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Questions for "A New Resolution for the Twin Paradox"

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Abstract and Figures

This is a response to Mike Fontenot's web page https://sites.google.com/site/cadoequation/cado-reference-frame This web-page has been under discussion at http://sciforums.com/threads/a-new-simultaneity-method-for-accelerated-observers-in-special-relativity.162684/page-11 and http://www.sciforums.com/threads/minguizzis-simultaneity-method.162805/page-3 I will post a link to this paper in those forums.
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Questions for “A New Resolution”
Jonathan Doolin
March 19, 2020
1 Intro
This paper refers to Mike Fontenot’s “Twin Paradox: A New Resolution”. In it, I
first confirm the calculations of coordinates he has calculated, then I question the final
calculations in Section 8.
2 Calculation of Points for Figure
Figure 1 was created with instruction from Mike Fontenot’s web page currently at
https://sites.google.com/site/cadoequation/cado-reference-frame.
While Mike includes some detailed calculations of the coordinates of points in his
paper, I have used a more general formula that requires the input of two points and two
slopes. This method is recommended if you have access to any sort of software that can
do repetitive calculations, such as a graphing calculator, or a spreadsheet program.
First, given an equation
(xx1) = m1(tt1)
(xx2) = m2(tt2)
We can re-write these with all variables on the left, and all constants on the right as
m1t+x=m1t1+x1
m2t+x=m2t2+x2
This can be converted into a matrix multiplication
m11
m21t
x=x1m1t1
x2m2t2
To solve, left-multiply both sides by the inverse matrix:
t
x=1
m2m111
m2m1x1m1t1
x2m2t2
Using this formula I have calculated the following points:
1
Figure 1: A figure drawn with instruction from Mike Fontenot
TA=(40, 23.094); the Turnaround event occurs at the intersection of t=40 and x=v t,
where v=tan(30)=0.577
tau3=(40, 0) Event “She” considers to be simultaneous with Turnaround
tau1=(16.906, 0.); “Her” event observed from “His” position when he Turns Around.
Lines intersecting here: (TA slope 1; the speed of light) and (E0 Slope 0) “Her” world-
line.
tau2=(26.6667, 0.); “Her” event calculated to be simultaneous with the TurnAround
event according to the OCMIO (Outgoing Co Moving Inertial Observer). Lines inter-
secting here: (TA slope 1/v); the line of simultaneity from the turn-around event for the
OCMIO, and (E0 Slope 0) “Her” world-line.
M=(53.3333, 0.) M was not in Mike Fontenot’s list, but it is similar to tau2. “Her”
event calculated to be simultaneous with the TurnAround event according to the ICMIO
(Incoming CoMoving Inertial Observer). Lines intersecting here (TA slope -1/v); the
line of simultaneity from the turnaround event for the ICMIO, and (E0 Slope 0) “Her”
world-line.
R=(40, 11.547) This is the point halfway between T3 and TA. An event that “She”
sees to be half the distance to “Him” when he makes his turnaround.
Q=(47.3205, 18.8675) The event where R is observed by “Him” and the ICMIO. Lines
intersecting here: (TA slope -v), and (R slope 1)
tau7=(28.453, 0.) “Her” event observed by “Him” at Q. Lines intersecting here: (TA
slope -v), and (E0 slope 0)
u3=(42.6795, 24.641) (Not included in Mike’s description) Event simultaneous with
“Her” photon-release at T7 according to OCMIO. Lines intersecting: (T7 slope 1/v),
and (TA slope v).
tau8=(33.3333, 0.) “Her” event simultaneous with R according to OCMIO. Lines
intersecting here (R slope 1/v) and (E0 slope 0)
2
u2=(50., 28.8675) This event was not calculated by Mike, but it is the event where the
OCMIO is when tau8 occurs. Lines intersecting here: (R slope 1/v) and (TA slope v)
tau6=(47.3205, 0) Event “She” considers to be simultaneous with Q.
tau9=(58.2137, 0.) Event “He” and ICMIO considers to be simultaneous with Q.
tau10=(46.6667, 0.) Event ICMIO considers to be simultaneous with R. Lines inter-
secting (R slope 1/v), (E0 slope 0)
u1=(30., 28.8675) (Not calculated by Mike) Event that happens to ICMIO at the same
time that R and tau10 occur. Lines intersecting (R slope 1/v), (TA slope -v)
P=(54.641, 14.641) “His” observation of event T3. Lines intersecting (T3 slope 1),
(TA slope -v)
tau4=(54.641, 0) Event “She” measures as simultaneous with P
tau5=(63.094, 0.) Event “He” measures to be simultaneous with P. Lines intersecting
(P slope -1/v), (E0 slope 0)
gamma=1.22474
3 left half + right half = total?
After all this construction, Mike has add together (T9-T10) and (T8-T7) to find the
amount of time that “She” ages while the photon is under way between events T7 to Q.
He says “During the portion of the pulse’s trip that was in the lef half of the diagram,
she aged by”
T8-T7 = 33.33 - 28.45 = 4.88 years
I feel that this is missing a qualifier, it should say “According to the OCMIO, during
the portion where the pulse went from T7 to R, she aged by 4.88 years” However, it
should also be noted that the OCMIO is traversing the space between u3 and u2 during
this time, events which never happen to “Him” at all.
He then says the ICMIO says that she aged by
T9-T10 = 58.21 - 46.67 = 11.54
years during the transit of the pulse from point R to point Q.
Again what’s missing here is the fact that these events occurred as the ICMIO traversed
the space between event u1 and Q. The events from u1 to TA did not happen to “Him”
at all.
4 The Fundamental Problem Remains
Mike Fontenot’s CADO method, if I’m not mistaken would have simply said that as
“He” turned around, “She” simply aged from 26.67 years (event T2) to age 53.33 years
(event M). Ostensibly this is the issue that this new method is meant to solve. However,
now “Her” world-line is broken up into three segments, (T7-T8), (T8-T10), (T10-T9),
and “He” is adding together the times from T7 to T8, and the times from T10 to T9,
but still skipping the middle time.
3
5 Why not CADO?
I would conclude that Mike should go back to his Current Age of Distant Objects formula.
It should be noted that any time above that I used a slope of 1/v or -1/v, I was using
something equivalent to CADO. Using a slope of 1/v gave the line of simultaneity for
an object with velocity, v, and using a slope of -1/v gave the line of simultaneity for an
object with velocity -v. When that intersection ws formed with “Her” worldline, it was
also calculating her Current Age.
4
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