Relativity & Simultaneity Convention: Meaning & Impact

[EclogiteFacies]

TL;DR Summary

Hi all, how 'real' is the simultaneity convention? Should we think of it as a mathematical tool rather than a physically definite description of the world?

Hi all,

I have been figuring out relativity. I was just wondering how meaningful the simultaneity convention is, how meaningful it is for say something a million light years away to be on our plane of simultaneity.

I'm guessing not very, we must really wait for light to get to use to have any idea of what's happening so far away.

Furthermore, us moving back and forth here can dramatically change our idea of simultaneity so far away. Of course this doesn't change the notion of now for an obsverer a million light years away..

So, I gather that simultentiety across huge distances is pretty meaningless and not something we should get hooked on. We instead acknowledge it as a mathematical representation but not a physical plane? Things so far away are not so simply organised into past and future.

Thanks!

 

[Dale]

TL;DR Summary: Hi all, how 'real' is the simultaneity convention. Should we think of it as a mathematical tool rather than a physically definite description of the world?

Things so far away are not so simply organised into past and future.

I think that is the key insight. We would like "now" to be a physically meaningful plane in spacetime that divides the past from the future, but nature doesn't seem to work that way. There is a physically meaningful causal past and a causal future which are the interior of the light cone. And everywhere else is not causally related so it is not in either the past or the future in any physically meaningful sense.

 

[FactChecker]

I have been figuring out relativity. I was just wondering how meaningful the simultaneity convention is, how meaningful it is for say something a million light years away to be on our plane of simultaneity.

You should get used to the Special Relativity concept of the local relativity of simultaneity before you worry about huge galactic distances.

I'm guessing not very, we must really wait for light to get to use to have any idea of what's happening so far away.

You are confusing the concept of two, spatially separated, events being simultaneous with the concept of our knowledge of what happened far away. Events can be simultaneous even if we never know about them.

Furthermore, us moving back and forth here can dramatically change our idea of simultaneity so far away. Of course this doesn't change the notion of now for an obsverer a million light years away..

The distortion of spacetime over long distances is the subject of General Relativity (GR), which is much more complicated than Special Relativity. Even Einstein struggled with GR for years. I suggest that you concentrate on Special Relativity for a while.

So, I gather that simultentiety across huge distances is pretty meaningless and not something we should get hooked on. We instead acknowledge it as a mathematical representation but not a physical plane? Things so far away are not so simply organised into past and future.

Not necessarily. There are often questions here regarding the speed with which gravitational changes far away are felt here and people ask if they are felt instantly, i.e. felt simultaneous to the change. (The answer is that the effects travel at the same speed as light in a vacuum.) There are similar questions regarding quantum entangled particles that are separated. These are not just questions of "mathematical representation", they have physical consequences.

 

[EclogiteFacies]

You should get used to the Special Relativity concept of the local relativity of simultaneity before you worry about huge galactic distances.

Local as in on Earth, as far as I can gather we are basically in the same frame and see everything at the same time.

Or is local as in something else?

 

[PeterDonis]

as far as I can gather we are basically in the same frame and see everything at the same time

No, we don't. It's just that the differences are too small for us to detect under ordinary circumstances. If you and I are in the same room, it only takes light a few nanoseconds to travel between us; that's about a million times smaller than the time scale on which our nervous systems processes signals, and about ten to a hundred million times smaller than the time it takes us to be consciously aware of something changing. And if we move relative to each other at a meter per second, that again only causes differences in our natural simultaneity conventions of a few nanoseconds for events we can both see.

Things are different if, say, you're a technician trying to analyze a problem with packets being transmitted over the Internet to the other side of Earth. Or consider the time delays involved during the Apollo missions when the flight controllers on Earth had to talk to the astronauts on the Moon. Under those conditions you can no longer use the common approximation that "we are basically in the same frame and see everything at the same time".

