that we will present (with a geometric development and with out algebra) how the Relativity Precept and the Velocity of Gentle Precept suggest the incompatibilities with Absolute Time and Absolute Area, and counsel the necessity for the time-dilation and length-contraction results (as side-effects), on the way in which to establishing the equality of light-clock diamond areas (which is basically the invariance of the sq. interval).