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he world is full of motion, but describing it (that's what kinematics does) requires two perspectives: (i) the perspective of that which is moving e.g. "the traveler", and (ii) the reference or bookkeeper perspective which ain't moving e.g. ``the map". To see this let's begin with Galileo's kinematics, which was based on quantitative motion studies.
Galileo's experiments around 1600 indicated that speed from rest is proportional not to distance traveled but to time elapsed, so that the causes of a traveling object's motion reside not in space but in time [cf. Epstein]. Moreover the time-based cause of motion (what Newton referred to as the force per unit mass which multiplies time-elapsed) has units of distance per unit time per unit time. Thus Galilean kinematics required a focus on not just velocity (the rate of position's change) but on acceleration (the rate of velocity's change) too, something many earlier approaches to describing motion did not consider.
What folks didn't realize until around 1900 was that time is local, and that motion's causes connect most directly to time-elapsed on the clocks of the traveler. Thus today we start by imagining a map-frame, defined by a coordinate-system of yardsticks say measuring map-position x with synchronized-clocks fixed to those yardsticks measuring map-time t, plus a traveler carrying her own clock that measures traveler (or proper) time τ. A definition of extended-simultaneity (i.e. not local to the traveler and her environs), where needed for problems addressed by this approach, is provided by that synchronized array of map-clocks.
Before we take this leap, however, we might spend a paragraph describing kinematics in terms of variables that allow one to describe motion locally regardless of speed and/or space-time curvature. These variables are frame-invariant proper-time τ on traveler clocks, synchrony-free proper-velocity w ≡ dx/dτ defined in the map-frame, and the frame-invariant proper-acceleration α experienced by the traveler, which for unidirectional motion in flat space-time equals (1/γ)dw/dτ where dw/dτ is the bookkeeper-acceleration in traveler-time units as seen in the map frame. Acceleration from the traveler perspective where the causes of motion are felt is key, because as Galileo and Newton demonstrated in the 17th century, those causes of motion are intimately connected to this second-derivative of position as a function of time. The relationships above allow us to write proper-acceleration as the proper-time derivative of hyperbolic velocity-angle or rapidity η, defined by setting c sinh[η] equal to proper-velocity w in the acceleration direction. These relationships in turn simplify at low speeds (as long as we can also treat space-time as flat) as shown at right below, because one can then approximate the proper-values for velocity and acceleration with coordinate-values v ≡ dx/dt and a ≡ dv/dt.
We can pretend that map-clocks on earth may be synchronized (as if time's rate of passage does not increase with altitude), but let's initially treat traveler-time τ as a local quantity that may or may not agree with map-time t. In that case the space-time version of Pythagoras' theorem says that in flat space-time, with lightspeed constant c, the Lorentz-factor or "speed of map-time" is γ ≡ dt/dτ = Sqrt[1+(dx/dτ)2/c2]. This indicates that for many engineering problems on earth (except e.g. for GPS and relativistic-accelerator engineering) we can ignore clock differences, provided that we also pretend that gravity arises not from variations in time's passage as a function of height (i.e. from kinematics) but from a "proper" force that acts on every ounce of an object's being. In that case we can follow Newton by pretending that time is global, and that accelerations all look the same to "inertial" observers i.e. whose velocity is fixed.
As mentioned above, treating space-time as flat requires that our map frame be seen as a free-float-frame (i.e. one experiencing no net forces). Much of the remainder of this course will therefore concentrate on drawing out uses for the kinematic equations on the right hand side above. We provide the ones on the left, to show that only a bit of added complication will allow one to work in curved-spacetimes and accelerated-frames where geometric-accelerations as well as force-related proper-accelerations have an impact on the bookkeeper-accelerations observed.
Related References
Galileo Galilei (1632/1953/1962/1967) Dialogue Concerning the Two Chief World Systems (Dialogo sopra i due massimi sistemi del mondo) translated by Stillman Drake (U. California Press, Berkeley CA).
Epstein, Lewis Carroll (1895/1994). Relativity Visualized, (Insight Press, San Fransisco), ch. 10,11,12.
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