A better way of talking about time?

[robphy]

These are great diagrams!
But "basic" is a subjective term. Your examples are exercises, very useful for studding, while practical problems are too messy to draw, harder to interpret graphically (e.g. 4D) and easier to handle algebraically. That's my limited experience anyway.

The examples provide explanations, both physical and geometrical.
Some with nice numbers so that one can focus on the physics, and less on the arithmetic.

By "practical problems", do you mean not-necessarily nice numbers?
Sure... once one understands the physics and the underlying math, then one can use the formulaic approaches.
The problem for the beginner is that, in my opinion,
the heavy-formulaic approaches are too detached from the physics and the geometry.

Try doing a euclidean geometry problem or analyzing a block on an incline
algebraically or formulaically alone (that is, without drawing an associated diagram).

 

[Orodruin]

I'm jealous of your students. When I was studying SR we were still recovering from "SR can't handle accelerating observers" thing.

I mean, this is another of these common misconceptions that you still see parroted in popular media and "modern" physics courses ... Every year there will be at least one or two students that have this preconception that needs to be rooted out. Again an issue of lower level courses being taught by people who never delved any deeper into relativity and parrot what they learned themselves in the same type of course.

Then we also have my favourite misconception: When considering muons produced in the atmosphere (and reaching the Earth due to being time dilated in the Earth frame) answering the question "how far did the muon travel in its rest frame" with a non-zero number. Nowadays I tend to ask exactly this question to students every year as it is great for hammering in the concept of relative motion when you follow up by telling them that the correct answer is zero.

Try [...] analyzing a block on an incline
algebraically or formulaically alone (that is, without drawing an associated diagram).

Been there, done that.

 

[robphy]

Try [...] analyzing a block on an incline
algebraically or formulaically alone (that is, without drawing an associated diagram).

Been there, done that.

For beginners? (the context of my suggestion).
Maybe you have some amazing students.

 

[PeterDonis]

Every non-trivial coordinate-free one contains a hidden coordinate frame.

I don't think this is true. For a fairly simple counterexample, see Misner, Thorne, & Wheeler, Section 2.8. For a more extended counterexample, see much of Wald's GR textbook, where Wald's abstract index notation is defined and used; the whole purpose of that notation is to be able to keep track of things like tensor indices (what MTW calls "slots") without having to choose any coordinates.