Written by:
BurtJordaan
Tuesday, June 17, 2008
This "perennial" is almost as old
as Einstein's special theory of relativity itself. Einstein did not invent the
paradox, although it was stimulated by his 1905 paper, where he spoke about the
fact that two clocks that were separated, one staying inertial and the other one
being moved away from the first one and then brought back again, will not read
the same time.
The popular press explained it in terms of twins of closely the same age, where
one twin sets off on a long, fast journey and eventually returns home. Special
relativity then predicts that the "away twin" will be younger than
the "at home" twin.
The "paradox" arises out of an abuse of special relativity's freedom
to choose any inertial frame as the reference frame and make all calculations
relative to it. The "abuse" on it's part arises out of choosing a
non-inertial frame (the away twin) as reference and then making wrong
conclusions---like that the home twin can just as well be considered as in
motion relative to the away twin. This then means that the home twin should
therefore suffer the same amount of time dilation as has been calculated for
the away twin and could thus be considered to end up the younger one (or at
least both twins still being the same age).
To confuse the issue even further, many explanations of the difference between
the two reference frames are very confusing and unconvincing---even some given
in reputable technical books. Search the web for "Too Many Explanations: a
Meta-Objection" (the author found it on "http://math.ucr.edu")
and see for yourself.
None of the one's discussed there is fully convincing either---which may be
just another "meta-objection" (what does "fully convincing"
mean anyway?) In my eBook "Relativity 4 Engineers" I give three
reasonable explanations: (i) a simple "hand waving" argument; (ii) a
very relativistic calculation and (iii) a more engineering-like representation
and calculation. I will concentrate on the "engineering" solution
here, using electromagnetic signals and relativistic Doppler shift.
Pam and Jim are twins that decided to put Einstein to the test. On January 1st,
2007 Pam quickly accelerated her spacecraft to a speed of 0.6c and flew away
inertially for 4 years, when she will quickly turn around and head back to
Earth at 0.6c again. Jim stays at home and the twins agreed to send each other
a New Years greeting on every January 1st until Pam returns safely.
This scenario is illustrated in the Minkowski spacetime diagram, Figure 1, a
split image for clarity. This shows a crucial non-symmetry in the signals that
the home twin and the away twin receive. Although the amount of stretch and
shrinkage of the received periods are the same, the amount of time that the
signals are stretched and shrunk is very different between the respective
twins.
Figure 1
The Doppler shift ratio (period of received signal (Tr) to period of
transmitted signal (T) for the outbound leg (opening velocity) is
Tr/T = sqrt[(1+0.6)/(1-0.6)] = 2.0
and the same ratio for the inbound leg (closing velocity) is
Tr/T = sqrt[(1-0.6)/(1+0.6)] = 0.5,
as can be clearly seen in figure 1. The non-symmetry comes from the fact that
the away twin receives compressed period signals immediately after turnaround,
while the home twin has to wait until the first signal after turnaround arrives
at home before noticing the event.
In the four years that Pam heads away from home, she will receive only two New
Year's messages from Jim. This is because Jim here represents the T period of 1
year. Pam is the receiver, with the Tr period of 2 years. On her calendar she
will receive "happy New Year 2008" only on Jan 1, 2009, and the next
one ("happy New Year 2009") on Jan 1, 2011. Weird, but this is due to
the increasing distance between them and the time light takes to cross the gap.
During her return trip, the situation is reversed, so in the last four
years, she will receive eight New Year's messages from Jim, one every six
months! She will receive the last (tenth) message, on New Year's Day 2015 on
her calendar, as she makes a close fly-by of Earth.
Does this solve the twin paradox? Not quite, yet.
That was Pam receiving Jim's messages. At what rate will Jim receive New
Year's messages from his sister? For the first eight years, he will also have
to wait two years for every 'happy New Year' message. This means that it will
be 2015 on Earth before Jim gets the message that his sister has sent on New
Year's Day 2011, with a note that she has just turned around for the home leg
of her trip.
Then, for the last two years, Jim will receive a New Year's message every
six months - four of them. Add them up and Jim will receive only eight messages
from Pam in the decade that he waited for her return. Conclusion: Pam recorded
only eight years during her voyage, while Jim recorded ten years.
I hope this clears up at least some of the confusion that surrounds the
"twin paradox".
Burt Jordaan (http://www.relativity-4-engineers.com)
Tags:
7 comments so far...
Re: The "Twin Paradox"
I'm sorry Burt, but this is about as "clear as mud". I consider myself to be an enlightened layman in the field of relativity, but I don't follow your reasoning. (I do accept the end result, however.) Could you perhaps pick one or more transmissions, both from Jim and from Pam, and calculate when they would be received in each other's "time". I was trying to visualize (say) Jim's 1st transmission. It happened on 1/1/08. Pam was 0.6 light years away at the time. Could you illustrate what her calendar said? I'm thinking that when Pam receives Jim's first signal, she is 0.6 + (1-0.6) = 1 light year away from Jim. I don't know the time dilation equation, but I'd assume Pam's calendar reads 1/1/10 when she gets the 1st signal. OK, I may be alright and may be able to work out the remaining details. I still don't understand this assymetry, however.
