Simultaneity hyperplanes "curved" while stationary?

[KyungMin]

I found this spacetime diagram at https://en.wikipedia.org/wiki/Proper_acceleration with an accelerating traveler, resulting in "curved" hyperplanes of simultaneity.

Notice how the simultaneity planes are angled "normally" (The faster the steeper the slope and no slope when stationary) but only when close to the traveler.
However the simultaneity planes get distorted in farther space, obviously due to acceleration.

Also in the pink area, where the traveler is stationary, the simultaneity planes are horizontal close to the traveler like a normal spacetime diagram, however in farther space the simultaneity planes are CURVED, which can only be explained by "the result of future acceleration that has not yet happened".

Also in the yellow area, the traveler finished the journey and is stationary, but the simultaneity planes are curved in farther space, if interpreted literally, due to past acceleration.

Now I see why Einstein told Besso's family that "time is a stubborn illusion" after he had died.

Or am I completely mistaken?

 

[Ibix]

Simultaneity is just a convention so, within reason, you can adopt any shape of plane you like. Those diagrams are most likely drawn using radar coordinates, which are exactly what they sound like. You send out a radar pulse and wait for its return. You say that the event it reflected off occurred half way between the emission and reception events (using your proper time) and at $c\Delta \tau/2$ distance.

Obviously the pulse return time depends on your velocity profile while the pulse is in flight. And you can't define simultaneity until it returns. So, the right way to state your conclusion is that the simultaneity planes are CURVED, which can only be explained by your decision to wait for "the result of future acceleration that has not yet happened" before choosing how you will define simultaneity in a dynamic situation (my additions in bold).

It's also worth asking: whose future do you mean? The diagram was drawn by the stay-at-home twin. But you can always find an inertial frame that will regard all of the accelerated observer's plane of simultaneity to the left to be in the past and all to the right in the future.

Radar coordinates:
https://arxiv.org/abs/gr-qc/0104077

 

[Orodruin]

What simultaneity is is purely conventional. You can pick any space-time foliation of space-like surfaces.

The simultaneity surfaces here have just been chosen in a particular way so that they are orthogonal to the workd line of the accelerating observer.

You also cannot talk about "future acceleration" without reference to a simultaneity convention...

 

[KyungMin]

What simultaneity is is purely conventional. You can pick any space-time foliation of space-like surfaces.

The simultaneity surfaces here have just been chosen in a particular way so that they are orthogonal to the workd line of the accelerating observer.

You also cannot talk about "future acceleration" without reference to a simultaneity convention...

The simultaneity planes are physically relevant and meaningful. In the light blue area the increased gaps in simultaneity planes perfectly explains slowing down of traveler's time compared to stationary twin! They really represent differing tickings of time

 

[Orodruin]

The simultaneity planes are physically relevant and meaningful.

No they are not. You can only directly compare clocks that are colocated. If they are not, their relative times depend on the simultaneity convention and therefore unphysical.

 

[KyungMin]

No they are not. You can only directly compare clocks that are colocated. If they are not, their relative times depend on the simultaneity convention and therefore unphysical.

But I assume the results will all be same despite varying clock comparing procedures... after all, the traveling twin will age less than the stationary twin when he returns no matter how you measure it!

 

[Orodruin]

But I assume the results will all be same despite varying clock comparing procedures... after all, the traveling twin will age less than the stationary twin when he returns no matter how you measure it!

The key point being "when he returns". When he returns they will be colocated and can compare clocks without a simultaneity convention. I suggest reading my Insight article on the geometrical view of the twin paradox and that you understand the relativity of simultaneity as it occurs between different Minkowski frames before you try understanding simultaneity conventions with bent foliations.

 

[KyungMin]

The key point being "when he returns". When he returns they will be colocated and can compare clocks without a simultaneity convention. I suggest reading my Insight article on the geometrical view of the twin paradox and that you understand the relativity of simultaneity as it occurs between different Minkowski frames before you try understanding simultaneity conventions with bent foliations.

Your so called "co-location" is impossible in real world. two observers must be at some distance apart, however small, or they would form a singularity in spacetime. So stationary twin's clock is 10 and traveler twin's clock is 8 in both frames of reference when they "meet" is merely an approximation, which only holds when two observers are INFINITELY CLOSE.

 

[Ibix]

If two clocks are 1m apart then the maximum possible error from ignoring the simultaneity convention is a little over 6ns. Next to a multi-year trip this is negligible. This is not the same situation at all as as comparing clocks situated three light years apart.

 

[Orodruin]

Your so called "co-location" is impossible in real world. two observers must be at some distance apart, however small, or they would form a singularity in spacetime. So stationary twin's clock is 10 and traveler twin's clock is 8 in both frames of reference when they "meet" is merely an approximation, which only holds when two observers are INFINITELY CLOSE.

