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Force-free runaway solution of the LAD equation

Physical scope

Counterexample within the adopted flat LAD effective ODE: a force-free, smooth future-timelike solution with exponentially growing proper acceleration. It is an equation-level pathology, not an observation or claim about an actual finite charge.

Statement refuted

Within the adopted flat LAD effective equation with $τ_{e} > 0$ and zero prescribed external field, every smooth future-timelike proper-time solution defined for all future proper times has bounded proper acceleration toward the future. This is an assertion about the adopted ODE, not an empirical statement about a finite charge.

Example

Assume the flat effective LAD postulate Flat Lorentz–Abraham–Dirac effective equation with $τ_{e} > 0$ and take $F \equiv 0$ . In Minkowski coordinates $(x^{0}, x^{1}, x^{2}, x^{3})$ , choose constants $θ_{0} \in R$ , $A \in R ∖ {0}$ with units $s^{- 1}$ , and $τ_{0} \in R$ . Define for all $τ \in R$

$θ (τ) : = θ_{0} + A τ_{e} e^{(τ - τ_{0}) / τ_{e}}, U (τ) : = c (\cosh θ (τ), \sinh θ (τ), 0, 0),$

and let $x (τ) = x_{0} + \int_{τ_{0}}^{τ} U (s) d s$ . Then $x$ is a smooth future-timelike worldline parametrized by proper time, and it solves the adopted LAD equation with zero external force. Its proper acceleration magnitude is

$\sqrt{η (a, a)} = c ∣ A ∣ e^{(τ - τ_{0}) / τ_{e}},$

so it grows exponentially as $τ \to + \infty$ . By contrast, the force-free reduced LL field in Reduced-order Landau–Lifshitz local ODE and mass shell is zero on the shell and has constant-velocity solutions. This is a counterexample about the adopted ODEs, not an observed phenomenon or a statement about finite-body electrodynamics.

Facts & Assumptions

Given: The flat branch of Charged and spinning test-body quantities, the adopted LAD equation in Flat Lorentz–Abraham–Dirac effective equation, Minkowski signature $(-, +, +, +)$ , $c > 0$ , $τ_{e} > 0$ , and the constants above.

[A1]

Adopt the flat LAD equation as a physical model assumption and set the prescribed external two-form to zero.

[F1]

The six hyperbolic functions and their natural domains and Addition formulas, identities, parity, and derivatives of the hyperbolic functions: $\cosh^{2} θ - \sinh^{2} θ = 1$ and the derivatives of $\cosh$ and $\sinh$ are $\sinh$ and $\cosh$ .

[F2]

The exponential function is smooth and $(\exp)^{'} = \exp$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , 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 $g (c) \neq 0$ , The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive, and A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: the displayed smooth functions differentiate by the chain/product rules, and the integral defining $x$ has derivative $U$ .

[F3]

Flat Lorentz–Abraham–Dirac effective equation: for $F = 0$ , LAD reads $a = τ_{e} (\dot{a} - η (a, a) U / c^{2})$ .

[F4]

Reduced-order Landau–Lifshitz local ODE and mass shell: when $F = 0$ , $f_{0} = B = 0$ , so the reduced LL acceleration field vanishes.

Counterexample

1.1givenA1F1F2algebra

The integral defining $x$ is well-defined on every compact $τ$ interval because $U$ is smooth by [F2]. The fundamental theorem [F2] gives $\dot{x} = U$ . By [F1], $η (U, U) = c^{2} (- \cosh^{2} θ + \sinh^{2} θ) = - c^{2}$ . Also $U^{0} = c \cosh θ > 0$ , so $U$ is future directed in the chosen inertial chart. Thus $τ$ is proper time and $x$ is future timelike.

2.1givenA1F1F2F3step 1.1algebra

Differentiating with [F1, F2] gives $a = c θ^{'} (\sinh θ, \cosh θ, 0, 0), \dot{a} = c θ^{''} (\sinh θ, \cosh θ, 0, 0) + c (θ^{'})^{2} (\cosh θ, \sinh θ, 0, 0), η (a, a) = c^{2} (θ^{'})^{2} .$ Consequently $\dot{a} - η (a, a) U / c^{2} = c θ^{''} (\sinh θ, \cosh θ, 0, 0)$ , and the force-free equation [F3] is equivalent to $θ^{'} = τ_{e} θ^{''}$ . The chosen $θ$ has $θ^{'} = A e^{(τ - τ_{0}) / τ_{e}}$ and $θ^{''} = θ^{'} / τ_{e}$ , so the equation holds identically. This proves that the curve solves the adopted LAD equation for every real proper time.

3.1A1F3F4step 2.1algebra∎

The invariant $η (a, a) = c^{2} (θ^{'})^{2}$ gives $\sqrt{η (a, a)} = c ∣ A ∣ e^{(τ - τ_{0}) / τ_{e}}$ , which is unbounded toward the future. For $F = 0$ , [F4] gives $a_{L L} = 0$ , hence its local solutions have constant $U$ . The displayed LAD solution is non-inertial since $A \neq 0$ , so it demonstrates that the force-free LAD equation admits runaway solutions even though the reduced LL force-free field does not. This is an ODE comparison, not an empirical conclusion.

Force-free runaway solution of the LAD equation

Statement refuted

Example

Facts & Assumptions

Counterexample

Depends on

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