I Understanding the phrase "simultaneity convention"

[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 $\partial_T$.

 

[PeterDonis]

You additionally want the foliation to be spacelike.

I think that would make the most sense, but there are examples of null foliations (for example Eddington-Finkelstein coordinates). I don't think it makes sense for the surfaces of the foliation to have any timelike tangent vectors, and AFAIK there are no examples of proposed foliations that do.

 

[PeterDonis]

I personally think both requirements should be met.

I agree that this case (timelike integral curves, spacelike foliation surfaces) will be the one that most clearly matches our intuitive sense of how things should work, and is therefore the most "natural" case.

 

[Dale]

don't think it makes sense for the surfaces of the foliation to have any timelike tangent vectors, and AFAIK there are no examples of proposed foliations that do.

Anderson specifically considers ordinary time zones and the date line as a valid foliation.

 

[PeterDonis]

Anderson specifically considers ordinary time zones and the date line as a valid foliation.

Yes, we discussed that before. To recap briefly, time zones and dates don't define a foliation that has timelike tangent vectors on its simultaneity surfaces. They just define different labelings of a foliation whose simultaneity surfaces are all spacelike. For example, if I adjust my clock from Eastern to Central time because I'm traveling, I'm changing coordinate charts, but I'm not changing foliations; both charts use the same foliation, they just label the surfaces differently (a given surface has a Central time label that is 1 hour earlier than its Eastern time label).

Anderson's convention for $\kappa > 1$, by contrast, has "simultaneity" surfaces that have timelike tangent vectors. I was never able to get a look at the actual paper to see what, if any, argument was made for this, but I don't see how it is the same as time zones and dates. But you are right that it does appear to be a foliation in the published literature that does have simultaneity surfaces with timelike tangent vectors; I had forgotten that example when I made my earlier post.

 

[Dale]

To recap briefly, time zones and dates don't define a foliation that has timelike tangent vectors on its simultaneity surfaces. They just define different labelings of a foliation whose simultaneity surfaces are all spacelike.

Reading his paper I don’t think that is what he intended.

For example, if I adjust my clock from Eastern to Central time because I'm traveling, I'm changing coordinate charts, but I'm not changing foliations; both charts use the same foliation, they just label the surfaces differently (a given surface has a Central time label that is 1 hour earlier than its Eastern time label)

I think that is not his intention. I think he intended to allow a foliation such that 9:00 am Eastern is simultaneous with 9:00 am Central. It is an odd approach, but I think that is exactly what he is proposing.

His only stated restriction allows considering 9:00 am Eastern to be simultaneous with 9:00 am Central and the surrounding text supports that message.

To be clear, I don’t like that approach. I have not seen it elsewhere.

 

[PeterDonis]

I think he intended to allow a foliation such that 9:00 am Eastern is simultaneous with 9:00 am Central.

Perhaps so, but in any event that is not the standard interpretation of time zones. Nobody actually treats 9 am Eastern and 9 am Central as simultaneous. Everybody understands that the latter is an hour later than the former. So if he is claiming that his proposal is just like the standard usage of time zones, I don't think that claim is valid.

 

[Dale]

that is not the standard interpretation of time zones. Nobody actually treats 9 am Eastern and 9 am Central as simultaneous

I agree.

 

[Ibix]

I think that is not his intention. I think he intended to allow a foliation such that 9:00 am Eastern is simultaneous with 9:00 am Central. It is an odd approach, but I think that is exactly what he is proposing.

Is that allowed? The two geographical regions share an edge where the time coordinate is either ill-defined or arbitrarily chosen to be one or the other. Doesn't that make the coordinates either ill-defined or defined on closed regions?

Edit: you could define a finite but narrow transition region between the zones, of course.

 

[cianfa72]

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 $\partial_T$.

Ok, so the Killing vector field $\partial_T$ is spacelike inside the horizon, Painleve observer's timelike curves are not integral curves of $\partial_t$, however they are orthogonal to the spacelike hypersurfaces of constant $T$.

 

[Dale]

Is that allowed? The two geographical regions share an edge where the time coordinate is either ill-defined or arbitrarily chosen to be one or the other. Doesn't that make the coordinates either ill-defined or defined on closed regions?

Edit: you could define a finite but narrow transition region between the zones, of course.

That is a good question. Carroll introduces the requirement that the chart be $C^\infty$. But then he almost immediately backs off on the requirement. So I am not sure what subsequent theorems rely on the continuity and how far they really rely on it. And he does use polar and spherical coordinates later. So I am just not sure how much that requirement can be relaxed.

 

[cianfa72]

Carroll introduces the requirement that the chart be $C^\infty$. But then he almost immediately backs off on the requirement. So I am not sure what subsequent theorems rely on the continuity and how far they really rely on it. And he does use polar and spherical coordinates later. So I am just not sure how much that requirement can be relaxed.

Sorry, which specific chapters/sections of Carroll lectures/book are your referring to ?

 

[Ibix]

That is a good question. Carroll introduces the requirement that the chart be $C^\infty$. But then he almost immediately backs off on the requirement. So I am not sure what subsequent theorems rely on the continuity and how far they really rely on it. And he does use polar and spherical coordinates later. So I am just not sure how much that requirement can be relaxed.