As stated by Ibix, you often have to make idealisations in physics. While my post contained subtextual idealisations, I would say that you need to take a few steps back to start by understanding relativity of simultaneity in the Minkowski setting if you really want to understand how simultaneity conventions work, as I pointed out in #7.

 

[KyungMin]

As stated by Ibix, you often have to make idealisations in physics. While my post contained subtextual idealisations, I would say that you need to take a few steps back to start by understanding relativity of simultaneity in the Minkowski setting if you really want to understand how simultaneity conventions work, as I pointed out in #7.

But simultaneity planes tilt completely according to velocity.. In a spacetime diagram where light paths are at 45°, simultaneity planes of a frame moving at half the speed of light tilt exactly at 22.5° compared to horizontal simultaneous planes of a stationary frame in order to preserve constant light speed. How on Earth would they depend on measurement convention?
And the diagram I found follows the same law in colored areas where all the grids are straight lines.

 

[Ibix]

How on Earth would they depend on measurement convention?

Nothing obliges me to use the Einstein coordinates in my rest frame. I can use another frame's coordinates just as well and, as you noted, preserve the constant speed of light. But now I have a different definition of simultaneity.

Other answers to this are possible. You may wish to read the article by Dolby and Gull that I linked in #2 post to see how the diagram you linked was generated.

 

[Dale]

The simultaneity planes are physically relevant and meaningful.

In what way is simultaneity physically meaningful? Two simultaneous events are not causally connected. How then can their simultaneity be physically meaningful?

 

[Orodruin]

But simultaneity planes tilt completely according to velocity.

Simultaneity surfaces depend on your choice of coordinates. What you say here is true in Minkowski coordinates, but not in general.

How on Earth would they depend on measurement convention?

That the simultaneity surfaces are tilted with respect to each other is the difference of simultaneity convention!

 

[A.T.]

...in order to preserve constant light speed.

Note that constant light speed only applies to inertial frames.

 

[KyungMin]

In what way is simultaneity physically meaningful? Two simultaneous events are not causally connected. How then can their simultaneity be physically meaningful?

Look at this spacetime diagram I made.
Length contraction is just caused by different perceptions of simultaneity.
You can only observe past so nothing is simultaneous?

 

[Mister T]

Your so called "co-location" is impossible in real world. two observers must be at some distance apart, however small,

We're using what's called the particle model in these analyses. In this model the sizes of objects are small enough compared to their separation distance that we can ignore them. If that were not the case you'd have to specify which part of the traveler your diagram is drawn for. Draw one for his feet and it would look different than the one for his head. Insist that the diagram is for his head, and you'd have to specify which side of his head. Specify that it it's for his left eyeball and you'd have to specify which side of his eyeball. And so on. In the limit, the traveler is modeled as a particle.

Note that in a statement of the twin paradox we just say the twins are together before the departure and together again at the reunion. We don't worry about the fact that they can't occupy the same space at the same time. We just say that the distance between them is small compared to the distances between them during the trip. How small? Small enough to ignore. As in, the error generated by doing so is negligible. That's the particle model and it's used throughout physics, not just in relativity. You use it, for example, when you study rectilinear motion with uniform acceleration during that first few weeks of your freshman physics class.

 

[Dale]

Look at this spacetime diagram I made.
Length contraction is just caused by different perceptions of simultaneity.
You can only observe past so nothing is simultaneous?

Multiple events can certainly be simultaneous in some frame, but it doesn't matter physically. Nature cares about causality, not simultaneity.

 

[KyungMin]

You also cannot talk about "future acceleration" without reference to a simultaneity convention...

It's also worth asking: whose future do you mean? The diagram was drawn by the stay-at-home twin. But you can always find an inertial frame that will regard all of the accelerated observer's plane of simultaneity to the left to be in the past and all to the right in the future.

Of course if you compared two different reference frames their future and past would get mixed up due to relativity of simultaneity but isn't it obvious I'm talking about only the traveler's frame, and as you see each simultaneity plane is above or below the other, no intersection, thus no mixup of past and future if you consider only one frame. So, before acceleration=past, after acceleration=future, no mixup unless you can time travel

You physicists only think about rigorous theories and lost sense of intuition...

 

[PeterDonis]

You physicists only think about rigorous theories and lost sense of intuition...

Please refrain from such comments; you are getting close to receiving a warning. Your claim that "the simultaneity planes are physically relevant and meaningful" is mistaken, and complaining about others telling you so and trying to explain why does not change that fact.

 

[KyungMin]

Please refrain from such comments; you are getting close to receiving a warning. Your claim that "the simultaneity planes are physically relevant and meaningful" is mistaken, and complaining about others telling you so and trying to explain why does not change that fact.

In this spacetime diagram with 3 frames, their relative velocities are ±0.6c and 1.2/1.36c and the dotted lines are the "imaginary" simultaneity planes.
Look how the intersections of simultaneity planes and time axes of other observers denote time dilation precisely.
So time dilation is unphysical too?