Twin Paradox Analysis in Absolute Terms
diagrams and corresponding equations
Cite the book, Relativity Trail
Luebeck, R. Relativity Trail. Mpls: L B Writ Publishing, (2008)
Cite this web page
See also: Relativity in Absolute Terms
It is a concise overview document.
Three pages from Relativity Trail: 106 R E L A T I V I T Y T R A I L ADDITION OF VELOCITIES AND THE CLOCK PARADOX
To check on the clock paradox more simply, let's use the situation as described on page 84, comparing the situation of clock A changing from a state of rest in the form of A' to catch B with the situation of a traveling B reversing motion in the form of B' to return to A.
In case 1, B passes by A, continues 1 light second, then reverses direction by virtue of transfering clock information to B'. tB = 1.333 at reversal. Final tB = 2.667. Final tA = 3.333. This means that in case 2, tA must equal 1.333 at the moment it transfers its clock reading to A', with A' then chasing down B. tB = 1.067 at that moment, since it has velocity of .6. Final tB must equal 3.333, meaning tB during the chase interval equals 2.267. This implies Ut during chase interval is 2.833. dB during chase interval thus equals 1.7 ls. We also note that tA = Ut = 1.333 at transfer implies B has traveled .8 ls. Thus dA = dB + .8 = 2.5. This implies actual vA' = .88235. 107 R E L A T I V I T Y T R A I L Since A was at rest when A' passed it, A measured vA' as .88235. We must now check to see what value was obtained by B for vB' as B' passed by B B makes timing of B' with clock towers .8 ls apart as established by the laying out of rods, so that B regards the towers to be 1 ls apart. We know actual vB' = .6, so we can read the familiar numbers off the diagram. CT2 meets B' at Ut = .167, CT2 time of .133. B adds 1 second for triggering of CT2 (initiated by CT1) to yield total time tB = 1.133. B thus calculates vB' = 1 ls / 1.113 s = .88235. We'll dispense with the trivial considerations of A and B taking stock of each other's velocity, as we've shown this elsewhere. 108 R E L A T I V I T Y T R A I L We do note however, that A' has contraction of .4706. Thus A' took stock of B's velocity in accordance with the diagram below: .6 t + .4706 = .88235 t , yielding t = 1.667. Since A' considers d = 1, A' uses d = vt to calculate velocity of B as .6 . In addition to demonstrating that the parties involved cannot determine their true motion status with respect to the universe, what we have just done on these pages is to verify the addition of velocities formula for relativity:
B + A'(B) B - B' A' = --------------- B(B') = ---------- 1 + B * [A'(B)] 1 - B * B'
where A' is velocity of A', A'(B) is the velocity of B according to A', B is velocity of B, B' is the velocity of B', and B(B') is the velocity of B' according to B. ---------------------------------------------------------------------
Download the free pdf file of the book Relativity Trail to see the diagrams and math for other phenomena, such as mutually measured clock rate contraction, mutually measured length contraction, mutually measured mass increase, the formal derivation of length contraction, and much more.