 

[EclogiteFacies]

No, we don't. It's just that the differences are too small for us to detect under ordinary circumstances. If you and I are in the same room, it only takes light a few nanoseconds to travel between us; that's about a million times smaller than the time scale on which our nervous systems processes signals, and about ten to a hundred million times smaller than the time it takes us to be consciously aware of something changing. And if we move relative to each other at a meter per second, that again only causes differences in our natural simultaneity conventions of a few nanoseconds for events we can both see.

I was definitely being naive hahaha.
Can I ask what you mean by natural simultaneity convention here, is it the coordinates we apply to the occurrence of an event?

 

[PeterDonis]

Can I ask what you mean by natural simultaneity convention here

The convention of an inertial frame in which the chosen observer is at rest.

 

[EclogiteFacies]

Ye and due to each obsverer having their own convention for simultaneity when applied to their own frame, if inertial lead to contrasting time coordinates for the occurrence of an event.
I gather though you don't apply too much physicality to these simultentiety conventions though?

 

[PeterDonis]

I gather though you don't apply too much physicality to these simultentiety conventions though?

I don't apply any "physicality" to them. They are conventions. They have no physical meaning.

 

[vanhees71]

Neveretheless the important thing is that these conventions are realizable physically, and that's why we can figure out by measurements whether the postulates, underlying special relativity, are consistent with the observations, i.e., you can indeed experimentally check, whether the special-relativistic spacetime-description or the Newtonian one describes the observations better. Of course, special relativity wins with very high accuracy.

A gedanken experiment for the standard simultaneity convention is that, given Einstein's postulates are valid, can be constructed as follows:

From Einstein's 1st postulate, which is in fact identical with Newton's 1st postulate (the special principle of relativity), is that there is a (global) inertial frame of reference, which is physically in principle realizable by choosing an event (a point in the spacetime manifold) as "the origin". Then you can assume that there are local ideal clocks, which always tick at a well-defined given rate for any inertial observer at the place of this clock being at rest relative to the clock. In practice this clock is given in the international system of units by a cesium atomic clock, whose frequency is defined (based on the stability of the hyperfine transition of Cs-137 atoms). Further it's assumed that any inertial observer's "space" is a Euclidean affine manifold, i.e., for the location of points at rest relative to the origin of the inertial frame the metrics (including distances and angles) of our standard Euclidean geometry are valid.

From Einstein's 2nd postulate, it's clear that the speed of light is a constant of Nature. In practice within the SI system of units, the speed of light is a fixed quantity defining the distance unit meter using the time unit seconds.

Now to establish a time not only in the origin of the coordinate system you place clocks, all at rest relative to the clock in the origin of the inertial frame, at any distant point you are interested in. You can determine the distance of each of these clocks to the clock at the origin by sending a light signal and measure the time it needs to be reflected back to you. By definition the distance between these clocks is given by half of the time it takes for the light signal to run to the distant clock and back. Note that for this you only need "local measurements", i.e., you measure the time the light signal needs with one clock at rest in the origin to measure this time. The assumption that it takes the same time for the light to travel to the distant clock as it takes for the reflected light to travel back to your clock is the arbitrary convention chosen to define the time it takes for the light signal to travel just from the clock at the origin to the clock. And after having conventionally established in this way the relation between the travel time of light signals and travelled distances of all the clocks from the clock at the origin you can synchronize each clock with the clock at the origin: You mount a clock at a distance $L$ from the origin and set its time to $L/c$. And then at an arbitrary time $t=0$ (defining the "time origin" at the origin of the inertial frame) you send a light signal to the distant clock which right at its arival starts to count the ticks. Then it's synchronized with the clock at the origin in Einstein's synchronization convention.

You can then show that all these clocks are not only at rest relative to the clock at the origin but also relative to each other, and that also any two of these clocks are synchronized with each other and with the clock at the origin. That makes the synchronization convention consistent for one inertial frame of reference.

