traveler-point kinematics

start with a map

To describe motion we often first define a bookkeeper coordinate-system, or "map with metric", made up of distributed clocks as well as position markers that will be used to locally describe the map position r = {x,y,z} and map time t of events. This coordinate-system might be simply be a set of identical synchronized clocks separated by standardized yardsticks, although this is not always possible even here on earth e.g. since time's rate of passage increases with altitude.

When two separately-located events show the same time t on their bookkeeper clocks, we say that those events are simultaneous. Folks using a different map-frame or metric, even one simply moving at a fixed speed with respect to the first, may determine that those two events are not simultaneous i.e. that one event occurs before the other. Hence map-time, map-position, and the meaning of simultaneity are all defined by our choice of coordinate-system or metric.

variables to describe motion

Motion may be described most generally, e.g. at any speed and in curved spacetimes, by describing it from one traveling object's perspective. That is because every one (regardless of metric, including our traveler) measures the same value for the separation between events, or proper-time τ elapsed, on a traveling object's "world line" through spacetime. To work with many traveling objects at once, particularly at low speeds, it may be useful to pretend that elapsed map-time δt and all traveler-times elapsed (δτ) are one and the same, but we don't want to do that just yet.

The increment of traveler-time δτ elapsed along a world line follows a kind of space-time Pythagoras' theorem. In it's simplest form, this can be written as (c δτ)² = (c δt)² - δs•δs, where the displacement-increment or hypotenuse δs, as usual in Cartesian {x,y,z) coordinates, obeys δs•δs = (δx)² + (δy)² + (δz)². Here the spacetime constant c, often referred to as lightspeed or "the number of meters in a second" is, like traveler-time, also a frame-invariant i.e. something on which observers from all vantage points can agree. This follows because the metric equation also defines invariants for derivatives of the map-coordinates r and t, with respect to proper-time τ.

This allows us to define the "first-derivative" traveler-point velocity variables named proper-velocity w ≡ δr/δτ and differential aging-factor γ = δt/δτ, as well as "second-derivative" traveler-point velocity variables which, when combined, allow one to track the traveler's proper-acceleration 3-vector α, whose magnitude (like that of δτ and c) is also frame-invariant i.e. something on which observers in all frames reference can agree. That proper acceleration 3-vector is also the acceleration measured by the accelerometer in the traveler's smartphone.

Proper-velocity w does not require synchronized map-clocks for its measurement, and it is equal to the momentum p carried by our traveler per unit traveler rest-mass m. Note that any traveling object's rest-mass is also, in this context, a frame-invariant and something on which observers from all frames can agree. Unlike the "coordinate-velocity" v ≡ δr/δt that we'll be using later in this course, proper-velocity also has no upper limit. This is great news for relativistic travelers, if they are wanting to get somewhere as fast as possible on their own clocks, i.e. regardless of how much time the trip takes from perspective of "couch-potatoes" in the map-frame.

This is the place to introduce constant proper-acceleration equations for proper-velocity w[τ] and displacement r[τ] in terms of the proper-acceleration 3-vector α and the initial proper-velocity "at turnaround" w_o given in our (3+1)D table, which at low speeds reduce to the familiar w_o+ατ and w_oτ+½ατ², respectively. In that limit also γ[τ] goes to 1+½(w/c)² (equal to rest plus kinetic energy, divided by mc²) and t[τ] goes to τ. For unidirectional motion at any speed, the trajectory looks like w[τ] = τ_osinh[τ/τ_o]α and r[τ] = τ_o² (cosh[τ/τ_o]-1)α, where characteristic time τ_o becomes simply c/α (about a year for 1-gee ≈ 1 ly/y² accelerations). These latter equations are quite useful e.g. for analyzing constant proper-accceleration roundtrips between two stars.