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Old January 7th 13, 03:41 PM posted to sci.physics.relativity,sci.physics,sci.math,sci.astro
Vilas Tamhane
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Default Simplified Twin Paradox Resolution.

On Jan 7, 11:28*am, Koobee Wublee wrote:
On Jan 6, 5:47 am, "Paul B. Andersen" wrote:









On 6/01/2013 3:59 PM, Koobee Wublee wrote:
Instead of v, let’s say (B = v / c) for simplicity. *The earth is
Point #0, outbound spacecraft is Point #1, and inbound spacecraft is
Point #2.


According to the Lorentz transform, relative speeds a


** *B_00^2 = 0, speed of #0 as observed by #0
** *B_01^2 = B^2, speed of #1 as observed by #0
** *B_02^2 = B^2, speed of #2 as observed by #0


** *B_10^2 = B^2, speed of #0 as observed by #1
** *B_11^2 = 0, speed of #1 as observed by #1
** *B_12^2 = 4 B^2 / (1 – B^2), speed of #2 as observed by #1


** *B_20^2 = B^2, speed of #0 as observed by #2
** *B_21^2 = 4 B^2 / (1 – B^2), speed of #1 as observed by #2
** *B_22^2 = 0, speed of #2 as observed by #2


When Point #0 is observed by all, the Minkowski spacetime (divided by
c^2) is:


** *dt_00^2 (1 – B_00^2) = dt_10^2 (1 – B_10^2) = dt_20^2 (1 – B_20^2)


When Point #1 is observed by all, the Minkowski spacetime (divided by
c^2) is:


** *dt_01^2 (1 – B_01^2) = dt_11^2 (1 – B_11^2) = dt_21^2 (1 – B_21^2)


When Point #2 is observed by all, the Minkowski spacetime (divided by
c^2) is:


** *dt_02^2 (1 – B_02^2) = dt_12^2 (1 – B_12^2) = dt_22^2 (1 – B_22^2)


Where


** *dt_00 = Local rate of time flow at Point #0
** *dt_01 = Rate of time flow at #1 as observed by #0
** *dt_02 = Rate of time flow at #2 as observed by #0


** *dt_10 = Rate of time flow at #0 as observed by #1
** *dt_11 = Local rate of time flow at Point #1
** *dt_12 = Rate of time flow at #2 as observed by #1


** *dt_20 = Rate of time flow at #0 as observed by #2
** *dt_21 = Rate of time flow at #1 as observed by #2
** *dt_22 = Local rate of time flow at Point #2


So, with all the pertinent variables identified, the contradiction of
the twins’ paradox is glaring right at anyone with a thinking brain..
shrug


- - -


*From the Lorentz transformations, you can write down the following
equation per Minkowski spacetime. *Points #1, #2, and #3 are
observers. *They are observing the same target.


** *c^2 dt1^2 – ds1^2 = c^2 dt2^2 – ds2^2 = c^2 dt3^2 – ds3^2


Where


** *dt1 = Time flow at Point #1
** *dt2 = Time flow at Point #2
** *dt3 = Time flow at Point #3


** *ds1 = Observed target displacement segment by #1
** *ds2 = Observed target displacement segment by #2
** *ds3 = Observed target displacement segment by #3


The above spacetime equation can also be written as follows.


** *dt1^2 (1 – B1^2) = dt2^2 (1 – B2^2) = dt3^2 (1 – B3^2)


Where


** *B^2 = (ds/dt)^2 / c^2


When #1 is observing #2, the following equation can be deduced from
the equation above.


** *dt1^2 (1 – B1^2) = dt2^2 . . . (1)


Where


** *B2^2 = 0, #2 is observing itself


Similarly, when #2 is observing #1, the following equation can be
deduced.


** *dt1^2 = dt2^2 (1 – B2^2) . . . (2)


Where


** *B1^2 = 0, #1 is observing itself


According to relativity, the following must be true.


** *B1^2 = B2^2


Thus, equations (1) and (2) become the following equations
respectively.


** *dt1^2 (1 – B^2) = dt2^2 . . . (3)
** *dt2^2 = dt1^2 (1 – B^2) . . . (4)


Where


** *B^2 = B1^2 = B2^2


The only time the equations (3) and (4) can co-exist is when B^2 = 0.