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:

Note also that figure captions may often be accessed by either mousing-over or clicking-on a given figure.

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.

Thus kinetic energy is directly related to the faster passage of map (t) with respect to traveler (τ) time when simultaneity is defined by the map-frame. However, strangely enough, in non-freefloat or "geometric force" settings (where forces materialize which act on every ounce of your being) a similar relationship linked to your position rather than your speed also materializes.

local time and rotation

An easy to understand example of a geometric force well-depth arises in the case of rotational artificial gravity, provided at least that we ignore azimuthal forces which also operate as one's radius is changed. It is simpler because the differential aging arises purely from the tradeoff between kinetic energy in a single freefloat frame, and geometric "potential energy" whose (negative energy) well-depth to co-rotating travelers is

W_{\text{depth}} = \left(\tfrac{\Delta t_\text{axis}}{\Delta t_\text{off-axis}}-1\right)m c^2 = \left(\tfrac{1}{\sqrt{1-(\tfrac{\omega r}{c})^2}}-1\right)m c^2 \simeq \tfrac12 m \omega^2 r^2

Taking the derivative of this, and dividing by m, gives a simulated g-force as a function of radius of the form g = γ3ω2r which reduces to the classical value of ω2r when r << c/ω. Note that the γ3 factor similarly relates proper-acceleration to coordinate-acceleration in (1+1)D flat spacetime. This "centrifugal acceleration" in an accelerated traveler's frame is perhaps the simplest case of a geometric-force (and associated "potential") linked to differential aging.

local time and work

A similar type of differential aging takes place when an extended object of length L in flat spacetime undergoes constant proper-acceleration α. One obtains (at least for αL << c2, with details depending on how you define simultaneity) for the "well-depth"  or 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. This example is complicated by the fact that no single freefloat frame can be used to see this geometric "potential-energy" as a direct tradeoff with kinetic energy, as was possible in the rotation case above.

local time and gravity

The accelerated ship example above is transition from the purely kinetic centrifugal potential, and the purely motionless gravitational potential around a spherical object. Here the traveler's separation r, between the center of a mass-M spherical object with gravitational-acceleration grM, puts them into a potential-energy well whose depth (i.e. "escape energy") is:

W_{\text{esc}} {\color{OliveGreen}[\text{Joule}]} \simeq \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 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.

mathematical summary 

In the table below we show for several "geometric-force" settings how the metric equation leads one to see both kinetic energy, and potential energy well-depths, as "clock-slowing debts" that must be "worked off" in returning a mass m traveler from her traveler-point perspective to that of the bookkeeper or map frame (that we're using to define simultaneity). Thus kinetic energy slows down a traveler's clock by increasing dt/dτ, while potential-energy (the negative of potential-well depth) speeds it up, so that "escape velocity" (if pointing in the correct direction) can be used to climb out of a potential well, in the process eliminating the clock-slowing effects of both. Differential aging is thus a measure not of total energy, written as kinetic (K) minus well-depth (U), but of the deviation of the traveler perspective from that of the map in terms of kinetic minus potential energy, here written as K+U=(dt/dτ-1)mc2.

frame-dependent energy needed to go from traveler-point to bookkeeper/map-point of view
using a diagonal metric with time-independent coefficients of the form (cδτ)2 = ξ[x](cδt)2-χ[x](δx)2.
Quantity\Setting flat spacetime
proper-velocity w≡dx/dτ
rotating habitat
ω radians/sec
accelerating rocket
acceleration α≥0 lag x≤0
static sphere of
mass M radius r≥R
coefficient ξ of
map-time t
1 1 e2xα/c2 1-2GM/(c2r)
coefficient χ of
map-position x or r
1 1/(1+(ωr/c)2) 1 1/(1-2GM/(c2r))
differential aging
γ ≡ dt/dτ
(1+(w/c)² 1/(1-(ωr/c)² e-xα/c2(1+(w/c)² γr1+(γrw/c)²
with γr≡1/1-2GM/(c²r)
potential well-depth
0 0 e-xα/c2-1 γr-1
kinetic energy
(1+(w/c)²-1 1/(1-(ωr/c)²-1 e-xα/c2((1+(w/c)²-1) γr(1+(γrw/c)²-1) 

Note that to the traveler in a rotating habitat, the kinetic energy seen from the map point-of-view is a potential energy as seen from the traveler's point-of-view. 

Since (dt/dτ-1)mc2 (easily obtainable from a diagonal metric) equals the map-frame's value for kinetic minus potential energy (T-V) in the above (1+1)D examples, we've begun to explore the possibility that this quantity can also serve as a "proper-time" or traveler-point Lagrangian L, for generating equations of motion as well as estimates of T-V. A detailed look at L=(dt/dτ-1)mc2 suggests that even although the acceleration-side of the Euler-Lagrange equation is not zero (e.g. as required for a free particle in flat spacetime) when written as a coordinate-time derivative ∂(∂L/∂w)/∂t (which works e.g. for a free particle in flat spacetime when L=T/γ), it is zero when written as a proper-time derivative ∂(∂L/∂w)/∂τ.

The non-zero force-relations that it yields in the above examples also look promising. In other words, L=T-V might qualify as a  Lagrangian for describing motion in curved spacetime, even though its generalized momentum (in the flat spacetime case p/γ = mw/γ = mdx/dt) is not useful for tracking momentum exchanges in the map-frame, since objects exchanging momentum generally don't share the same proper-time. For momentum-exchanges we therefore stick with the standard momentum p = mw = dx/dτ, which is also a proper-time derivative. The next step may be to check this out in some classical (3+1)D cases.

Related (and more recent) notes in pdf form:

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