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, lets 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.
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