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One-tenth-light-speed kinetic remainder

Physical scope

A conditional pointwise calculation for a massive particle in one flat inertial frame when the model state is stipulated to have speed at most one tenth of c. It reports no measurement, apparatus precision, or trajectory approximation.

Example

Let a massive particle in a named inertial frame have constant mass $m > 0$ , speed $∣ v ∣ \leq c / 10$ , Lorentz factor $γ = (1 - ∣ v ∣^{2} / c^{2})^{- 1 / 2}$ , observer energy $E = γ m c^{2}$ , and kinetic energy $T = E - m c^{2}$ as in Mass shell inversion, observer work and variable mass. Treat the speed bound as a stipulated state condition. Then

$0 \leq T - \frac{1}{2} m ∣ v ∣^{2} \leq \frac{3 m ∣ v ∣^{4}}{8 c^{2} (99 / 100)^{5 / 2}} .$

Facts & Assumptions

Given: A massive inertial-frame state with constant $m > 0$ , $c > 0$ , and the stipulated speed bound $∣ v ∣ \leq c / 10$ .

[L1]

In the constant-mass flat inertial model, $γ = (1 - ∣ v ∣^{2} / c^{2})^{- 1 / 2}$ , $E = γ m c^{2}$ , and $T = E - m c^{2}$ (Mass shell inversion, observer work and variable mass). This supplier is a draft result in batch 23; this example uses its exact kinematic definitions conditionally.

[L2]

For every $0 < δ < 1$ and $∣ v ∣ / c \leq δ$ , the kinetic-energy remainder is nonnegative and at most $3 m ∣ v ∣^{4} / [8 c^{2} (1 - δ^{2})^{5 / 2}]$ (Explicit low-speed momentum and energy remainders).

Verification

1.1givenL1L2algebra

The hypothesis $∣ v ∣ \leq c / 10$ is the low-speed lemma's condition with $δ = 1 / 10$ . Substituting $δ^{2} = 1 / 100$ into [L2] gives $1 - δ^{2} = 99 / 100$ and therefore the displayed lower and upper bounds. The lower bound is the same nonnegativity conclusion in [L2]; no division by $∣ v ∣$ is made, so the case $v = 0$ is included.

2.1givenL1step 1.1∎

The bound is conditional on the stipulated state speed and the definitions in [L1]. It is a pointwise model calculation, not an observation and not an estimate comparing two trajectories.

One-tenth-light-speed kinetic remainder

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources