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Simplified Twin Paradox Resolution.
On Jan 7, 6:41 am, Vilas Tamhane wrote:
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 coexist is when B^2 = 0. Thus, the twins paradox is very real under the Lorentz transform. shrug Stuff is so simple that it is a big surprise when all the socalled bright minds in the scientific communities have so much trouble understanding. Stuff is simple and they dont have any trouble in understanding. But they cannot go against the politics of people belonging to one race and same people who happen to hold the strings of economy of great America. After all these years trying to talk sense out of the likes, Koobee Wublee would have to disagree with you. They are indeed fvcking morons who cannot understand basic algebra and basic concept of what science is all about. The strong language is indeed very appropriate. shrug Coming back to Paul, he will never consider the example you suggested, that, A and B both go away from C and come back to C. In this case, there is no way to show difference in time in the clocks of A and B, though, according to Einstein, each others clock was running slow and still their own were running at proper rate. This proposed reciprocity is exactly the foolishness that SR is based on. Koobee Wublee would tend to agree on this one. paul is so busy justifying why he has said the Doppler effect from GR needed to be corrected for the carrier frequencies of downlink signals. The recent excuse is that he did not meant what he said in the past. We will see what other excuses that he can come up with. In the meantime, try not to hold your breath for paul to modify his JAVA applet to allow for both twins to travel with identical acceleration profile and also an arbitrary time period for the twins to coast (with no acceleration). It would be interesting to see how paul copes with the mutual time dilation building up. If he had any brain at all, it would be like similar to Zed holding his hands to his forehead yelling no in the movie Zardoz after the Wizard of Oz was exposed to him. shrug http://en.wikipedia.org/wiki/Zardoz 
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