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There are some threads about this, but some answers seem to disagree.

First, this is what Einstein said on this matter:

The light rays emitted by the flashes of lightning A and B would reach him simultaneously", and again: " the observer will see the beam of light emitted from B earlier

and this is the reason why he says:

Events which are simultaneous with reference to him are not simultaneous with respect to the train, and vice versa.

(Source)

He basically says that it is the observation of two event that can either be simultaneous or not depending on the frame of reference. But do the observers disagree on the simultaneity of the events after the "post-processing" (tracing back the actual events in spacetime)? It is at this point that things start to become difficult: after the "post processing", will the observers disagree on the order of the actual events, or will they agree?. Which is correct and why? And what does a Cauchy Hypersurface have to do with this?

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  • $\begingroup$ I think your phrasing in the question might be a bit misleading, your first quote is: "If an observer [...] in the train did not possess this velocity, [...] the flashes of lightning A and B would reach him simultaneously." It then continues, "Now in reality he is hastening towards the beam of light coming from B [...]. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A." So your two quotes in your first block quote refer to different physical situations. $\endgroup$
    – David
    Commented Feb 9, 2023 at 21:07
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    $\begingroup$ On the topic of raw observations vs measurements after post-processing to account for light travel time delays: physics.stackexchange.com/a/556942/123208 Also related (kind of): physics.stackexchange.com/a/597516/123208 $\endgroup$
    – PM 2Ring
    Commented Feb 10, 2023 at 9:52

5 Answers 5

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do the observers disagree on the simultaneity of the events after the "post-processing" (tracing back the actual events in spacetime)?

Yes. The relativity of simultaneity specifically refers to the disagreement about simultaneity that remains after the observers have accounted for the finite speed of light.

In other words, if I receive light from two novas tonight, one 100 light years distant and one 200 light years distant, I will say that they did not occur simultaneously. Similarly, if I receive the light from one 200 light years distant and read that a 100 light year distant nova was observed here 100 years ago, then I will say those did occur simultaneously.

Someone traveling with respect to me will make the same computations and disagree about both results.

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  • $\begingroup$ How is that possible? That is not clear to me. Now imagine this: the novas are actually at rest with each other, stationary but distant 50 light years from each other. One nova will see the other explode after exactly 50 years from its own explosion, and vice versa. After "post-processing" the novas agree that they are simultaneous. How is possible that others disagree on the simultaneity (or even the order) of the events after adjusting for all the possible variables (light-bending, light-speed travel, etc.)? $\endgroup$
    – GennaroMa
    Commented Feb 8, 2023 at 20:09
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    $\begingroup$ Indeed, that is the single most difficult concept to learn when studying special relativity for the first time. In my opinion, the lesson to learn is that nature doesn’t care about simultaneity. It only cares about causality. Causes always come before effects, in all reference frames. But otherwise nature just doesn’t care $\endgroup$
    – Dale
    Commented Feb 8, 2023 at 20:26
  • $\begingroup$ @GennaroMa This may be hard to accept, but the question "how is this possible?" is, in this specific instance, not a scientific one, but a metaphysical, thus philosophical, question. From a purely scientific perspective, the only answer we can give to that question is: "Dunno, that's just the way nature is. Nature doesn't care about our intuitions". $\endgroup$ Commented Feb 10, 2023 at 7:58
  • $\begingroup$ @GennaroMa With the "post-processing" you are prescribing a global simultaneity where there is a frame of reference that can observe everything, everywhere instantaneously when one does not exist. $\endgroup$
    – DKNguyen
    Commented Feb 10, 2023 at 19:37
  • $\begingroup$ @Vercassivelaunos I am not sure that "we don't know" is entirely accurate. The other fundamental observation, after Dale's very well put remark about causality, is that the future is unknown. This is often seen as a paradox in relationship with causality, but it is like that only in a Galilean setting. I would instead start with two fundamental observations: causality AND an open future. In a relativistic setting that even determines the structure of quantum mechanics if my intuition is not entirely wrong. The important thing to understanding is to pick the right starting points. $\endgroup$ Commented Feb 1 at 4:21
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It is better to think of the relativity of simultaneity as being a property of the geometry of spacetime, independent of any observers. Take this analogy... in 3D space there is no absolute 'up' direction. On Earth, we take 'up' by convention to mean a direction normal to local sea level, so 'up' in Australia is a quite different direction to 'up' in the US. We say two places are level if they have the same altitude, but that is only true in a given frame of reference. Sydney is above sea level in Australia, and New York is above sea level in the US, but New York is below sea level if you consider sea level to be an infinitely extended horizontal plane centred on Sydney, and Sydney is below sea level if you consider sea level to be an infinitely extended horizontal plane through New York. So the idea of 'level' is entirely frame-dependent.

In 4d spacetime, there is no absolute time direction. If you and I are moving relative to each other, our respective time axes are tilted. In your frame of reference, a plane of simultaneity is a level slice through spacetime normal to your time axis, which means that it is a sloping slice through time in my frame. That is like Sydney sea level being a sloping slice through space for a person in New York. It is just a property of reference frames. You don't need to bring observers into the picture. Indeed, mentioning observers in explanations of special relativity is, in my opinion, the cause of half of the misunderstandings that are raised by visitors to this site.

