## 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

It is a concise overview document.

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

```

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.

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.