A traveler-centered intro to kinematics
by P. Fraundorf of Physics & Astronomy/Center for NanoScience at University of Missouri -St Louis (63121)
Treating time as a local variable permits robust approaches to
kinematics that forego questions of extended-simultaneity, which because
of their abstract nature might not be addressed explicitly until a
first relativity course and even then without considering the dependence
of clock-rates on position in a gravitational field. For example we
here use synchrony-free ``traveler kinematic" relations to construct a
brief story for beginning students about: (a) time as a local quantity
like position that depends on ``which clock", (b)
coordinate-acceleration as an approximation to the acceleration felt by a
moving traveler, and (c) the geometric origin (hence mass-independence)
of gravitational acceleration. The goal is to explicitly rule out
global-time for all from the start, so that it can be returned as a
local approximation, while tantalizing students interested in the
subject with more widely-applicable equations in range of their math
Sections of this paper include:
Note also that figure captions may often be accessed by either mousing-over or clicking-on a given figure.
Some familiar equations re-framed:
local time and motion
An object's momentum p = mw is the product of its frame-invariant rest-mass m, and its synchrony-free proper-velocity w ≡ Δx/Δτ = γv where Lorentz-factor or "speed-of-map-time" γ ≡ Δt/Δτ = 1/Sqrt[1-(v/c)2] ≥ 1 and coordinate-velocity v ≡ Δx/Δt. Here τ is frame-invariant proper-time elapsed on the clocks of our traveler, and t is map-time measured on clocks fixed to the yardsticks used to measure map-position x.
Greek letter "Delta" (Δ) in front of any thing means "change in" i.e. the difference between final and initial values of that thing. At low speeds, p ≈ mv.
Similarly an object's translational kinetic energy (e.g. in Joules) is
approximately half its mass m times the square of its speed v2, i.e.
where c ≈ 3×108[m/s] is the space-time constant that we call lightspeed for historical reasons.
local time and gravity
External proximity, by separation r to the center of a mass-M spherical object with gravitational-acceleration grM, puts you into a potential-energy well whose depth is:
where here gravity's acceleration is grM = GM/r2 and speed-of-far-time is Δtfar/Δτ = 1/Sqrt[1-2GM/(c2r)] ≥ 1, Δtfar is time elapsed on "Schwarzschild" map-clocks far away from our object, and Δτ
(also used above) is the "proper" time-elapsed on observer clocks
nearer to the object surface. Put another way, acceleration due to
gravity is grM = (Δtfar/Δτ-1)c2/r. In other words: On earth stuff falls because when standing "your head ages faster than your feet".
Thus momentum, kinetic energy, and potential energy are all intimately linked to local differences
in the rate at which time passes, and in particular to the
"speed-of-reference-time" ratio (greater than or equal to one) between
some "bookkeeper time" and frame-invariant proper-time on the clocks of a