This question pertains to the section § 2. On the Relativity of Lengths and Times of Einstein's original 1905 paper "ON THE ELECTRODYNAMICS OF MOVING BODIES".

I'm trying to figure out exactly what Einstein is saying in his demonstration that simultaneity is relative. The conclusion is very familiar to me, so I'm not asking for an alternative demonstration. I want to understand the passage quoted below; and, in particular, the footnote.

Apparently he is saying that when the system at rest measures the length of the rod by "simultaneously" recording the positions of the ends, each moving observer is to set his clock to match the rest-frame time of the measurement event at his location.

Then it gets weird. He seems to be saying that the clocks moving with $\rm A$ and $\rm B$ are to continue to match the coinciding rest-frame clocks as they pass them, into the future. But for that to happen, the moving clocks will need a different unit of time than clocks at rest.

Furthermore, for my interpretation to be correct, the statement "[T]hese observers apply to both clocks the criterion established in § 1 for the synchronization of two clocks." will have to mean that the moving observers are not setting their clocks by this method, but are merely checking to see if they are synchronized; and those moving clocks are "slaved" to local rest-frame time. I say this because the footnote indicates that both the rest-frame and moving clock at, say the reflection event, are to read time $t_{\rm B}.$

Am I reading this correctly?

We imagine further that at the two ends $\rm A$ and $\rm B$ of the rod, clocks are placed which synchronize with the clocks of the stationary system, that is to say that their indications correspond at any instant to the “time of the stationary system” at the places where they happen to be. These clocks are therefore “synchronous in the stationary system.”

We imagine further that with each clock there is a moving observer, and that these observers apply to both clocks the criterion established in § 1 for the synchronization of two clocks. Let a ray of light depart from $\rm A$ at the time [footnote] $t_{\rm A}$, let it be reflected at $\rm B$ at the time $t_{\rm B}$, and reach $\rm A$ again at the time $t^\prime_{\rm A}$. Taking into consideration the principle of the constancy of the velocity of light we find that

$$t_{\rm B}-t_{\rm A}=\frac{r_{\rm AB}}{c-v} \text{ and } t^\prime_{\rm A}-t_{\rm B}=\frac{r_{\rm AB}}{c+v}$$

where $r_{\rm AB}$ denotes the length of the moving rod—measured in the stationary system. Observers moving with the moving rod would thus find that the two clocks were not synchronous, while observers in the stationary system would declare the clocks to be synchronous.


“Time” here denotes “time of the stationary system” and also “position of hands of the moving clock situated at the place under discussion.”

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    $\begingroup$ You have to bear in mind that this was a long time ago that he wrote that paper. The polished treatment of the subject you see in textbooks has evolved over time. In this paper, he is figuring stuff out for the first time as he goes along. $\endgroup$
    – Tom
    Nov 1, 2022 at 17:00
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    $\begingroup$ I wasn't criticizing. If I were to critique the presentation, it wouldn't be for the method of proof (which is quite clever). I would suggest clearer naming, and stressing that "resting" clocks are not assumed to run at the same rate as "moving" clocks, contrary to what is subsequently assumed. $\endgroup$ Nov 1, 2022 at 18:27
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    $\begingroup$ I guess you can alternatively frame it like this: imagine there are, at every point in spacetime floating clocks (an abstract, mathematical field of clocks, if you will), all synchronized with the stationary system (they tell the time of the stationary system, at every point in spacetime), and the moving observers are always reading the display from the clock that happens to be where they are at (so the particular "readout clock" changes as they move; or equivalently, they have a device onboard that can calculate and display that time, and that's what they are looking at). $\endgroup$ Nov 1, 2022 at 21:22
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    $\begingroup$ That's plagiarism! (just kidding). It is exactly how I was thinking about it. In my personal notes I use the terms clock field and field time (also called frame-proper time) as distinct from particle time, which is proper time along the world-line of a mass-point. Einstein's added concept of local moving clocks did make me stretch my understanding just a bit. $\endgroup$ Nov 1, 2022 at 21:34
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    $\begingroup$ I'm kind of sad it's fallen out of favor, it's actually quite a nice mental model, cause it lets you see intuitively (at least once you wrap your head around it) that the simultaneity hyperplane of the moving observer is tilted compared to what the stationary observer considers simultaneous (i.e. if you are moving with the rod, the clocks of this imaginary field all have different values along the length of the rod - one end is more "in the future"). Clever fella, this Einstein. $\endgroup$ Nov 1, 2022 at 21:36

4 Answers 4


Then it gets weird.

Yes, that is an apt description. Remember, this was brand new. There weren’t any of the standard pedagogical techniques then and he couldn’t ask for any help in making a better explanation. So this specific section was a little weird and no subsequent author (including himself) ever used this argument again.

