Understanding the phrase "simultaneity convention"

[PeterDonis]

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.

In fact, I can sharpen this by considering the case $\kappa = 2$ [Edit: should be sqrt(2), see post #54], for which the metric is competely symmetric in $x$ and $t$: [Edit: this is incorrect, see post #54]

$$
ds^2 = -dt^2 - dx^2 - 2 dx dt + dy^2 + dz^2
$$

Now which coordinate counts as "spatially coincident"?

 

[PeterDonis]

what can I say, it is in the scientific literature

Yes, I understand that, but that doesn't mean it's immune from criticism. Or, if you like, you can take my posts as pointing out what one is committed to if one decides to adopt a chart with $|\kappa| > 1$.

 

[PeterDonis]

Causality doesn't.

Well, the paper is using the word "causality". Why are they using it if that's not what they mean?

 

[Dale]

Now which coordinate counts as "spatially coincident"?

Either one, clearly.

Yes, I understand that, but that doesn't mean it's immune from criticism. Or, if you like, you can take my posts as pointing out what one is committed to if one decides to adopt a chart with $|\kappa| > 1$.

Clearly, but the issue is that you are simply assuming, a priori, that a synchronization convention must establish a global causal ordering. Anderson explicitly does not make that assumption. He instead explicitly makes the weaker assumption that a synchronization convention must establish a local causal ordering.

IF we require the global ordering THEN we can show that Anderson's convention only meets the requirement for $|\kappa|<1$. However, simply asserting that assumption is begging the question.

What defines a synchronization convention is a matter of definition, not a pre-existing fact of the world. The fact is that different authors can and do use different definitions. The OP should be aware of that and not assume that there is a "one size fits all" definition in use.

Now, let me be clear, I am not Anderson and I have no interest in defending him from criticism. I am answering the OP's question and pointing out that your answer is not the unique answer in the literature. Frankly, I prefer your definition, but Anderson's definition also exists.

 

[Mister T]

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.

Yes, but the very premises which predict time dilation also predict that a clock can't travel at light speed. Just because it's tempting to think about something a certain way doesn't mean it's valid.

When we say things like a light beam can't experience time we don't mean it experiences zero time. We mean that the very notion of the passage of time doesn't exist for a light beam. But to an observer that light beam does take time to travel from one place to another.

 

[vanhees71]

In fact, I can sharpen this by considering the case $\kappa = 2$, for which the metric is competely symmetric in $x$ and $t$:

$$
ds^2 = -dt^2 - dx^2 - 2 dx dt + dy^2 + dz^2
$$

Now which coordinate counts as "spatially coincident"?

Well this doesn't define valid coordinates, because the eigenvalues of the corresponding quadratic form are -2, 1, 1, 0, i.e., it can't be components of a Lorentzian fundamental form.

 

[PeterDonis]

Either one, clearly.

But is Anderson saying that?

the issue is that you are simply assuming, a priori, that a synchronization convention must establish a global causal ordering

No, I'm investigating the implications of Anderson's convention, at least as I understand it. To me, if Anderson wants to draw a distinction between "spatially coincident causality" and "distant causality", he needs to justify that, since as far as I know the rest of the literature just has one concept of "causality". So I'm wondering if there is any justification for this distinction anywhere in the paper.

What defines a synchronization convention is a matter of definition, not a pre-existing fact of the world.

Yes, I agree, that's why I said in post #32 in response to @Freixas that this is really a matter of words, not physics.

 

[PeterDonis]

this doesn't define valid coordinates

How about the case $\kappa = 1.1$, where the metric would be:

$$
ds^2 = - dt^2 - 0.21 dx^2 - 0.2 dx dt + dy^2 + dz^2
$$

 

[Dale]

To me, if Anderson wants to draw a distinction between "spatially coincident causality" and "distant causality", he needs to justify that, since as far as I know the rest of the literature just has one concept of "causality". So I'm wondering if there is any justification for this distinction anywhere in the paper

As far as I can tell, he only mentions it there. However, it is a VERY long paper and I could easily have "skimmed" over some section where he attempts to justify it in more detail.

