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

[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".

 

[NoahsArk]

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.

I know that it will take time to create the collage of photos. My point is, though, that eventually it will be possible to create the collage of photos of whatever was going on at a certain point t in a given observer's frame. Is it true that all collages from the same IFR will be identical, and all collages from different IFRs will be different? In other words, is the line of simultaneity the same for all observers who are in the same IFR, and is each line of simultaneity different for each separate IFR?

Regarding the question about whether or not something which has happened in the staying twin's frame could have already happened in the traveling twin's frame:

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.)

Given that this is true, I'm assuming that, although our line of simultaneity may contain events from another observer's future or past, we can only ever observer their future and past from a great distance. We can never actually "visit" their past or future because in order to do so we'd have to switch frames of reference, and as soon as we caught up to their location we would have ended up in their present. Is this something like how it works?

Also, what does it mean that it's not a "physical" thing?

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.

But isn't it true that even if an event is happening at a very far distance from me, it's still happening in my now, I'll just observe it later because it will take the light from the event time to travel to me?

Regarding the question about T1 seeing S2's future, let's say that S2 is a block of ice that has already melted in T1's reference frame. If S2 is coming inbound while T1 is coming outbound, and T1 sticks his hand out of the window as S2 passes, I'm assuming T1 won't feel any ice because in his frame it already melted? If that's true, what would S2 see as T1 sticks his hand out the window? I'm assuming he won't see T1 hit his hand against an piece of ice?

 

[Nugatory]

I know that it will take time to create the collage of photos. My point is, though, that eventually it will be possible to create the collage of photos of whatever was going on at a certain point t in a given observer's frame. Is it true that all collages from the same IFR will be identical, and all collages from different IFRs will be different? In other words, is the line of simultaneity the same for all observers who are in the same IFR, and is each line of simultaneity different for each separate IFR?

Quick answer: Yes, to all three questions.
Longer answers:
Yes, all collages constructed by all observers who are at rest relative to one another for long enough to assemble their collages will be identical. These collages will also differ from those collected by other groups of observers at rest relative to one another but moving relative to the first group of observers.

Yes, each line of simultaneity (that is, the set of all events that have the same time coordinate) will be will be the same for everybody who chooses to use the same frame to assign time coordinates to events.

Yes, if a different frame is used to assign time coordinates to events, then different sets of events will have the same time coordinate so the lines of simultaneity will be different.

Note that I have very carefully avoided saying that anyone or anything is "in" a frame. You'll hear that phrase all the time because it is convenient (note how stiff and awkward the last three sentences sound because I'm not using that natural-sounding phrase), but it is sloppy and tends to confuses beginners about what a frame is. A frame is a convention for assigning time and space coordinates to events, and when we say that an observer is "in his" frame, that's just a convenient shorthand for the more precise but clumsy-sounding "the observer has chosen to assign coordinates to events using a convention in which his own spatial coordinates are not changing and therefore he is at rest". So anytime you find yourself thinking or writing about things happening or being "in" a frame... Stop and restate it, at least for yourself, to be sure that you yourself really understand what you're saying. You may find that this clarifies some of the other questions you've asked in the previous post.

 

[Nugatory]

Regarding the question about T1 seeing S2's future, let's say that S2 is a block of ice that has already melted in T1's reference frame. If S2 is coming inbound while T1 is coming outbound, and T1 sticks his hand out of the window as S2 passes, I'm assuming T1 won't feel any ice because in his frame it already melted? If that's true, what would S2 see as T1 sticks his hand out the window? I'm assuming he won't see T1 hit his hand against an piece of ice?

Using your idea of collages assembled after the fact to allow for light travel time: when you work through the math, it will turn out that any collage constructed by any observer anywhere in the universe, no matter how fast they're moving and how they change their speeds, will be consistent. If the collage contains a photo of an unmelted block of ice, it will also contain a photo of T1 before he's reached S2 and stuck out his hand. If it includes a photo of T1 after he's passed S2, in that collage T1 will have a wet hand and the collage will also include a photo of a puddle of water where the block of ice had been.

Thus, the physical story is the same no matter what frame you use to assign coordinates to the various events in the history of S2, T1, and the rest of the universe: there was a block of ice, it melted, S2 passed by and stuck his hand in the puddle of meltwater.

 

[Stephanus]

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

Ahh, lightlike. I think it's 45 degree. Thanks PF Forum.

