Proof of Special Relativity Length Contraction Assumption
It is generally acknowledged that the Special Relativity (SR) time dilation equation has been proven experimentally. However, it is also generally acknowledged that the SR length contraction assumption has never been proven experimentally or mathematically. There is now undisputable proof of the validity of the SR length contraction equation L=Lo*sqrt(1-v^2/c^2) where the at rest length of a path between two points is Lo and where L is the contracted length of the path when the two points are moving away from point ‘a’ at a velocity v according to a stationary observer at point ‘a’ - see example in Figure I at http://www.ken-More.com/uploads/Figure_I.doc. Finally, we can justifiably disregard all of the “cranks” who have been saying that “SR has not been proven”. Also, we will not need to throw away all of our text books on relativity as the “cranks” have been saying.
This proof starts with a premise that the length (L) of a path between two moving points ‘b’ and ‘d’ satisfies the equation L = (ETa * c) - (ETa * v); where L is the length Lo after contraction and
ETa is Elapsed Time at point ‘a’. This premise states that any “assumed” contracted length L = (ETa * c) - (ETa * v). If you agree that this premise is true when points ‘b’ and ‘d’ are moving away from point ‘a’, answer “Yes” for question (2) below. If you have determine that the premise is correct then answer “Yes” or “No” to questions (4), (6) & (8) that are based upon SR assumptions. If your answer is “Yes” to all four questions then the SR length contraction equation must be valid because of the proof shown in equations (9) through (18) below.
The questions below apply to point 'a', point 'b', and point 'd' on the x axis with 'b' and 'd' moving at velocity v on the x axis away from point 'a' as shown in Figure I. A stationary observer is at point ‘a’. Elapsed Time at ‘a’ = ETa = 0 at the beginning of a light flash sent from ‘b’ when ‘b’ was at ‘a’ to ‘d’. The flash is received at the destination‘d’ at Elapsed Time ETa = 1 sec in Figure I.
(1) L = (ETa * c) - (ETa * v); where L is the length of Lo after contraction.
(2) Is (1) true at the end of the send in Figure I according to the stationary observer?
(3) ETa = (ETbd' + OT1) * F; where F=1/sqrt(1-v^2/c^2) and OT = Lo*(v/c^2)
(4) Is (3) true at the end of the send in Figure I according to the stationary observer?
(5) ETbd' = Lo/c for the traveler from the perspective of the traveler’s clock.
(6) Is (5) always true at the end of a send transmission for the traveler at point 'b' ?
(7) From (1): ETa = L / (c - v)
(8) Is (7) true at the end of the send in Figure I according to the stationary observer?
(9) From (3) and (7): (ETbd'+OT) * F = L / (c - v)
(10) From (9): L = ((ETbd'+OT) * F) * (c - v)
(11) From (10) and (5): L = (((Lo/c)+OT)*F) * (c - v)
(12) From SR : L = Lo*sqrt(1 – v^2/c^2)
(13) Therefore: Lo*sqrt(1 – v^2/c^2) = (((Lo/c)+OT)*F)*(c - v)
(14) From (13): Lo*sqrt(1 – v^2/c^2) = (((Lo/c)+OT)*(c-v))/(sqrt(1 – v^2/c^2)))
(15) From (14): Lo*(1 – v^2/c^2) = (((Lo/c)+OT)*(c-v)) = (((Lo/c)+(Lo*(v/c^2))*(c-v)
(16) From (15): (1 – (v^2/c^2)) = ((1/c) + (v/c^2))*(c-v) = ((1/c) + (v/c^2))*c*(1-(v/c))
(17) From (16): (1 + (v/c))*(1-(v/c)) = ((1/c) + (v/c^2))*c*(1-(v/c))
(18) From (17): 1+(v/c) = ((1/c) + (v/c^2))*c = 1+(v/c)
The above proof demonstrates that the length L as computed by equation (11) above is the same as the length L as computed by the SR equation (12) above but only when ETa = (ETbd'+OT)*F. The concept of simultaneity explains why the distance L that is computed by equation (11) will be the same as that computed by equation (12) when ‘b’ and ‘d’ are moving away from point ‘a’ as well as why the stationary observer “thinks” that OT must be added to ETbd’. Therefore, the time dilation equation ETa = ETbd’*F is correct even though we must “think” that ETa = (ETbd'+OT)*F in order to prove that the SR length contraction assumption is correct.