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  • $\begingroup$ But your example doesn't rule out absolute simultaneity. It seems to me GR doesn't care about the actual events, just about perception. Imagine a big box in spacetime that contains a set of balls and two synchronized clocks. At time t0, everything is at rest with respect to each other. From time t1, the balls will move at different constant speeds with respect to the box. $\endgroup$
    – GennaroMa
    Commented Feb 11, 2023 at 12:16
  • $\begingroup$ If you're born on a ball at t1, because you cannot know your past history of accelerations (which are absolute from the box's frame of reference), you cannot tell what your absolute speed is -> so you consider yourself at rest and the two clock are not simultaneous. Isn't this what's happening here? $\endgroup$
    – GennaroMa
    Commented Feb 11, 2023 at 12:18
  • $\begingroup$ No. My explanation does rule out absolute simultaneity, so you must have mis-understood what I said. $\endgroup$ Commented Feb 11, 2023 at 16:58
  • $\begingroup$ Sorry, you said that since every inertial frame has its own plane of simultaneity, then simultaneity is relative. But each inertial frame consider itself at rest: this is an inaccuracy (as i showed in the example, you can also check the Twin Paradox in my other comment) since each inertial frame has had an history of accelerations that brought to that constant motion through space greater that zero. If this was the case, then relativity of simultaneity would be an "observer issue", because the observer cannot exactly determine its motion through space. $\endgroup$
    – GennaroMa
    Commented Feb 11, 2023 at 18:20
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    $\begingroup$ `Wrong again. You are assuming there is an absolute stationary frame relative to which there are absolute speeds- what evidence do you have for that? All of modern physics is based on the assumption that there is no absolute frame of rest. $\endgroup$ Commented Feb 11, 2023 at 19:12
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I want to address an ambiguity around the use of the word 'disagree' here.

I propose the following about thought demonstrations:
all of the protagonists featured in the narrative are to be physicists who are fluent in application of relativistic physics.

In a thought demonstration of relativity of simultaneity the two observers are both aware of relativity of simultaneity. Both observers will agree that the point of view of the other observer is self-consistent.

So: in that sense the two observers do not disagree with each other.



The actual message is: in Minkowski spacetime simultaneity is underdetermined.

As pointed out in a comment by stackexchange contributor Dale, the limits are that the same causality relations must still obtain.

Within the boundaries of still obtaining the same causality relations there is a certain leeway in adopting a plane of simultaneity.

Within that leeway Einstein synchronization procedure is one of the available options. For a given inertial coordinate system the Einstein synchronization procedure is the symmetrical approach. As a general rule, when there is leeway, pick the symmetrical choice as your convention.

(You do need to pick one choice, in order to represent physics taking place. Once you have made your pick - for a given inertial coordinate system - you must stick with that choice, otherwise you would introduce self-contradiction.)



To the two observers:
The two observers have a velocity relative to each other, so for each the co-moving inertial coordinate system is a different one. For each inertial coordinate system the Einstein synchronization procedure arrives at a different plane of simultaneity.

As stated earlier, we should assume the two protagonists of the thought demonstration are professional physicists, so they are aware of the reason why for two different inertial coordinate systems the Einstein synchronization procedure arrives at two different planes of simultaneity.

So that is why I find it awkward to see assertions that the two observers will disagree. Disagree about what? We should assume both are professional physicists: they will not disagree about the proper application of special relativity.



As pointed out in other answers: 'relativity of simultaneity' refers exclusively to that what remains after transmission delays have been taken into account.

(Therefore, if you read a discussion of relativity of simultaneity, and the author suggests that transmission delay effects are part of the story of relativity of simultaneity, then stop reading that author.)


Further viewing:
The series of animated gif's by Andrew Hamilton for special relativity, including discussion of relativity of simultaneity

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  • $\begingroup$ "In Minkowski spacetime simultaneity is underdetermined". It seems to me that the problem of simultaneity appears when two observers are in motion in respect to each other and since "there is no absolute motion", then each plane of simultaneity is as valid as the other. But doesn't twin paradox show exactly that movement is absolute? From the Minkowski diagram, it follows that clock A is slower than clock B, and also that clock B is slower than clock A. But this would mean that no time-dilation should be measured when comparing the two clocks. $\endgroup$
    – GennaroMa
    Commented Feb 11, 2023 at 13:04
  • $\begingroup$ But in reality, one clocks ticks slower -> the axiom of SR that there is no preferred inertial frame is not exactly correct (and acceleration doesn't solve the problem), because the slower clock is moving with respect to the other absolutely. So is relativity of simultaneity just a mask for the fact that "we cannot know our inertial frame's past acceleration, so each frame that is in motion at constant speed consider itself exactly at rest, therefore we will disagree wether two events in space are simultaneous"? $\endgroup$
    – GennaroMa
    Commented Feb 11, 2023 at 13:05
  • $\begingroup$ @GennaroMa Given your questions: I recommend that you use a threaded forum instead, for example: physicsforums. Stackexchange is specifically designed to not be a threaded forum. The stackexchange policy is that the comment section is not for discussion, and I endorse that policy; I think it is healthy. Other than that: indeed the Twin scenario. For the stay-at-home twin there is a single plane of simultaneity from start to end. For the traveling twin there is no a single plane of simultaneity, since that journey must involve making a U-turn. $\endgroup$
    – Cleonis
    Commented Feb 11, 2023 at 13:46
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The answer is yes only for the rather silly definition of "observer" (and "disagree") that is used in discussions of special relativity.

