Understanding the phrase "simultaneity convention"

[Freixas]

I have a number of questions about some S.R. basics. The answer to one question may affect the next, so I will try to focus on one question at a time. The question here is about the possible range of “simultaneity conventions.”

We place an observer at location A. This observer releases a photon at time $t_1 = 0$ in the direction of a mirror at B, located at distance $d$ from A. The mirror is at rest with respect to the observer and aligned to reflect the photon back to A. The photon returns to A at time $t_2$. The two-way speed of light is then $2d / t_2$. This is an invariant and is represented by $c$.

The mirror at B has a clock that is set when the photon reaches the mirror. How might we choose the clock’s setting, $t$?

1. We could allow any function $t = f(d)$.
2. We could allow only functions where the time values are monotonically nondecreasing (for every event that forms the worldline of light, the time value of the event must be greater than or equal to that of any event occurring earlier).
3. We could further limit the functions to ones where the velocity in one direction is a constant. I believe this limits the possible formulas to $t = 2{\epsilon}d/c$, where $\epsilon$ ranges from 0 to 1.
4. Finally, we could limit the function to only $t = d/c$, which is Einsteinian synchronization, where the one-way speed in every direction is the same as the two-way speed ($\epsilon$ = ½).

Some notes:

1. If we allow using any function, some functions may make some calculations difficult or impossible. If a choice is dysfunctional, that is a good reason not to choose it, but not clearly a reason for forbidding it. Is that correct?
2. This limitation would still allow functions such as $t = floor(d)/c$ (I believe this satisfies the causal requirement).
3. I believe math currently exists to handle physics in which the possible simultaneity conventions match the formula $t = 2{\epsilon}d/c$, for any valid value of $\epsilon$.
4. David Malament proposed a theorem that he used to argue that the only valid simultaneity convention was the one Einstein chose. Others think that Malament’s Theorem is invalid. I don’t believe he has anything testable.

How arbitrary can a simultaneity convention be? Note that you are not restricted to options 1–4. I presented them just to show the possibilities I had come up with.

Simultaneity is an odd concept. Simultaneity is clear when we are talking about events that are co-located, but not when events are separated by space. If simultaneity is somewhat (or totally) arbitrary, then things that are commonly discussed, such as the length or velocity of a moving object, become equally arbitrary.

I don’t have any preferred answer—I am just trying to understand the meaning and any issues relating to the term “simultaneity convention”. I did enough research to realize there is a philosophical war around the conventionality of simultaneity (summarized here: https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/significance_conv_sim/index.html). Physics is not philosophy, though—I’d like the viewpoint of physicists.

As a reminder, I am discussing this strictly with respect to S.R. I know little about G.R., but I’ve heard it messes with the concept of simultaneity even more.

 

[Dale]

How arbitrary can a simultaneity convention be?

A coordinate chart is a mapping between events in spacetime and points in R4. There are very few requirements. The mapping must be smooth (diffeomorphic) and one-to-one (invertible). Other than that you are not restricted.

The simultaneity convention is then just the convention you used to choose which events share the same t coordinate. The usual implication is that the coordinate basis vector for t is timelike, but I am not certain even that is actually required.

 

[PeterDonis]

The question here is about the possible range of “simultaneity conventions.”

Before even considering that, you need to make sure you understand what a "simultaneity convention" is. A spacetime is a 4-dimensional set of events. A "simultaneity convention" is a way of breaking up the spacetime into disjoint 3-dimensional subsets, such that all of the events in each subset are defined to happen "at the same time". This requires that, for each subset, all of the events in the subset are spacelike separated from each other (meaning that no two events can be connected by a timelike or null curve).

Given the above, the possible range of "simultaneity conventions" is obvious: any division of 4-dimensional spacetime into 3-dimensional subsets that meets the above requirements is a valid "simultaneity convention".

 

[PeterDonis]

We place an observer at location A. This observer releases a photon at time $t_1 = 0$

Time according to what? The answer I think you intend is time according to the observer's own clock, but you should not leave that for others to guess. You should make it explicit.

in the direction of a mirror at B, located at distance $d$ from A.

Distance according to what? The answer I think you intend is proper distance, i.e., arc length along a spacelike geodesic from the event at location A at which the photon is released, to the worldline of the mirror at B. But again, you should not leave this for others to guess. You should make it explicit.

A good rule when framing scenarios in relativity is to never use the words "time" or "distance" without an explicit specification of how the time or distance is to be measured.

We could allow any function $t = f(d)$.

No, you can't. The time $t$ you assign to this event on the mirror's worldline has to satisfy $t_1 < t < t_2$ for it to define a valid simultaneity convention. That is because only events on the worldline of the observer at location A whose clock readings lie in that interval are spacelike separated from the event on the mirror's worldline where the photon is reflected.

We could allow only functions where the time values are monotonically nondecreasing (for every event that forms the worldline of light, the time value of the event must be greater than or equal to that of any event occurring earlier).

"Monotonically nondecreasing" is not enough. The times of events along any timelike worldline, including the mirror's worldline, must be strictly increasing.

We could further limit the functions to ones where the velocity in one direction is a constant. I believe this limits the possible formulas to $t = 2{\epsilon}d/c$, where $\epsilon$ ranges from 0 to 1.

You could, but there is no need to. The first two requirements above (only spacelike separated events can happen at the same time, and times along any timelike worldline must be strictly increasing) are sufficient to ensure that you are defining a valid simultaneity convention.

Finally, we could limit the function to only $t = d/c$, which is Einsteinian synchronization, where the one-way speed in every direction is the same as the two-way speed ($\epsilon$ = ½).

Obviously this is the most common choice, but it is of course not the only possible choice.

How arbitrary can a simultaneity convention be?

See above.

If simultaneity is somewhat (or totally) arbitrary, then things that are commonly discussed, such as the length or velocity of a moving object, become equally arbitrary.

Only if you insist on defining such things relative to a choice of simultaneity convention. But you don't have to do that. There are invariant ways of defining these things.

I know little about G.R., but I’ve heard it messes with the concept of simultaneity even more.

Not really; the general concept works the same way in GR as I have described above.

 

[Dale]

@Freixas note that the coordinate-based description I used and the geometrical description @PeterDonis used are equivalent for most “typical” setups. There are some edge cases where they differ. You should be aware that different authors may choose different meanings, so you need to understand how a specific author uses the term.

 

[PeterDonis]

The usual implication is that the coordinate basis vector for t is timelike, but I am not certain even that is actually required.

AFAIK this is not required; as long as each surface of simultaneity is spacelike, there is no requirement that the fourth coordinate basis vector be timelike, it just has to be linearly independent from all tangent vectors to each surface of simultaneity.

 

[PeterDonis]

There are some edge cases where they differ.

Can you give an example?

 

[Dale]

Can you give an example?

Inside the horizon for standard Schwarzschild coordinates. There the standard t coordinate would be a valid synchronization convention under my description but not under yours.

Anderson’s convention I am not certain about. I would need to check, but it might violate yours for $1<|\kappa|$

 

[PeterDonis]

Inside the horizon for standard Schwarzschild coordinates. There the standard t coordinate would be a valid synchronization convention under my description

How can it be a valid synchronization convention when curves of constant $t$ are timelike? You would have timelike separated events being labeled with the same time. That seems invalid.

 

[PeterDonis]

Anderson’s convention

Can you give a reference?

 

[Dale]

How can it be a valid synchronization convention when curves of constant $t$ are timelike? You would have timelike separated events being labeled with the same time. That seems invalid.

In this approach t is just a coordinate, so as long as it is a valid coordinate it is not excluded a priori. Simultaneity conventions are just labels, not physical. So you can use weird non-physical labels.

Can you give a reference?

Yes, but it will have to be later when I am on my computer instead of my phone

 

[Freixas]

No, you can't. The time you assign to this event on the mirror's worldline has to satisfy ... for it to define a valid simultaneity convention. That is because only events on the worldline of the observer at location A whose clock readings lie in that interval are spacelike separated from the event on the mirror's worldline where the photon is reflected.

I think I am missing the physics definition of "simultaneity". The dictionary definition isn't useful: "at the same time". What is the principle that excludes timelike-separated events from being simultaneous?

Let's use a Taylor/Wheeler lattice. The lattice is normally a 3D "jungle-gym", with one clock at each intersection. The clocks are initialized per Einsteinian synchronization. Once initialized, we run some experiment. Each lattice junction records nearby events, logging their time. When the experiment completes, we download the data for analysis.

To this, let's add multiple clocks at each location, each initialized using a different f(d) formula. We can download data sets for each collection of clocks that use the same f(d) initialization.

Nothing seems to limit my ability to use any f(d) formula I want. Using the dictionary definition, simultaneous events would be ones where the recorded clock values are the same. Some f(d) choices will pair events separated by a timelike interval. Events could appear to precede their causes, but that would just be an artifact of the choice of clock-initialization formula.

So I assume that, in physics, the word "simultaneous" has a more detailed definition, one that is tied to causality. The causality addition seems to be there to make physics easier and not necessarily an intrinsic part of the concept of "simultaneity". At least, I can't see the connection--I'm hoping you can help point it out.

"Monotonically nondecreasing" is not enough. The times of events along any timelike worldline, including the mirror's worldline, must be strictly increasing.

Hmm... Let's use the formula t = 0 (this is $t = 2{\epsilon}d/c$, where $\epsilon$ = 0) This seems not to be monotonically increasing but appears to be permitted.

Let's use the lattice again. For simplicity, we will reduce it to one dimension. From the lattice's origin, I release a photon in the positive X direction. As the photon passes each junction, the clock is set to 0. This is equivalent to an instantaneous one-way speed of light in the positive direction and a one-way speed of c/2 for light headed back.

Is this incorrect?

 

[PeterDonis]

What is the principle that excludes timelike-separated events from being simultaneous?

The principle that two different events on a single observer's worldline cannot happen at the same time. For any pair of timelike separated events, there will be some possible observer that has both of them on their worldline. (Note that this requirement is also the basis for "strictly increasing" instead of "nondecreasing" in my earlier specifications.)

(Note that a similar principle also excludes pairs of null separated events from being simultaneous: in this case, that any such pair of events represents a possible emission and reception of a light ray that travels a nonzero distance, and such emission and reception can never happen at the same time.)

 

[Dale]

Can you give a reference?

It is:

Anderson, Vetharaniam, and Stedman. "Conventionality of synchronisation, gauge dependence, and test theories of relativity". Physics Reports. 295 (1998) 93-180.

In the last paragraph on page 106 he explicitly rejects the restriction to $|\kappa|<1$. So that means that the metric in Anderson coordinates is $$ds^2 = -dt^2 + (1-\kappa^2)\ dx^2+dy^2+dz^2 -2 \kappa \ dx \ dt$$ with $\kappa$ potentially being larger than 1.

So a surface of constant $t$ would have the metric $$ds^2 = (1-\kappa^2)\ dx^2+dy^2+dz^2 $$ which would not be spacelike for $1<|\kappa|$

 

[PeterDonis]

Anderson, Vetharaniam, and Stedman. "Conventionality of synchronisation, gauge dependence, and test theories of relativity". Physics Reports. 295 (1998) 93-180.

Looks like it's paywalled:

https://www.sciencedirect.com/science/article/abs/pii/S0370157397000513

 

[Freixas]

The principle that two different events on a single observer's worldline cannot happen at the same time.

You explained it short, sweet, and comprehensible (again!). Bonus points! Have you ever thought of writing a book on physics?

 

[PeterDonis]

In the last paragraph on page 106 he explicitly rejects the restriction to $|\kappa|<1$. So that means that the metric in Anderson coordinates is $$ds^2 = -dt^2 + (1-\kappa^2)\ dx^2+dy^2+dz^2 -2 \kappa \ dx \ dt$$ with $\kappa$ potentially being larger than 1.

So a surface of constant $t$ would have the metric $$ds^2 = (1-\kappa^2)\ dx^2+dy^2+dz^2 $$ which would not be spacelike for $1<|\kappa|$

For $|\kappa| > 1$, this would violate the condition I gave, that no two distinct events on the same timelike curve can have the same time. Since a line of constant $t$ is itself timelike for $|\kappa| > 1$, it is a timelike curve with an infinity of points all having the same time.

Does the paper address this concern anywhere?

 

[PeterDonis]

Does the paper address this concern anywhere?

Btw, just to be clear, I'm not questioning whether the chart in question is a valid coordinate chart; of course it is. I'm just questioning whether its time coordinate for the case $|\kappa| > 1$ can validly be described as defining a "synchronization convention", given that it allows multiple events on the same timelike curve to have the same coordinate time (whereas of course they won't have the same proper time).

 

[Dale]

this would violate the condition I gave, that no two distinct events on the same timelike curve can have the same time

Yes, which is why I thought that it was an example of an “edge case” where your condition and mine disagreed (but I had to look up the metric to be sure)

Does the paper address this concern anywhere?

They don’t see it as a concern. They liken it to the international date line.

I'm just questioning whether its time coordinate for the case |κ|>1 can validly be described as defining a "synchronization convention"

Well, that is the focus of the paper, so it seems to me that Anderson believes that it is a valid synchronization convention.

 

[Freixas]

The principle that two different events on a single observer's worldline cannot happen at the same time. For any pair of timelike separated events, there will be some possible observer that has both of them on their worldline. (Note that this requirement is also the basis for "strictly increasing" instead of "nondecreasing" in my earlier specifications.)

I'll reply to the note in parenthesis this time.

If one sets clocks based on the worldline of light using a formula such as t = f(d) = 0, which is nonincreasing, then this says that two events on a lightlike interval could be simultaneous. Since only light can travel at the speed of light, there could never be any observer with two simultaneous and separate events on their worldline. Therefore, this specific nonincreasing formula does not appear to violate your rule.

 

[PeterDonis]

They don’t see it as a concern. They liken it to the international date line.

I don't see the comparison. My clock doesn't read the same time permanently if I cross the international date line or travel along it. It just has to be adjusted forward or backward occasionally. But according to their $|\kappa| > 1$ case, if I'm understanding their claim correctly, to be properly "synchronized" my clock would have to read the same time forever if I traveled along a constant $t$ timelike curve. That doesn't make sense to me.

I guess I'll have to try and get access to a copy of the paper to see what their approach is.

 

[PeterDonis]

I'll reply to the note in parenthesis this time.

Before doing that, you should have read the part of my post that you didn't quote, which specifically talks about the lightlike case.

 

[PeterDonis]

My clock doesn't read the same time permanently if I cross the international date line or travel along it.

Also, when I cross the International Date line and have to adjust my clock (or indeed whenever I switch time zones and have to adjust my clock), I am switching between different coordinate charts (since that's what the time coordinates of the different time zones are). Moreover, those charts all share the same synchronization (simultaneity) convention, which is not a convention like the $|\kappa| > 1$ case.

In the $|\kappa| > 1$, by contrast, I am using the same coordinate chart; I'm not "synchronizing" my clock to always read the same time forever because I keep switching charts, but because the chart I'm using doesn't (IMO) define a valid simultaneity convention.

So again I don't see as valid the comparison the paper appears to be claiming. Unless I'm missing something that is explained elsewhere in the paper.

 

[Dale]

So again I don't see as valid the comparison the paper appears to be claiming.

I don’t think that they care much about this edge case. They essentially acknowledge it and move on. They don’t spend effort validating or expanding the analogy.

But according to their |κ|>1 case, to be properly "synchronized" my clock would have to read the same time forever if I traveled along a constant t timelike curve. That doesn't make sense to me

To me it seems that they just don’t see that as a problem.

