A traveler-centered intro to kinematics

by P. Fraundorf of *P**hysics & 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
background.

Sections of this paper include:

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## Some familiar equations re-framed:

### local time and motion

An object's momentum **p** = m**w** is the product of its __frame-invariant__ rest-mass m, and its __synchrony-free__ *proper-velocity* **w** ≡ Δ**x**/Δτ = γ**v** where *Lorentz-factor* or "speed-of-map-time" γ ≡ Δt/Δτ = 1/Sqrt[1-(v/c)^{2}] ≥ 1 and *coordinate-velocity* **v** ≡ Δ**x**/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** ≈ m**v**.
Similarly an object's translational kinetic energy (e.g. in Joules) is
approximately half its mass m times the square of its speed v^{2}, i.e.

- .

where c ≈ 3×10^{8}[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* g_{rM}, puts you into *a potential-energy well* whose depth is:

- ,

where here *gravity's acceleration* is g_{rM} = GM/r^{2} and *speed-of-far-time* is Δt_{far}/Δτ = 1/Sqrt[1-2GM/(c^{2}r)] ≥ 1, Δt_{far} is time elapsed on "Schwarzschild" map-clocks far away from our object, and Δ*τ*
(also used above) is the "proper" time-elapsed on observer clocks
nearer to the object surface. Put another way, acceleration due to
gravity is g_{rM} = (Δt_{far}/Δτ-1)c^{2}/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
traveler.

#### Related (but earlier) notes in pdf form:

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