 

[Nugatory]

Ahh, lightlike. I think it's 45 degree.

Whether the path of the light forms a 45 degree angle or not depends on the choice of coordinates. It always will when using the simplest coordinates in flat spacetime, so that's what you usually see drawn in spacetime diagrams but it doesn't have to be that way.

It's much better to describe a lightlike path in coordinate-independent terms: The spacetime interval along it is zero. This will be true no matter what coordinate system you choose.

 

[Herman Trivilino]

Given that this is true, I'm assuming that, although our line of simultaneity may contain events from another observer's future or past, we can only ever observer their future and past from a great distance.

You can never observe an event in their future. It hasn't yet happened to them. An event is something that occurs at a particular position and at a particular clock-reading. In different frames of reference different positions and clock-readings can be assigned to an event, but that has no effect on the reality of the event occurring at a particular place and time. After the event occurs news of it can be transmitted outward at a speed no faster than $c$. Therefore no one can ever observe it before it happened.

Also, what does it mean that it's not a "physical" thing?

A convention is an agreed-upon way of doing something. It's part of the modelling process that is physics, but it's not part of the physical world that the physics is modelling.

 

[Stephanus]

So,
Timelike,
More than 45 degree.
Order is invariant.

Spacelike
Less than 45 degree.
Order can change.

Likelight.
Exactly 45 degree.
Order can't change.
Even though $\Delta x$ and $\Delta t$ can change.
But, always $\Delta x = \Delta t$

...but it's not part of the physical world that the physics is modelling.

Physical or not, Now I know what "now" is.

 

[Micheth]

One simple approach to the paradox is to trust the stay-at-home twin's calculations (you know you can trust them, because you've nailed the concepts of inertial reference frames and time dilation). And, you can simply say that you don't know how to do the calculations for the moving twin because of the periods of accelerating and decelerating reference frames
...
In any case, this verifies what you know from the analysis in A's frame: that 10 years have passed on A's clock and only 6 years on B's during the out and back journey that B made.

Does it completely destroy the utility of SR to simply take the viewpoint that it really is only valid to talk about accelerated bodies as moving?
In other words, to say that the moving twin is in fact unjustified in saying the whole universe moved away from him and then toward him, which would have resulted in less time passing for the staying twin than himself (creating a paradox), but then doing (relatively) complex calculations with his frame of reference to show how the paradox is mathematically resolved.
That is to say, there would be no paradox in the first place if one were to accept the (wild?) proposition that when a spaceship undergoes events that cause its acceleration, less time actually passes for it than for the relevant entities that were not accelerated, and that that is the only valid viewpoint.
Therefore, 8 years passed for the spaceship and 10 for the Earth, period.
I understand (albeit loosely) that the time frame rotation considerations allow the paradox to be resolved, but doesn't that make the situation much more complex than it needs to be in order to be explained?
Why not merely make the reasonable interpretation that if an entity acclerates, and a certain time period passes until it decelerates and returns, then yes, *it* was the entity that actually moved, and therefore time dilated for it, and nothing else (respect to the IFR)...

 

[Ibix]

You can easily enough set up variants where both twins undergo the same accelerations at different times and end up with different elapsed times. So, no, it isn't to do with who accelerates. It's that the two twins follow different paths through spacetime and those paths have different "lengths", which turn out to be directly related to their measured elapsed times.

When a twin is accelerating he can say that he is changing his motion. He can't say in any absolute sense whether he was moving or not before or after the acceleration because there is no absolute motion.

 

[PeroK]

Does it completely destroy the utility of SR to simply take the viewpoint that it really is only valid to talk about accelerated bodies as moving?
In other words, to say that the moving twin is in fact unjustified in saying the whole universe moved away from him and then toward him, which would have resulted in less time passing for the staying twin than himself (creating a paradox), but then doing (relatively) complex calculations with his frame of reference to show how the paradox is mathematically resolved.
That is to say, there would be no paradox in the first place if one were to accept the (wild?) proposition that when a spaceship undergoes events that cause its acceleration, less time actually passes for it than for the relevant entities that were not accelerated, and that that is the only valid viewpoint.
Therefore, 8 years passed for the spaceship and 10 for the Earth, period.
I understand (albeit loosely) that the time frame rotation considerations allow the paradox to be resolved, but doesn't that make the situation much more complex than it needs to be in order to be explained?

