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.


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?

  • $\begingroup$ 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
    Feb 9 at 21:07
  • $\begingroup$ 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
    Feb 10 at 9:52

4 Answers 4


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.

  • $\begingroup$ 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
    Feb 8 at 20:09
  • 2
    $\begingroup$ 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
    Feb 8 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". $\endgroup$ Feb 10 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
    Feb 10 at 19:37

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.

  • $\begingroup$ 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
    Feb 11 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
    Feb 11 at 12:18
  • $\begingroup$ No. My explanation does rule out absolute simultaneity, so you must have mis-understood what I said. $\endgroup$ Feb 11 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
    Feb 11 at 18:20
  • 1
    $\begingroup$ `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. $\endgroup$ Feb 11 at 19:12

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

  • $\begingroup$ "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
    Feb 11 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
    Feb 11 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
    Feb 11 at 13:46

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.

  • $\begingroup$ 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
    Feb 11 at 14:43

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