I Understanding the phrase "simultaneity convention"

[Herman Trivilino]

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.

 

[martinbn]

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

And not just any foliation. For instance a foliation of timelike hypersurfaces may be of interest but I still think it is strange to call it a synchronization convention. In fact if it is not a family of Cauchy surfaces, so that you can study initial value problems, it would be a strange choice of words to call them surfaces of simultaneity.

ps To be fair the paper may be listing that convention just for completeness.

 

[PeterDonis]

In fact if it is not a family of Cauchy surfaces, so that you can study initial value problems, it would be a strange choice of words to call them surfaces of simultaneity.

While I don't disagree with this, I would point out that it implies that only globally hyperbolic spacetimes can even have "simultaneity conventions" defined in them in the first place. I personally don't think that's much of an issue, since all of the spacetimes we actually use in models of our actual universe or things in it are globally hyperbolic, but it is still a very restrictive condition mathematically.

 

[Ibix]

While I don't disagree with this, I would point out that it implies that only globally hyperbolic spacetimes can even have "simultaneity conventions" defined in them in the first place. I personally don't think that's much of an issue, since all of the spacetimes we actually use in models of our actual universe or things in it are globally hyperbolic, but it is still a very restrictive condition mathematically.

I think spacetimes that aren't globally hyperbolic have issues with causality anyway, don't they? Things like CTCs, or removed points? That would make "now" a slippery concept, so perhaps "now is a Cauchy surface" is appropriately restrictive from a physics perspective.

 

[martinbn]

While I don't disagree with this, I would point out that it implies that only globally hyperbolic spacetimes can even have "simultaneity conventions" defined in them in the first place. I personally don't think that's much of an issue, since all of the spacetimes we actually use in models of our actual universe or things in it are globally hyperbolic, but it is still a very restrictive condition mathematically.

Yes, but I meant it for the initial value problem. What are other reasons to have a simultaneity convention?

 

[martinbn]

I think spacetimes that aren't globally hyperbolic have issues with causality anyway, don't they? Things like CTCs, or removed points? That would make "now" a slippery concept, so perhaps "now is a Cauchy surface" is appropriately restrictive from a physics perspective.

If one believes in the strong cosmic censorship conjecture then the non globally hyperbolic ones are the exceptions.

 

[PeterDonis]

I think spacetimes that aren't globally hyperbolic have issues with causality anyway, don't they? Things like CTCs, or removed points?

Not quite, there are a number of conditions in between globally hyperbolic and spacetimes with those kinds of pathologies. The gory details are in Hawking & Ellis.

 

[Ibix]

The gory details are in Hawking & Ellis.

I think I'll finish Wald first...

 

[PeterDonis]

I meant it for the initial value problem. What are other reasons to have a simultaneity convention?

Clock synchronization conventions are useful for all kinds of things. For example, the GPS system has one that doesn't match Einstein clock synchronization (it's impossible to have a global Einstein clock synchronization convention for a rotating congruence of worldlines), but works fine for its intended purpose. Astronomers use a different synchronization convention for labeling events in the solar system, which also works fine for its intended purpose. One might say that part of the intended purpose of at least the latter is solving initial value problems, since the solar system convention is used for things like launching space probes and predicting when they will reach particular mission milestones, but it's certainly not limited to that. But the spacetime in question is globally hyperbolic in any case.

 

[PeterDonis]

I think I'll finish Wald first...

Wald Chapter 8 discusses causality conditions, though not to anything like the level of detail that Hawking & Ellis does.

 

[Ibix]

Wald Chapter 8 discusses causality conditions, though not to anything like the level of detail that Hawking & Ellis does.

Yeah. Chapter 8 is definitely one I need to revisit...

 

[vanhees71]

Clock synchronization conventions are useful for all kinds of things. For example, the GPS system has one that doesn't match Einstein clock synchronization (it's impossible to have a global Einstein clock synchronization convention for a rotating congruence of worldlines), but works fine for its intended purpose. Astronomers use a different synchronization convention for labeling events in the solar system, which also works fine for its intended purpose. One might say that part of the intended purpose of at least the latter is solving initial value problems, since the solar system convention is used for things like launching space probes and predicting when they will reach particular mission milestones, but it's certainly not limited to that. But the spacetime in question is globally hyperbolic in any case.

Well, sometimes pretty local synchronization conventions are sufficient, which should be the case for the GPS and the astronomer's one.

 

[PeterDonis]

Well, sometimes pretty local synchronization conventions are sufficient, which should be the case for the GPS and the astronomer's one.

Yes, it all depends on what "pretty local" means. GPS is useful in or near the Earth. The astronomers' convention is useful in the solar system. Those are large regions for us humans, but of course they're extremely small when compared to the universe as a whole.

 

[vanhees71]

Sure, but it shows that "FAPP" we only need local concepts like simultaneity conventions, etc.

 

[cianfa72]

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).

So there is actually a restriction on the type of "constant coordinate time" hypersurfaces allowed to define "a simultaneity convention" i.e. events "at the same time" (i.e. they must be spacelike).

On the other hand any other type of set of 3d hypersurfaces foliating spacetime is good from the point of view of defining a coordinate chart.

 

[PeterDonis]

So there is actually a restriction on the type of "constant coordinate time" hypersurfaces allowed to define "a simultaneity convention" i.e. events "at the same time" (i.e. they must be spacelike).

Yes.

On the other hand any other type of set of 3d hypersurfaces foliating spacetime is good from the point of view of defining a coordinate chart.

Strictly speaking, you don't even need a foliation to define a coordinate chart.

 

[cianfa72]

Strictly speaking, you don't even need a foliation to define a coordinate chart.

So in case that coordinate chart has just a local/finite extension in spacetime.

 

[Ibix]

Strictly speaking, you don't even need a foliation to define a coordinate chart.

Doesn't a coordinate chart imply a foliation, by picking one coordinate and holding it constant? Or does "foliation" imply spacelike planes?

 

[PeterDonis]

Doesn't a coordinate chart imply a foliation, by picking one coordinate and holding it constant?

Except in some pathological cases (spacetimes with "holes" in them, etc.--such examples are discussed in Hawking & Ellis), yes, I believe so. But you don't need to define a foliation first in order to define a coordinate chart.

Or does "foliation" imply spacelike planes?

In some contexts, yes, but I don't believe the general definition does.

 

[PeterDonis]

does "foliation" imply spacelike planes?

It's worth noting that in globally hyperbolic spacetimes, a foliation by spacelike 3-surfaces always exists--in fact these surfaces are Cauchy surfaces, which means every timelike or null curve intersects the surface exactly once. Many physicists believe that all spacetimes that are actually realizable physically are globally hyperbolic.

 

[vanhees71]

Doesn't a coordinate chart imply a foliation, by picking one coordinate and holding it constant? Or does "foliation" imply spacelike planes?

A coordinate chart is just a mathematical description of some neighborhood of a differentiable manifold, i.e., a continuous bijective map between an open subset of a Hausdorff point manifold and $\mathbb{R}^n$ with the standard topology (e.g., induced by the Euclidean metric).

It think to do physics you need a bit more, i.e., some notion of a local reference frame and some notion of causal time ordering. This means to do physics you need in fact at least in some local neighborhood a "foliation". That's in order to be able to describe some physical system like "point particles" and "fields" as an initial-value problem of equations of motion describing the "dynamics" of this system.

 

[cianfa72]

I think that nobody wants to allow a coordinate with spacelike integral curves to define a synchronization convention.

Sorry to resume this old thread. I think you mean that nobody wants to allow that a such coordinate (with the feature of having spacelike integral curves) defines a synchronization convention w.r.t. the set of events that share a given value of that coordinate.

 

[PeterDonis]

I think you mean that nobody wants to allow that a such coordinate (with the feature of having spacelike integral curves) defines a synchronization convention w.r.t. the set of events sharing a given value of that coordinate.

That is what using a coordinate to define a synchronization convention means.

 

[PeterDonis]

Since the thread has been reopened:

I think that nobody wants to allow a coordinate with spacelike integral curves to define a synchronization convention.

Actually that doesn't necessarily indicate a problem. I would say the real problem would be if the surfaces of constant value of the coordinate are not spacelike. (I suppose null surfaces might be acceptable as an edge case, but certainly not timelike.) Spacelike integral curves don't necessarily imply that: for example, inside the horizon of a black hole, the integral curves of the Painleve "time" coordinate $T$ are spacelike, but so are surfaces of constant $T$, so $T$ can still be, and is, used to define a synchronization convention.

 

[Dale]

I honestly don’t remember the point I was trying to make with that wording. Sorry.

 

[cianfa72]

Actually that doesn't necessarily indicate a problem. I would say the real problem would be if the surfaces of constant value of the coordinate are not spacelike. (I suppose null surfaces might be acceptable as an edge case, but certainly not timelike.) Spacelike integral curves don't necessarily imply that

Yeah, agreed. That was my concern too.

 

[Dale]

I would say the real problem would be if the surfaces of constant value of the coordinate are not spacelike.

Ok, I went back and see what I was saying. I was not speaking of what you are describing here. I was talking about the integral curves, not the surfaces of constant time coordinate, the foliation.

I believe that you agree that the integral curves must be timelike. You additionally want the foliation to be spacelike. The author I cited agrees with the first requirement, but not the second.

So I merely was saying that the first was a consensus requirement, while the second is not. I tend to agree with the spacelike foliation, but I wasn’t expressing my opinion there. I was just expressing the lowest requirement that I think everyone agrees must hold.

 

[cianfa72]

I believe that you agree that the integral curves must be timelike.

@PeterDonis made the example of Painleve coordinate time $T$ inside the black hole horizon: the integral curves are spacelike, however the hypersurfaces of constant Painleve coordinate time $T=cost$ are spacelike as well defining a synchronization convention.

 

[Dale]

If you are understanding @PeterDonis position then there may be no consensus requirements at all.

I personally think both requirements should be met.

 

[cianfa72]

I personally think both requirements should be met.

I.e. that both the integral of coordinate curve $\alpha$ (described by other coordinates fixed and varying $\alpha$) must be timelike and the hypersurfaces of constant $\alpha$ spacelike.

 

[PeterDonis]

I believe that you agree that the integral curves must be timelike.

No, I don't. I gave a counterexample: Painleve coordinates inside the horizon of a black hole. The integral curves of $T$ there are spacelike, but so are the surfaces of constant $T$, so $T$ can be, and is, used to define a simultaneity convention there. That convention is the "natural" one for Painleve observers, i.e., observers free-falling into the hole from rest at infinity, to use; the surfaces of constant $T$ are everywhere orthogonal to the worldlines of Painleve observers (which of course are timelike). But those worldlines are not integral curves of the Killing vector field $\partial_T$.