### 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".**

### 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 << c^{2}, 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:

- .

Here t_{trailing} 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/mc^{2}.
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, 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.