Simplified Twin Paradox Resolution.

On Jan 7, 6:41 am, Vilas Tamhane wrote:
Koobee Wublee wrote:

Instead of v, let’s say (B = v / c) for simplicity. The earth is
Point #0, outbound spacecraft is Point #1, and inbound spacecraft is
Point #2.

According to the Lorentz transform, relative speeds a

** B_00^2 = 0, speed of #0 as observed by #0
** B_01^2 = B^2, speed of #1 as observed by #0
** B_02^2 = B^2, speed of #2 as observed by #0

** B_10^2 = B^2, speed of #0 as observed by #1
** B_11^2 = 0, speed of #1 as observed by #1
** B_12^2 = 4 B^2 / (1 – B^2), speed of #2 as observed by #1

** B_20^2 = B^2, speed of #0 as observed by #2
** B_21^2 = 4 B^2 / (1 – B^2), speed of #1 as observed by #2
** B_22^2 = 0, speed of #2 as observed by #2

When Point #0 is observed by all, the Minkowski spacetime (divided by
c^2) is:

** dt_00^2 (1 – B_00^2) = dt_10^2 (1 – B_10^2) = dt_20^2 (1 – B_20^2)

When Point #1 is observed by all, the Minkowski spacetime (divided by
c^2) is:

** dt_01^2 (1 – B_01^2) = dt_11^2 (1 – B_11^2) = dt_21^2 (1 – B_21^2)

When Point #2 is observed by all, the Minkowski spacetime (divided by
c^2) is:

** dt_02^2 (1 – B_02^2) = dt_12^2 (1 – B_12^2) = dt_22^2 (1 – B_22^2)

Where

** dt_00 = Local rate of time flow at Point #0
** dt_01 = Rate of time flow at #1 as observed by #0
** dt_02 = Rate of time flow at #2 as observed by #0

** dt_10 = Rate of time flow at #0 as observed by #1
** dt_11 = Local rate of time flow at Point #1
** dt_12 = Rate of time flow at #2 as observed by #1

** dt_20 = Rate of time flow at #0 as observed by #2
** dt_21 = Rate of time flow at #1 as observed by #2
** dt_22 = Local rate of time flow at Point #2

So, with all the pertinent variables identified, the contradiction of
the twins’ paradox is glaring right at anyone with a thinking brain.
shrug

- - -

From the Lorentz transformations, you can write down the following
equation per Minkowski spacetime. Points #1, #2, and #3 are
observers. They are observing the same target.

** c^2 dt1^2 – ds1^2 = c^2 dt2^2 – ds2^2 = c^2 dt3^2 – ds3^2

Where

** dt1 = Time flow at Point #1
** dt2 = Time flow at Point #2
** dt3 = Time flow at Point #3

** ds1 = Observed target displacement segment by #1
** ds2 = Observed target displacement segment by #2
** ds3 = Observed target displacement segment by #3

The above spacetime equation can also be written as follows.

** dt1^2 (1 – B1^2) = dt2^2 (1 – B2^2) = dt3^2 (1 – B3^2)

Where

** B^2 = (ds/dt)^2 / c^2

When #1 is observing #2, the following equation can be deduced from
the equation above.

** dt1^2 (1 – B1^2) = dt2^2 . . . (1)

Where

** B2^2 = 0, #2 is observing itself

Similarly, when #2 is observing #1, the following equation can be
deduced.

** dt1^2 = dt2^2 (1 – B2^2) . . . (2)

Where

** B1^2 = 0, #1 is observing itself

According to relativity, the following must be true.

** B1^2 = B2^2

Thus, equations (1) and (2) become the following equations
respectively.

** dt1^2 (1 – B^2) = dt2^2 . . . (3)
** dt2^2 = dt1^2 (1 – B^2) . . . (4)

Where

** B^2 = B1^2 = B2^2

The only time the equations (3) and (4) can co-exist is when B^2 = 0.
Thus, the twins’ paradox is very real under the Lorentz transform.
shrug

Stuff is so simple that it is a big surprise when all
the so-called bright minds in the scientific communities
have so much trouble understanding.

Stuff is simple and they don’t have any trouble in understanding. But
they cannot go against the politics of people belonging to one race
and same people who happen to hold the strings of economy of great
America.

After all these years trying to talk sense out of the likes, Koobee
Wublee would have to disagree with you. They are indeed fvcking
morons who cannot understand basic algebra and basic concept of what
science is all about. The strong language is indeed very
appropriate. shrug

Coming back to Paul, he will never consider the example you suggested,
that, A and B both go away from C and come back to C. In this case,
there is no way to show difference in time in the clocks of A and B,
though, according to Einstein, each other’s clock was running slow and
still their own were running at proper rate. This proposed reciprocity
is exactly the foolishness that SR is based on.

Koobee Wublee would tend to agree on this one. paul is so busy
justifying why he has said the Doppler effect from GR needed to be
corrected for the carrier frequencies of downlink signals. The recent
excuse is that he did not meant what he said in the past. We will see
what other excuses that he can come up with. In the meantime, try not
to hold your breath for paul to modify his JAVA applet to allow for
both twins to travel with identical acceleration profile and also an
arbitrary time period for the twins to coast (with no acceleration).
It would be interesting to see how paul copes with the mutual time
dilation building up. If he had any brain at all, it would be like
similar to Zed holding his hands to his forehead yelling “no” in the
movie “Zardoz” after “the Wizard of Oz” was exposed to him. shrug

http://en.wikipedia.org/wiki/Zardoz