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Conditional radius loss of a prescribed slowly orbiting charge in a reduced quasistatic model

Physical scope

Reduced quasistatic circular-orbit model with an adopted Larmor energy-balance premise (H): the radius ODE $r^{'} = - A / r^{2}$ , the cubic solution $r^{3} = r_{0}^{3} - 3 A (t - t_{0})$ and the formal finite-time collapse limit as a property of (H). Excluded: any claim that a physical orbit reaches the centre or an atom decays in finite time; coupled self-force dynamics; derivation of lifetimes; measurements, statistical models and every empirical claim.

Statement

When invoking the cited point-source emitted-power theorem as an established interpretation, retain its Countable Choice premise ( ${A C}_{ω}$ , The Axiom of Countable Choice ( ${A C}_{ω}$ )). The reduced loss-rate formula in (H) below is independently adopted as a modelling hypothesis; the conditional finite-calculus orbit computation from (H) uses no choice.

Adopt the following idealized, hypothetical setup of a prescribed slowly orbiting charge; it is a thought experiment with explicit premises, not a reported measurement. (H) Let $m > 0$ be an externally supplied rest mass and $q \neq 0$ a charge parameter (Framework-local SI units, the vacuum constants and the exact/measured status of their values, The Lorentz coupling: external force on a test charge and the continuum force and power densities), let $κ > 0$ carry J m, and let $J_{t} \subseteq R$ be a nonempty open interval, $t_{0} \in J_{t}$ , and $r : J_{t} \to (0, \infty)$ be $C^{1}$ , with $r_{0} : = r (t_{0}) > 0$ . The reduced model assigns to each radius the circular-orbit values about a fixed laboratory centre (Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields) for the attractive potential $V (r) = - κ / r$ : the circular-orbit energy is $E_{o r b} (r) = - κ / (2 r)$ and the circular orbit with speed $v$ satisfies $m v^{2} = κ / r$ , with centripetal acceleration $a = κ / (m r^{2})$ . These are instantaneous circular reference values, not the exact kinematics of a curve with varying radius. To specify the intended validity window, choose dimensionless bounds $0 < δ_{v}, δ_{a} < 1$ and require $v (t) / c \leq δ_{v}$ and $T_{o r b} (t) ∣ r^{'} (t) ∣ / r (t) \leq δ_{a}$ , where $v (t) : = \sqrt{κ / (m r (t))}$ and $T_{o r b} (t) : = 2 π r (t) / v (t)$ ; the second ratio measures fractional radius change in one reference period. Independently adopt that the electromagnetic energy emitted per unit time is instantaneously taken to be the Larmor value $q^{2} a (t)^{2} / (6 π ε_{0} c^{3})$ of Instantaneous-rest Larmor power and uniform low-speed comparison for a prescribed point charge evaluated at the instantaneous centripetal acceleration, so that the adopted energy balance of the reduced model reads

$\frac{d}{d t} E_{o r b} (r (t)) = - \frac{q^{2} a (t)^{2}}{6 π ε_{0} c^{3}} .$

Then, with $A : = q^{2} κ / (3 π ε_{0} m^{2} c^{3}) > 0$ :

(a) the reduced model implies $r^{'} (t) = - A / r (t)^{2}$ for all $t$ where the model applies, and the circular orbit radius in the reduced model obeys $r (t)^{3} = r_{0}^{3} - 3 A (t - t_{0})$ on any interval containing $t_{0}$ on which $r > 0$ ;

(b) $r$ is strictly decreasing on $J_{t}$ . The scalar balance ODE, considered without the validity-window inequalities, has the unique maximal positive continuation $\hat{r} (t) : = (r_{0}^{3} - 3 A (t - t_{0}))^{1 / 3}$ on $(- \infty, t_{*})$ , where $t_{*} : = t_{0} + r_{0}^{3} / (3 A)$ . It restricts to $r$ on $J_{t}$ and satisfies $\hat{r} (t) ↓ 0$ as $t ↑ t_{*}$ ; assigning zero at $t_{*}$ is only a continuous endpoint extension, not a positive-radius solution;

(c) [Scope and qualifications] the deduction is conditional on (H). The low-speed premise $v^{2} = κ / (m r) ≪ c^{2}$ necessarily fails as $r \to 0$ , so the model's own hypotheses break down near the collapse time and the computation is not a proof that a physical orbit reaches $r = 0$ ; it is not a derivation of atomic lifetimes, not a coupled point-particle self-force calculation (that dynamics is not available in this framework), and it supplies no empirical evidence. The reduced-model finite-time collapse is retained as a mathematical property of (H), while all conclusions about nature remain conditional on (H) and on its regime of validity. No measurement, dataset, statistical model or uncertainty is introduced.

Facts & Assumptions

Given: the hypothetical setup (H), including the $C^{1}$ map $r : J_{t} \to (0, \infty)$ on a nonempty open time interval, $t_{0} \in J_{t}$ and $r_{0} = r (t_{0}) > 0$ : external parameters $m > 0$ , $q \neq 0$ , $κ > 0$ ; a circular orbit of radius $r (t) > 0$ about a fixed centre in the laboratory chart with $E_{o r b} (r) = - κ / (2 r)$ , $m v^{2} = κ / r$ and centripetal acceleration $a = κ / (m r^{2})$ ; the adopted energy balance $\frac{d}{d t} E_{o r b} (r (t)) = - \frac{q^{2} a (t)^{2}}{6 π ε_{0} c^{3}}$ ; $A : = q^{2} κ / (3 π ε_{0} m^{2} c^{3}) > 0$ ; ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Units and model reading: the parameters $m, q$ carry the SI meanings of Framework-local SI units, the vacuum constants and the exact/measured status of their values and The Lorentz coupling: external force on a test charge and the continuum force and power densities, while $κ$ has the independently declared unit J m, and the orbit is read in the laboratory chart of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields. (Framework-local SI units, the vacuum constants and the exact/measured status of their values, The Lorentz coupling: external force on a test charge and the continuum force and power densities, Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields)

