2. Time as merely-local

In the first part of the 20th century it was discovered that time is a local variable, linked to each clock's location through a space-time version of Pythagoras' theorem i.e. the local metric equation. Both height in the earth's gravitational field, and clock-motion, affect the rate at which time passes on a given clock. Both of these effects must, for example, be taken into account in the algorithms used by handheld global positioning systems. The fact that time is local to the clock that's measuring it means that we should probably address the question of extended-simultaneity (i.e. when an event happened from your perspective if you weren't present at the event) only as needed, and with suitable caution. Care is especially needed for events separated by ``space-like" intervals i.e. for which space Δx and time Δt separations obey Δx > cΔt where c is the space/time constant sometimes called ``lightspeed".

Recognizing that traveling clocks behave differently also gives us a synchrony-free (Shurcliff1996) measure of speed with minimal frame-dependence, namely proper-velocity (SearsBrehme1968) w ≡ dx/dτ = γv, which lets us think of momentum as a 3-vector proportional to velocity regardless of speed. Here τ is the frame-invariant proper-time elapsed on the traveling object's clock, Lorentz-factor γ ≡ dt/dτ, and as usual coordinate-velocity v ≡ dx/dt. 

Recognition of the height-dependence of time as a kinematic (i.e. metric-equation) effect, moreover, allows us to explain the fact that free-falling objects are accelerated by gravity at the same rate regardless of mass. Hence gravity is now seen as a ``geometric force"  instead of a proper one, which is only felt from the vantage point of non ``free-float-frame" coordinate-systems like the shell-frame normally inhabited by dwellers on planet earth. 

By starting with a traveler-kinematic that uses only frame-invariant and/or synchrony-free variables, and then showing that at large radii and low speeds we can treat gravity as a force that acts on every ounce of an object's being, we set the stage for Galilean kinematics without the implicit assumption about global-time that we now know to be wrong. Global time, where used in application of Newton's laws, can then be introduced from the start as the useful approximation that it is.

The concept of local-time also leads naturally to the illustration-triplet of x-ct (cf. Minkowski 1907), x-cτ (cf. Epstein 1985) & ρ-cτ (cf. Dolby & Gull 2001) nomograms, useful for the visual learner to quantitatively examine geodesy and extended-simultaneity in situations that range from constant-speed and accelerated motion in flat space-time, through the visualization of these same effects in the presence of both earth-like, and extreme, gravitational curvature.