Then it gets weird. He seems to be saying that the clocks moving with A and B are to continue to match the coinciding rest-frame clocks as they pass them, into the future. But for that to happen, the moving clocks will need a different unit of time than clocks at rest

Yes, that is correct. A similar thing is actually done with GPS satellites. They are moving in the earth centered inertial frame (ECIF), and so their clocks are adjusted so that they do not keep correct proper time but rather they match the ECIF time.

the moving observers are not setting their clocks by this method, but are merely checking to see if they are synchronized

Yes, you are reading it correctly.

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    $\begingroup$ I the method find it compelling, in the sense that it now stands as unconventional; so it gives an alternative perspective. It's certainly worthy of a section in a book on SR. I have to wonder how many people have actually figured out what he was saying in those paragraphs. I only understood it when I tried to rewrite it for my personal notes. That's how I learn physics. I try to dumb it down to the point that even I can understand it. $\endgroup$ Nov 1, 2022 at 18:16

The following is a comment; I'm posting it here because comment space is too small for this message.
(I endorse that the comment space is small.)

In the preceding paragraph, 'definition of simultaneity', Einstein presents the synchronization procedure that would later be named 'Einstein synchronization procedure'. Then Einstein wrote about that procedure:

We assume that this definition of synchronism is free from contradictions

Well, for sure: in Minkowski spacetime the Einstein synchronization procedure does not give rise to self-contradiction.

However, it is interesting to consider how Einstein sychronization procedure fares in newtonian space&time

Let me discuss the case of propagation of sound.
Take a train, with three emitters/receivers of sound.

Run the synchronizatio procedure with a range of different velocities of the train, and compare the results.

The procedure:
The emitter in the middle sends pulses of sound to the left and right. As each pulse is detected at the end of the train a sound pulse is sent back to the middle.

As we know: for each velocity of the train the two reply-pulses arrive at the same time back at the middle of the train.

However, the duration of a cycle is not the same in each case. The duration of a cycle is shortest when the train has no velocity with respect to the air through which the sound pulses are propagating.

(In the two images below the yellow lines are hardly visible, my apologies for that. The images still convey the message, though.)

Diagram: newtonian space&time I

The diagram above shows a plot of the sound pulses (yellow) as they travel throught the air, for the case when the train is stationary with respect to the air. The red lines plot the motion of the three emitters/detectors on the train.

Diagram: newtonian space&time II

The diagram above shows a plot for the case when the train has a velocity relative to the air.

What this shows that if your apply the Einstein synchronizatio procedure in newtonian space&time, trying a range of velocities, then you can home in on the velocity where you are stationary with respect to the medium through which the waves are propagating.

In Minkowski spacetime however, you get a very different result.

When the motion is taking place in Minkowski spacetime then you have to take time dilation effects into account.

As we know: if you do the Einstein synchronization procedure in Minkowski spacetime, using pulses of light, and you do that for a range of velocities, then a co-moving clock will report the same duration for the cycle for every velocity.

Returning to the assertion by Einstein:

We assume that this definition of synchronism is free from contradictions

In retrospect we see that to assume that no self-contradiction will arise is already sufficient to necessitate Minkowski spacetime, instead of Newtonian space&time

Anyway, what is most striking to me is the tone of voice of Einstein in the 1905 article. I assume he was fully aware how profound his proposal was. But his demeanour is casual.

Of course, in a scientific article it is standard to use a neutral voice. But still.

I like to think Einstein was being pragmatic here. Einstein did not state upfront: "This will change all of physics!". Instead Einstein presented the concept piece by piece, and at no point does he let on that he is ushering in a revolution.

  • $\begingroup$ Einstein's ideas were not quite as novel as we are typically led to believe. For example, the method of determining an inertial frame on the basis of observing test particles, and without consideration of "absolute space" had been clearly enunciated by Lange who wrote in 1885 "Newton’s absolute space is a phantom that should never be made the basis of an exact science." Einstein had certainly been exposed to these ideas through Mach's writings. Lange also introduced the notion of a frame-dependent inertial time dimension. See Pfister and King's Inertia and Gravitation. $\endgroup$ Nov 3, 2022 at 14:55
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    $\begingroup$ @StevenThomasHatton Of course: before 1905 it was established in the physics community that there is a principle of relativity of inertial motion. The issue was this: if propagation of light is wave propagation in a medium (the medium being referred to as Lumineferous Aether), and if Galilean transformation holds good, then there is a conflict. The solution was to replace Galilean transformation with Lorentz transformation. The concept of an equivalence class of inertial coordinates systems carried over from newtonian to relativistic. The change: replace Galilean with Lorentz transf. $\endgroup$
    – Cleonis
    Nov 3, 2022 at 16:47
  • $\begingroup$ Again. Lorentz had already published his transformations. Einstein claimed he was unaware of the paper by Lorentz, but that doesn't mean Michele Besso was unfamiliar with Lorentz's work, or that Einstein could have otherwise be influence by it. Certainly Einstein's paper was significant. But much of what has been attributed to Einstein did not originate with him. Pretty much every piece of special relativity had already been enunciated. Both Lorentz and Poincaré had established the mathematical structure of SR. Ludwig Lange had clearly enunciated the idea of inertial frame relativity. $\endgroup$ Nov 3, 2022 at 18:25
  • $\begingroup$ @StevenThomasHatton Dude - there is general concensus that special relativity was floating in the air, and if Einstein would not have taken the Lorentz transformations as the universal transformations then within 5 years or so someone else would have introduced that (most likely Minkowski). The big one is General Relativity. Without Einstein's influence that might only have been formulated half a century later. That is why the physics community gravitates to associating Einstein with the introduction of special relativity. Priority disputes about relativistic physics are a waste of time. $\endgroup$
    – Cleonis
    Nov 3, 2022 at 18:50
  • $\begingroup$ physics.stackexchange.com/a/56896/117014 $\endgroup$ Nov 3, 2022 at 21:37

When he's saying that simultaneity is relative, he's saying that the speed $v = ∞$ is not absolute, but relative. In Newtonian physics, it's absolute. Infinity is the speed at which simultaneous happens. So the term "simultaneity" is just a back-door way of saying "infinite speed".

It's also the speed at which gravity, in Newton's universe, happens. So, in Newtonian physics, the only absolute speed is infinity, while all finite speeds are relative. In contrast, the absolute speed $c$ in Special Relativity is finite and non-zero.

You could also conceive of a universe where the absolute speed is zero, while all finite speeds (including the infinite speed) are relative. That's the "Carrollian" world, because in it things have momentum without going anywhere. And you could even conceive of a universe where all speeds are absolute. It's associated with what's known as the "static group".

In Special Relativity, a motion at speed $v$ in a given direction, relative to a system that's going at speed $u$ in a collinear direction, where $|v| < c$, is going at $(v - u)/(1 - vu/c^2)$.

If $v = ∞$ then in the new frame of reference it is $(∞ - u)/(1 - ∞u/c^2)$, which you can treat as the previous expression taken in the limit $v → ∞$. That works out to $-c^2/u$. So, you'll see "simultaneous" happen as going toward you at a "reciprocal" speed; i.e. if you're going half the speed of light in a given direction, then what was simultaneous now happens at double the speed of light, coming in a direction toward you.

Conversely, for every speed $|v| > c$, there is a frame of reference in which it is infinite, namely, the one where $u = c^2/v$.

So, all faster-than-light's are actually simultaneous'es that are being seen in a moving frame of reference, while all slower-than-light's are at-rest's that are being seen in a moving frame of reference.

In a Carrollian world, in place of the Galilean or Lorentz transform would be the transform $(x,t) → (x, t - Λx)$, where (now) the moving frame of reference's speed is measured in "marathoner"'s units (minutes per mile) as $Λ$, or "slowness". Simultaneity is also relative there. But, this time $v = 0$ stays absolute, while all other non-zero speeds transform as $v → v/(1 - vΛ)$ ... or as "marathoner" speeds: $1/v → 1/v - Λ$. So, simultaneity happens at 0 marathoner speed and transforms to $-Λ$, while a finite speed $v$ has a transform to $∞$ in a frame of reference that's moving at slowness $Λ = 1/v$. For this world $c = 0$, so all actually moving objects would be like tachyons, while the stationary objects are a merger of luxons (or "light-speed" systems) and bradyons ("slower-than-light" systems).

In contrast, in the Galilean world, where the transforms are $(x,t) → (x - vt, t)$, the finite speed systems are bradyons, while the infinite speed systems are a merger of luxons and tachyons. I've termed them the "synchrons". They include the action-at-a-distance "lines of force", like that for gravity, in the Newtonian world.


Einstein gives us the reverse view of what is usually adopted in modern presentations of SR. Today we would normally consider clocks that are always synchronised in their rest frame, so that when you compare clocks in one frame with clocks in another moving relative to the first they are out of synch. Instead, Einstein assumes that clocks in the 'moving' frame always show the local time in the stationary frame, and thus they seem out of synch to the observers moving with the clocks. It is the same physics but presented in a different way.

As an aside, almost all of the basic questions you find on this site concerning SR, and all of the many so called refutations of SR that I have encountered, stem from a failure to cotton on to the meaning and importance of the relativity of simultaneity. I overlooked it myself when I first encountered SR, and went down the usual rabbit hole of finding the symmetry of time dilation to be utterly baffling because I had the wrong conception of what it was.


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