In any case, I am not going to defend it too much. I think that your definition makes more sense. That would then simply forbid the $1<|\kappa|$ case, which is fine.

My main point is that Anderson’s idea of a synchronization convention is in the published literature. I assume yours is too. So the OP won’t be able to find “the” meaning of synchronization convention. They will have to figure out how each author is using the term.

It might be good to post a reference for your approach, to kind of balance the thread.

 

[Freixas]

Before doing that, you should have read the part of my post that you didn't quote, which specifically talks about the lightlike case.

I did read that, but it was not written as clearly.

I was basing my $\epsilon$ on Hans Reichenbach's formula. The articles I had looked at just said that $\epsilon$ was (ambiguously) between 0 and 1. I finally found that a paper clarifying that $0 < \epsilon < 1$.

It is not clear exactly why emission and reception cannot happen simultaneously, but in this case, I'm willing to accept this as a fact. It sounds reasonable, but massless particles behave in odd ways, so if someone had said that emission and reception could be simultaneous, that wouldn't sound unreasonable, either.

Thanks for pointing out this error.

 

[PeterDonis]

It is not clear exactly why emission and reception cannot happen simultaneously

If it doesn't make sense to you that your clock should have the same reading at multiple events on your worldline, i.e., at the start and end of a timelike curve that covers multiple events, it should equally not make sense to you that your clock should have the same reading at both the emission and the reception of a light ray that covers a nonzero distance, i.e., at the start and end of a null curve that covers multiple events. Only for spacelike curves should this make sense, since nothing, neither light nor you nor I, can have a spacelike curve as our worldline.

 

[vanhees71]

In this approach t is just a coordinate, so as long as it is a valid coordinate it is not excluded a priori. Simultaneity conventions are just labels, not physical. So you can use weird non-physical labels.

Yes, but it will have to be later when I am on my computer instead of my phone

But inside the event horizon according to the signature of the metric $t$ is a spacelike and $r$ a time-like coordinate, right?

 

[martinbn]

Well, that is the focus of the paper, so it seems to me that Anderson believes that it is a valid synchronization convention.

It is a valid choice of coordinates, but do the say anything about a synchronization convention? Just because one coordinate is labeled $t$, it doesn't mean that the imply that constant $t$ surfaces are to be considered a synchronization convention. Can you quote more from the paper?

 

[Dale]

Does the paper address this concern anywhere?

Can you quote more from the paper?

I read a little more in depth. In the section I was referring to earlier Anderson says

Accepting any such restriction on $\kappa$ has then to be done for more subtle reasons, for example ensuring that coordinates are assigned so as to ensure the formal respectability of a globally causal ordering of events, i.e., ensuring that the local coordinate times for reception always are later than those for transmission (see Sections 1.4.1, 2.3.2).

So I went and looked at section 2.3.2 which explains a little more of his reasoning:

it should be pointed out that one should distinguish “spatially coincident causality” from “distant causality”; there is no contradiction if an occurrence at $P$ at time $t$ causes another occurrence at $Q \ne P$ at time $t’ < t$ because the two different times are measured at spatially different locations. Indeed, as mentioned in Section 1.5.1, such apparent inconsistencies are familiar consequences of the International Date Line for airline travellers.

According to Anderson then, it is explicitly not necessary for a synchronization convention to establish a causal global ordering. By causal ordering I mean that for a cause $P$ at time $t_P$ and an effect $Q$ at time $t_Q$ that $t_P<t_Q$. For a global causal ordering that must be true for any $P$ and $Q$. Anderson, in contrast, requires only a local causal ordering. Meaning that it is only necessary that $t_P<t_Q$ if $P$ and $Q$ are at the same location.

Anderson's convention respects that condition. Starting with his metric expression $$ds^2 = -dt^2 + (1-\kappa^2)\ dx^2+dy^2+dz^2 -2 \kappa \ dx \ dt$$ and requiring the point to be the same ($dx=dy=dz=0$) gives us $$ds^2 = -dt^2 $$ so at a fixed location time runs normally regardless of $\kappa$. That is the condition that he uses.