As far as what the rest of the literature says, it would be nice to actually produce a reference that you feel is representative of the rest of the literature. I don't know the literature well enough to have a good feel for what is "generally accepted" in the rest of the literature and if Anderson is typical or an outlier.

 

[Freixas]

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.

Thanks. FWIW, I did say (at the bottom):

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

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.

I am not "bound and determined" to look at a wider range of [weird] synchronization conventions. I am trying to understand what it means for two events to be "simultaneous".

The precise bounds on simultaneity influences some questions I have lined up for the future.

At this point, your meaning has become more opaque; my best guess is that you are saying that some conventions might be valid (in the sense of a being valid coordinate charts), but invalid (in the sense of violating certain useful principles).

I should add that I don't know what a coordinate chart is, nor did a quick lookup on the web help since the explanations were usually full of other terms that I did not understand. I am not asking for an explanation unless you have one that is short and immediately comprehensible to someone whose advanced math skills have faded from more than 40 years of disuse.

 

[Dale]

I should add that I don't know what a coordinate chart is, nor did a quick lookup on the web help since the explanations were usually full of other terms that I did not understand. I am not asking for an explanation unless you have one that is short and immediately comprehensible to someone whose advanced math skills have faded from more than 40 years of disuse.

A coordinate chart is a smooth and invertible mapping between events in spacetime and points in R4.

 

[PeterDonis]

I am not "bound and determined" to look at a wider range of [weird] synchronization conventions. I am trying to understand what it means for two events to be "simultaneous".

Understanding what it means for two events to be "simultaneous" requires looking at synchronization conventions, since that is what simultaneity is. And IMO considering any convention that makes the speed of light infinite in a particular direction qualifies as "bound and determined to look at a wider range of [weird] synchronization conventions".

The precise bounds on simultaneity

Depend on what kinds of synchronization conventions you are willing to accept. As the subthread between @Dale and me in this thread should make clear, there is not one single answer to this.

my best guess is that you are saying that some conventions might be valid (in the sense of a being valid coordinate charts), but invalid (in the sense of violating certain useful principles).

No, my point was simply that not every valid coordinate chart necessarily defines a valid synchronization convention. The two terms are not synonymous.

 

[Freixas]

A coordinate chart is a smooth and invertible mapping between events in spacetime and points in R4.

Thanks for trying, Dale! What, exactly, is "smooth"? I think I know what invertible means. I don't know what "R4" is. Originally you said smooth was "diffeomorphic". I looked that up and opened another can of worms.

Originally, my first proposal for setting $t$ was $t = f(d)$, where $f(d)$ could be any function. Presumably, this does not define a valid coordinate chart unless $f(d)$ is smooth and invertible, so option 1 was never a contender (because it didn't have this limitation).

@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.

I finally went through your discussion with Peter. I can't really follow all of what you are discussing. The sense I get is that there are coordinate maps, some of which people define as valid simultaneity (or synchronization) conventions. Different people may define different coordinate maps as valid--this is a choice of definition. Physics doesn't care, but you have to live with your choice and some choices might make one's life more difficult (in terms of doing useful physics) than others.

So if I limit myself to conventions in which different simultaneous events can be connected by spacelike or lightlike intervals and someone else picks only conventions where only spacelike intervals are allowed, physics is silent on which is "correct." I just have to make clear what my definition is and live with any computational issues resulting from my choice.

Is this summary correct, in your opinion?

By the way, in reading the "Suggested for" section below, I saw that you made this comment back in 2014: "At the extremes the speed of light is infinite in one direction and c/2 in the other direction. The difference between 0 and 1 is just which direction is infinite." So you seem to allow for the possibility of infinite light speed in one direction.

 

[PeterDonis]

it would be nice to actually produce a reference that you feel is representative of the rest of the literature

I can't say whether it's representative of all of the rest of the literature, but I found the previous thread I mentioned before:

https://www.physicsforums.com/threads/global-simultaneity-surfaces.734070/

The Sachs & Wu definition of a "synchronizable reference frame" referred to there is what I was thinking of. Now I need to go back and check that that definition actually has the properties I was thinking it has.

