I How Does the Twin Paradox Illustrate Time Dilation in Special Relativity?

[NoahsArk]

When you say "the rocket instantaneously starts flying at .8c" you are basically saying that all parts of the rocket start moving at the same time

That's something I didn't consider, and that fact, that the bottom of the rocket starts moving first, makes my example more complicated:)

But that's not because the two distant clocks jumped forward/backward in time or their reading shifted, they always ticked one second every second their entire existence, no jumps at all, it's your notion of "now" that changed

That's the subtle point that I'm trying to wrap my head around, that my notion of "now" has changed, but so far have not been able to do. I have the notion (which is false based on the responses here) that when I read X = .6, after it earlier read X=1, that I've traveled into the past of that clock! That probably sounds silly, but its how I'm seeing it. Likewise, when the clock went from -1 to -.6, I feel like I've jumped into that clock's future.

With time dilation, I can at least kind of see that the clocks in the other IFR are just running slower relative to mine, and it doesn't seem as far fetched as what's happening in the scenarios above.

Stephanus in your example it seems like both A and B will have received the same number of signals. They both moved the same distance relative to each other.

 

[Nugatory]

With time dilation, I can at least kind of see that the clocks in the other IFR are just running slower relative to mine, and it doesn't seem as far fetched as what's happening in the scenarios above.

Even though your clock is also running slower than the clock in the other frame? That's a question, not an argument, but as I've pointed out above... Time dilation is derived from relativity of simultaneity, so it's hard to see how you can accept time dilation yet find relativity of simultaneity more far-fetched.

 

[Herman Trivilino]

Even though your clock is also running slower than the clock in the other frame? That's a question, not an argument, but as I've pointed out above... Time dilation is derived from relativity of simultaneity, so it's hard to see how you can accept time dilation yet find relativity of simultaneity more far-fetched.

Well, strictly speaking, it's the symmetry of time dilation that requires relativity of simultaneity for it to make sense.

It's very easy to start with the derivation of time dilation (using the light clock) and not address the issue of relativity of simultaneity until later. This is of course a mistake, because it leaves the symmetry of time dilation either ignored or a mystery to be solved later.

 

[NoahsArk]

Even though your clock is also running slower than the clock in the other frame? That's a question, not an argument, but as I've pointed out above... Time dilation is derived from relativity of simultaneity, so it's hard to see how you can accept time dilation yet find relativity of simultaneity more far-fetched.

Time dilation at least makes visual sense for me- yes, even though my clock is also running slower in the other frame. A helpful comparison for me from the "Relativity Visualized" book was the comparison to perspective in art, or how I see someone standing far away to be shorter, but they also see me to be shorter.

Relativity of simultaneity makes visual sense too, and the example you used with the two painters and the brushes in the earlier post was helpful. Its when someone is in one frame and has one view about the simultaneity (or lack thereof), between two events, and then switches frames and has another view about the simultaneity of the events that is hard to accept. If events A and B happened at the same time in my frame, its hard to accept then when I switch frames I'll conclude they happened at different times.

 

[m4r35n357]

That's the subtle point that I'm trying to wrap my head around, that my notion of "now" has changed, but so far have not been able to do.

Changing of "now" is just an inevitable consequence of the concept of simultaneity changing with velocity (boost). Are you familiar with the Andromeda Paradox? That is about as clear an explanation as I can find ATM.

It is what it is, and doesn't necessarily describe any sort of "reality". Think of it as accounting if that helps, and just accept it ;)

 

[Herman Trivilino]

That's the subtle point that I'm trying to wrap my head around, that my notion of "now" has changed, but so far have not been able to do.

I know that you think you've properly accounted for the delays due to the travel of light signals, but in fact you have only partially done so. Yes, you accept the fact that the delay exists, and that it has to be accounted for when an observer makes a claim as to when a distant event must have occurred, but you are ignoring the effect it has on simultaneity. It is the delay due to the transmission of information that accounts for the fact that the shift in the notion of a simultaneous distant event is unlike the shift in clock readings between distant events.

