Greetings dear fellow friends of science, today we are facing -
- the twin paradox, a theme broadly discussed in internet on the subject -
- of special relativity. If you are here, I take for granted that you know well -
- the concepts of relativity and therefore I am not resuming them, except those that -
- will specifically serve me for this demonstration, demonstration that, -
- unlike what you generally see on the internet, will not resort to -
- neither the concept of general relativity nor the -
- space time Minkowski diagrams neither will go to see -
- Lorentz transformation formulas, which, by the way, would all be correct approaches.
We will try to give you a more -
- intuitive view, for that too, more upsetting. We hope to succeed.
So let's start right away by resuming what the -
- twin paradox is; and so let's go see this drawing: well in the twin paradox -
- we have two twins here represented by two watches, one of which travels -
- with respect to the other, takes a certain speed, goes in a certain-
- direction -
- and then comes back; where is the paradox? The paradox is in -
- comparing two results of relativity which apparently -
- seem to be in contradiction with each other; and what are these results? the first result is -
- time dilation. Please recall that special relativity says -
- if the blue is moving with respect to the red, the blue clock does not just move into space -
- but it will even run slower. Therefore special relativity says the blue watch -
- at the end of this journey will be younger than the red one. But the other result -
- of Galileo's relativity, not even Einstein's, is that there is no -
- preferred reference system, so if according to the red, the blue was -
- moving, according to blue the be red is moving, obviously in this other direction,
back and forth, hence according to the red, sorry the blue, the red will -
be younger. So the twin paradox says how can each one -
- assume to be both younger and older of the other? Well to fix this -
- we will resort to a third feature of special relativity, -
- usually is overlooked, and that is the feature of the relativity of -
- simultaneity; let's see what I mean -
- with that.
Okay in order to explain the relativity of -
- simultaneity we refer to this drawing. Here we see, I drew -
- a reference system, here in red, with a clock that is this blue one.
A light beam starts going in both directions and -
- at the same time a clock starts. Let's see the effect. The light beam expands, -
- the clock turns, as you see. What do we get?
Here, when the beam reaches these two points,-
- I've made sure that two other clocks, which I had placed in these two -
- points, began to move, naturally by marking exactly this time. How -
- did I mark this time? Because I know the speed of light, -
- I know the distance, so I know how long it takes for light to get -
- from here to there and from here to there, which for me it's, obviously, the same -
- amount of time. Hence I can already predict what my central -
- watch will mark when light gets here. So from this moment on, my -
- watches travel as you see, synchronized, right? They always mark the same time.
So I have my own concept of simultaneity because I can say that -
- light reaches this point at the same time as light reaches -
- this point, and this applies to any type of equidistant point from,
from the central point. But now let's see, here we keep going, obviously, what we -
- may say when this scenario is seen by -
- someone moving around with me in this direction, so he sees my -
- world move in this other direction. Let us remember that for special -
- relativity, light travels with the same speed for any reference -
- system, and what are the consequences? The consequences are, that as we can see,
my system travels, the clock keeps turning, the beam in turn -
- continues to expand, now as light expands with the same -
- speed of light, for him as it was for me, it is clear that -
- the disk is not moving, seen from this new point of view. The center of the disk is not -
- moving but it stays where it was, it is simply the beam that's expanding; and so -
- what do we get? He's observing what happens to me, we say -
- that he sees my light beam reaching the right notch, that is -
- the one that's trailing along with my movement, long before the left notch,
so once again, what do we get? That he sees, that I set the clock -
- obviously in the reading I set it in, that is when it marks 15! Say, -
- at this moment when the beam comes here, and hence long before the central -
- clock has reached the quarter-turn, and even more, let's go -
- ahead, you see the watches move,
once again, when then the beam, from his point of view, has reached my third -
- notch he will see what I do at that moment, that is, again I set -
- the ' quarter-turn clock. So what have we experienced? First thing -
- that events that for me were simultaneous, that is, light comes here and -
- light comes here, for him they are not simultaneous, but in the second most important -
- feature, is that the clocks are no no longer synchronized, as you see,
the more I shift towards the head of my movement, remember that I moved in this -
- direction, and the more the clocks are behind in time, the more I shift to the tail of -
- my movement, that is, the more I shift in space toward the tail of my movement, and -
- the more the clocks are ahead in time; and this will be the key to solving the -
- twin paradox. Let's see why, indeed.
Let us see a first peculiarity of the twin paradox,
the forward journey. Let's analyze the first half of the journey, and -
- starting from this first drawing; well, sorry, in this drawing we have -
the two twins, the blue departs and goes up to a point. Now it does not matter -
- how far away to a very distant star, the concept is still valid, so -
- let's keep it simple, let's just get it here and then eventually will go back ; I have -
- also incidentally sketched the watch of my reference -
- system, which is solid with mine, located at the point where my traveling -
- twin will arrive. Well, for what we have said, we start from this -
- scenario, all watches mark -
- the same time, this is for convenience.
When he reverts we will have, from my point of view, my watches -
- will have turned beyond the quarter turn, say, while his watch will -
- have turned less than the quarter of a turn; on the return trip the same thing will happen, but -
- let's just focus on the trip forward, because already there, on the trip -
- forward, we have the chance to deal with a piece of the resolution -
- of the twin paradox. Let's look at this same movie seen from -
- the split point of the blue clock, so let's move on this diagram here.
Again here I redrew the two red and blue watches and this time -
- obviously will be my watch, I will be moving in this -
- direction, I also redrew the equivalent of my orange watch,
which is at the point where, from my RED point of view, I could see the blue twin -
- to stop and come back. But please note that this orange watch,
seen from my, ehm, from the blue point of view, is no longer synchronous with the red watch!
And as it is, we have seen before, in the tail of my motion, I move this way,
this clock is definitely further ahead !!!
