How Does the Twin Paradox Affect Aging in Relativity?

[sylas]

To 'atyy' and 'sylas': you are assigning preference to one inertial frame over another.
Ron

Ron, this is not up for debate. It is not a case of two inertial frames.

If you want to understand this better, we can help; but in my experience it is almost impossible to help someone with this if they insist they already understand it but continue to make all the same errors. As you are doing.

The twin that turns around is not inertial. It is in a different frame on the outward journey, and on the inward journey. As I said previously, it is the non-inertial twin which experiences less elapsed time between departure and when they meet up again.

Cheers -- sylas

 

[Fredrik]

Recall that the 'Twins Paradox' arose specifically out of Special Relativity as an effect of Time Dilation. It is quite simple; does Time Dilation cause one clock to tick more slowly than another? the consequence of this phenomenom is the paradox, that one clock will count less time than another if they meet up after 'one of them' has traveled for a while. Answer: no.

Ah, then I was wrong about what you had misunderstood. SR definitely predicts that the twin that goes on the trip is younger when they meet again. This is an immediate consequence of the axiom that says that a clock measures the proper time of the curve that represents its motion.

That's the coordinate independent resolution of the paradox. If you'd like to see it resolved completely in terms of co-moving inertial frames, then see the spacetime diagram that I linked to in #51 (in the last quote box).

 

[JesseM]

To 'atyy' and 'sylas': you are assigning preference to one inertial frame over another.
Ron

No, you can analyze the problem from any inertial frame and all will have the same answer about the age of the inertial twin and the age of the non-inertial twin when they reunite. Let's call the inertial (Earth-bound) twin "Terence" and the traveling twin "Stella", following the Twin Paradox FAQ. First let's look at the numbers in Terence's rest frame. Suppose that in this frame, Stella travels away from Terence inertially at 0.6c for 10 years, at which point she is at a distance of 0.6*10 = 6 light-years from Earth in this frame, then she turns around (i.e. she accelerates, a non-inertial motion which will cause her to experience G-forces that show objectively that she wasn't moving inertially) and heads back towards Terence at 0.6c, finally reuniting with Terence after 20 years have passed since her departure in this frame. Since Terence is at rest in this frame, he has aged 20 years. But since Stella was moving at 0.6c in this frame, the time dilation formula tells us her aging was slowed down by a factor of \sqrt{1 - 0.6^2} = 0.8, so she only aged 0.8*10 = 8 years during the outbound leg of her trip, and another 0.8*10 = during the inbound leg, so she has only aged 16 years between leaving Earth and returning.

Now let's analyze the same situation in a different inertial frame--namely, the frame where Stella was at rest during the outbound leg of her trip (she can't also be at rest during the inbound leg in this frame, since this is an inertial frame while Stella accelerated between the two legs of the trip). In this frame, Terence on Earth is initially moving away from Stella at 0.6c while she remains at rest. In Terence's frame, remember that Stella accelerated when she was 6 light-years away from Earth, so we can imagine she turns around when she reaches the far end of a measuring-rod at rest in Terence's frame and 6 light-years long in that frame, with Terence sitting on the near end; in the frame we're dealing with now, the measuring-rod will therefore be moving along with Terence at 0.6c, so it'll be shrunk via length contraction to a length of only 0.8*6 = 4.8 light-years. So, Stella accelerates when the distance between her and Terence is 4.8 light-years in this frame, and since Terence as moving away from her at 0.6c in this frame, they will be 4.8 light-years apart after 4.8/0.6 = 8 years have passed. During these 8 years, it is Terence's aging that is slowed down by a factor of 0.8, so while Stella ages 8 years during this leg, Terence only ages 0.8*8 = 6.4 years. Then Stella accelerates to catch up with Terence, while Terence continues to move inertially at 0.6c. Using the relativistic velocity addition formula, if Stella was moving at 0.6c in Terence's frame and Terence is moving at 0.6c in the same direction in this frame, then in this frame Stella must be moving at (0.6c + 0.6c)/(1 + 0.6*0.6) = 0.88235c during the inbound leg. And since Terence is still moving at 0.6c in the same direction, the distance between Stella and Terence will be closing at a "closing speed" of 0.88235c - 0.6c = 0.28235c. Since the distance was initially 4.8 light years at the moment Stella accelerated, in this frame it will take 4.8/0.28235 = 17 years for Stella to catch up with Terence on Earth. During this time Terence has aged another 0.8*17 = 13.6 years, so if you add that to the 6.4 years he had aged during the outbound leg, this frame predicts he has aged 20 years between Stella leaving and Stella returning, same as in Terence's frame. And since Stella is traveling at 0.88235c her aging is slowed by a factor of \sqrt{1 - 0.88235^2} = 0.4706, so during those 17 years in this frame she only ages 0.4706*17 = 8 years during the inbound leg. If you add that to the 8 years she aged during the outbound leg, you find that this frame predicts she has aged 16 years between departing and returning, which again is the same as what was predicted in Terence's frame.

