0. Abstract

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 background. 

Sections of this paper include:

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Some familiar equations re-framed:

Energies associated "with the indicated frame only" include the kinetic energy of motion, as well as potential energies associated with "geometric forces" which arise in accelerated frames and in curved spacetime. Strangely enough, all of these frame-specific energies seem to connect to "differential-aging" γ ≡ dt/dτ i.e. via a relation of the form (γ-1)mc2. One thing we will try to deal with in these pages is to explore other places where (γ-1) comes in handy for describing "indicated frame-only" effects.

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.

K {\color{OliveGreen} [\text{Joule}]} = (\tfrac{\Delta t_\text{map}}{\Delta \tau} - 1)m c^2 \overset{v \ll c}{\cong} \frac{1}{2} m {\color{OliveGreen} [\text{kilogram}]} v^2 {\color{OliveGreen}\left[ \tfrac{\text{meter}^2}{\text{second}^2}\right]} \propto v^2.

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:

U_g {\color{OliveGreen}[\text{Joule}]} = \left(\tfrac{\Delta t_\text{far}}{\Delta \tau}-1\right)m c^2 \overset{r \gg r_s}{\cong}  m g_{rM} r = G  \frac{m  M }{r }  \propto \frac{1}{r},

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".

local time and work

This differential aging ratio is also what modifies terms in the flat space metric equation to give us curved spacetime. A similar type of differential aging moreover takes place when an extended object of length L in flat spacetime undergoes constant proper-acceleration α, so that (at least for αL << c2, with details depending on how you define simultaneity) one has for the work needed to climb up from the trailing to the leading edge, e.g. of your accelerating space-ship, something like:

W_{\alpha L} \simeq \left(\tfrac{\Delta t_\text{leading}}{\Delta t_\text{trailing}}-1\right)m c^2 \simeq m \alpha L = \Sigma F_o L .

Here ttrailing might also be thought of as time τ on the clocks of a passenger getting ready to "make the climb" from the trailing to the leading end of her accelerating home.

local time and energy overview

Given an energy-difference ΔE between states, one can sometimes at least imagine an associated differential-time ratio like dt/dτ ≈ 1+ΔE/mc2. Note that the higher and lower energy states change positions in the kinetic energy case, which is why global-positioning-system (GPS) satellite clocks are sped up by increasing orbital-altitude, but slowed down by increasing orbital-speed.

Thus kinetic-energy, and for geometric ("connection-coefficient") forces potential-energy and work, may all be intimately linked to local differences in the rate at which time passes! In other words, very small amounts of differential aging (like the dt/dτ ≈ 10-9 associated with gravity here on earth's surface) can have very-significant physical consequences. 

Related (but earlier) notes in pdf form:

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