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## A trivial refutation of one of Dingle's Fumbles (Ref: Talk:Herbert Dingle Archive)

On page 230 in this appendix to "Science At the Crossroads", Dingle writes:

(start quote)
Thus, between events E0 and E1, A advances by $\color{ForestGreen}{t_1}$ and B by $\color{Blue}{t'_1 = a t_1}$ by (1). Therefore
$\frac{\color{ForestGreen}{\text{rate of A}}}{\color{Blue}{\text{rate of B}}} = \frac{\color{ForestGreen}{t_1}}{\color{Blue}{a t_1}} = \frac{1}{a} > 1 \qquad \text{(3)}$
...
Thus, between events E0 and E2, B advances by $\color{Brown}{t'_2}$ and A by $\color{Red}{t_2 = a t'_2}$ by (2). Therefore
$\frac{\color{Red}{\text{rate of A}}}{\color{Brown}{\text{rate of B}}} = \frac{\color{Red}{a t'_2}}{\color{Brown}{t'_2}} = a < 1 \qquad \text{(4)}$
Equations (3) and (4) are contradictory: hence the theory requiring them must be false.
(end quote)

Dingle should have written as follows:

(start correction)
Thus, between events E0 and E1, A, which is not present at both events, advances by $\color{ForestGreen}{t_1}$ and B, which is present at both events, by $\color{Blue}{t'_1 = a t_1}$ by (1). Therefore
$\frac{\color{ForestGreen}{\text{rate of clock not present at both events E0 and E1}}}{\color{Blue}{\text{rate of clock present at both events E0 and E1}}} = \frac{\color{ForestGreen}{\text{coordinate time of E1}}}{\color{Blue}{\text{proper time of E1}}} = \frac{\color{ForestGreen}{\text{rate of A}}}{\color{Blue}{\text{rate of B}}} = \frac{\color{ForestGreen}{t_1}}{\color{Blue}{t'_1}} = \frac{\color{ForestGreen}{t_1}}{\color{Blue}{a t_1}} = \frac{1}{a} > 1 \qquad \text{(3)}$
...
Thus, between events E0 and E2, B, which is not present at both events, advances by $\color{Brown}{t'_2}$ and A, which is present at both events, by $\color{Red}{t_2 = a t'_2}$ by (2). Therefore
$\frac{\color{Brown}{\text{rate of clock not present at both events E0 and E2}}}{\color{Red}{\text{rate of clock present at both events E0 and E2}}} = \frac{\color{Brown}{\text{coordinate time of E2}}}{\color{Red}{\text{proper time of E2}}} = \frac{\color{Brown}{\text{rate of B}}}{\color{Red}{\text{rate of A}}} = \frac{\color{Brown}{t'_2}}{\color{Red}{t_2}} = \frac{\color{Brown}{t'_2}}{\color{Red}{a t'_2}} = \frac{1}{a} > 1 \qquad \text{(4)}$
Equations (3) and (4) are consistent and say that any event's coordinate time is always larger than its proper time:

hence there is no reason to say that the theory requiring them must be false.

(end correction)

DVdm 12:18, 6 August 2007 (UTC)