 

[stevmg]

Sports Fans:

Here is that Yale lecture series I spoke of. I think it is lecture 13 or 14. Start with lecture 12 and step forward.

http://www.youtube.com/watch?v=pHfFSQ6pLGU&feature=SeriesPlayList&p=FE3074A4CB751B2B

Yea, everyone is moving BUT start with twin A on the Earth and look at the second twin B in mtion in space relative to A and it is B who age slower than A. all clocks slow down under time dilation including biological clocks.

Listen to this guy - he explains it a lot better than I can and a lot better than I saw on the earlier posts on this blog.

 

[JesseM]

Yea, everyone is moving BUT start with twin A on the Earth and look at the second twin B in mtion in space relative to A and it is B who age slower than A. all clocks slow down under time dilation including biological clocks.

Only in A's rest frame. In B's rest frame, it is A who ages slower than B (assuming both are moving inertially).

 

[stevmg]

JesseM -

You are the person who straightened my brain out before..
Oh well,

Maybe it has something to do with General Relativity...

If you think I knew nothing about Special Relativity, I won't disappoint you with my less than zero knowledge of GR.

Under GR, time slows down by acceleration and gravity. The guy on Earth is attached to something with more mass than a spaceship.. The B guy has to accelerate and turn (centripetal force), so I guess that is the explanation.

I read Einstein's book "Relativity" and if I wasn't lost and confused after SR, I was clueless and pulseless after reading GR.

Loved that bending the light demonstration and the perihelon{?sp) of Mercury.They even did a movie about a group of guys in 1919 who went to Brazil and did some picture taking of an ecclipse and made it with some local Brazilian girls. Kept my interest up.

Too bad that they didn't believe the 1919 pictures and Uncle Al had to wait until 1923 when a more "official version" was taken and Einstein became God on Earth.

 

[stevmg]

The name of that movie is "House of Sand" - great flick!

 

[Dale]

Not a clue about that. But my mind is sufficiently blown for the night. Google is my friend tomorrow.

Hope you had happy Googling Sorry about my terse response earlier, I was in a rush.

Anyway, a spacetime diagram is simply a graphical plot of the path of a particle. By convention, it is drawn so that the time axis is vertical and the space axis is horizontal. Since paper is traditionally 2D usually we just include one spatial axis and politely ignore the other two, but if you had a 3D graphics system you could easily add a second spatial axis. Also by convention, the time axis is scaled by c (i.e. ct instead of just t) so that even though it is "time" the units have dimensions of length (ct -> L/T T = L).

Using these conventions you can think of the path of particles as geometric figures. A particle at rest would be represented by a vertical "worldline". A pulse of light would be represented by a line at 45 degrees. A particle moving inertially at some v<c would be a straight line with some slope in between. An accelerating particle would have a curved worldline. Etc.

In such a diagram a clock measures the length of the worldline, except instead of using the normal Euclidean metric, the metric that a clock measures is called the Minkowski metric ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2. Then it is no longer very surprising that one clock measures a different time than another since they are measuring the lengths of different lines.

 

[Jackslap]

HA! I'm such a moron for not posting that link. Can't believe I made that mistake. Here it is:

http://www.youtube.com/watch?v=ksbgDJF9bsM&feature=related

Anyway, now that I've officially ruined relationships on this board I suppose I'll continue this pattern by stating that I was confused by the difference of opinions, and quite frankly still am a little bit.

I am thinking that Ron's input must be taking something into consideration. Something that I'm not yet capable of understanding. I see that some of you have suggested such a thing. I shall try to wade through the wealth of information you all have left me so far today. Specifically the Yale lectures. I'll probably only be able to get through one part tonight though.

You guys are really awesome for helping. This site is such a resource for me lately. Such a convenient way to share knowledge and I'm humbled that some of you would share your learning which probably cost a ton of cash with some dude like me that you don't know and probably will never meet.

 

[JesseM]

JesseM -

You are the person who straightened my brain out before..
Oh well,

Maybe it has something to do with General Relativity...

If you think I knew nothing about Special Relativity, I won't disappoint you with my less than zero knowledge of GR.

Under GR, time slows down by acceleration and gravity. The guy on Earth is attached to something with more mass than a spaceship.. The B guy has to accelerate and turn (centripetal force), so I guess that is the explanation.

Well, usually when people discuss the twin paradox they ignore GR--you can imagine a world where all objects, including the Earth, have negligible mass (or you can imagine that the inertial twin is not an Earth but on a space station deep in interstellar space, with too little mass of its own to cause any measurable gravitational time dilation).

 

[stevmg]

You are right, Jesse M. If one did a circular route 10 light years in diameter (30 light years in circumference) it would take only 1/10 g to create the circle to get back to the origin and that won't slow anything down to speak of.

 

[stevmg]

Using the 4-space equation, a different path would change the t although the sum of the x^2, -(ct^2) = sum of the (x')^2, (-ct')^2 is is invariant.

I have to accept the 4-space equation on faith alone.

 

[Dale]

I have to accept the 4-space equation on faith alone.

In science the unproven assumptions of a theory are called postulates. They are not accepted on faith alone, but they are verified experimentally. When a particular experiment agrees with the logical consequences of a postulate or set of postulates then the theory based on those postulates is said to be experimentally verified.

In the case of the Minkowski metric, the experimental evidence is ample and such statements about faith are rather absurd.
http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html

 

[TcheQ]

Using the 4-space equation, a different path would change the t although the sum of the x^2, -(ct^2) = sum of the (x')^2, (-ct')^2 is is invariant.

Keep reviewing relevant literature, various lectures on youtube etc etc.

Alternatively you can always take a postgrad theoretical physics class

 

[stevmg]

With all due respect, it is this blog that is absurd...

I am using ordinary figures of speech and not literal meanings...

Am signing off, folks...

Hasta la vista...[figure of speech-wise, not literally Spanish]

Steve G

 

[JesseM]

You are right, Jesse M. If one did a circular route 10 light years in diameter (30 light years in circumference) it would take only 1/10 g to create the circle to get back to the origin and that won't slow anything down to speak of.

Actually gravitational time dilation only occurs in spacetime with "real" gravity due to mass and energy curving spacetime. The G-forces you feel when accelerating in flat spacetime (like taking a circular path in deep space far from any massive objects) don't cause any gravitational time dilation, at least not if you analyze things from the perspective of an inertial frame (if you analyze things in a non-inertial frame you can have a pseudo-gravitational field which leads to pseudo-gravitational time dilation, as discussed in this section of the twin paradox page, but when dealing with accelerating objects in flat spacetime it's much easier to just use inertial frames to calculate how much time goes by on their clocks).

 

[sylas]

Actually gravitational time dilation only occurs in spacetime with "real" gravity due to mass and energy curving spacetime. The G-forces you feel when accelerating in flat spacetime (like taking a circular path in deep space far from any massive objects) don't cause any gravitational time dilation, at least not if you analyze things from the perspective of an inertial frame (if you analyze things in a non-inertial frame you can have a pseudo-gravitational field which leads to pseudo-gravitational time dilation, as discussed in this section of the twin paradox page, but when dealing with accelerating objects in flat spacetime it's much easier to just use inertial frames to calculate how much time goes by on their clocks).

Actually, JesseM, you do get time dilation in a pseudogravitational field.

You can show this by consideration of a long spaceship which is experiencing a constant acceleration. First, take a single particle (the spaceship captain) and work out their world line so that they experience a constant acceleration. Then get another particle representing the rear of the spaceship, which has a world line defined so that the pilot can send a light signal from their location in the cockpit to a mirror at the back of the ship, and receive a reflection, and this is always a constant time by the pilot's clock.

This will give you a world line for the rear of the space ship, so that everyone at any point on the ship is experiencing constant acceleration, and everyone on board agrees that the ship is not stretching or doing anything weird like that.

What you will find is that different parts of the ship experience a different acceleration! And there is a measurable time dilation difference all along the ship, exactly analogous to the time dilation differences you find in a gravitational field.

Cheers -- sylas

 

[JesseM]

Actually, JesseM, you do get time dilation in a pseudogravitational field.

I agree, that's what I meant when I said "if you analyze things in a non-inertial frame you can have a pseudo-gravitational field which leads to pseudo-gravitational time dilation". My point was that both the pseudo-gravitational field and the pseudo-gravitational time dilation are absent if you analyze the accelerating object from the point of view of an inertial frame, which is always possible in flat spacetime; from the perspective of the inertial frame, all the differential aging of the accelerating object can be accounted for in terms of velocity-based time dilation, even though the non-inertial frame accounts for the differential aging differently (but as always, all frames agree on local predictions like what ages two observers will be when they meet at a single point in spacetime).

 

[sylas]

... all the differential aging of the accelerating object can be accounted for in terms of velocity-based time dilation ...

Ah! Glad we are on the same page, sorry I misunderstood you.

The above comment is a hugely important point, IMO, which lots of people get wrong. Your comment is succinct and spot on. I'm going to steal your phrasing when I am explaining it for others! Kudos.

 

[Jackslap]

lol, this has gone WAY beyond my ability to compute. I watched the first half of the Yale lecture number 12 that was linked to, but it froze up at 29 minutes in. I found myself understanding the instructors hypothetical situations and things that he said in ENGLISH. But he quickly starts jotting down formulas and using tons of variables for which I have no place for right now in my physics infancy.

I think I'd need to start at his lecture number 1, Newtonian Mechanics. However, it's freaking Yale and I'm thinking even then I don't have enough background to begin there. For your info, the highest level math class I've ever taken was an Algebra 2 class in college as a pre-req for the x-ray program. That's it. However I was encouraged when the formulas he was using in lecture 12 were fairly simply manipulated using algebra anyway. But at some point he started using sin, cos, which is trigonometry stuff. I aint's got a clue 'bout none o' that.

Not your guys job to teach me though, I must begin at the beginning if I am going to learn this. And my beginning will be quite slow due to work and family schedules. Baby steps as Bob would say.

 

[TcheQ]

I think I'd need to start at his lecture number 1, Newtonian Mechanics. However, it's freaking Yale and I'm thinking even then I don't have enough background to begin there. For your info, the highest level math class I've ever taken was an Algebra 2 class in college as a pre-req for the x-ray program. That's it. However I was encouraged when the formulas he was using in lecture 12 were fairly simply manipulated using algebra anyway. But at some point he started using sin, cos, which is trigonometry stuff.

It doesn't matter what university it is, they all teach the same thing (assuming it's maths or physics :P). The Yale guy is particularly good. As for maths, you need to learn calculus, differentials and matrices for relativity. If you want to learn quantum mechanics you need some form of statistics and computing, signal interpretation (such as Fourier transforms) and a background in computing. Nothing that's beyond a normal human being. It's always best to start at the beginning :)

 

[stevmg]

Cut me a break DaleSpam as you have been most infolrmative otherwise...

"Faith alone" does not refer to a religious or mystical approach. My reference is that I have seen the Minkowski 4-space but I do not understand the mathematical derivation of it. I do not challenge but do accept it. This is allegory.

I do not understand how it is shown that the Twin B does not age as fast as the twin A.
I do understand that if one takes a different course in the (twin B) xyz coordinates then the ct coordinate is different than the original (twin A). It is the Minkowski 4-space that establishes that I am told but I do not understand. Given that the Minkowski 4-space is true ("faith alone" statement) then I accept the outcome.

Don't get too freakin' literal as I do write in conversational language. To wit - when we used to write fitness reports on officers, we would always write "water-walkers" [reference to Jesus] to say that this guy or gal was OK. We weren't really saying they were like Jesus. Again, that's a "figure of speech" or allegory.

Let's get on the same page.

 

[W.RonG]

Well, this has certainly been enlightening. Since I have only been 'dabbling' in Physics for forty years or so, I was aware only of the 'classical' Twins Paradox. That's the one that came out of Special Relativity, where 'B' flies around in space while 'A' watches, and since neither is in a preferred inertial frame neither clock can run slower than any other.
There seems to be a new Twins Paradox. This one allows clocks to be assigned 'inertial' and 'non-inertial' so it exists outside the realm of Special Relativity. A 'GR' neo-twins-paradox perhaps? that's the only reasonable explanation for the dilemma.
Thanks again,
Ron

 

[sylas]

Well, this has certainly been enlightening. Since I have only been 'dabbling' in Physics for forty years or so, I was aware only of the 'classical' Twins Paradox. That's the one that came out of Special Relativity, where 'B' flies around in space while 'A' watches, and since neither is in a preferred inertial frame neither clock can run slower than any other.

As you have been told now many times, you are incorrect. This is not meant to be an insult; it is an attempt to help.

One twin is inertial. The other twin is not. The twin that is in an inertial frame is the one that WILL experience less time, when they have come back together again.

This is the only "twin" paradox. It is the same as explained by Einstein in his original works on the subject. The effect is confirmed by direct measurement.

Your reasoning is incorrect, because you have failed to note that there is a real difference between being in an inertial frame and NOT being in an inertial frame. The two twins are not symmetrical, and the only reason this is a paradox is because some people find it hard to understand; not because there is any actual inconsistency.

You also don't understand this yet -- despite 40 years of dabbling. This, by the way, is not all that unusual, and I mean no offense. But this is something that is taught in first year physics classes or even high school in some cases. You will do better to accept that you might need a teacher rather than rely on your own dabbling.

Cheers -- sylas

 

[Dale]

Cut me a break DaleSpam as you have been most infolrmative otherwise...

"Faith alone" does not refer to a religious or mystical approach.

...

Don't get too freakin' literal as I do write in conversational language. To wit - when we used to write fitness reports on officers, we would always write "water-walkers" [reference to Jesus] to say that this guy or gal was OK. We weren't really saying they were like Jesus. Again, that's a "figure of speech" or allegory.

I will gladly cut you a break if you will stop with the religious and Biblical allusions. They are simply not appropriate to the forum, even in context. Being a person of both science and faith I don't like to see either misunderstood or treated so lightly and dismissively. I know you do not intend your comments to be provocative in this way, but to me they make a real impediment to actually understanding what your scientific or technical question is. I appreciate that to you they are "conversational" expressions, but the internet is a medium where such colloquial expressions do not come through and misunderstandings can easily result. It requires more care in communication than a casual conversation with friends where everyone knows your background, can read your body language, and understand immediately your intent.

 

[yuiop]

Well, this has certainly been enlightening. Since I have only been 'dabbling' in Physics for forty years or so, I was aware only of the 'classical' Twins Paradox. That's the one that came out of Special Relativity, where 'B' flies around in space while 'A' watches, and since neither is in a preferred inertial frame neither clock can run slower than any other.
There seems to be a new Twins Paradox. This one allows clocks to be assigned 'inertial' and 'non-inertial' so it exists outside the realm of Special Relativity. A 'GR' neo-twins-paradox perhaps? that's the only reasonable explanation for the dilemma.
Thanks again,
Ron

Hi Ron,

You are right that if two twins moving relative to each other with purely inertial motion (I.e. in a straight line and never accelerating, decelerating or turning around) then each measures the others clock to be running slower than their own clock and there is no way to determine which twin is "really" ageing less than the other. However, the classic twin's paradox has a non inertial element because one twin has to turn around and this involves acceleration. There is a popular conception that SR can not handle acceleration but this is not true. The introduction of acceleration into purely inertial SR is very easy to do because it turns out that acceleration has no effect on the instantaneous proper rate of a clock which is determined purely by the instantaneous velocity of the moving/accelerating clock. The twin's paradox is called a paradox because of the apparent paradoxical (contradictory) conclusions that inertial considerations say there is no real differential ageing of the twins while the round trip thought experiment says they will age differently. Of course it is not really a paradox because it can be resolved and SR predicts only one outcome (The twin's age differentially). The paradoxical nature is compounded because as I said before, acceleration has no direct effect on the clock rates! The solution and full understanding of the twin's paradox is subtle and involves differences in paths through spacetime that can not be transformed away from any point of view. The subtlety of the resolution to the paradox is the reason that there are literally hundreds of threads and thousands of posts on the subject. Once you accept that differential ageing occurs as has been demonstrated by actual experiments I strongly recommend that you do not jump to the conclusion that acceleration is the cause of differential ageing because that is a false conclusion and this is also a fact proven by experiments.

Here is a demonstration of how acceleration does not effect clocks with reference to the relativistic rocket equations here: http://www.phys.ncku.edu.tw/mirrors/physicsfaq/Relativity/SR/rocket.html

We see that the instantaneous velocity of the accelerating rocket is:

v = at / sqrt(1 + (at/c)^2)

which can be solved for acceleration (a) to give:

a = v/t*1/sqrt(1 - v^2/c^2 )

The instantaneous time dilation factor (gamma) of a clock on the accelerating rocket (T) relative to the initial inertial clock (t) is given as:

t/T = sqrt(1 + (at/c)^2)

Substitution of the value found for a into the above equation gives:

t/T = sqrt(1 + (v/c*1/sqrt(1 - v^2/c^2 )^2) which simplifies to:

t/T = 1/sqrt(1 - v^2/c^2 )

which is the familiar time dilation gamma factor of SR with no dependence on acceleration whatsoever.

 

[Fredrik]

Well, this has certainly been enlightening. Since I have only been 'dabbling' in Physics for forty years or so, I was aware only of the 'classical' Twins Paradox. That's the one that came out of Special Relativity, where 'B' flies around in space while 'A' watches, and since neither is in a preferred inertial frame neither clock can run slower than any other.
There seems to be a new Twins Paradox. This one allows clocks to be assigned 'inertial' and 'non-inertial' so it exists outside the realm of Special Relativity. A 'GR' neo-twins-paradox perhaps? that's the only reasonable explanation for the dilemma.
Thanks again,
Ron

One of them has to accelerate after they separate. Otherwise the distance between them would just keep increasing, and they would never meet again.

The scenario that goes by the name "the twin paradox" involves two twins that are are both present at two events E and F. One of the twins has been doing inertial motion from E to F, and the other twin hasn't. Special relativity predicts that if they're both the same age at E, the one who stayed inertial is older at F.

You said that "since neither is in a preferred inertial frame neither clock can run slower than any other". This is incorrect. The fact that neither of the frames is preferred implies that if A's clock is slow in B's rest frame, then B's clock is slow in A's rest frame. But this doesn't imply that A's clock can't be slow in B's rest frame or that B's clock can't be slow in A's rest frame.

 

[Fredrik]

One twin is inertial. The other twin is not. The twin that is in an inertial frame...
...
being in an inertial frame and NOT being in an inertial frame.

I'm just going to nitpick your choice of words a bit. (It wouldn't be fair to Mentz114 if I let this one slide ). I think it's fine to say that "one twin is inertial". That just means that he isn't accelerating. But the phrase "is in an inertial frame" doesn't really mean anything. I know that you meant the same thing as when you said that the twin "is inertial", and in this context it's probably clear to everyone else as well, but it's still a meaningless phrase that novices in particular should be discouraged from using.

