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1
A trivial refutation of one of Dingle's Fumbles (Ref: Talk:Herbert Dingle Archive)
User:DVdm
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On page 230 in this appendixto"Science At the Crossroads", Dingle writes:
-
(start quote)
-
-
-
Thus, between events E0 and E1, A advances by
and B by
by (1). Therefore
-
![{\displaystyle {\frac {\color {ForestGreen}{\text{rate of A}}}{\color {Blue}{\text{rate of B}}}}={\frac {\color {ForestGreen}{t_{1}}}{\color {Blue}{at_{1}}}}={\frac {1}{a}}>1\qquad {\text{(3)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbe8599cf2cae6557cdc9fcecf6ee2238171138a)
-
...
-
Thus, between events E0 and E2, B advances by
and A by
by (2). Therefore
-
![{\displaystyle {\frac {\color {Red}{\text{rate of A}}}{\color {Brown}{\text{rate of B}}}}={\frac {\color {Red}{at'_{2}}}{\color {Brown}{t'_{2}}}}=a<1\qquad {\text{(4)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2dc47144dc1426b1cbe3be500c1241c272aaae6)
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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
and B, which is present at both events, by
by (1). Therefore
-
![{\displaystyle {\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}{at_{1}}}}={\frac {1}{a}}>1\qquad {\text{(3)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dd83f0f3b37dc46402c53d42de650477de5e2c8)
-
...
-
Thus, between events E0 and E2, B, which is not present at both events, advances by
and A, which is present at both events, by
by (2). Therefore
-
![{\displaystyle {\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}{at'_{2}}}}={\frac {1}{a}}>1\qquad {\text{(4)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5cc7ffa5b797fb272dffc3458ab1aa40a5821a9)
-
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)
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