Relativity Lite

ATrip to Alpha Centauri | 37 in flight ( yellow , green , cyan , and blue ), yet to be received, and a fifth is about to be emit- ted. Those extra three ( green , cyan , and blue ), almost four, in-flight pulses each mark the passage of about 0.728 years of Earth time in 3 seconds of saucer time. We should thus expect the Earth to have experienced almost 2.9 years more than the saucer during those 3 seconds. The saucer must be shifting from one reference frame to another reference frame for this to be possible. Note that exactly two pulses ( red and then orange ) have already been received in both the outgoing and incoming saucer frames: the saucer cannot go back and change its own reality by such frame shifting under acceleration. The s ymmetry o f t he t rip a llows u s to i nfer t hat t he i nitial i nstantaneous j ump f rom stillness to motion at the start of the saucer’s outbound trip added to the instantaneous jump from motion to stillness at the end of the saucer’s return trip would speed up the Earth’s clocks by another 2.9 years as seen by the saucer, for a total acceleration-induced ex- tension of Earth’s clocks of 5.8 years . But time dilation in the unaccelerated portion of the trip slows the Earth’s clocks down, as perceived by the saucer, precisely negating this extra 2.9 years. We should thus have Earth clocks reading 2.9 + 2.9 − 2.9 = 2.9 more years than the saucer, and we have seen that they do. At 6.552 years , the first (count 1) of Earth-pulses is received by the saucer on the return trip. The others come at 7.28 , 8.008 , 8.736 , 9.464 , 10.192 , and 10.92 (counting 2 , 3 , . . . and 7), and at 11.64 years , the planets come to rest and need not send the eighth pulse. The twin on the traveling Earth is back. We will not display the corresponding figures, as not that much changes between them, but you should play the full movie of the trip, available at https://doi.org/10.15760/pdxopen-2 9 . The s aucer s ees t wo pulses f rom E arth on t he outbound t rip, a t 2 .912 y ears b etween pulses, for a subtotal of 5.824 years . The s aucer s ees e ight p ulses o n t he i nbound t rip, at 0.7284 years between pulses, for a subtotal of 5.824 years . Thus, t he t otal t rip t ook 2 × 5.824 year = 11.648 years , the duration we expected from time dilation. The difference between the Earth and saucer clocks is 14.56 − 11.648 = 2.912 years , just about the factor we predicted from the new Earth pulses that appeared as the saucer shifted through a series of accelerated frames of reference as it changed directions at the Alpha Centauri end of the trip (since the Earth-end acceleration and deceleration time effect a nd t he t ime-d ilation effect while coasting essentially nullify each other). Note that we are required to have the saucer see the Earth’s proper time dilated as Earth moves relative to the saucer at rest (figure 13) . * Without t he E arth’s c locks s lowed down when seen from the saucer, the forward skipping of Earth time as the saucer shifts from one accelerated reference frame to another accelerated reference frame would have produced too much Earth time having passed. Mutual time dilation is not a paradox of relativity but a requirement . * Had the saucer not seen the Earth’s proper time dilated, the distance the Earth would move in the saucer frame during the τ = 1.456 yr until the first pulse would be sent would be 0.6 c × 1.456 yr = 0.874 c yr . The return trip for the light would be 0.874 years . Add τ = 1.456 yr for the Earth’s outbound proper time and you get a return time for the pulse of 2.33 yr after the Earth left. This is the wrong frequency. It is 4/5 (1/1.25) of the correct result of 2.912 years !