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Exact quasistatic and finite-interval mass-change bounds

Statement

Assume ${A C}_{ω}$ as defined by The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $G, c, ℏ, η, M_{0}, M_{\min} > 0$ be the relevant constants and initial/cutoff data of Exact ideal emitter mass ODE and entropy accounting, with $0 < η \leq 1$ and $0 < M_{\min} < M_{0}$ . Let

M (t_{\infty}) = (M_{0}^{3} - 3 α t_{\infty})^{1 / 3}, α = \frac{η ℏ c^{4}}{15360 π G^{2}}, τ = \frac{M_{0}^{3} - M_{\min}^{}}{3 α}

be the positive solution and cutoff supplied by that proposition. Use the diagnostic of Quasistatic diagnostic for the ideal emitter model. Then

ϵ_{q s} (t_{\infty}) = \frac{2 G α}{c^{3} M (t_{\infty})^{2}} \leq \frac{2 G α}{c^{3} M_{\min}^{}} .

For any $ϵ_{0} > 0$ , the condition $M_{\min}^{} \geq 2 G α / (c^{3} ϵ_{0})$ therefore implies $ϵ_{q s} (t_{\infty}) \leq ϵ_{0}$ throughout $[0, τ]$ .

There is also an exact finite-interval bound. If $0 \leq t_{\infty} \leq t_{\infty} + Δ t \leq τ$ and

δ : = \frac{3 α Δ t}{M (t_{\infty})^{3}} \leq \frac{1}{2},

then

\frac{M (t_{\infty} + Δ t)}{M (t_{\infty})} = (1 - δ)^{1 / 3}, ∣ \frac{M (t_{\infty} + Δ t)}{M (t_{\infty})} - 1 ∣ \leq \frac{2^{2 / 3}}{3} δ .

Both bounds are exact internal consequences of the adopted ODE. They give no error bound between that ODE and an unconstructed Einstein–quantum backreaction solution.

Facts & Assumptions

Given: ${A C}_{ω}$ , the fixed-reference ODE, its positive solution, the cutoff and constants of Exact ideal emitter mass ODE and entropy accounting, with $0 < M_{\min} < M_{0}$ , $0 < η \leq 1$ , and $α > 0$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ means every countable family of nonempty sets has a choice function; this is the exact choice assumption carried by the mathematical photon-flux supplier.

[F1]

Exact ideal emitter mass ODE and entropy accounting: its Statement defines $r_{H} = 2 G M / c^{2}$ , $α = η ℏ c^{4} / (15360 π G^{2})$ , and the exact positive solution $M (t_{\infty}) = (M_{0}^{3} - 3 α t_{\infty})^{1 / 3}$ until $M_{\min} > 0$ .

[F2]

Quasistatic diagnostic for the ideal emitter model: $t_{c r o s s} = r_{H} (M (t_{\infty})) / c$ and $ϵ_{q s} = ∣ M^{'} ∣ t_{c r o s s} / M$ ; the latter is a dimensionless ODE diagnostic only.

[F3]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : if a function is continuous on a closed interval and differentiable in its interior, its endpoint difference equals its derivative at an interior point times the interval length.

Proof

Given: A positive solution $M$ of the stated ODE on $[0, τ]$ , positive $G, c, α$ , and the definitions in [F1]–[F2].

1.1givenA1F1F2algebra

Differentiate the exact solution in [F1] on $0 \leq t_{\infty} \leq τ$ ; since $M (t_{\infty}) > 0$ , this gives $M^{'} = - α / M^{2}$ . Also [F1] gives $r_{H} = 2 G M / c^{2}$ , so substituting these identities and $t_{c r o s s} = r_{H} / c$ into [F2] yields

ϵ_{q s} = \frac{(α / M^{2}) (2 G M / c^{3})}{M} = \frac{2 G α}{c^{3} M^{2}} .

All factors are positive on the interval.

1.2givenA1F1algebra

From [F1], $M (t_{\infty} + Δ t)^{3} = M (t_{\infty})^{3} - 3 α Δ t$ . Divide by the positive $M (t_{\infty})^{3}$ and take positive cube roots to get $M (t_{\infty} + Δ t) / M (t_{\infty}) = (1 - δ)^{1 / 3}$ .

2.1givenA1step 1.1algebra

Since $M (t_{\infty}) \geq M_{\min} > 0$ , one has $M (t_{\infty})^{- 2} \leq M_{\min}^{}$ . The formula in step 1.1 gives the uniform upper bound; if $M_{\min}^{} \geq 2 G α / (c^{3} ϵ_{0})$ , multiplying by the positive denominator yields $ϵ_{q s} \leq ϵ_{0}$ .

3.1givenA1F3step 1.2algebra∎

If $δ = 0$ , step 1.2 gives a ratio of one and the bound is equality. If $0 < δ \leq 1 / 2$ , $h (x) = (1 - x)^{1 / 3}$ is continuous on $[0, δ]$ , differentiable on its interior, and satisfies $∣ h^{'} (x) ∣ = \frac{1}{3} (1 - x)^{- 2 / 3} \leq 2^{2 / 3} / 3$ . Apply [F3] on this nondegenerate interval and use $h (0) = 1$ to obtain $∣ h (δ) - 1 ∣ \leq (2^{2 / 3} / 3) δ$ . Step 1.2 gives the claimed ratio bound.

Exact quasistatic and finite-interval mass-change bounds

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources