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Worked uniform Galilean chart bound on a bounded region

Example

Take $v = e_{1}$ (so $∣ v ∣ = 1$ ), $T = 1$ , $R = 2$ and the point $(t, x) = (1, 2 e_{1})$ in Galilean limiting chart maps and bounded chart regions, and let $Φ_{v, C} (1, 2 e_{1}) = (t^{'}, x^{'})$ . For $C > 1$ , write $g_{C} = C / \sqrt{C^{2} - 1}$ , so $t^{'} = (C^{2} - 2) / (C \sqrt{C^{2} - 1})$ and $x^{'} = g_{C} e_{1}$ , while $G_{v} (1, 2 e_{1}) = (1, e_{1})$ . The exact point deviations and whole-region bounds are $d_{t} (C) : = 1 - \frac{C^{2} - 2}{C \sqrt{C^{2} - 1}} \leq B_{t} (C) : = g_{C} - 1 + \frac{2 g_{C}}{C^{2}}, d_{x} (C) : = g_{C} - 1 \leq B_{x} (C) : = 3 (g_{C} - 1) .$ Their rounded values are:

$C$	$g_{C}$	$t^{'}$	$d_{t} (C)$	$B_{t} (C)$	$d_{x} (C)$	$B_{x} (C)$
$10$	$1.0050378153$	$0.9849370590$	$0.0150629410$	$0.0251385716$	$0.0050378153$	$0.0151134458$
$100$	$1.0000500038$	$0.9998499937$	$0.0001500063$	$0.0002500138$	$0.0000500038$	$0.0001500113$

Here $g_{10} = 10 / \sqrt{99} = 10 / (3 \sqrt{11})$ . The computed deviations shrink by factors $d_{t} (10) / d_{t} (100) \approx 100.4154227043$ and $d_{x} (10) / d_{x} (100) \approx 100.7487483984$ , or about $100.4$ and $100.7$ to one decimal place, consistent with the $O (C^{- 2})$ rate of Uniform Galilean limit of the boost charts on bounded regions, and its failure on the whole chart domain,(c).

Finally, for tolerance $ε = 0.01$ , take $z_{0} = 1 / 4$ . Then $K_{0} = 4 \sqrt{3} / 9$ and $A_{0} = 1 / 2 + \sqrt{3} / 9 < 3 / 4$ . Choose exactly $z_{1} = 1 / 900$ . With $H = T + R + ∣ v ∣ T = 4$ , one has $z_{1} < z_{0}$ , $A_{0} z_{1} < 1 / 1200 < 1 / 2$ and $A_{0} H z_{1} < 1 / 300 < ε / 2 = 1 / 200$ , so every threshold inequality is satisfied. The sufficient threshold is $C_{ε} = \max {1 / \sqrt{z_{1}}, \sqrt{3 ∣ v ∣ R / ε}} = \max {30, \sqrt{600}} = 30 > 1 = ∣ v ∣ .$ Thus $C = 30$ is sufficient uniformly on $D_{1, 2}$ . At the specified point $γ_{30} = 30 / \sqrt{899}$ , $∣ t^{'} - t ∣ = 1 - 449 / (15 \sqrt{899}) \approx 0.0016674389$ and $∣ x^{'} - (x - t v) ∣ = 30 / \sqrt{899} - 1 \approx 0.0005560189$ , both below $0.01$ . The whole-region bound values at $C = 30$ are approximately $0.0027794768$ and $0.0016680568$ , also below $0.01$ . Computed values are rounded to ten places after the decimal point, except the explicitly stated one-place shrink factors; terminating rational data are exact and all inequalities compare exact values. No physical interpretation of the auxiliary $Φ_{v, C}$ family is asserted.

Facts & Assumptions

Given: $v = e_{1}$ , $T = 1$ , $R = 2$ , the point $(t, x) = (1, 2 e_{1})$ , the maps $Φ_{v, C}$ and $G_{v}$ and the region $D_{1, 2}$ of Galilean limiting chart maps and bounded chart regions, and the bounds and threshold of Uniform Galilean limit of the boost charts on bounded regions, and its failure on the whole chart domain.

[L1]

In the frame data, $∣ v ∣ = ∣ e_{1} ∣ = 1$ and $δ (v, x) = δ (e_{1}, 2 e_{1}) = 2$ , and $Φ_{v, C} (t, x) = (γ_{C} (t - δ (v, x) / C^{2}), x + \frac{γ_{C} - 1}{∣ v ∣^{2}} δ (v, x) v - γ_{C} t v)$ for $v \neq 0$ while $G_{v} (t, x) = (t, x - t v)$ (Galilean limiting chart maps and bounded chart regions).

[L2]

With these data the lemma gives: the exact differences $t^{'} - t = (γ_{C} - 1) t - γ_{C} δ (v, x) / C^{2}$ and $x^{'} - (x - t v) = (γ_{C} - 1) (x_{∥} - t v)$ ; the bounds $∣ t^{'} - t ∣ \leq (γ_{C} - 1) T + γ_{C} ∣ v ∣ R / C^{2}$ and $∣ x^{'} - (x - t v) ∣ \leq (γ_{C} - 1) (R + ∣ v ∣ T)$ ; and the sufficient threshold $C_{ε} = \max {∣ v ∣ / \sqrt{z_{1}}, \sqrt{3 ∣ v ∣ R / ε}}$ for any admissible $z_{1}$ (Uniform Galilean limit of the boost charts on bounded regions, and its failure on the whole chart domain).

[L3]

Rational powers: $γ_{C} = (1 - 1 / C^{2})^{- 1 / 2} = (C^{2} / (C^{2} - 1))^{1 / 2} = C / \sqrt{C^{2} - 1}$ for $C > 1$ , and $(3 / 4)^{- 5 / 2} = (4 / 3)^{5 / 2} = (4 / 3)^{2} (4 / 3)^{1 / 2}$ (Rational powers $a^{r}$ of a positive base, Laws of rational exponents).

[L4]

Square roots: $\sqrt{}$ is the nonnegative root of Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , and for $0 \leq a < b$ one has $\sqrt{a} < \sqrt{b}$ ; in particular $\sqrt{3} < 9 / 4$ because $3 < 81 / 16$ , and $\sqrt{600} < 30$ because $600 < 900$ . Also $(4 / 3)^{5 / 2} = (16 / 9) (2 / \sqrt{3}) = 32 / (9 \sqrt{3})$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , Monotonicity of $r \mapsto a^{r}$ and of $a \mapsto a^{r}$ ).

[L5]

Order arithmetic in $R$ : inequalities may be multiplied by positive numbers and added, and $0 < 1 / 300 < 1 / 200$ (Ordered field): $(b + d) - (a + c) = (b - a) + (d - c)$ and $λ b - λ a = λ (b - a)$ are nonnegative when $a \leq b$ , $c \leq d$ , $λ > 0$ , by positive-cone closure, with equality handled separately.

Verification

technique · direct

1.1givenL1L3L5algebra

The exact deviations at the point. With $v = e_{1}$ , $∣ v ∣ = 1$ , $δ (v, x) = 2$ , $x_{∥} = 2 e_{1}$ and $x - t v = e_{1}$ , [L1] and [L3] give $γ_{C} = C / \sqrt{C^{2} - 1}$ , $t^{'} = γ_{C} (1 - \frac{2}{C^{2}}) = (C^{2} - 2) / (C \sqrt{C^{2} - 1})$ and $x^{'} = 2 e_{1} + 2 (γ_{C} - 1) e_{1} - γ_{C} e_{1} = γ_{C} e_{1}$ , so the exact deviations are $d_{t} (C) = 1 - (C^{2} - 2) / (C \sqrt{C^{2} - 1})$ and $d_{x} (C) = γ_{C} - 1$ . Indeed, if $1 < C^{2} \leq 2$ then $t^{'} \leq 0 < 1$ ; if $C^{2} > 2$ , both numerator and denominator of $t^{'}$ are positive and $(C^{2} - 2)^{2} < C^{2} (C^{2} - 1)$ since $4 < 3 C^{2}$ , so $t^{'} < 1$ again. Thus $d_{t} (C) > 0$ for every $C > 1$ , including $C = 10, 30, 100$ , and the absolute value may be read as written.

1.2givenL3L4L5algebra

The $ε$ -threshold data. Take $z_{0} = 1 / 4$ . By [L3] and [L4], $K_{0} = \frac{3}{8} (3 / 4)^{- 5 / 2} = \frac{3}{8} \cdot \frac{32}{9 \sqrt{3}} = \frac{4 \sqrt{3}}{9}$ and $A_{0} = \frac{1}{2} + K_{0} z_{0} = \frac{1}{2} + \frac{\sqrt{3}}{9}$ ; since $\sqrt{3} < 9 / 4$ by [L4], $A_{0} < \frac{1}{2} + \frac{1}{4} = \frac{3}{4}$ . With $z_{1} = 1 / 900$ one has $z_{1} < z_{0}$ , $A_{0} z_{1} < \frac{3}{4} \cdot \frac{1}{900} = \frac{1}{1200} < \frac{1}{2}$ and $A_{0} H z_{1} = 4 A_{0} z_{1} < \frac{1}{300} < \frac{1}{200} = ε / 2$ for $ε = 1 / 100$ , so $z_{1}$ satisfies the hypothesis of [L2],(c).

2.1step 1.1L2algebra

The whole-region bound values. Instance of [L2],(b) with $T = 1$ , $R = 2$ , $∣ v ∣ = 1$ : the bounds read $B_{t} (C) = (γ_{C} - 1) + 2 γ_{C} / C^{2}$ and $B_{x} (C) = 3 (γ_{C} - 1)$ , and $d_{t} (C) \leq B_{t} (C)$ , $d_{x} (C) \leq B_{x} (C)$ hold at every $C > 1$ by the lemma; the statement records the ten-place rounded values of the exact expressions $d_{t} (C), d_{x} (C), B_{t} (C), B_{x} (C)$ at $C = 10$ and $C = 100$ .

2.2step 1.2L2L4algebra

The sufficient threshold. By [L2],(c) applied with $z_{1} = 1 / 900$ , $∣ v ∣ = 1$ , $R = 2$ and $ε = 1 / 100$ , $C_{ε} = \max {1 / \sqrt{z_{1}}, \sqrt{3 ∣ v ∣ R / ε}} = \max {30, \sqrt{600}} = 30,$ because $\sqrt{600} < 30$ by [L4]; since $30 > 1 = ∣ v ∣$ , every $C \geq 30$ meets both error bounds of [L2],(b) on the whole region $D_{1, 2}$ .

3.1step 1.1step 2.1L3algebra

The evaluations and ratios at the named scales. For $C = 10$ one has $γ_{10} = 10 / \sqrt{99} = 10 / (3 \sqrt{11})$ , $d_{t} (10) = 1 - 98 / (10 \sqrt{99})$ , $d_{x} (10) = γ_{10} - 1$ ; for $C = 100$ one has $γ_{100} = 100 / \sqrt{9999}$ and the corresponding expressions; for $C = 30$ one has $γ_{30} = 30 / \sqrt{899}$ , $t^{'} = 898 / (30 \sqrt{899}) = 449 / (15 \sqrt{899})$ and the bound value $B_{t} (30) = 451 / (15 \sqrt{899}) - 1$ . The two ratios of exact deviations are $\frac{d_{t} (10)}{d_{t} (100)} = \frac{(10 \sqrt{99} - 98) \cdot 10 \sqrt{9999}}{\sqrt{99} (100 \sqrt{9999} - 9998)}, \frac{d_{x} (10)}{d_{x} (100)} = \frac{γ_{10} - 1}{γ_{100} - 1},$ evaluated from these exact radicals before rounding; the statement records the ten-place rounded values of all these quantities, and each bound comparison at the named scales is the instance of [L2],(b) already recorded in step 2.1.

4.1step 2.2step 3.1L2given∎

Conclusion of the worked instance. The exact differences, the uniform bounds at $C = 10, 30, 100$ , the ten-place rounded evaluations, the two ratios of exact deviations and the $ε = 1 / 100$ threshold $C_{ε} = 30$ have been checked above from the exact radicals, and the constants $K_{0} = 4 \sqrt{3} / 9$ , $A_{0} = 1 / 2 + \sqrt{3} / 9 < 3 / 4$ and the inequalities $z_{1} < z_{0}$ , $A_{0} z_{1} < 1 / 1200$ , $A_{0} H z_{1} < 1 / 300 < ε / 2$ hold with exact values; the auxiliary family $Φ_{v, C}$ is mathematical and no physical claim is made.

Worked uniform Galilean chart bound on a bounded region

Example

Facts & Assumptions

Verification

Depends on

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