It's not that much more work to type "X is at rest in an inertial frame" instead of just "X is in an inertial frame".

 

[sylas]

It's not that much more work to type "X is at rest in an inertial frame" instead of just "X is in an inertial frame".

I appreciate a good suggestion and this is a good suggestion and a legitimate clarification of poor phrasing.

Thanks!

 

[stevmg]

I will gladly cut you a break if you will stop with the religious and Biblical allusions. They are simply not appropriate to the forum, even in context. Being a person of both science and faith I don't like to see either misunderstood or treated so lightly and dismissively. I know you do not intend your comments to be provocative in this way, but to me they make a real impediment to actually understanding what your scientific or technical question is. I appreciate that to you they are "conversational" expressions, but the internet is a medium where such colloquial expressions do not come through and misunderstandings can easily result. It requires more care in communication than a casual conversation with friends where everyone knows your background, can read your body language, and understand immediately your intent.

Dale -

Sorry - the military will continue to talk in their superlative and metaphorical terms which involves a whole lot of swearing for years to come. Saying "someone is a genius" just means that "he comes to work on time." "Has a knack for tracking details" means "he follows the rules and is not a serial killer." How about the old Irish phrase - "May the Lord take a liking to you - but not too soon!" I never had anybody ever misunderstand me before - even when I spoke in German and my German wasn't very good. Of course, neither is my English.

But the original statement and question about taking the Minkowski 4-space as it relates to this twin paradox is "If you look at x,y,z,ct coordinants and the x',y',z',ct' coordinates and they are the same at the beginning and end the the journey for the twins - aren't the t and t' going to be the same (i.e. - same age)?

Other posts on this thread all ascribe to supposition that you can't tell which frame of reference (FOR) is the basis so neither twin will be older or younger - or is it that one twin will always see the second twin as younger? (Choose one twin - the other guy is younger. Choose the other twin, and the first guy is younger.)

By the way (an aside) - you can have same embryo twins of different sex - did you know that? You will floor me if you know how that is possible.

I know that the GPS satellites atomic clocks run slower than their Earth counterparts (hence are factored up by the gamma factor to offset time dilation.) Don't ask me where I heard that - I can't remember. Yes, it is only nanoseconds but didn't Eddington (May 1919) and later the Aussies of 1922 "confirm" Einstein's GR with trivial differences in where stars were and where they were supposed to be ("trivial" is not a put-down but is a reference to something really, really, really tiny.)

Seems like if Eddington and those Aussies in 1922 (I don't remember their names) both had the Earth as the FOR, so maybe there is a way to select the proper FOR.

Now, again, I don't know how Minkowski came up with a four-space so I do take it on "faith" (NOT IN A RELIGIOUS SENSE) alone - meaning I trust Minkowski and all you folks who understand it and know the confirmatory evidence to be giving me a "good" equation; good enough that I can plug in numbers and come up with a numeric answer.

The word "faith" has many meanings and not just a religious one, so I am not incorrect in its usage in my statement.

Miriam Webster Dictionary (I had to use it since my English is bad)
Main Entry: faith
Pronunciation: \ˈfāth\
Function: noun
Inflected Form(s): plural faiths \ˈfāths, sometimes ˈfāthz\
Etymology: Middle English feith, from Anglo-French feid, fei, from Latin fides; akin to Latin fidere to trust — more at bide
Date: 13th century
1 a : allegiance to duty or a person : loyalty b (1) : fidelity to one's promises (2) : sincerity of intentions
2 a (1) : belief and trust in and loyalty to God (2) : belief in the traditional doctrines of a religion b (1) : firm belief in something for which there is no proof (2) : complete trust
3 : something that is believed especially with strong conviction; especially : a system of religious beliefs <the Protestant faith>

synonyms see belief

— on faith : without question <took everything he said on faith>

On this blog, and since you are a science contributor and because of space limitations, can't I take certain things YOU say "on faith?"

But, please, let's address the scientific question I asked.