Think about the version of the twin paradox solution where at the turn-around point the traveling twin jumps from an outbound ship to an inbound ship. He makes his jump when the two ships pass each other. During that jump there's a shift in what he considers to be simultaneous events for his stay-at-home twin, but it's not a leap in the staying twin's clock reading.

Let's say the staying twin watches three television shows while his twin is traveling. Starting at 12:00 noon he watches the first three episodes of "Fun with Flags". Each lasts for 30 minutes. When he gets to the end of the first episode his clock reads 12:30. The traveling twin makes his leap when, according to him, it's 12:30 back at home. But when he lands on the inbound ship he reckons it's 1:00 at home, and his twin has finished watching the second episode. So what happened during that second episode? Did the staying twin's clock suddenly jump from 12:30 to 1:00, or did he spend a half hour watching the second episode? Of course he spent a half hour watching it. There was no sudden leap of 30 minutes according to the traveling twin, either. It's just that after allowing for the travel time of light signals, a person in the outbound ship reckons it's 12:30 when he passes the inbound ship, but a person in the inbound ship reckons it's 1:00 at home when he passes the outbound ship.

 

[NoahsArk]

Well, strictly speaking, it's the symmetry of time dilation that requires relativity of simultaneity for it to make sense.

Time dilation makes sense to me w/o relativity of simultaneity. The way I think of it is that time dilation is due to the way velocities add: anything that's moving on a train will have a certain speed with respect to people on the train, but people on the ground will measure that same speed to be less. If I throw a ball on a train and I measure the ball moving at 5 miles an hour w respect to me, someone on the ground will view it as moving something less than 5mph with respect to me.

I do see, though, how relativity is causing length contraction- i.e. to measure something both places have to be at a certain point at the same time. I also see how that's related to time dilation. E.g. I'll measure the time it takes the front of a moving train to cross the bridge I'm standing on to be more than the time someone on the train will measure because, for the person on the train, the bridge is smaller. So, I can see how time dilation, length contraction, and R.O.S are related. But, with respect to time dilation, for me the rules of addition velocity are enough to explain it (unless that's a misconception).

Changing of "now" is just an inevitable consequence of the concept of simultaneity changing with velocity (boost). Are you familiar with the Andromeda Paradox? That is about as clear an explanation as I can find ATM.

I've heard of it but don't know what it says. I will look into it:) Thanks for the tip.

 

[robphy]

That's the subtle point that I'm trying to wrap my head around, that my notion of "now" has changed, but so far have not been able to do.

From my Insight,

the notion of "now" according to an observer
is encoded by the line joining his or her "half-ticks" (reflections at that observer's light-clock mirrors [whose worldlines are not shown]).
[In my terminology, that line is along the diagonal of the "light-clock diamond".]
So, when the observer changes velocity, that sense of now--extended to events off his or her worldline--changes.

That diagonal of the "light-clock diamond" is actually tangent to a hyperbola, which defines what "perpendicular to the worldline" means in Minkowski spacetime.
Here is version 0.5 of a geogebra visualization:
Light Clock Diamond (robphy) - v0.5
https://www.geogebra.org/m/Jq4jDMRW
https://www.geogebra.org/files/00/03/71/96/material-3719659-thumb.png

 

[NoahsArk]

When he gets to the end of the first episode his clock reads 12:30. The traveling twin makes his leap when, according to him, it's 12:30 back at home. But when he lands on the inbound ship he reckons it's 1:00 at home, and his twin has finished watching the second episode. So what happened during that second episode? Did the staying twin's clock suddenly jump from 12:30 to 1:00, or did he spend a half hour watching the second episode? Of course he spent a half hour watching it. There was no sudden leap of 30 minutes according to the traveling twin, either. It's just that after allowing for the travel time of light signals, a person in the outbound ship reckons it's 12:30 when he passes the inbound ship, but a person in the inbound ship reckons it's 1:00 at home when he passes the outbound ship.

Hmmm. Well that raises some questions.

It's just that after allowing for the travel time of light signals, a person in the outbound ship reckons it's 12:30 when he passes the inbound ship, but a person in the inbound ship reckons it's 1:00 at home when he passes the outbound ship.

Traveling twin when he passes the outbound ship is the same distance from the staying twin as the person on the inbound ship is from the staying twin. So, it takes light the same amount of time to travel from the staying twin to both ships. Correct me if I'm wrong, but you're not saying that if it's 1:00 at home according to the inbound ship, but 12:30 at home according to the outbound ship, that the outbound ship is closer to home?

The traveling twin makes his leap when, according to him, it's 12:30 back at home. But when he lands on the inbound ship he reckons it's 1:00 at home, and his twin has finished watching the second episode. So what happened during that second episode? Did the staying twin's clock suddenly jump from 12:30 to 1:00, or did he spend a half hour watching the second episode? Of course he spent a half hour watching it. There was no sudden leap of 30 minutes according to the traveling twin, either.

For me to understand this portion better, let's say that the staying twin, instead of being home on earth, is home on some rectangular shaped planet, if one existed, that doesn't spin. The traveling twin while going outbound would observe things on opposite ends of the rectangle happening in the opposite order from the person coming inbound if I'm not mistaken? So, if the staying twin were watching the episodes on the end of the planet that is farthest from the traveling twin, the traveling twin would observe events happening on his brothers side of the planet after events that happened on the other side, and the person coming inbound would see the opposite. Does this have something to do with why the outbound twin observes it to be 12:30 (on the part of the planet where the staying twin is) when the person going inbound observes it to be 1?

Robphy, are the half ticks the blue rectangles?

 

[robphy]

Robphy, are the half ticks the blue rectangles?

The blue rectangles ("clock diamonds") are Bob's ticks as traced out by his light clock.
From O to Q to Z, Bob logged 8 ticks.

His half-ticks are the events halfway along one tick, where the light rays he emitted would reflect off his mirrors. They are the two corners of the clock diamond not on Bob's worldline. To Bob, those events are simultaneous.

As long as Bob maintains his velocity, these lines of simultaneity are parallel to each other ( and Minkowski-perpendicular to his worldline). Try drawing a line parallel to one of these line through Q (the turn around event)... extended over to Alice's worldline.

When he changes velocity, the family of these lines change direction. Through Q draw a parallel to the new family of lines.

 

[Stephanus]

I think the key to Twins Paradox is asymmetry.
If a rocket leaves the earth, so the Earth and the rocket both observers the other clock is slow. But..., WHEN the rocket reaches its destination, say 100 light years away and turns back. There's no way for the Earth to "know" that the rocket has already turned back. Earth has to wait 100 years, but that's not the case with the rocket. The rocket "knows" once it turns back.
And the asymmetry of doppler effect and time dilation can cause the Twins Paradox.

 

[Herman Trivilino]

Correct me if I'm wrong, but you're not saying that if it's 1:00 at home according to the inbound ship, but 12:30 at home according to the outbound ship, that the outbound ship is closer to home?

You are not wrong. They are the same distance from home when they pass each other. They share the same location, so they are the same distance from home, or any other location you happen to choose! Clock readings have no effect on the validity of these statements about distances.

The traveling twin while going outbound would observe things on opposite ends of the rectangle happening in the opposite order from the person coming inbound if I'm not mistaken?

Let's say there are two televisions. One on the near side and one on the far side. Moreover, let the transmitter be equidistant from them, so that in their rest frame the episodes of "Fun with Flags" are appearing simultaneously.

In that case, what you say is true.

So, if the staying twin were watching the episodes on the end of the planet that is farthest from the traveling twin, the traveling twin would observe events happening on his brothers side of the planet after events that happened on the other side, and the person coming inbound would see the opposite. Does this have something to do with why the outbound twin observes it to be 12:30 (on the part of the planet where the staying twin is) when the person going inbound observes it to be 1?

If the events are things happening on the televisions, then yes. In other words, people on the inbound ship will reckon that the episodes are appearing on the far side first. But people on the outbound ship will reckon that episodes are appearing first on the near side.

Does this have something to do with why the outbound twin observes it to be 12:30 (on the part of the planet where the staying twin is) when the person going inbound observes it to be 1?

Depends on what you mean. Certainly both are examples of the same phenomenon. Events that are separated along the line of relative motion and are simultaneous in one frame are not simultaneous in the other. Moreover, the order in which the events occur can be reversed for frames that move in opposite directions (for events like these that have spacelike separation).

 

[NoahsArk]

When he gets to the end of the first episode his clock reads 12:30. The traveling twin makes his leap when, according to him, it's 12:30 back at home. But when he lands on the inbound ship he reckons it's 1:00 at home

I've been thinking about this. So, from the staying twins perspective, as he just begins watching the 1:00 episode, could he rightly say "for my brother whose traveling in space away from me, and for everyone in his ship, to them this moment that's happening for me right now hasn't happened for them yet. For them I'm still watching the earlier episode. Also, to people who are right now coming in a ship towards me, this moment already happened?

a person in the outbound ship reckons it's 12:30 when he passes the inbound ship, but a person in the inbound ship reckons it's 1:00 at home when he passes the outbound ship

That leads to a related question: From the staying twins perspective, at the moment the inbound and outbound ship pass each other, what time is it for him? Is it 12:30, 1, or some other time? Just like there is a proper time and a proper length, does it make sense to ask what the proper "now" is?

 

[PeterDonis]

So, from the staying twins perspective, as he just begins watching the 1:00 episode, could he rightly say "for my brother whose traveling in space away from me, and for everyone in his ship, to them this moment that's happening for me right now hasn't happened for them yet. For them I'm still watching the earlier episode. Also, to people who are right now coming in a ship towards me, this moment already happened?

Yes and no. He can say it, but only because it makes no difference physically. What relativity of simultaneity is really telling you is that simultaneity--the concept of what is happening "now" at some place distant from you--is not a physical thing; it's just an artifact of how you choose coordinates.

 

[Stephanus]

Yes and no...

Now and again, I've been seeing the concept of "now" in SR.

I've been thinking about this. So, from the staying twins perspective, as he just begins watching the 1:00 episode, could he rightly say "for my brother whose traveling in space away from me, and for everyone in his ship, to them this moment that's happening for me right now hasn't happened for them yet. For them I'm still watching the earlier episode. Also, to people who are right now coming in a ship towards me, this moment already happened?

The concept of "now" for the staying twin and for the traveling twin is difference.
Let's say S1: is the staying twin
T1: is the traveling twin.
I think to understand the concept of "now" can be described as this.
T1 has a friend, T2 who is comoving with T1, that is at the same speed and at the same direction and behind T2.
T2 who happens to meet S1 at the same location at S1) when S1 is watching episode 1:00, T2 photographed what S1 is watching. And write the time stamp at the photograph. Later on..., T2 informs T1 what T2 sees, of course this event takes some time for the light from T2 to reach T1, plus if T2 doesn't directly send the photograph. He drinks coffe or anything.
Now T1 can conclude that, WHEN S1 is watching the episode 1:00, T1's clock read something... That is "now" for T1.
Of course T1 also "knows" what is T2 distance from T1. T2 could have sent its own clock reading, too. And of course because both (T1 and T2) are comoving, they can synchronize their clock.

Of course S1 can also have a companion (S2) who happens to meet T1 WHEN S1 is watching episode 1:00
Okay, let's forget about the episode, I'm confused.
Let's just say that S1 and S2 are synchronizing their clock.
And T1 and T2 are synchronizing their clock, separately from the S group.
S1 and S2 clocks match.
T1 and T2 clocks match.
It doesn't make any sense for S group (S1 and S2) to match their clocks with T group.
Okay, now let's do this experiment again.

Of course S1 can also have a companion (S2) who happens to meet T1 WHEN S group's clock are at 1:00
T1 travels at the direction from S1 to S2
And also at the same time with respect to S group, that is when S1 clocks read 1:00. S1 meets T1's friend, let's call him T3 and S1 photographed T3 clock.
S2 photographed T1 clock and sometimes later S2 informs S1 what S2 sees.
Now S1 compared both photographs (T1 and T3).
T1 clock might shows 03:00 (S group and T group clock might not match but that's ok, because T group are moving with respect to S group)
Guess what T3 clock in the photograph? It might shows 03:30. A time after T1 time! So, S1 actually is seeing T1's future!
And guess what. What is actually happening at T group?
T1 has a friend T2 who happens to meet S1 when T1 meets S2 at T group frame. T2 photographed S1 time and sends it to T1.
So later on when T1 compares both photographs..., the clock reading at S1 might show 00:30, a time before S2 time. T2 is seeing S2 past!

I think the concept of "now" can only be applied to comoving parties. And for moving parties, the concept of "now" is rather vague.
And I might quote.

...-the concept of what is happening "now" at some place distant from you--is not a physical thing; it's just an artifact of how you choose coordinates.

 

[Nugatory]

Just like there is a proper time and a proper length, does it make sense to ask what the proper "now" is?

You'd have to define what you mean by "proper now" before we could decide whether it made sense. What, exactly, do you mean when you say that something is happening "now"? What do all the events that occur "now" have in common that makes you say they're the ones that are happening now?

But for any definition that matches what we mean by "now" in our ordinary day-to-day non-relativistic understanding of time, the answer is going to be that it makes no sense to talk about it, or it has no physical significance.

 

[Herman Trivilino]

I've been thinking about this. So, from the staying twins perspective, as he just begins watching the 1:00 episode, could he rightly say "for my brother whose traveling in space away from me, and for everyone in his ship, to them this moment that's happening for me right now hasn't happened for them yet. For them I'm still watching the earlier episode. Also, to people who are right now coming in a ship towards me, this moment already happened?

Yes. (Adding the caveat that we're using the Einstein convention for simultaneity, which is just a convention. As PeterDonis points out, it's not physical.)

That leads to a related question: From the staying twins perspective, at the moment the inbound and outbound ship pass each other, what time is it for him? Is it 12:30, 1, or some other time? Just like there is a proper time and a proper length, does it make sense to ask what the proper "now" is?

If I think about proper time as being "my time" and proper length as being "my length" in the sense that it's the length I measure, then I don't really understand why you couldn't refer to it as "my now" (given the above-mentioned caveat).

An interesting observation is that in Einstein's native German, proper time is called eigenzeit which I understand is translated literally as "my time".

 

[PAllen]

Yes. (Adding the caveat that we're using the Einstein convention for simultaneity, which is just a convention. As PeterDonis points out, it's not physical.)
If I think about proper time as being "my time" and proper length as being "my length" in the sense that it's the length I measure, then I don't really understand why you couldn't refer to it as "my now" (given the above-mentioned caveat).

An interesting observation is that in Einstein's native German, proper time is called eigenzeit which I understand is translated literally as "my time".

Even in SR, for non-inertial motions, there are different, equally plausible notions of "my now". One would be to use the literal Einstein synchronization from (e.g.) a rocket, sending and receive radar signals and treating the reflection event as simultaneous with half way between the sending and reception events on my rocket. Another would be to use the 'now' of a momentarily comoving inertial frame (in some sense, adopting the synchronization of an observer following a different world line that is momentarily at rest with respect to my rocket). Which do you pick as "my now"? They are completely different, and both plausible. There is no such ambiguity for proper time. There is, actually, a messy set of issues for proper length, in that only very specialized motions admit a well defined notion of rigidity which is needed for a definition of proper length.

 

[NoahsArk]

Thank you again for the great responses.

You'd have to define what you mean by "proper now" before we could decide whether it made sense.

the concept of what is happening "now" at some place distant from you--is not a physical thing; it's just an artifact of how you choose coordinates

Defining "now" is not easy. The way I would try to do it is: "the set of all events which, for a particular observer, are happening at the same time". If someone made a collage of their "now", from their frame of reference, it would contain photos of all the events happening at that moment for them (assuming they could get instant access to those photos). E.g. it would have photos related to the election, and would have pictures of everything that anyone happened to be doing at that moment, along with things going on elsewhere in the universe. Am I correct to say that every person who is in the same inertial frame would have an identical collage, and no two observers who are in motion relative to one another would have the same collage?

Something interesting would happen, I think, when two people in different inertial frames came together with their collages. Say, for example, when traveling twin is born, he takes off at a speed such that when staying twin is 30 in his own reference frame he observes traveling twin to have aged only 15 years. When staying twin is 30, he has a collage made of his "now". One of the photographs of his collage will include his brother being 15 and he being 30. If at that same moment the traveling twin had a collage done, the traveling twin's collage would include a photograph of the staying twin being only 7.5 years old, and himself being 15! If they some how managed to get back together and compare collages, the staying twin could say "this, as my collage shows, is how things in the world were when you were 15, but the traveling twin would say "no, things were the way that my collage shows".

Is it true, though, that there would be one thing in common between the two collages: the photo of the traveling twin being 15? That seems like it leads to a paradox, though: why is it the 15 year old traveling twin that appears in both collages and not the 30 year old staying twin? According to the traveling twin, when he is 60, his brother is 30. So, why when they get back together shouldn't the staying twin's collage have a photograph of him being 30 and his brother 15, while the traveling twin has a collage of himself being 60 and the staying twin being 30? (After writing the rest of this post and coming back to think about it, I think the answer might be that traveling twin wouldn't even agree with the staying twin that the two collages were made at the same time).

Guess what T3 clock in the photograph? It might shows 03:30. A time after T1 time! So, S1 actually is seeing T1's future!

T2 is seeing S2 past!

Is it also true that T1 is seeing S2's future?

 

[Nugatory]

Defining "now" is not easy.

On the contrary, it is quite easy. There are two good definitions, and they are equivalent.
1) All events that have the same $t$ coordinate as the event corresponding to my current position are happening now; none others are.
2) All events that have an $x$ coordinate that differs from my $x$ coordinate by a quantity $\Delta{x}$, and for which a light signal emitted at that event will reach me at an event that is separated from the event that is my current position by an amount $\Delta{t}$, and $\Delta{t}=\Delta{x}$ are happening now; none others are.

#2 is actually a specific procedure (the only sensible one in flat spacetime, and equivalent to Einstein clock synchronization) for assigning the $t$ coordinates that we'd use in #1.

(assuming they could get instant access to those photos)

But they can't - it's physically impossible. Either you're ignoring the need for light to travel from the event to the camera, or you're placing the camera close to the remote event and we need to carry the film to the observer. Either way, the access is not "instant".

In fact, assuming the possibility of instant access is just overlooking (again!) the relativity of simultaneity. To say that I have instant access to a snapshot of an event at a distant location is equivalent to saying that I have access to the snapshot at the same time that the event happens.

 

[PeterDonis]

(assuming they could get instant access to those photos)

It's never a good idea to assume something that's contrary to the laws of physics.

The issue is not that defining "now" is hard; as Nugatory points out, it's easy. The issue is that "now" is not a physical thing; it's a convention. Asking what is happening on the Moon, or Alpha Centauri, or in the Andromeda galaxy "now" is not asking about physics; it's asking about how you want to define coordinates. If your intent is to ask about physics, then you need to retrain yourself not to ask about or think about "now" at all.

 

[Stephanus]

Is it also true that T1 is seeing S2's future?

Short answer is: Yes.
T1 can see S2's future (and S2's past).

Okay, let's discuss this further.
I decide to use "spoiler" to divide my post into some sections. If only PF forum has sub sections tools.
They are not spoilers after all. Just click any of them to see.

Spoiler: Seeing S2 FUTURE

From T point of view or frame of reference.
And from T frame of reference it's S group who are moving. T can consider them at rest.
T1 has a companion T2.
T1 and T2 synchronize their clocks. So do S group. S2, S3, S4, ...S9.

Spoiler: There are two events at the picture.

Event 1: T2 meets S2
Event 2: T1 meets S9
Both events happen at the same time in T frame of reference. But in S frame of reference T2 meets S2 before T1 meets S9

Before the experiment began, T1 ordered T2 to wait and photograph [Edit: S2's] clock WHEN T2 meets S2.
While T1 is photographing every S member who passes T1.
Later when the experiment is done. Perhaps over dinner (over dinner here is not a joke. See causality violation), when T1 and T2 meets T1 would ask T2.
T1: What time did you meet S2 and what is S2's clock?
T2: I meet S2 when my clock showed 03:30 and I see that S2 clock showed 01:30.
later T1 consulted T1's photographs and see that when T1 clock showed 03:30 T1 met [Edit: S17] and [Edit: S17's] clock showed 02:00.
So you can say that T1 is seeing [Edit: S2] future.

Spoiler: Seeing S2 PAST

Here, as in above. S group members are synchronizing their clock. T group as well.
T1 ordered T0 don't forget to record and take a picture when T0 meets S2.
T1 is photographing every S member who passes T1.
Later, after dinner (see causality of events) T1 asks T0.
T1: What time when you (T0) met S2?
T0: My clock showed 02:30 and I saw S2's clock showed 00:30.
Then T1 consults its photographs and finds that at 02:30, T1 meets S9 and S9's clock showed 00:00.

Spoiler: Causality violation

Because there's no way in both scenarios above that T1 knows at the same time (in the afternoon) when its companion meets S2. Perhaps this can be done over dinner?
So what if both T1 and its companion are able to know at once when the companion meets S2?
There will be causality violation.
Consider this.
In S frame. Event T2 meets S2 happens before T1 meets S9. in T frame both happen at the same time.

T2 is photographing what S2 is reading, and it shows that S2 is reading page 45. When Tintin finds a chest and is wondering what is inside.
And in page 49, professor Calculus concludes that it's not treasure but old documents.

Here what it looks in T frame:

So, if T2 can send message instantly to T1 and T1 can see that S9 is reading page 49 and T1 tells T2 instantly what is inside the chest, and T2 informs S2 that it's old documents not treasures, then S2 must have wondering. How could T2 knows before at S frame, all of them are reading page 45, they haven't read page 49 yet. Where do the information come from? The future?

Sorry for the spoiler buttons.

 

[phyti]

NoahsArk;

Since light serves as a universal measurement tool, the light clock is a logical place to start for understanding of Special Relativity.
Experimental evidence has shown the propagation speed of light in space (vacuum) is constant and (more importantly) independent of its source.
The clock cycle is defined as the motion of a photon from an emitter to a mirror m, and return to a detector adjacent to the emitter, a distance of 2w.
In the drawing, with w = 1.0, the clock moves at speed v in the x direction of the U frame, with the mirror oriented in the p direction, which is perpendicular to the x direction. The 'time' is equal to the line ct, i.e. a light-distance. If the mirror is not moving, the time for light to travel the distance 2w is t. If the mirror is moving, the motion of the successful photon in the x-p plane can be resolved into two components. The first equal to vt, compensates for the motion of the mirror. The second equal to ut, becomes the active component of the clock. As v increases, vt increases, and ut decreases, and the clock runs slower.
Time dilation, the decreasing rate of processes involving light interactions, is then a motion induced phenomenon, and a function of the clock speed. This would include the biological processes of the observer moving with the clock, resulting in an altered sense of time, and thus preventing detection of the slower clock and any accompanying devices. For the observer the clock is still functionally equivalent by definition to the U clock with one cycle equal to 2w/c.
If no questions, we will move on to the next issue.

Re: "now"
Einstein did define 'now' as the local clock event assigned to the event of interest. 1905 paper, par 1.
The question of what event is happening 'now' at a remote (atypical long distance,and not doing particle physics) location, is meaningless, since you are not there. That is the purpose of the clock synchronization, assignment by definition, since you cannot be certain of the reflection events, which are classified as remote.

 

[PeterDonis]

Short answer is: Yes.
T1 can see S2's future (and S2's past).

No. At least, this is IMO an extremely misleading way of putting it.

T1, at a given event on his worldline, can see whatever is in the past light cone of that event. That is the physical fact. The same is true of every other observer at every other event; they can see whatever is in the past light cone of that event.

But whether what T1, or any other observer, sees in his past light cone at a given event on his worldline, counts as the "future" or "past" of some other observer depends on what event we pick on the other observer's worldline, and what simultaneity convention we adopt. Neither of those things are physics; they are human conventions about how we use words and construct models.

 

[Stephanus]

On the contrary, it is quite easy. There are two good definitions, and they are equivalent.
1) All events that have the same $t$ coordinate as the event corresponding to my current position are happening now; none others are.

Is Event_1 "now" for Event_0 wrt Blue?

But for Event_1, "now" is Event_2 wrt Green?
"None other are" So, Event_0 is not "now" wrt Green?