Now you can use Einstein's 2nd postulate, according to which the speed of light in any direction of light propagation is the same and independent of the velocity of the light source relative to any inertial frame of reference to show that if you synchronize the clocks in one inertial frame $\Sigma$ according to Einstein's convention and do the same for the clocks in another inertial frame $\Sigma'$, which is moving relative to $\Sigma$ with some constant velocity, the clocks in $\Sigma$ are synchronized within $\Sigma$ but not with the clocks in $\Sigma'$ (although these are synchronized within themselves). This is the "relativity simultaneity", which is the key for the understanding of all the socalled "kinematical paradoxes" like length contraction, time dilation, etc.

Formally from the synchronization convention it follows that the transformation between space and time coordinates as established with the Einstein synchronization convention, is given by (proper orthochronous) Lorentz transformations (or if you include also shifts of the spacetime origins relative to each other the corresponding Poincare transformations) rather than the Galilei transformations we are used to in Newtonian physics.

A direct consequence is time dilation, and this can, e.g., checked experimentally using unstable particles. You measure the mean life time in an inertial frame when they are at rest and compare it to the mean lifetime when they move with some velocity $v$. This can be done in storage rings with radioactive nuclei or with elementary unstable particles like the muon, and all these measurements confirm with high precision the prediction of time dilation, following from Einstein's postulates and the Lorentz transformation, i.e., if $\tau$ is the mean lifetime for the particle at rest, then for an observer, relative to which the particle moves with speed $v$, the lifetime is measured to be $t'=\gamma \tau=\tau/\sqrt{1-v^2/c^2}$.

 

[FactChecker]

I don't apply any "physicality" to them. They are conventions. They have no physical meaning.

I think that we could say more than this. Suppose there is an inertial reference frame in which two spatially separated events are simultaneous. Then there can be no other inertial reference frame in which the precedence of one event is enough to allow a cause-effect relationship between the two. They are too far separated for one to have physically caused the other. In other words, if the events are simultaneous in one IRF, then they are "essentially" simultaneous in all IRFs. I am sure that there is a better terminology for this.

 

[Nugatory]

I am sure that there is a better terminology for this.

The phrase that you are looking for is "spacelike separated", as opposed to "timelike separated" (one of them will have happened before the other in all frames and therefore there can be a causal relationship between them), and "lightlike separated" (a light signal could get from one to the other, but nothing slower).

 

[Frank Wappler]

[...] So, I gather that simultaneity across huge distances is pretty meaningless and not something we should get hooked on. [...]

Regardless of whether the distances under consideration happen to be huge, or tiny, or in the range of human everyday experience:

The (conventional) stipulation, how and under which conditions simultaneity (or dis-simultaneity) should be determined at least in principle, is (for instance) essential in definition and evaluation (incl. determination of systematic uncertainties) of "speed"; namely explicitly concerning the "denominator term" in the symbolic/mnemonic expression

$$\text{average speed} := \frac{\text{distance}}{\text{duration}}.$$

As a concrete illustrative example consider the case of a projectile $P$ having propagated from a gun muzzle $A$ to the corresponding target region $B$. If $A$ and $B$ had been, and remained, suitably at rest wrt. each other (suitably having been and remained together members of the same inertial frame), then the (average) speed $\overline v_{AB}[ \, P \, ]$ of projectile $P$ wrt. inertal system $AB$ is explicitly defined as

$$\overline v_{AB}[ \, P \, ] := \frac{\text{distance}[ \, A, B \, ]}{\text{duration}\tau A[ \, \text{from having indicated }P\text{'s departure}, \text{until simultaneous to }B\text{'s indication of }P\text{'s arrival} \, ]}$$

which by the simultaneity relation (as applicable to $A$ and $B$) is likewise

$$\overline v_{AB}[ \, P \, ] = \frac{\text{distance}[ \, A, B \, ]}{\text{duration}\tau B[ \, \text{from simultaneous to }A\text{'s indication of }P\text{'s departure}, \text{until having indicated }P\text{'s arrival} \, ]}.$$