 

[PeterDonis]

The Sachs & Wu definition of a "synchronizable reference frame" referred to there is what I was thinking of.

To amplify this a bit: Sachs & Wu actually give four different concepts related to synchronization. I'll just list them along with (what I think are) examples of each:

(1) Locally synchronizable: The natural frame of an observer rotating around a circular path in flat spacetime.

(2) Locally proper time synchronizable: The standard coordinate chart on Godel spacetime. (? this one might actually be globally proper time synchronizable)

(3) Synchronizable: Schwarzschild coordinates on the exterior (outside the horizon) region of Schwarzschild spacetime, with reference to "hovering" observers (observers with constant $r$, $\theta$, $\phi$.

(4) Proper time synchronizable: FRW coordinates in FRW spacetime, with reference to "comoving" observers.

 

[Mister T]

I am trying to understand what it means for two events to be "simultaneous".

That the events have a spacelike separation. In other words, the same burst of light can't be present at both events.

 

[vanhees71]

How about the case $\kappa = 1.1$, where the metric would be:

$$
ds^2 = - dt^2 - 0.21 dx^2 - 0.2 dx dt + dy^2 + dz^2
$$

If I understand it right the proposed forms of the metric are given by
$$\mathrm{d}s^2 =-\mathrm{d} t^2 +(1-\kappa^2) \mathrm{d} x^2-2 \kappa \mathrm{d} x \mathrm{d} t +\mathrm{d} y^2 + \mathrm{d} z^2.$$
The eigenvalues of the metric are
$$\frac{1}{2} (-\kappa^2 \pm \sqrt{4+\kappa^4}),1,1,$$
i.e., the metric has for any real $\kappa$ the correct signature (-+++) for a Lorentzian manifold.

It's in fact just Minkowski space since using the coordinates
$$\begin{pmatrix} t' \\ x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix}t+\kappa x \\ x \\ y \\ z \end{pmatrix}$$
leads to
$$\mathrm{d} s^2 = -\mathrm{d} t^{\prime 2} + \mathrm{d} x^{\prime 2} + \mathrm{d} y^{\prime 2} + \mathrm{d} z^{\prime 2}.$$
I've not looked at the paper, but I guess it's simply related to the idea to define different "one-way speeds of light" in different directions along $\pm x$ with the two-way speed of light kept at 1.

It's of course a bit complicated to formulate the causality structure in terms of the somewhat idiosyncratic coordinates $(t,x,y,z)$ instead of the "Minkowskian" (pseudo-Cartesian) ones, $(t',x',y',z')$.

I think this kinds of considerations go back to some philosophical debates related to Reichenbach, and as expected it leads to confusion without much physical insight ;-). SCNR.

 

[PeterDonis]

It's in fact just Minkowski space

Yes, the Anderson paper @Dale linked to says that this family of coordinate charts are all just alternate charts for Minkowski spacetime.

 

[PeterDonis]

the case $\kappa = 2$, for which the metric is competely symmetric in $x$ and $t$:

$$
ds^2 = -dt^2 - dx^2 - 2 dx dt + dy^2 + dz^2
$$

I see after looking at post #52 that I got this wrong, the completely symmetric case is $\kappa = \sqrt{2}$, and the correct metric for that case is

$$
ds^2 = - dt^2 - dx^2 - 2 \sqrt{2} dx dt + dy^2 + dz^2
$$

which does have the correct signature.

 

[Dale]

I think this kinds of considerations go back to some philosophical debates related to Reichenbach, and as expected it leads to confusion without much physical insight

So true.

 

[Freixas]

A lot of the points made here seem relevant to my question. It's a shame a lot of it goes over my head.

I think this kinds of considerations go back to some philosophical debates related to Reichenbach, and as expected it leads to confusion without much physical insight ;-)

I'm still curious as to what part of simultaneity is limited by physics and what part is philosophy. Even @PeterDonis, who doesn't normally spend much time on philosophy, has added phrases such as "IMO" and "But that's really a matter of words, not physics," so there's a hint that some of his comments might be philosophical ones.

I think geometrically, because I don't have the advanced math that the mentors here have. So I picture a Minkowski spacetime diagrams with events and worldlines. On this I can overlay arbitrary curves that define simultaneous events. As long as these curves have properties that allow me to map back to the usual Minkowski space, it appears to me that physics doesn't care how I draw the lines. I think this might be what Dale was saying as the initial response to this thread.

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.

From there, I think we get into philosophy in that the dictionary definition of "simultaneous" ("at the same time") acquires reasonable causality baggage that implies we might only want to choose conventions which maintain spacelike intervals between events.

That's fine. I just want to make sure that this is the point where we step from physics to philosophy.

 

[vanhees71]

I meant the attempt to physically interpret arbitrarily chosen coordinates. Coordinates usually do not have a direct physical meaning. They are mathematical descriptions that enable you to calculate physical observables, which are always independent on the choice of coordinates. That's why we use tensor calculus to define physical observables.

 

[Ibix]

I'm still curious as to what part of simultaneity is limited by physics and what part is philosophy.

It would be odd to define "simultaneity" so that I could throw a ball and you could catch it simultaneously or even earlier. Given that definition, the physics restricts surfaces of simultaneity to being everywhere spacelike. Also, whatever surface of simultaneity I choose now, the one I choose now may not cross it anywhere, and the ones between those two nows must smoothly progress through the gap. Finally, the surfaces must be achronal, which means they must never circle around and enter the future lightcone of any of their events. (Imagine a (2+1)d Minkowski diagram and draw a shallow sloped helix on it spiralling around the time axis. The helix can be everywhere spacelike but still passes through the future lightcone of its earlier events. That is not achronal.)

I admit I haven't been following the conversation about Anderson particularly closely. I think he essentially relaxes the definition of simultaneous a bit, allowing the planes to be timelike and only insisting that if plane $t$ crosses some worldline then plane $t+1$ must cross later. As coordinates go this is fine (you can have 'em all timelike or all null if you want), but I wouldn't call that a simultaneity plane since it does allow things like a target being hit at an earlier "now" than the one where the trigger is pulled (edit: although it does enforce that, if the target returns fire, the return hit is after the first trigger pull).

 

[Ibix]

Perhaps more crisply: given a definition of simultaneity, physics dictates what families of planes can be considered simultaneity planes by that definition. But the definition of simultaneity is a matter of taste, at least to some extent, and you can do physics without bothering to define simultaneity.

 

[martinbn]

Just to point out that the Anderson's paper is very long and the section discussed here is very short and titled "Nonconventional synchrony".

 

[Dale]

I just want to make sure that this is the point where we step from physics to philosophy.

My personal view is that all of simultaneity is philosophy. The physics is causality, but simultaneity is purely philosophy. Even simultaneity conventions that provide a global causal ordering are still philosophy

 

[Nugatory]

I think geometrically, because I don't have the advanced math that the mentors here have. So I picture a Minkowski spacetime diagrams with events and worldlines. On this I can overlay arbitrary curves that define simultaneous events. As long as these curves have properties that allow me to map back to the usual Minkowski space, it appears to me that physics doesn't care how I draw the lines.

Not completely arbitrary: the curves must be spacelike if we’re going to stay within the generally accepted meaning of “happens at the same time”.

Mapping back to Minkowski space is an unnecessarily strong constraint, limiting us to just the special case that is Special Relativity. Generalize by considering spacelike curves in a non-flat spacetime and we’ll be doing General Relativity.

 

[Freixas]

I meant the attempt to physically interpret arbitrarily chosen coordinates. Coordinates usually do not have a direct physical meaning. They are mathematical descriptions that enable you to calculate physical observables, which are always independent on the choice of coordinates. That's why we use tensor calculus to define physical observables.

Thanks.

