Twin Paradox Analysis in Absolute Terms

diagrams and corresponding equations

This short document exists to accompany my other online documents concerning special relativity in absolute terms. Links to the other documents are found at the bottom of this page. There you will also find a link to the free pdf file of the book Relativity Trail.

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.

What we should find is that A and B will regard each other's velocities, as well as their prime operatives (A' and B'), in the same manner in either case.

Assuming a relative velocity of .6 between A and B (as measured by each of them) for both the out and in trip, we'll present two possibilities for the round trip:

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.

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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.

Other documents which are recommended reading before reading the book:


Relativity in Absolute Terms. My most comprehensive online document. A concise overview of why special relativity must be diagrammed in absolute terms.


Symmetry of Measuring. Diagrams and equations demonstrating the symmetry of clock rate and length measures across inertial frames.


Twin Paradox Animation on youtube. Contains textual description not found in the document below.


Twin Paradox Animation. Alternative text, and animation of the twin paradox. (Embedded youtube animation.)


Free pdf file of the book:

Relativity Trail, free pdf format, with 192 pages, 65 diagrams and 75 illustrations, will provide you with complete detailed algebraic derivations of all the kinematical effects of special relativity. Everything is charted out in absolute terms against a system at rest with respect to the totality of the universe for perfect clarity as well as soundness of theoretical basis. It is the totality of the universe that imparts the inertial properties of clock rates and lengths which generate the effects of relativity. This is explained in detail in Relativity Trail.



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