The "observers" in special relativity are not people or cameras or anything of the sort. They are networks of clocks and metersticks, large enough in space and time to reach whatever is being "observed" (spanning galaxies if necessary), and synchronized in a certain way. To "observe" an event means to note the position and time reading of the clock in that network that is nearest the event.

So when they say that two relatively moving observers have different notions of simultaneity, they are technically correct: two constellations of Einstein-synchronized clocks that are in relative motion will report different time readings where they coincide.

In reality, we don't construct coordinates in which we are personally at rest at all times. We use coordinate systems that are available to us. For example, on cosmological scales, the averaged Hubble flow can be used as a reference, giving what are called comoving coordinates. The solar system has a largish speed (about $0.001c$) in those coordinates. We don't define an alternate cosmological coordinate system in which we are at rest, because they would be inconvenient and would serve no purpose. On Earth, when you're driving to work, you normally think in coordinates affixed to the Earth, in which your workplace is stationary and you are moving toward it. It makes sense to define a car-centered coordinate system for some purposes, but those coordinates extend only as far as your car; it wouldn't make sense to extend them even to your workplace, much less to the Andromeda galaxy, because the car doesn't extend that far.

If driving speeds were large enough that time dilation noticeably affected wristwatch readings, people would adjust their wristwatches to match a reference clock when they arrived at work, because the whole point of having a wristwatch is to coordinate times with other people. They wouldn't claim that their personal wristwatch reading is the correct one and "disagree" with people who live a different distance from work.

I strongly dislike Dale's answer because it doesn't clarify the above points, but I like this comment he made:

In my opinion, the lesson to learn is that nature doesn’t care about simultaneity. It only cares about causality. Causes always come before effects, in all reference frames. But otherwise nature just doesn’t care.

"Observer" is a misnamed abstraction. The Earth and the Hubble flow are "observers"; you aren't. You can use any notion of simultaneity you want, as long as it respects light-cone causality. Nature doesn't care.

The relevance of Cauchy hypersurfaces is that in any global coordinate system that respects light-cone causality, the surfaces of constant time are Cauchy surfaces.

See also this answer.

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  • $\begingroup$ To answer your comment, please read the answers i gave to other comments. How can simultaneity have different notion? It is clear what it means and should not depend on one's plane of simultaneity since every inertial frame considers (incorrectly) itself at rest. $\endgroup$
    – GennaroMa
    Commented Feb 11, 2023 at 14:43
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You're getting close, but you're not there yet. You have set up a straw man-actually, the observer is superfluous. The error lies elsewhere. The quandary in the text is very close to what you highlight. I found it a long time ago (40 years).

It is the phrase "fallt zwar...zusammen." THAT is where the problem lies. The English translation is "naturally coincides."

It is a flaw in the construction of the argument. The logical problem is that there is no definition, in the argument, of a "natural" coincidence of points. Nor does Einstein's idea of this play any logical role in the argument. It is simply a loose cannon in the relativity of simultaneity.

But since the "natural" coincidence of points is still in the argument, it prevents us from moving forward with the argument. We know the definition of the coincidence of points (get out your Euclid), and that definition is a definition in the relativity of simultaneity. But there is no role anywhere in the argument for the "natural" coincidence of points. Note that you may wish to give some meaning to a natural coincidence of points. However, the term plays no logical role in the argument--and that is the problem with it.

And yet we can't get rid of it! If you keep it, you can't get any further in the argument because it has no meaning-it cannot get you to the next step. Nor can you get around it. If you simply drop it, then M and M' coincide, but then the two Cartesian coordinate systems coincide in one system. What's the problem with that? The problem is that two such systems are an assumption of the argument. If we drop the "natural" we conclude with one system, and that contradicts the assumption of two systems. See? How often have you read the train experiment and failed to see the sleight of hand! If you ever had a good opinion of yourself as a physicist, please go right now to the toilet and flush away that opinion.

Whatever sentiments led Einstein to intrude this notion into his argument, it causes the argument to be incapable of establishing the relativity of simultaneity, or general relativity, or the standard model.

There are two lessons here:

  1. Do not share the sympathies of theorists. Be dispassionate when you read.

  2. Examine arguments carefully.

By the way, the 1905 paper is not the train experiment, but you are aware that you can mechanically translate the one into the other.

It is interesting to read how people who restate this train experiment, try to correct for Einstein's improper inclusion, without realizing that that is what they are attempting to do. Read, for example, the 1920 Italian translation of the small book in which Einstein presents the train experiment. You will know that you understand the logical problem with the relativity of simultaneity when you locate the "correction" in that Italian translation.

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