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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
t
1
{\displaystyle \color {ForestGreen}{t_{1}}}
t
1
′
=
a
t
1
{\displaystyle \color {Blue}{t'_{1}=at_{1}}}
rate of A
rate of B
=
t
1
a
t
1
=
1
a
>
1
(3)
{\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)}}}
...
Thus, between events E0 and E2, B advances by
t
2
′
{\displaystyle \color {Brown}{t'_{2}}}
t
2
=
a
t
2
′
{\displaystyle \color {Red}{t_{2}=at'_{2}}}
rate of A
rate of B
=
a
t
2
′
t
2
′
=
a
<
1
(4)
{\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)}}}
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
{\displaystyle \color {ForestGreen}{t_{1}}}
present at both events, by
t
1
′
=
a
t
1
{\displaystyle \color {Blue}{t'_{1}=at_{1}}}
rate of clock not present at both events E0 and E1
rate of clock present at both events E0 and E1
=
coordinate time of E1
proper time of E1
=
rate of A
rate of B
=
t
1
t
1
′
=
t
1
a
t
1
=
1
a
>
1
(3)
{\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)}}}
...
Thus, between events E0 and E2, B, which is not present at both events, advances by
t
2
′
{\displaystyle \color {Brown}{t'_{2}}}
present at both events, by
t
2
=
a
t
2
′
{\displaystyle \color {Red}{t_{2}=at'_{2}}}
rate of clock not present at both events E0 and E2
rate of clock present at both events E0 and E2
=
coordinate time of E2
proper time of E2
=
rate of B
rate of A
=
t
2
′
t
2
=
t
2
′
a
t
2
′
=
1
a
>
1
(4)
{\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)}}}
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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and B by 
by (1). Therefore
and A by 
by (2). Therefore