For the sake of completeness, I also drew the blue reference system clock -
- at the point where, from the blue point of view, the red twin -
- will come to a stop, so what do we see? We see that we get to the end,
we get to the end, and from the point of view of the blue clock, the red clock has -
- turned less! Exactly as we said from the RED point of view that -
- the blue would have turned less. So from the point of view of blue the red has -
- turned less, you can see it by doing this comparison, this clock -
- here marks the same time that the blue marks, which is the same time marked by -
- this one, because in this system all watches are solid with each other, but -
- it is clear that the red has turned less than blue. What about the yellow one? Even -
- the yellow turned less, but as it started ahead, it stays ahead, and -
- keeps on with exactly the same result that we have here. This turns out to be -
- a graphic representation of the result of the formulas that are in -
- Lorentz's transformation; so we can say, halfway through, that this -
- clock here, from BLUE point of view, is further ahead, and even from RED point of -
- view, and mark exactly the same, the same picture.
This is a piece of resolution already, because it is obvious right? So the point is,
we cannot say that there is incompatibility! These, these
results are all compatible with each other, because both from the point -
- of view of the red system and from the blue system point of view, the -
- results are the same when you compare the watches in same -
- point in space, okay? The key point is the comparison between the -
- clocks in the same point of the space !!!
Let's see if we can see it even better with a wider perspective.
So here we have all three watches, let's say. Here the world is seen from the point -
- of view of the red system, we have that the twin departs, the twin departs, in the -
- blue twin, seen from red, the clocks are again not synchronized, the clock that -
- is trailing in space is ahead in time, the clock leading in space is behind in time.
Then we will forget this purple watch here because it is not involved. When -
- the blue gets to the edge, it turned less than the red ones, the dotted one too -
- turned less, but stays ahead of this.
The blue is behind the yellow, and and from the standpoint of the ro,
of the blue, the situation is diametrically opposite
but actually identical, because again this time for Blue -
- its three clocks are synchronized, those that are not synchronized -
- are my "RED" clocks, so this clock that is at the head of my movement is -
- behind in time, this clock that is in the tail of my -
- movement is ahead, let's see what, what blue says when -
- RED gets down there, he will see again that my -
- red clock is behind his dotted watch there, or all its watches,
and this yellow one here is ahead than all his watches, exactly as -
- much as I saw it, we described it, from the point of view -
- of red. Now, however, you may say, okay this is a trick you use to -
- solve the theorem, sorry, the twin paradox,
based on the fact that you compare watches in different points in space;
but in the twin paradox eventually there is also the return trip!
Right. So let's see what we get on the way back.
Again here I represented the scenario,
the complete scenario, so from the point of view of the red one his clocks -
- are all synchronized, so the blue ones are not synchronous, the blue departs,
we get to this situation here;
from the point of view of the blue, like I said, same movie, these are -
- synchronized, this time these are not synchronized,
but the result here is the same. But now, the important -
- thing is, what is the important thing? The important thing is that,
while talking about uniform motions there is total symmetry, this -
- symmetry does not apply when someone changes direction! And here only one of them actually changes -
- direction; who is the one who changes direction? is the blue clock! that is, the blue -
- stops and goes back, or we might even think it jumps from a -
- train, from one spaceship to another, right? The blue up to here was on a ship -
- traveling in this direction, and then jumps and is on a ship -
- traveling in this other direction. Why is this important here? And -
- why am I saying that it's just the blue that realizes it? It is only blue to realize that -
- because the blue, if he did his experiments in his world, at some point -
- he would realize that if he was pouring water on a glass, the water -
- would suddenly fall out of the glass, just like we when we are in a -
- car and we are sitting seated, we have no problem sitting,
but if the car suddenly stops, we do feel it, don't we!
So much so that we need seat belts to avoid -
- banging. The same reasoning applies only to blue,
so what happens? So while the red one can keep saying: from my point of view,
sorry, we return from the point of red, from my point of view I have -
- the blue clock that turns slower, arrives here -
- and then, now I did not draw it, when it comes back again, it will continue to -
- be slower and will continue to be behind in time;
so when it's there it's behind and when it's back it is even further behind, from -
the Blue point of view of what's going on? The blue in turn, yes, says: it is the red -
- who departs and when it gets there is behind in time, but at this point he feels that he himself changes -
- the reference system; and what matters is what happens right here,
that is, let's remember that in this point in space both of them say -
- the same thing, you see, the ro, the the blue is behind with respect to the yellow, the -
- blue is behind with respect to the yellow. Seen form both points of, from -
- both reference systems. Now what was I saying? When the blue jumps on the -
- new ship for him the world, the universe has completely changed, for him the universe -
- is a universe like that, where obviously here nothing can change that, right?
We are in the same spot in space and therefore even in time scenarios, but now from its -
- point of view, everything works as if the red one had always, say, been -
- on a ship traveling in this other direction !!
And then again, we come up with the concept we saw when we spoke about -
- the relativity of simultaneity, from its point of view it means that -
- the red clock is far ahead of this other clock, which -
- in turn is already ahead of Blue; therefore, when, at the -
- end of the trip, we will find ourselves at the same point in space,
even for the blue his own clock will continue to be further behind -
- the red clock. Here is the definitive solution -
- to the twin paradox. As you see, we did not have to use neither -
- general relativity, nor Lorentz's formulas, nor Minkowski's diagrams,
we simply used these moving clocks and their -
- positions and readings in various positions;
the key concept is the concept of the relativity of simultaneity, which is -
- a really absurd concept, is it not? Let's remember how we express it? If two events for me are -
- simultaneous, for someone who moves over me -
- they are not, and that is the concept that helps us get out of the deadlock of -
- the twin paradox. Thank you for listening, see you -
- next time.
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