The traveling twin is perfectly justified to say that the home twin moved away from him, but he knows that he (the traveller) underwent proper acceleration and therefore needs a more advanced analysis to determine what exactly went on during his acceleration. In particular, he cannot use a single spacetime diagram for the entire journey, as the home twin can.

Note that in classical physics if you accelerate in a car, you will see a lampost accelerate towards or away from you. But, you cannot measure any force on the lampost that caused its acceleration. Your view is perfecly valid, but you cannot use Newton's second law $F = ma$ in your reference frame to explain the motion of the lampost. Newton's second law is only valid in an IRF (Inertial Reference Frame). To analyse motion in an accelerating reference frame in classical physics you must add a so-called fictitious force. This is analogous to an accelerating observer in SR not being able simply to use a single spacetime diagram.

Why not merely make the reasonable interpretation that if an entity acclerates, and a certain time period passes until it decelerates and returns, then yes, *it* was the entity that actually moved, and therefore time dilated for it, and nothing else (respect to the IFR)...

You can take this interpretation, but it doesn't get you very far. The issue of relative motion is that although you can say that the accelerating twin is definitely at times changing his velocity, you cannot give him an absolute velocity at any time. Also, while he is in constant motion, he sees time dilation in the home twin exactly as the home twin sees in him. If the traveling twin were absolutely moving at whatever speed and the stay at home twin were absolutely at rest, then only one would see time dilation. But this is not the observed experimental case.

 

[pervect]

Does it completely destroy the utility of SR to simply take the viewpoint that it really is only valid to talk about accelerated bodies as moving?
In other words, to say that the moving twin is in fact unjustified in saying the whole universe moved away from him and then toward him, which would have resulted in less time passing for the staying twin than himself (creating a paradox), but then doing (relatively) complex calculations with his frame of reference to show how the paradox is mathematically resolved.
That is to say, there would be no paradox in the first place if one were to accept the (wild?) proposition that when a spaceship undergoes events that cause its acceleration, less time actually passes for it than for the relevant entities that were not accelerated, and that that is the only valid viewpoint.
Therefore, 8 years passed for the spaceship and 10 for the Earth, period.
I understand (albeit loosely) that the time frame rotation considerations allow the paradox to be resolved, but doesn't that make the situation much more complex than it needs to be in order to be explained?
Why not merely make the reasonable interpretation that if an entity acclerates, and a certain time period passes until it decelerates and returns, then yes, *it* was the entity that actually moved, and therefore time dilated for it, and nothing else (respect to the IFR)...

I'm not quite sure how to answer this. In a hypothetical SR universe, It's equally consistent for the moving twin to regard himself as stationary, and the universe moving, as it is for the moving twin to consider the universe stationary, and himself moving. I'm a bit uneasy about the word "unjustified" here, the moving twin can adopt either point of view, as long as she sticks with it.

If you have two twins each driving a car, you don't expect them to both have driven the same distance when they reunite. Likewise, there is no inherent paradox to imagine two twins progressing through space-time, and being different ages when they reunite. There isn't any distance paradox, because you never expected the twins to drive the same distance in the first place.

In the case of the car driving through space, we say that the shortest distance between two points is a straight line, and thus we expect anyone who doesn't follow a straight line to drive a longer distance.

In the case of the space-time twins, the one that journeys through space-time in the space-time equivalent of a straight line, which is called a "geodesic", takes the longest time. This is perhaps confusing (because of the difference where "shortest" changed to "longest"), but it's not "paradoxical". If one draws a space-time diagram (which is highly recommended, but seems surprisingly difficult to get people to actually do for some unknown reason), the geodesic path are in fact straight lines on the space-time diagram.

The twin that ages less ages less not because they accelerated so much as because they didn't follow a geodesic path (the equivalent of a straight line path for the twins driving through space).

 

[Herman Trivilino]

Why not merely make the reasonable interpretation that if an entity accelerates, and a certain time period passes until it decelerates and returns, then yes, *it* was the entity that actually moved, and therefore time dilated for it, and nothing else (respect to the IFR)...

Because it doesn't provide a satisfactory quantitative answer at the most basic of levels. The magnitude of the difference in the twins' ages depends on the amount of time spent traveling while not accelerating. The easiest way to see this is to plot their paths through spacetime on a spacetime diagram. The longer the path lengths the greater the difference in their ages.