Acceleration and momentarily comoving inertial frames

At (coordinate) time t in K, let v be the velocity of a non-uniformly moving particle. Let t' denote the corresponding time in K' that is read off at the particle's momentary location in K at time t.

(Definition) K' is called a momentarily comoving inertial frame of the particle at time t.

It is comoving because in K' the particle is momentarily at rest at time t'; one can see that by applying the velocity addition formulas. (The derivation of the formulas can be adjusted in a straightforward manner to account for the case of non-uniform motion too.)

Let the particle's acceleration be a' ≠ 0 relative to K' during the time interval [t' – Δt'; t' + Δt'], and let a'x', a'y', and a'z' denote the (signed) x', y', and z' components of a', respectively. That is, a'2 = a'x'2 + a'y'2 + a'z'2. To obtain the corresponding ax, ay, and az in K, we take the [t'; t' + Δt'] segment of the particle's trajectory in K', and calculate the corresponding Δt, wx, wy, and wz values in K:

w'x' = a'x' · Δt'

Δx' = (1 / 2) · a'x' · Δt'2 = (1 / 2) · w'x' · Δt'

Δt = (Δx' · √1 – v2   / c2    · (v / c2) / √1 – v2   / c2    + Δt') / √1 – v2   / c2    = (1 + (1 / 2) · w'x' · v / c2) · Δt' / √1 – v2   / c2   

wx = (w'x' + v) / (1 + w'x' · v / c2)

w'y' = a'y' · Δt'

wy = w'y' · √1 – v2   / c2    / (1 + w'x' · v / c2)

w'z' = a'z' · Δt'

wz = w'z' · √1 – v2   / c2    / (1 + w'x' · v / c2)

We get to the acceleration components when Δt' → 0:

ax = limΔt'→0 (wx – v) / Δt = a'x' · (1 – v2 / c2)3/2

ay = limΔt'→0 (wy – 0) / Δt = a'y' · (1 – v2 / c2)

az = limΔt'→0 (wz – 0) / Δt = a'z' · (1 – v2 / c2)

(Theorem) a < a' · (1 – v2 / c2) < a'.

(Theorem) If v → c, then a → 0.

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