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.
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".
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:
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, 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.