I Proper (and coordinate) times re the Twin paradox

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Mister T

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So which of them is the proper time?
if Δτ2 = (Δct2 - (Δx2)
then surely τ must be the vertical cathetus, x the other catheus and ct the hypotenuse
##\Delta \tau## is the proper time. Note that the hypotenuse is not the longest side of the triangle!
 
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Yet this is confusing if τ is the proper time, it is the temporal displacement along the rotated worldline which must be ct - the temporal displacement...

Please help me I am confused again...
You would probably be a bit less confused if you used the standard terminology. “Temporal displacement” is ambiguous and non standard, discard it. The standard terms are “coordinate time” and “proper time”.

“Coordinate time” is just the value of the time coordinate in a given coordinate chart. In an inertial frame it represents a system of clocks that are Einstein-synchronized. In other frames it may have little or no physical meaning.

“Proper time” is the time given by a clock following a given worldline. It is only defined on the worldline so there is no synchronization involved. It always has a clear physical meaning.
 
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You would probably be a bit less confused if you used the standard terminology. “Temporal displacement” is ambiguous and non standard, discard it. The standard terms are “coordinate time” and “proper time”.
Yes, sorry about that but I was responding to this quote:
This is Minkowski geometry, not Euclidean geometry. Increasing the spatial displacement (the difference between starting and ending x coordinates) while holding the temporal displacement (the difference between starting and ending t coordinates) constant reduces the length of the (timelike) interval between the end points.
so I was trying to use the same terminology; it is part of what makes things confusing when different terms are used as it becomes impossible to be sure what is being referred to...

I am going to try and find a way of understanding Minkowski diagrams. It seems they are more different than I have understood from Wiki - I know it isn't the best place but as a pensioner in the highlands of Scotland I am limited to what I can find on the internet.
Has anyone any better suggestions for my level of learning?
 
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I have found the relativity Wikipedia entries to be pretty reasonable. Not 100%, but better than 90%.

My recommendation is to work problems. The only way to understand this stuff is to apply it and practice it. This material is not intuitive, so you have to rely on math. Come back here frequently and we can check your math and offer recommendations.

Regarding ##d\tau^2=dt^2-dx^2##. The quantity that is invariant is ##d\tau##. Both ##dt## and ##dx## are frame variant.

In Euclidean geometry you have ##ds^2=dx^2+dy^2##. Here ##ds## is invariant and both ##dx## and ##dy## are frame variant under rotations. If you do a whole series of rotations you will find that after each rotation you have a different ##dx## and a different ##dy## and the same ##ds##. You will also find that the set of all points with the same ##ds## traces out a circle, meaning that in Euclidean geometry distance is defined by circles and circles are unchanged under rotations.

In spacetime, if you do a whole series of boosts you will find that after each boost you have a different ##dx## and ##dt## and the same ##d\tau##. You will also find that the set of all points with the same ##d\tau## traces out a hyperbola, meaning that in Minkowski geometry timelike distance (proper time) is defined by hyperbolas and hyperbolas are unchanged under boosts.
 
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I am limited to what I can find on the internet.
Has anyone any better suggestions for my level of learning?
I also liked the Susskind lectures on SR:
 

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