On the contrary, it is quite easy. There are two good definitions, and they are equivalent.
2) All events that have an $x$ coordinate that differs from my $x$ coordinate by a quantity $\Delta{x}$, and for which a light signal emitted at that event will reach me at an event that is separated from the event that is my current position by an amount $\Delta{t}$, and $\Delta{t}=\Delta{x}$ are happening now; none others are...

#2 is actually a specific procedure (the only sensible one in flat spacetime, and equivalent to Einstein clock synchronization) for assigning the $t$ coordinates that we'd use in #1.

Is it like this? "Now" for Event_0 is Event_1 and "now" for Event_1 is Event_2?
Just like should the sun explode "now" we'll see the effect (darkness, and also gravity changes) eight minutes later?

But they can't - it's physically impossible. Either you're ignoring the need for light to travel from the event to the camera, or you're placing the camera close to the remote event and we need to carry the film to the observer. Either way, the access is not "instant".

In fact, assuming the possibility of instant access is just overlooking (again!) the relativity of simultaneity. To say that I have instant access to a snapshot of an event at a distant location is equivalent to saying that I have access to the snapshot at the same time that the event happens.

Yes, this I already have got the picture, although not instantly.

 

[Stephanus]

I think now I've seen the light.
I think the concept of "now" is a matter of coordinates.

If two events, each is not in the light cone of the other. The t coordinate of those events can change.
T0 on the left picture is below T1 on the right.
Because those events can't affect each other. They can't have cause and effect.
According to blue coordinate (left). Event_0 happens before Event_1.
Green (right) will see that Event_1 happens before Event_0.

But if two events are in the light cone.

There's no way to alter the order of the events.
For instance the sun explode "now" and a solar cell loss its power source 10 minutes later. So, there is now way to alter the event, solar cell loss its power before (no matter which coordinate you choos) the sun exploding.

 

[PeterDonis]

I think the concept of "now" is a matter of coordinates.

Indeed it is. But you don't seem to realize what that means. See below.

Is Event_1 "now" for Event_0 wrt Blue?

If you choose those coordinates, yes.

But for Event_1, "now" is Event_2 wrt Green?

If you choose those (different) coordinates, yes.

Is it like this? "Now" for Event_0 is Event_1 and "now" for Event_1 is Event_2?

No. "Now" depends on what coordinates you choose. There is no absolute "now". "Now" is just a convention. It's not a physical thing.

Please read and re-read the above until it sinks in.

If two events, each is not in the light cone of the other. The t coordinate of those events can change.

That's true of any events; if you change coordinates, the coordinates (including the t coordinate) can change.

The correct way of saying what you are trying to say in this and the next part of your post is: if two events are spacelike separated, then their time ordering (which one happens first) is not invariant; it can change if you change coordinates. It is also possible to choose coordinates such that neither happens first: they both happen at the same time.

But if two events are timelike or null separated, then their time ordering is invariant; it is independent of how you choose coordinates. And it is impossible to choose coordinates such that they happen at the same time; one will always happen before the other (and which one happens first is invariant).

Notice how I never used the word "now" at all in the above.

 

[Stephanus]

Please read and re-read the above until it sinks in.

Yes it does sink in. I just don't know how to express "it" in language. But "now" is not real. Even if you are face to face with someone, you can't see him "now", it takes some nanosecond (pico?) for light to his/her face to your eyes. (Much less talking, some microsecond for the sound wave to reach your ear). As you said, it's just a convention, not real thing.

If two events, each is not in the light cone of the other. The t coordinate of those events can change..

That's true of any events; if you change coordinates, the coordinates (including the t coordinate) can change.

After I read it again, I realize. What actually I wanted to say is "The order of the events can change".

The correct way of saying what you are trying to say in this and the next part of your post is: if two events are spacelike separated...

Yep, spacelike and timelike. Thanks to PF Forum, I know that.
Yes, timelike -> the order is invariant.
Spacelike -> the order can change.
Thanks a lot,

 

[PeterDonis]

timelike -> the order is invariant.

Strictly speaking, timelike or null (lightlike) -> order is invariant.

 

[robphy]

Hence, timelike or lightlike are subsets of "causal"... And "order" is more fully "causal order".