[F2]

The Larmor value $q^{2} a^{2} / (6 π ε_{0} c^{3})$ of Instantaneous-rest Larmor power and uniform low-speed comparison for a prescribed point charge is the exact instantaneous-rest emitted power of the point-source model; in (H) it is adopted as the instantaneous loss rate of the reduced model, an additional modelling assumption. (Instantaneous-rest Larmor power and uniform low-speed comparison for a prescribed point charge)

[F3]

One-variable calculus on intervals: the power rule for $u \mapsto u^{n}$ and the chain rule, the algebra of derivatives, the fundamental theorem for a continuous derivative, and the interval conventions of Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length. (For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Continuity and derivatives of positive-base real powers, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length)

Proof

technique · direct conditional computation from (H); differentiate the orbital energy along the prescribed radius, substitute the circular-orbit acceleration, integrate the resulting ODE, and read off the finite-time collapse with its scope qualifications

1.1givenF1F3algebra

The radius ODE (a, first part). Along $t \mapsto r (t)$ , the orbital energy satisfies $\frac{d}{d t} E_{o r b} (r (t)) = \frac{d}{d t} (- \frac{κ}{2 r (t)}) = \frac{κ}{2} \frac{r^{'} (t)}{r (t)^{2}}$ by the power rule and the chain rule [F3]. Equating this to the adopted balance in (H), $\frac{κ r^{'}}{2 r^{2}} = - \frac{q^{2} a^{2}}{6 π ε_{0} c^{3}}$ , and substituting the centripetal acceleration $a = κ / (m r^{2})$ gives $\frac{κ r^{'}}{2 r^{2}} = - \frac{q^{2} κ^{2}}{6 π ε_{0} c^{3} m^{2} r^{4}}$ , hence $r^{'} = - \frac{q^{2} κ}{3 π ε_{0} m^{2} c^{3} r^{2}} = - \frac{A}{r^{2}}$ , which is the displayed ODE; here $κ > 0$ and $r > 0$ legitimize the divisions [F1].

2.1step 1.1F3algebra

The cubic (a, second part). Multiplying the ODE of step 1.1 by $3 r^{2}$ gives $\frac{d}{d t} (r (t)^{3}) = 3 r^{2} r^{'} = - 3 A$ , a constant right-hand side; applying the fundamental theorem on any interval containing $t_{0}$ on which $r > 0$ [F3] yields $r (t)^{3} - r (t_{0})^{3} = - 3 A (t - t_{0})$ , that is the displayed cubic. Conversely, differentiating the cubic on that interval recovers the ODE, so the two descriptions are equivalent within the reduced model. This is (a).

3.1step 2.1F3algebra

Monotone decrease and formal maximal continuation (b). Since $A > 0$ and $r > 0$ , $r^{'} = - A / r^{2} < 0$ . For $t_{1} < t_{2}$ in $J_{t}$ , step 2.1 gives $r (t_{2})^{3} - r (t_{1})^{3} = - 3 A (t_{2} - t_{1}) < 0$ , and positivity implies $r (t_{2}) < r (t_{1})$ . Put $g (t) : = r_{0}^{3} - 3 A (t - t_{0})$ , which is positive exactly for $t < t_{*} : = t_{0} + r_{0}^{3} / (3 A)$ . The positive real-power function $\hat{r} = g^{1 / 3}$ is $C^{1}$ on $(- \infty, t_{*})$ by [F3] and has derivative $- A g^{- 2 / 3} = - A / {\hat{r}}^{2}$ . The cubic identity forces $J_{t} \subset (- \infty, t_{*})$ and $r = \hat{r}$ there: positive cubes are injective since $x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})$ and the second factor is positive. Every positive solution through $(t_{0}, r_{0})$ satisfies the same cubic by step 2.1, so this is the unique maximal positive solution of the scalar balance ODE; it cannot extend positively across $t_{*}$ where $g = 0$ . As $t ↑ t_{*}$ , given $ε > 0$ , $0 < g (t) < ε^{3}$ eventually, whence $0 < \hat{r} (t) < ε$ . This formal limit belongs to the extended scalar ODE, not to an arbitrary original validity interval, and no negative radius or positive solution at $t_{*}$ is asserted.

4.1step 1.1step 3.1F1F2algebra∎

Scope and qualifications (c). The deduction is conditional on (H) throughout: the balance is the adopted premise, and the Larmor expression is only the prescribed-source rest-event emitted power of [F2], converted into a mechanical loss rate solely by the assumption (H) and not by any derived equation of motion. The quantified low-speed bound $v / c \leq δ_{v}$ in (H) fails along the formal continuation as $r \to 0$ because $v^{2} = κ / (m r) \to + \infty$ with $κ > 0$ , so the model's own regime of validity excludes the collapse point: $t_{*}$ is a formal finite-time limit of the reduced model, not the predicted time at which a physical orbit reaches the centre. No coupled self-force dynamics, no lifetime derivation, no measurement, dataset, statistical model or uncertainty is introduced, and nothing here contradicts a theorem: the scalar balance is an independent hypothesis, and the prescribed-field radiation theorems supply no mechanical balance or coupled motion law. This is (c), and with steps 1.1-3.1 it completes the conditional deduction from (H).

Conditional radius loss of a prescribed slowly orbiting charge in a reduced quasistatic model

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources