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

[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.