That is a very weak condition, weaker than yours, but it would nonetheless rule out using the standard Schwarzschild $t$ coordinate inside the event horizon as a synchronization convention, contrary to what I said in posts 8 and 11.

 

[Freixas]

If it doesn't make sense to you that your clock should have the same reading at multiple events on your worldline, i.e., at the start and end of a timelike curve that covers multiple events, it should equally not make sense to you that your clock should have the same reading at both the emission and the reception of a light ray that covers a nonzero distance, i.e., at the start and end of a null curve that covers multiple events. Only for spacelike curves should this make sense, since nothing, neither light nor you nor I, can have a spacelike curve as our worldline.

Let me explain why it is less obvious to me.

The obvious analogy for my clock and my wordline is light's clock and light's worldline. It is tempting to think that, if light had a clock, the clock would be frozen during any travel, so that a photon would experience emission and reception at the same time.

But you said my clock and light's worldline. But if I define light's speed to be instantaneous, then emission and reception would occur at the same time. since that is what instantaneous means. So your statement is equivalent to saying: "It should be obvious that you cannot define light to move instantaneously."

I have been defining light to move instantaneously for a while and haven't run into any big problems. I have mainly looked at how this definition affects lengths and clock rates. I assume there must be some other physics equations that break down with the assumption of infinite speed for light, but I don't know what those are.

I am not arguing that you're wrong; I am just explaining why it is not obvious that there is a problem with defining light to move instantaneously (in one direction, anyway).

 

[PeterDonis]

your statement is equivalent to saying: "It should be obvious that you cannot define light to move instantaneously."

To move instantaneously in one particular direction. It won't move instantaneously in other directions. But taking that to its logical conclusion would lead to requiring isotropy of the speed of light, which is a much stricter condition (it basically leads you to Einstein synchronization as the only allowable convention), so yes, I can see where, if someone is bound and determined to look at a wider range of synchronization conventions, they are going to need to relax the isotropy requirement.

 

[PeterDonis]

I have been defining light to move instantaneously for a while and haven't run into any big problems.

That's because the coordinate charts you are defining are perfectly valid coordinate charts. I have never said otherwise. Note my comment in post #18.

My issue is simply with describing all of these weird charts as defining valid "synchronization conventions" or "simultaneity conventions". But that's really a matter of words, not physics.

 

[PeterDonis]

That is a very weak condition, weaker than yours,

Indeed, and as stated it doesn't make sense to me, because "causality" shouldn't depend on how I decide to pick my coordinates. Saying that, in a chart with $|\kappa| > 1$, an observer moving along a timelike line that is an integral curve of $\partial / \partial x$ can have his clock read the same forever, while an observer moving along a timelike line that is an integral curve of $\partial / \partial t$ can't, because the latter is a case of "spatially coincident causality" but the former is not, seems nonsensical to me. Both coordinates are timelike in this chart, so both sets of curves define valid worldlines for observers, and neither set of observers is privileged physically over the other.

 

[PeterDonis]

It might be good to post a reference for your approach

We had a thread quite a while ago discussing a paper that looked at various concepts of synchronization. I'll see if I can find it.

 

[Dale]

Indeed, and as stated it doesn't make sense to me, because "causality" shouldn't depend on how I decide to pick my coordinates.

Causality doesn't depend on your coordinates.

Saying that a synchronization convention establishes only a local causal ordering and not a global causal ordering in no way changes causality.

Saying that, in a chart with |κ|>1, an observer moving along a timelike line that is an integral curve of ∂/∂x can have his clock read the same forever, while an observer moving along a timelike line that is an integral curve of ∂/∂t can't, because the latter is a case of "spatially coincident causality" but the former is not, seems nonsensical to me.

Well, what can I say, it is in the scientific literature. You may feel it is nonsensical, but it is nonetheless part of the established literature. Probably it would be best just to find a reference that does it the way you prefer and then say "I follow _____'s approach".

Giving as your reason that ______'s approach establishes a global causal ordering while Anderson's establishes only a local one is a good reason.