Footnote 1. Out-of-sync Time (OT) is added to ETbd’ because the stationary observer must “think” that the time at the destination point ‘d’ is out of sync with the time at points ‘b’ and ‘a’ according to SR theory (see Synchronizing Clocks )
This proof starts with a premise that the length (L) of a path between two moving points ‘b’ and ‘d’ satisfies the equation L = (ETa * c) - (ETa * v); where L is the length Lo after contraction and
ETa is Elapsed Time at point ‘a’. This premise states that any “assumed” contracted length L = (ETa * c) - (ETa * v). If you agree that this premise is true when points ‘b’ and ‘d’ are moving away from point ‘a’, answer “Yes” for question (2) below. If you have determine that the premise is correct then answer “Yes” or “No” to questions (4), (6) & (8) that are based upon SR assumptions. If your answer is “Yes” to all four questions then the SR length contraction equation must be valid because of the proof shown in equations (9) through (18) below.
The questions below apply to point 'a', point 'b', and point 'd' on the x axis with 'b' and 'd' moving at velocity v on the x axis away from point 'a' as shown in Figure I. A stationary observer is at point ‘a’. Elapsed Time at ‘a’ = ETa = 0 at the beginning of a light flash sent from ‘b’ when ‘b’ was at ‘a’ to ‘d’. The flash is received at the destination‘d’ at Elapsed Time ETa = 1 sec in Figure I.
(1) L = (ETa * c) - (ETa * v); where L is the length of Lo after contraction.
(2) Is (1) true at the end of the send in Figure I according to the stationary observer?
(3) ETa = (ETbd' + OT1) * F; where F=1/sqrt(1-v^2/c^2) and OT = Lo*(v/c^2)
(4) Is (3) true at the end of the send in Figure I according to the stationary observer?
(5) ETbd' = Lo/c for the traveler from the perspective of the traveler’s clock.
(6) Is (5) always true at the end of a send transmission for the traveler at point 'b' ?
(7) From (1): ETa = L / (c - v)
(8) Is (7) true at the end of the send in Figure I according to the stationary observer?
(9) From (3) and (7): (ETbd'+OT) * F = L / (c - v)
(10) From (9): L = ((ETbd'+OT) * F) * (c - v)
(11) From (10) and (5): L = (((Lo/c)+OT)*F) * (c - v)
(12) From SR : L = Lo*sqrt(1 – v^2/c^2)
(13) Therefore: Lo*sqrt(1 – v^2/c^2) = (((Lo/c)+OT)*F)*(c - v)
(14) From (13): Lo*sqrt(1 – v^2/c^2) = (((Lo/c)+OT)*(c-v))/(sqrt(1 – v^2/c^2)))
(15) From (14): Lo*(1 – v^2/c^2) = (((Lo/c)+OT)*(c-v)) = (((Lo/c)+(Lo*(v/c^2))*(c-v)
(16) From (15): (1 – (v^2/c^2)) = ((1/c) + (v/c^2))*(c-v) = ((1/c) + (v/c^2))*c*(1-(v/c))
(17) From (16): (1 + (v/c))*(1-(v/c)) = ((1/c) + (v/c^2))*c*(1-(v/c))
(18) From (17): 1+(v/c) = ((1/c) + (v/c^2))*c = 1+(v/c)
The above proof demonstrates that the length L as computed by equation (11) above is the same as the length L as computed by the SR equation (12) above but only when ETa = (ETbd'+OT)*F. The concept of simultaneity explains why the distance L that is computed by equation (11) will be the same as that computed by equation (12) when ‘b’ and ‘d’ are moving away from point ‘a’ as well as why the stationary observer “thinks” that OT must be added to ETbd’. Therefore, the time dilation equation ETa = ETbd’*F is correct even though we must “think” that ETa = (ETbd'+OT)*F in order to prove that the SR length contraction assumption is correct.
Footnote 1. Out-of-sync Time (OT) is added to ETbd’ because the stationary observer must “think” that the time at the destination point ‘d’ is out of sync with the time at points ‘b’ and ‘a’ according to SR theory (see Synchronizing Clocks )
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Figure I can be viewed at the following link:
http://www.ken-more.com/uploads/Figure_I.docPosted Feb14-09 at 02:16 PM by Ken More 
