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

(start quote)

Thus, between events E0 and E1, A advances by

t_{1}

and B by

t'_{1} = a t_{1}

by (1). Therefore

\frac{rate of A}{rate of B} = \frac{t_{1}}{a t_{1}} = \frac{1}{a} > 1 (3)

...

Thus, between events E0 and E2, B advances by

t'_{2}

and A by

t_{2} = a t'_{2}

by (2). Therefore

\frac{rate of A}{rate of B} = \frac{a t'_{2}}{t'_{2}} = a < 1 (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

t_{1}

and B, which is present at both events, by

t'_{1} = a t_{1}

by (1). Therefore

\frac{rate of clock not present at both events E0 and E1}{rate of clock present at both events E0 and E1} = \frac{coordinate time of E1}{proper time of E1} = \frac{rate of A}{rate of B} = \frac{t_{1}}{t'_{1}} = \frac{t_{1}}{a t_{1}} = \frac{1}{a} > 1 (3)

...

Thus, between events E0 and E2, B, which is not present at both events, advances by

t'_{2}

and A, which is present at both events, by

t_{2} = a t'_{2}

by (2). Therefore

\frac{rate of clock not present at both events E0 and E2}{rate of clock present at both events E0 and E2} = \frac{coordinate time of E2}{proper time of E2} = \frac{rate of B}{rate of A} = \frac{t'_{2}}{t_{2}} = \frac{t'_{2}}{a t'_{2}} = \frac{1}{a} > 1 (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)