By DougJ on
Wednesday, June 25, 2008
|
Re2: The "Twin Paradox"
Hi Doug, you wrote:
"I was trying to visualize (say) Jim's 1st transmission. It happened on 1/1/08. Pam was 0.6 light years away at the time. Could you illustrate what her calendar said? I'm thinking that when Pam receives Jim's first signal, she is 0.6 + (1-0.6) = 1 light year away from Jim."
The relativistic Doppler ratio solves this scenario without any pain, but if you want to go through it the hard way, just to confirm the Doppler way, here it is:
If Jim could somehow read Pam's calendar when he sent that 1/1/08 signal, he would have seen 10/20/07, being 1/1/07 + 0.8 x 365 days. The 0.8 is Pam's time dilation relative to Jim. You are right that at that time, Jim will find Pam 0.6 ly away from him. In Jim's frame, he will know that his greeting signal travels at 1-0.6=0.4c relative Pam, so it will take 0.6/0.4=1.5 years to reach Pam. That means Jim's calendar will read 1/1/08 + 1.5 y = 7/1/09 when the signal reaches Pam. However, Pam's 0,8 time dilation factor means that on her calendar it takes only 0.8 x 1.5 = 1.2 years. Add that to her 10/20/07 and you get 1/1/09.
Easy, isn't it? ;) Not quite, it can be very confusing working it out this way, while those Minkowski diagrams show it at a glance!
Doug: "OK, I may be alright and may be able to work out the remaining details. I still don't understand this assymetry, however".
The asymmetry is caused the moment Pam ignites her rocket to turn around (year 2011 on her calendar). She will immediately notice a change in the Doppler shift of signals from Jim. Jim's calendar will read 2012 at that time, but he will not notice any change in Doppler shift of Pam's signals until his calendar time has passed the year 2015. Because of the distance, that delay is caused by the time light takes to travel to him from the turnaround point.
Shout if you need more help in wrapping your head around it.
Regards, Burt Jordaan (http://www.relativity-4-engineers.com)
By BurtJordaan on
Thursday, June 26, 2008
|
Re: The "Twin Paradox"
For those people who cannot follow this, which is the easy way to explain this, I think that you could be in the wrong place, mentally. This is very easy to understand if you an engineer as well as the other post with the harder (as he calls it) way to explain it. I find that asking questions when you do not understand is a great way of getting a better understanding. It never hurts to ask, someone to explain themselves.
great article
By attagirl on
Friday, January 23, 2009
|
Re: The "Twin Paradox"
This solution, like every other solution for the past 100+ years, is wrong. The twins end up the same age. It's so obvious that I can't for the life of me understand why no one else sees the mistake that's been made over and over and over. The problem is always the same...the traveling twin never travels far enough.
Jim see’s Pam’s time dilate, but also see’s her space contract. No, not just her ship...her SPACE! That’s the key. Pam’s clock is ticking slower, but she’s also not going as far. Moving forward one meter in her frame is really only a portion of a meter to the frame at rest. In relation to the frame at rest, Pam will travel further to get to the turn around point. However, Pam’s clock was also ticking slower. The result is that the slower clock cancels the longer travel distance. The same thing happens on the return trip, and she ends up being the same age.
If you treat the scenario in this way, and also add the transformations that Einstein missed (time speeds up and space extends opposite the direction of travel,) then an amazing thing happens...no matter which direction you send a beam of light through the two frames, both observers will measure speed as the speed of light, and length of the beam to be the same.
By Graystar on
Monday, June 01, 2009
|
Re: The "Twin Paradox"
Hi Graystar.
Sadly, your refutation, "like every other [one] for the past 100+ years", is simply wrong.
Misconception 1: "Jim see’s Pam’s time dilate, but also see’s her space contract. No, not just her ship...her SPACE! That’s the key."
Jim does not 'see' Pam's space contract. All he 'sees' in the scenario that I've sketched above is the greeting signals that arrive. And he gets only 8 in the 10 years that he reads on his clock. He can also check Pam's clock and find it to be consistent with the 8 years. If it weren't like this, your GPS would not have worked! Ask NASA...
Misconception 2: "If you treat the scenario in this way, and also add the transformations that Einstein missed (time speeds up and space extends opposite the direction of travel,) ..."
Einstein perfectly explained the isotropy of the speed of light. He missed no stitches when he knitted that theory of his...
Regards, Burt Jordaan (http://www.relativity-4-engineers.com)
By BurtJordaan on
Monday, June 01, 2009
|
Re: The "Twin Paradox"
I like this article. This is called a great article. I am new here. I like your site too. This is pretty awesome. i found some useful info here. anyways thanks for sharing with [url=http://www.hello-hotel.com/thailand.php]us[/url]. I am looking foreword your next post. Thanks. I’m just going to shear this site all my friend’s and i hope they live this site.
By Mr. Thailand Hotels on
Saturday, January 23, 2010
|
Re: The "Twin Paradox"
Thanks for the comments - not too many comments on this engineering site, which is why I post articles mostly in my Blog on Globalspec's CR4 (http://cr4.globalspec.com/blog/22/Relativity-and-Cosmology), which is quite a lively engineering forum.
I do occasionally put something here as well, so keep on watching! :)
Regards, Burt Jordaan (http://www.relativity-4-engineers.com)
By Burt Jordaan on
Sunday, January 24, 2010
|
|