I looked up tensor calculus. It sounds interesting ("physics equations in a form that is independent of the choice of coordinates" says Wikipedia). I regret not maintaining the calculus I learned back in the '70s, but unless you're in certain fields, there's not much call for it. I try to understand the basics of S.R. by using a one-dimensional universe and applying geometry, which I retained much more than calculus. I haven't looked into G.R. as I feel S.R. is keeping me busy enough.

 

[PeterDonis]

I'm still curious as to what part of simultaneity is limited by physics and what part is philosophy.

IMO simultaneity is neither. It's a convention that we humans adopt in order to help us in doing physics. Most commonly a simultaneity convention is defined as part of choosing a coordinate chart in which to write down equations.

 

[PeterDonis]

It would be odd to define "simultaneity" so that I could throw a ball and you could catch it simultaneously or even earlier. Given that definition, the physics restricts surfaces of simultaneity to being everywhere spacelike. Also, whatever surface of simultaneity I choose now, the one I choose now may not cross it anywhere, and the ones between those two nows must smoothly progress through the gap. Finally, the surfaces must be achronal, which means they must never circle around and enter the future lightcone of any of their events. (Imagine a (2+1)d Minkowski diagram and draw a shallow sloped helix on it spiralling around the time axis. The helix can be everywhere spacelike but still passes through the future lightcone of its earlier events. That is not achronal.)

I agree with all of this, but the Anderson paper that was referenced does not limit its discussion to "synchrony" conventions that obey these restrictions.

I admit I haven't been following the conversation about Anderson particularly closely. I think he essentially relaxes the definition of simultaneous a bit, allowing the planes to be timelike and only insisting that if plane $t$ crosses some worldline then plane $t+1$ must cross later.

No, he doesn't even restrict it to that. He includes "synchrony" conventions in which clock time along a single timelike worldline can go backwards. He claims that this is no different than setting your clock backwards when you cross the International Date Line. I don't personally think such choices of coordinates should be taken as defining a valid simultaneity convention, but Anderson seems to think there are physicists who do.

 

[Dale]

I don't personally think such choices of coordinates should be taken as defining a valid simultaneity convention,

I agree with you on this point. After all, nobody scheduling a Zoom meeting with European, American, and Asian participants will think that if everyone joins at 9:00 local time they will be in the meeting simultaneously. The paper by Anderson is generally good, but I think that it would have been better served to simply state the reasonable restriction to $|\kappa|<1$

 

[Ibix]

I don't personally think such choices of coordinates should be taken as defining a valid simultaneity convention

Agreed. Coordinate planes, yes, simultaneity planes, no.

Anderson seems to think there are physicists who do.

I would love to see the justification, barring exotic spacetime geometries and toplogies that don't admit global notions of past and future.

 

[vanhees71]

It's not exotic spacetime geometries but a special choice of coordinates in Minkowski space (or in a Lorentzial manifold of GR). I don't know, what one has to discuss about this. Coordinates are labels for points in spacetime which don't need to have a direct physical meaning.

 

[Dale]

Coordinates are labels for points in spacetime which don't need to have a direct physical meaning.

I think that the only discussion is whether all valid coordinates define a synchronization convention or if only some coordinates define valid synchronization conventions.

I think that nobody wants to allow a coordinate with spacelike integral curves to define a synchronization convention. Anderson encodes this requirement in his local causal ordering idea. But for him as long as the integral curves are timelike the synchronization between different curves is arbitrary.

@PeterDonis wants a more restrictive definition where a synchronization convention produces a spacelike foliation of the spacetime. That is more restrictive than Anderson, but is a simple matter of restricting his $|\kappa|<1$.

Under Anderson’s convention our global time zones are a valid synchronization convention. Each integral curve has strictly increasing coordinate times as a function of proper time. So he would allow that it is OK to say that 9:00 eastern time is simultaneous with 9:00 mountain time. @PeterDonis would not permit that as a valid synchronization convention but would require that 9:00 eastern time would be simultaneous with 7:00 mountain time plus or minus $d/c$.

 

[vanhees71]

I agree with @PeterDonis, i.e., a useful physical definition of synchronizity should be given by a foliation.