I think it's more a problem with the discontinuity than the differentiability. If coordinates aren't defined on the borders between time zones then you actually have 24 non-overlapping coordinate patches. What does it even mean for two things to be similtaneous if you can't draw a line of constant coordinate time from one to the other?

That isn't really a problem with polar coordinates because there's only a half-infinite line that's problematic and you can always go around it. But in the time zones case I think you end up having to appeal to some kind of global coordinate system, or at least another set of patches that do cover the joins. That doesn't feel like "a" coordinate system so much as several partially overlapping ones.

 

[Dale]

Sorry, which specific chapters/sections of Carroll lectures/book are your referring to ?

It is the first couple of pages of chapter 2 in the Lecture Notes on General Relativity

 

[Dale]

That isn't really a problem with polar coordinates because there's only a half-infinite line that's problematic and you can always go around it. But in the time zones case I think you end up having to appeal to some kind of global coordinate system, or at least another set of patches that do cover the joins. That doesn't feel like "a" coordinate system so much as several partially overlapping ones.

I don't see a difference between the two. I think the problem is the same, but again, this is not my preferred definition but it is in the literature. I am disinclined to argue either for or against it. Unfortunately, I don't know an actual reference that gives a better definition.

To me, this is a definition, so an author can choose a different definition. The real problem in my opinion is what @PeterDonis mentioned: it does not capture the idea that people usually mean by the word. So using this definition will cause confusion at some points, even if it is mathematically OK.

I think that a time coordinate should both be timelike everywhere and foliate the spacetime into spacelike simultaneity surfaces. Any other definition will not be what people generally mean.

 

[cianfa72]

It is the first couple of pages of chapter 2 in the Lecture Notes on General Relativity

I took a look there. Indeed Carroll claims for example that Mercator map doesn't include both North, South pole and International Date Line. As @Ibix said one needs a set of coordinate patches (an atlas) since the domain and the target of each coordinate patch/map must be an open set.

 

[PeterDonis]

the Killing vector field $\partial_T$ is spacelike inside the horizon

Yes.

Painleve observer's timelike curves are not integral curves of $\partial_t$, however they are orthogonal to the spacelike hypersurfaces of constant $T$.

Yes. (But note that it should be $\partial_T$ in the quote above, since I was using $T$ for the Painleve time coordinate.)

 

[PeterDonis]

I think that a time coordinate should both be timelike everywhere and foliate the spacetime into spacelike simultaneity surfaces. Any other definition will not be what people generally mean.

While this is true, it is also true that there will be cases where doing this will end up with something that isn't "what people generally mean" about that specific spacetime.

For example, if you want a chart with these properties on Schwarzschild spacetime including the black hole region, you will need to use something like the Kruskal chart. But the integral curves of that chart's time coordinate aren't the worldlines of any observer that would seem natural, and the foliation surfaces defined by that chart don't correspond to any foliation that would seem natural. The "natural" chart on Schwarzschild spacetime is the Schwarzschild chart, which fails both requirements at and inside the horizon.

 

[Dale]

For example, if you want a chart with these properties on Schwarzschild spacetime including the black hole region, you will need to use something like the Kruskal chart. But the integral curves of that chart's time coordinate aren't the worldlines of any observer that would seem natural, and the foliation surfaces defined by that chart don't correspond to any foliation that would seem natural. The "natural" chart on Schwarzschild spacetime is the Schwarzschild chart, which fails both requirements at and inside the horizon.

I think it is better to simply admit that.

We have a "natural" chart, and we have a common understanding of the requirements for a "time coordinate". The chart that is both natural has the common "time coordinate" is limited to outside the horizon. There are charts that have a common "time coordinates" throughout the manifold. There are charts that are "natural" throughout the manifold. It is only the combination of both features that becomes incompatible at or below the horizon.

I would rather not change the meaning of a "time coordinate" to capture this. After all, not all charts are required to have a time coordinate. And simply using the variable $t$ doesn't mean that the thing represented by the variable must be time.

But again, I don't know of a source that uses the definition I would prefer.

 

[cianfa72]

Since we're talking about coordinates, I'd ask for a clarification about the Schwarzschild metric in Schwarzschild coordinates. At the horizon $r=r_s$ the metric component $g_{rr}$ blows up while $g_{tt}$ goes to zero. My impression is that such coordinate singularity is alike what happens in polar coordinates along the negative (left) half-line starting from a fixed point O in Euclidean plane. In the target codomain of polar map $(r, \theta)$ points along the segment $r=0$ and along the lines $\theta = - \pi, \theta = \pi$ are actually excluded. This way polar map satisfies Carroll's definition of chart since it is a one-to-one map between open sets (and the composition with the identity map is a diffeomorphism where it is defined).

Having said that, is $r=r_s$ in Schwarzschild coordinates a coordinate singularity alike to the above?

 

[PeterDonis]

My impression is that such coordinate singularity is alike what happens in polar coordinates

Not really. In polar coordinates the metric coefficient $g_{rr}$ vanishes at $r = 0$. That means the metric has a vanishing determinant, but it doesn't make it undefined.

In Schwarzschild coordinates, the metric coefficient $g_{rr}$ is undefined at $r = r_s$ (zero in the denominator). That's not the same thing as the above.