What, exactly, are invariants?

[PeterDonis]

I thought that since the starting velocity was 0, there would be no red/blue shift at the start so that it must increase with time. But light doesn't travel instantly, so by the time the light from an endpoint at the start of the worldline reaches the other end, the receiving end is moving.

Yes. That is basically the argument Einstein originally used to show that there must be a "gravitational" blueshift (or redshift in the opposite direction) in an accelerating elevator.

 

[PeterDonis]

At the point where it unwinds onto the ship, it would experience a rapid change of velocity and so its length would rapidly expand.

It's not just that. The tape might deform in this process, because it has only a finite material strength, elasticity, etc., and those might be exceeded during the process. This means the distance between the markings on it when the process is finished might not be the same as it was when the tape was originally manufactured and wound up. Specifying that the unwinding is done very slowly is a way of avoiding all such issues, so that we can be reasonably sure the tape doesn't deform and the lengths it shows after the process are the same as the ones it was originally manufactured with.

 

[Ibix]

Then there's the mystery of how two observers can maintain simultaneity with each other at every instant yet have different elapsed time. If observers move inertially and have different clock rates, they also have different lines of simultaneity.

Those "nested" hyperbolae you've drawn are the Minkowski equivalent of concentric circles, in the sense that the "distance" to the center, $\sqrt{x^2-c^2t^2}$, is constant along each one, just as $\sqrt{x^2+y^2}$ is constant along a circle.

Here's a statement about Euclidean geometry: If we have straight lines, lines perpendicular to those lines are only parallel if the original lines were parallel. The equivalent in Minkowski geometry is: If we move inertially, we only share simultaneity planes orthogonal to our worldlines if we have the same velocity.

Euclid: If we have circles, lines perpendicular to those circles are only parallel if the circles are concentric, and the arc length between one radial line and another is different on different circles. Minkowski: if we follow hyperbolic worldlines we share simultaneity only if our paths are concentric, and the proper time between one shared simultaneity plane and the next is different along different hyperbolae.

 

[robphy]

Then there's the mystery of how two observers can maintain simultaneity with each other at every instant yet have different elapsed time. If observers move inertially and have different clock rates, they also have different lines of simultaneity.

Since our astronaut is now non-inertial,
durations measured with radar are now different from
durations measured with a wristwatch.

 

[Freixas]

As I said, it's a simple consequence of spacetime geometry and the geometry of the hyperbolas.

Those "nested" hyperbolae you've drawn are the Minkowski equivalent of concentric circles

Ok, I think I get it using the analogy to concentric circles. Concentric circles have different circumferences. A line sweeping from the common center to a point on the other circle will intersect the inner circle. The line is orthogonal to the tangents of both points (common simultaneity), but the distance swept through the outer circle is more than is swept through the inner circle (different elapsed time).